Imago Cosmi : The Vision of the Cosmos and the History of Astronomical Machines 9783031309434, 9783031309441

Table of contents :
Preface
Acknowledgments
Contents
About the Author
1: Introduction
2: Mathematics and Astronomy from Origin to Eighteenth Century
The Origin
Ancient Mathematics and Astronomy Until the End of the Roman Empire
Arab Mathematics and Astronomy during the Middle Age
From Renaissance to XVII Century
The XVIII Century
Academia and University
3: Ancient Visions of the Cosmos: Orienting, Classifying and Modeling
Orienting in the Sky: The Constellations
Stars Catalogs
The Motion and the Shape of the Cosmos
Eratosthenes, Eudoxos, Callippus and Aristoteles
Hipparchus and Apollonius
Claudius Ptolemy
Humanism and the Rebirth of the Studies
The Invention of the Solar System: Nicolaus Copernicus
4: A Lucky Astral Conjunction
In Search of Precision: Tycho Brahe
Johannes Kepler: A Journey in the Space
Galileo: Looking into the Deep Sky
The Cause of Celestial Motion: Isaac Newton
5: The New Vision of the Cosmos
Shape and Motion of the Earth
The Satellites of Jupiter
The Motion of the Moon
The Tides
The Speed of Light
The Planetary Periods
The Astrological Vision of the Cosmos
6: The Instruments
Mechanics
Instruments to Guide the Observation
Measuring the Time
A Challenge Between French and English: Measuring the Meridian and the Time in Navigation
7: The First Astronomical Machine: Antikythera
The Discovery
The Dating
Structure and Functions of the Mechanism
The Missing Parts: The Motion of the Moon, the Sun, and the Planetarium
The Planetarium
Construction Technology and Materials
Copies and Simulations
8: Astronomical Machines and Clocks from the Arab Times to the Renaissance
Astronomical Instruments of the Caliphate
Renaissance
Sphères Mouvant and Globes of Sixteenth Century. Eberhard Baldewein and Oronce Finé
Jost Bürgi: Mathematician, Mechanic, Clock Maker, Astronomer
9: Toward Planetary Machines
Paving the Way to the New Cosmos: Kepler’s Planetary Machine
Ole Rømer: Jovilabium and Planispherium
Christiaan Huygens
10: Orreries and Astronomical Clocks
The Planetary Machine by Thompion and Graham
The Dissemination of the New Astronomy in England
George Adams
Thomas Wright, Thomas Heath, Benjamin Martin
James Ferguson
The Cometarium
Italian Astronomical Machines
Francesco Generini and Bartolomeo Ferrari
Bernardo Facini: The Planisferologio Farnese
Francesco Borghesi and Bartolomeo Antonio Bertolla
11: France and Switzerland
Claude Simeon Passemant
Antide Janvier
François Ducommun
12: Blossoming in Germany and Austria: The Priestermechaniker
Bernard Stuart
Johan Georg Neßtfells
Johannes Klein
David Rutschmann, Frater David a S. Cajetano
Michael Fras – Frater Aurelius a S. Daniele
Engelbert Wenzel Seige
Alexius Johann
Philip Mathäus Hahn
The Universal Chronology
13: Chinese Philosophical and Mathematical Thought
Kingdoms and Dynasties
Philosophical Schools
Mathematics
14: Chinese Astronomy and Astronomical Machines
The Image of the Cosmos
The Arrival of the Jesuits and Western Astronomy
Mechanics
Astronomical Instruments and Machines
Chinese and Western Mathematics and Astronomy
15: Design of a Simple Planetary Machine
Background
Design Constraints
Gear Computation
The Structure and Construction of the Machine
Computer Control System
16: Conclusion
The Cosmos as a Celestial Machine
A Classification of Astronomical Machines
When and Where
Appendix A
Positional Astronomy
On the Measure of Time
Equation of Time
The Julian Day
Elementary Astronomical Computations
Interpolation
The Computation of the Date of Easter
Algorithms for the Computation of Continued Fractions
Planetary Orbits
Appendix B
Credits of the Illustrations
References
Index

Citation preview

Daniele L. R. Marini

Imago Cosmi

The Vision of the Cosmos and the History of Astronomical Machines

Astronomers' Universe

Series Editor Martin Beech, Campion College, The University of Regina Regina, SK, Canada

The Astronomers’ Universe series attracts scientifically curious readers with a passion for astronomy and its related fields. In this series, you will venture beyond the basics to gain a deeper understanding of the cosmos—all from the comfort of your chair. Our books cover any and all topics related to the scientific study of the Universe and our place in it, exploring discoveries and theories in areas ranging from cosmology and astrophysics to planetary science and astrobiology. This series bridges the gap between very basic popular science books and higher-level textbooks, providing rigorous, yet digestible forays for the intrepid lay reader. It goes beyond a beginner’s level, introducing you to more complex concepts that will expand your knowledge of the cosmos. The books are written in a didactic and descriptive style, including basic mathematics where necessary.

Daniele L. R. Marini

Imago Cosmi The Vision of the Cosmos and the History of Astronomical Machines

Daniele L. R. Marini Department of Computer Science University of Milan Milan, Italy

This work contains media enhancements, which are displayed with a “play” icon. Material in the print book can be viewed on a mobile device by downloading the Springer Nature “More Media” app available in the major app stores. The media enhancements in the online version of the work can be accessed directly by authorized users. ISSN 1614-659X     ISSN 2197-6651 (electronic) Astronomers' Universe ISBN 978-3-031-30943-4    ISBN 978-3-031-30944-1 (eBook) https://doi.org/10.1007/978-3-031-30944-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Foreground: Tellurium by the Author; photo by Corrado Crisciani. Background: Pillars of Creation, Science: NASA, ESA, CSA, STScI; Image Processing: Joseph DePasquale (STScI), Alyssa Pagan (STScI). This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Maria Cristina, Beniamino, Tommaso, and especially little Antonio, may have from me what my father gave to me. La più sublime, la più nobile tra le Fisiche scienze ella è senza dubbio l’Astronomia. L’uomo s’innalza per mezzo di essa come al di sopra di sé medesimo, e giunge a conoscere la causa dei fenomeni più straordinari. Giacomo Leopardi The most sublime, the most noble among the Physical sciences it is with no doubts the Astronomy. The man elevates by its mean above himself, and arrive to know the cause of the most extraordinary phenomena Giacomo Leopardi

Preface

On February 15, 1961, my father took us to the mountains north of Genova to see the only total solar eclipse of the twentieth century visible in Italy, my home country. As a high school student studying math and physics the event left a huge impression on me, even though I only had a very basic and dim refractor telescope. While it managed to show the Andromeda galaxy and Orion nebula, the simple device did not let me observe the planets very well. Inheriting my father’s manual dexterity and interest in repairing clocks, I also attempted to build a much brighter imaging reflecting telescope, and began to design a parabola to determine the new telescope mirror dimensions. Then school study commitments intervened and prevented me from pursuing those activities any further. A few years later in 1964, I tried to read Radioastronomy by the Italian astrophysicist Margherita Hack, and while failing to completely understand it, the book inspired me to enroll in university to study physics. But then the field of computer science caught my interest and became the focus of my career as a university professor. Upon retirement in 2014 I was finally able to return to my earlier interest in astronomy, purchasing the long dreamed of reflecting telescope, and resuming work on complicated mechanisms by designing and building a pendulum clock. That same year while visiting the world-famous clock museum in La Chaux-de-Fonds, Switzerland, I became fascinated by the astronomical clocks and mechanical planetariums on display. Standing like a child in front of these complex mechanical models of the cosmos, I not only fell in love with them but longed to build my own Tellurium, which is a working model of the Sun, Earth, and Moon. This in turn drove my need to understand in detail how to design and build one. vii

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Using my professional skills, I researched numerous scientific documents, publications, and books, visited museums exhibiting these machines, and consulted people with far more experience than myself in clockmaking and precision mechanics. They gave me some useful suggestions as well as many skeptical reactions. Rather than discouraging me, it motivated me to prove them wrong as my complicated mechanical device came together over the next couple of years. Upon its successful completion I decided to write this book both to reflect on how enormously satisfying the work was and to share my enthusiasm with all I had learned and experienced regarding the historical events and astronomical theories that led to these wonderful instruments. No previous book combined the history of mechanical planetariums with the associated astronomy, mathematics, mechanical engineering, science, technology, and two-way impact of religion, politics, and society. I tried to answer some of the many questions that have been rattling around in my head over the years. How is a gear train calculated? How is that calculation linked to modern number theory? What astronomical phenomena can this kind of mechanism simulate and predict? How did astronomers and mathematicians understand celestial motion? What was contributed by the great inventors and clockmakers who joined the scientists? What was the significance of these machines in the history of science? Why were these models built that depicted the shape of the cosmos? What is the difference between a mechanical planetarium compared to a modern projector planetarium governed by sophisticated computer software? And finally, the key question is what formed our present-day mental image of the solar system and the universe? The ambitious title I have chosen for this book, Imago Cosmi—Image or Visions of the Cosmos, is in recognition of the vast scientific and ideological implications of these planetary devices. The word cosmos comes from an ancient Greek term meaning order, harmony, and beauty. It referred to the order governing the universe, which followed the earlier chaotic birth of the world. Today, while derivative terms such as cosmetic and cosmetics refer simply to the improvement of appearance, the word cosmos still denotes the entire universe as a harmonious and ordered whole, including the earth and the forces that govern it and the space beyond. Semantic linkages also involve the origin of life, its narrative, and the emergence of knowledge and science, bringing attention to themes that touch on the deepest aspects of human experience on which religions and ideologies have been founded. To limit this enormous field, my work looks only at how our view of the solar system took form, by focusing on astronomical machines and their

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builders, with concise descriptions of how they led to modern astronomy. This book covers the most interesting examples I was able to study, extending up to the early 1800s, before present-day astronomy, astrophysics, and space exploration. One of several examples which I found extraordinary is about Johannes Kepler, who was key to early understanding of the solar system’s organization and planetary motion. In the late 1500s he attempted to design and build an early mechanical planetarium. The attempt failed, but a record of his design survived in his letters, and it allowed me to make and show for the first time a three-dimensional model of it using CAD and virtual reality techniques. I have included anecdotes, brief biographies, and social context to enrich the work of some of the most important scientists and thinkers. In my opinion, episodes from their lives are essential to understanding the context in which they operated and how it affected their thinking. Important religious, ideological, moral, and psychological connections necessary to historically frame the evolution of astronomical and scientific thought are also included. It is my hope that not only specialists in the history of technology but also those simply interested in the history of science, scientific thought, and technology will enjoy my passion and love for these marvelous machines and their stories. Please note that some additional insights which did not fit in the space available in this book will be available on the website associated with this work. For ease of reading, the credits of the illustrations have been collected in Appendix B. Milano, Italy December 2022

Daniele L. R. Marini

Acknowledgments

I am grateful to my wife Maria Cristina, first for her patience during the construction of the Tellurium and then for sharing with me visits to the museums that preserve some of the most beautiful works: tolerating a mono-maniac is certainly not easy. But I am especially grateful to her because she was able to bear the fatigue of reading and reviewing a work that to her, a mathematical mind, often appeared on the verge of generality and imprecision, if not incomprehensibility, and many times she found the many errors I ran into. To my sons, Beniamino and Tommaso, special thanks for reading the very first versions of this book, helping me to focus on the final readers. Finally, to little Antonio not only goes my thanks for showing the enthusiasm and interest that my planetary machine solicited in him, but above all goes my wish that he might draw inspiration from his grandfather's work to form his own critical thinking. Fulvia Cattaneo has been the first reviewer and editor of the Italian version of this work; to her goes an undying gratitude. Special thanks go to Roberto Moro, who as a historian has often steered me into a discipline not my own. I thank Maria Soresina for her help in German translation and Roberto Capuzzo Dolcetta, Gianfranco Prini, and Marco Santambrogio for reading the first draft and encouraging me. In my research, I had the opportunity to exchange views with other researchers, including Lino De Martino, Giuseppe Di Stefano, Roberto Fanciulli, Stefano Gattei, Michel Hayard, Maria Morigi, Luca Natali, Ludwig Oechslin, Giancarlo Truffa, and Brigitte Vinzens. For the construction of the Tellurium, I am grateful to Angelo Moretti, and before that to his father, who initiated me into some of the secrets of lathe operations. Angelo was my advisor, introducing me to the various craftsmen xi

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in Brianza, who in turn helped me to calibrate my lathe and suggested a workshop for tempering of a tool and a workshop for laser cutting of brass wheels. I also thank Giovanni Corti and Flavio Teruzzi, who work with Angelo and assisted me on a few occasions with machining that I could not have done at home. Carlo Croce, who introduced me into subtleties of clockmaking and tried to discourage me from making the tellurium. Christian Slanzi has been my programming consultant; Andrea Papini assisted me on the preparation of the wooden base of the machine. I would like to thank all my enthusiast watchmaker friends who followed and encouraged me during the construction of the Tellurium, in particular the Orologiko group 1 who had the courage to award me and the association Amici di Hora 2 whose journal has been a valuable bibliographic source. Citing the names of everyone is not possible; I would risk forgetting some who would only take offense: so, a collective thank you! Finally, I do not know how to thank the universe of people who are working to make the Internet the most extraordinary place to do research work. Wikipedia editors, scientists who maintain public sites with documentation of their research, and above all librarians who by putting online catalogues of books and digital versions of ancient manuscripts have allowed me to compress the time of bibliographic research from years to days, leaving me comfortably seated in front of my computer. It does not seem a contradiction that I am also grateful for the service of publishing books in digital form, which has allowed me to get a quoted text in order to delve into it or check it out in a matter of minutes. Without the web, I could not even have started this work. This book was written in Italian. English not being my mother tongue, I had the invaluable help of two anonymous friends, to whom goes my gratitude. Lisa Scalone was patient, professional, and generous with advice that allowed me to tailor the text according to the principles of a high-quality scientific publication.

 http://www.orologiko.it  https://hora.it

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Contents

1 I ntroduction  1 2 Mathematics  and Astronomy from Origin to Eighteenth Century  7 The Origin   7 Ancient Mathematics and Astronomy Until the End of the Roman Empire  10 Arab Mathematics and Astronomy during the Middle Age   19 From Renaissance to XVII Century   24 The XVIII Century   26 Academia and University   29 3 Ancient  Visions of the Cosmos: Orienting, Classifying and Modeling 33 Orienting in the Sky: The Constellations   33 Stars Catalogs  36 The Motion and the Shape of the Cosmos   37 Eratosthenes, Eudoxos, Callippus and Aristoteles   38 Hipparchus and Apollonius   41 Claudius Ptolemy  43 Humanism and the Rebirth of the Studies   49 The Invention of the Solar System: Nicolaus Copernicus   51

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4 A  Lucky Astral Conjunction 59 In Search of Precision: Tycho Brahe   60 Johannes Kepler: A Journey in the Space   71 Galileo: Looking into the Deep Sky   92 The Cause of Celestial Motion: Isaac Newton  102 5 The  New Vision of the Cosmos107 Shape and Motion of the Earth  108 The Satellites of Jupiter  110 The Motion of the Moon  112 The Tides  114 The Speed of Light  115 The Planetary Periods  116 The Astrological Vision of the Cosmos  117 6 T  he Instruments119 Mechanics 119 Instruments to Guide the Observation  121 Measuring the Time  136 A Challenge Between French and English: Measuring the Meridian and the Time in Navigation  152 7 The  First Astronomical Machine: Antikythera161 The Discovery  162 The Dating  163 Structure and Functions of the Mechanism  164 The Missing Parts: The Motion of the Moon, the Sun, and the Planetarium 171 The Planetarium  172 Construction Technology and Materials  172 Copies and Simulations  176 8 Astronomical  Machines and Clocks from the Arab Times to the Renaissance179 Astronomical Instruments of the Caliphate  179 Renaissance 182 Sphères Mouvant and Globes of Sixteenth Century. Eberhard Baldewein and Oronce Finé  188 Jost Bürgi: Mathematician, Mechanic, Clock Maker, Astronomer  195

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9 T  oward Planetary Machines203 Paving the Way to the New Cosmos: Kepler’s Planetary Machine  204 Ole Rømer: Jovilabium and Planispherium  219 Christiaan Huygens  222 10 Orreries  and Astronomical Clocks229 The Planetary Machine by Thompion and Graham  229 The Dissemination of the New Astronomy in England  231 George Adams  232 Thomas Wright, Thomas Heath, Benjamin Martin  233 James Ferguson  236 The Cometarium  238 Italian Astronomical Machines  239 Francesco Generini and Bartolomeo Ferrari  240 Bernardo Facini: The Planisferologio Farnese  243 Francesco Borghesi and Bartolomeo Antonio Bertolla  251 11 F  rance and Switzerland257 Claude Simeon Passemant  257 Antide Janvier  261 François Ducommun  282 12 Blossoming  in Germany and Austria: The Priestermechaniker285 Bernard Stuart  286 Johan Georg Neßtfells  288 Johannes Klein  292 David Rutschmann, Frater David a S. Cajetano  294 Michael Fras – Frater Aurelius a S. Daniele  297 Engelbert Wenzel Seige  300 Alexius Johann  301 Philip Mathäus Hahn  304 The Universal Chronology  316 13 Chinese  Philosophical and Mathematical Thought321 Kingdoms and Dynasties  321 Philosophical Schools  333 Mathematics 339

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14 Chinese  Astronomy and Astronomical Machines357 The Image of the Cosmos  358 The Arrival of the Jesuits and Western Astronomy  371 Mechanics 374 Astronomical Instruments and Machines  376 Chinese and Western Mathematics and Astronomy  386 15 Design  of a Simple Planetary Machine389 Background 389 Design Constraints  392 Gear Computation  395 The Structure and Construction of the Machine  409 Computer Control System  416 16 C  onclusion423 The Cosmos as a Celestial Machine  423 A Classification of Astronomical Machines  425 When and Where  427 A  ppendix A435 A  ppendix B471 R  eferences479 I ndex493

About the Author

Daniele L. R. Marini graduated in physics. He has been Professor of Computer Science at the Università degli Studi di Milano and Politecnico di Milano until 2014 and is currently retired. He pioneered computer graphics research and education in Italy and founded a start-up company, Eidos, in 1980 (led until 1989) to offer computer animation services. His research field encompasses digital imaging, virtual reality, color processing, and color perception. His commitment with cultural institutions dates back to 1995, when he co-­ designed the first website of Teatro alla Scala, and in 1998, he was a consultant of Museo della Scienza e della Tecnica in Milano, the largest science and technology museum in Italy. From 1997 to 2000, he was curator of the multimedia activities at the Triennale in Milano. In 2016, he presented a novel approach to visual rendering of astro-photographs, and in the last few years, he published several articles on astronomical photography. From 2020 to 2021, he built a mechanical planetarium, and received an award from Orologiko, the Italian forum of horological experts. He authored more than 200 peer-reviewed articles and conference communications. He published the textbooks: Marini D., Corso di Eidomatica: Introduzione alla Computer Graphics in BASIC, Fratelli Fabbri Editore, Milano, (1985), Ferrario M. & Marini D., Computer e Immagini, Ghedini Editore, Milano, (1988), Marini D., et al., Comunicazione visiva digitale: fondamenti di Eidomatica, Addison Wesley, Milano, (2001), and a research book: Rasheed S., Marini D, Rizzi A. Recognition of Colors in EEG: Planning Towards Brain-Computer Interface Applications, LAP Lambert Academic Publishing, Germany (2012). Among recent publications: Marini D.L.R., et al., Perceptual contrast enhancement in visual rendering of astrophotographs, J. Electron. xvii

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About the Author

Imaging 26(3), (2017), Marini, D.L.R. The Planetary machine designed by Johannes Kepler, XLI Symposium of Scientific Instruments Commission, (2022), Marini, D.L.R. La Prima Macchina Astronomica: Antikythera. La Voce di Hora, (2022). He is life member of IEEE and member and fellow of IS&T and has been member of ACM. He is also member of the British Horological Society and of the Scientific Instruments Society.

1 Introduction

The present-day view of the Cosmos is the end result of a long history beginning with the earliest observations of the night sky aimed at understanding the flow of time and the rhythm of seasons. Alexander von Humboldt was the first scientist in modern times to write about the cosmos as a whole and to put forward an integrated view encompassing both the sky and the Earth and its life. Following extensive travelling in South America, Central America, and Asia, across the steppes on the border of China, he developed a comprehensive view as no scientists had ever thought of before. Using the information he gathered about the geography, geology, botany and climate of the places he visited, Humboldt created a picture of the entire natural world, from the sidereal depths to the details of the Earth’s surface. In 1845, he published the first volume of his great work in five volumes Kosmos. Entwurf einer physikalischen Weltbeschreibung (A. von Humboldt, 1845) [Cosmos. Design of a Physical Description of the World]— a veritable Weltanschauung. 1 In this book, I shall not be putting forward any Weltanschauung. The mere amount of knowledge made available today is mind-boggling, so a comprehensive description of the cosmos is beyond the reach of any individual scientist. I mainly focus on the solar system. The contribution made by both religious and scientific thinking is writ large in the history of how we view it. The same holds for the progress of technology and the impact of social organizations such as universities and scientific societies. From the early stages of astronomical observations aiming at merely describing the sky and giving  A worldview.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_1

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names to stars and constellations that expressed complex mythologies, we have moved on to measurements precisely locating the stars on the celestial sphere and to rich star catalogues. The diurnal and annual cycle of the sun and the monthly cycle of the moon were the first celestial motions carefully observed and recorded. They formed the basis for measuring time. As social organization progressed, the commitment to observe the sky was taken by shamans and tribal leaders. They were responsible for governing religious rites and festivities. Accurate observations made it possible to detect more subtle phenomena, more regularities in the movements of celestial bodies, seasonal anomalies, and recurring eclipses over multi-year cycles. Greek mathematics and geometry were instrumental in producing the first proper theories about celestial motions. It was thought that celestial bodies followed perfectly circular paths in the sky. Then the first guesses were put forward concerning their causes. From the very beginning, the mechanic-technician accompanied the scientist, astronomer and priest in charge of religious rites. Manufacturing the instruments for observing and measuring was the task of technicians who sometimes anticipated scientists and always empowered them. They were not mere craftsmen and could design and manufacture sophisticated machines. Unfortunately, since the beginning of Western philosophy, the intellectual prowess and social value of mechanician were underestimated. As Paolo Rossi, a distinguished physicist, pointed out, “The Greek term banausía means mechanical art or manual labour. Callicles, in Plato’s ‘Gorgias’, says that the machine builder is to be despised and called bánausos (Rossi, 2016, pos. 490) so as to offend him. No one would like to give his own daughter in marriage to one of them.” Only in the Renaissance, this attitude was reversed. Only philosophers such as Giordano Bruno, Francis Bacon, and René Descartes— Rossi remarks—could appreciate and value the work of technicians. Still, some misunderstanding of the intellectual and scientific content of mechanical work, as opposed to the liberal arts, remains to the present day. It surfaces, for instance, in some philosopher’s contrasting the sciences and humanities and downplaying the former as being too close to the technician’s job. In this book, I intend to show that the scientists who built the modern view of the cosmos were the first to appreciate what technicians were able to do and were happy to work in tandem with them, which had the effect of generally upgrading their social status. The relationship between mathematics and geometry, on the one hand, and mechanics, on the other, has long remained in the dark. It was Newton who clearly saw that coupling mechanics and mathematics could bridge the gap.

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The notion of rational mechanics, later developed into a fully fledged discipline, is due to him. According to Greek mythology, Prometheus steals technology from the goddess Athena and gives it to man together with fire. Thus, homo faber could master nature against the will of the gods. In a different vein, recent work by the Chinese philosopher Hui Yuk (2021) highlights the relationship between technology and science in the Chinese tradition. Daoist thought postulates perfect harmony between the gods, man, and nature. In Hellenistic times, Greek thinkers collected and systematized astronomical knowledge originated with the Egyptians and Babylonians. Euclid’s geometry was the basis for the construction of astronomy. Their bequest is still with us today. An interruption in this path occurred when the expansion of the Roman Empire caused Hellenistic science to be almost forgotten in the West. Finally, the Renaissance revamped it, when the writings of Hellenistic scientists were rediscovered and translated from their Arabic sources. Thus, mathematical and astronomical science could start again on a new basis. The discoveries of Nicolaus Copernicus, Tycho Brahe, Galileo Galilei, Johannes Kepler, and Isaac Newton were no longer limited to describing the cosmos. Rather, they sought to explain its workings and predict its future. Even though the Catholic church put all its might into fostering the cosmological views in the religious texts, the clergy often engaged in astronomical research. At that time, there was a widespread belief that the world was the product of divine creation and would forever remain mysterious. Many scientists, however, argued that trying to understand the motions of celestial bodies is a step toward finding the very meaning of creation. As the ability to describe those motions improved, they searched the Scriptures for some foundations, only to find inconsistencies between religious faith and scientific thought. Some strove to make the latter compatible with the Scripture, while others simply ignored the apparent inconsistencies. Attempts at resolving this conflict mostly failed. Galileo, for one, was condemned by the Holy Office. 2 The leading characters in this quest were the great technicians who could produce instruments, such as quadrants, astrolabes and armillary spheres, and particularly clocks, ever more finely tuned. Time-measuring instruments embody a vision of the cosmos in as much as they correspond, at least in their early stages, to a model of the Earth’s rotation cycle marked by hours, minutes, and seconds. Planetary machines, too, represent portions of the cosmos restricted to some celestial bodies and their  Religious congregation founded by Paul III in 1542 to combat the Lutheran heresy.

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motions. These devices were also tools for learning, computing, predicting, and popularizing. As a computer scientist, I consider that the Antikythera machine shows the Hellenistic conception of a computable cosmos. Confronted with some of the masterpieces of great mechanical and artistic values, such as the astronomical clocks of the Baroque age preserved in museums, one wonders why such mechanisms were needed to depict the complexity of the cosmos. The drive to represent the cosmos mechanically was probably due to two reasons. The first reason was the need to predict celestial phenomena, such as a solar or lunar eclipse, that were believed to have a direct bearing on human lives. Knowing them in advance could have a great impact on political and religious affairs. Astrology was, for a long-time part and parcel of astronomy. Physicians were required to be steeped in the discipline of studying the effects on a person’s character of the configuration of planets at his or her birth. Accurate calendars were also needed for practical purposes, e.g., in order to know when exactly to collect wood before the first snow fall and when to begin sowing in spring. Machines that could greatly simplify the lengthy calculations required and supersede the perusing of astronomical tables were highly valued. The primary reason for inventing and building astronomical machines is to be found in the need to simplify the scientist’s work. This was certainly the motivation that drove Johannes Kepler—to whom we owe the first basically accurate picture of the solar system and its motions—to make the first attempt in modern times to design and build a mechanical planetarium. The second reason lies in the need to give laymen some knowledge of the cosmos. In the age of the Enlightenment, the power of the higher aristocracy began to make room for the bourgeoisie and lower aristocracy. Wealth and force were no longer sufficient to preserve the traditional privileges, and knowledge acquired a new social meaning. In the sixteenth and seventeenth centuries, the problem of educating the young was acutely felt. The response was mainly to hire private teachers to educate one or two pupils at a time. The following century saw the rise of public education alongside religious institutions. Planetary machines were an effective tool for teaching the principles of astronomy. At first, only the very rich could afford such expensive objects, which were also items of conspicuous consumption. Subsequently, schools and universities, especially in England, became equipped with them. In 1812, Antide Janvier (1812, p. IX), an accomplished watchmaker, edited Huygens’ description of a planetary machine. Being neither a scholar nor a scientist, he wrote in the preface

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I pay my tribute to the Fatherland by publishing machines from which some help may come for the instruction of the Son of the King, toward whom the hearts of all the French are turned.

The industrial revolution, spurred by scientific progress, opened up new fields of research. Advances in optics led to telescopes and optical instruments of better quality as well as to the design and construction of planetary projection machines, some of which are still in use today in public planetariums. Eventually, however, the epoch of mechanical astronomical clocks and star projectors came to an end. Fully computational simulations have taken over. They are still models, though not mechanical and fully mathematical. But, of course, they are also much more accurate. Mechanical planetary machines, also known as Orrery, are now of purely historical and artistic interest. Putting the evolution of scientific and technical thought in historical perspective is a challenging task. One has to keep track of the many stages undergone by theories and artefacts without losing sight of their origin and the frame of mind of those who first invented them. Especially important are the thoughts of those who challenged the received beliefs. For example, how did Copernicus come to question the Ptolemaic foundations of geo-centrism? What reasoning led Kepler to frame the laws of planetary motion? To answer these questions, I have made place for biographic narratives concerning the most notable scholars, who were often living in times of war and amid religious conflicts, so as to place their work in context. Chapter 2 provides a comprehensive overview of the history of mathematics and astronomy, tracing its roots from ancient civilizations such as Mesopotamia and Egypt, through the contribution of Arab culture during the Middle Ages, and the most significant advancements in mathematical science up to the eighteenth century. Chapter 3 delves into the early theories on the motion of the Sun, Moon, and planets by Greek astronomers, from the Classical and Hellenistic periods until the Renaissance, culminating in the formulation of Copernicus’ model of the solar system. Chapter 4 focuses on the contributions of the four major astronomers of the seventeenth and eighteenth centuries, including Tycho Brahe, Johannes Kepler, Galileo Galilei, and Isaac Newton, whose work ushered in a turning point in understanding the solar system. This is further summarized in Chap. 5. Chapters 6–12 provide a detailed account of the astronomical instruments used by ancient astronomers, including the most important astronomical clocks and planetary machines leading up to the beginning of the nineteenth

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century. Chapter 9 specifically describes a planetary machine designed by Kepler but never built, which is interpreted using virtual reality techniques. Chapter 13 seeks to rectify the lack of knowledge on East Asian contributions to astronomy by presenting a synthetic overview of ancient Chinese culture and mathematics. Chapter 14 explores the evolution of Chinese astronomy and astronomical machines up to the last decades of the Ming dynasty when Jesuits brought scientific knowledge and methods from Western countries to China. Chapter 15 delves into the practical aspects of designing and manufacturing an astronomical machine, specifically a Tellurium, reflecting the author’s initial interest in planetary machines. Finally, Chap. 16 concludes the book by outlining a classification of astronomical machines, discussing their common features, differences, scientific and social functions, and a final chronology. Appendix A provides basic concepts of astronomy and a series of tables with the characteristics of the machine examined in the book. This book is ideal for anyone interested in the history of science and technology, particularly astronomers, mathematicians, physicists, and historians of science and technology.

2 Mathematics and Astronomy from Origin to Eighteenth Century

The Origin The construction of astronomical machines is the fruit of three disciplines: mathematics, astronomy, and mechanics. Throughout history, these three areas have developed and grown together, although at times there have been long periods of stasis and decadence, both scientific and technical. Moments of decadence and stasis can be linked to major political upheavals, such as the alternating power struggles in the Middle East between Mesopotamian, Persian, and Greek civilizations, the formation of the Roman Empire and its fall, plagues and countless wars between European kingdoms and princedoms. In modern times, however, European wars were also a great stimulus for scientific research and the development of new techniques up to the first industrial revolution, as they provided a continuous demand for building weapons and machines. During the many centuries of religious and political conflicts, the community of scholars, often religious scholars, for whom the study was reserved, created and maintained contacts and relations either through travels, writing, or, as we shall see, the establishment of Academies. At the same time, these scholars attracted the interest of lords and princes for the help they could offer in enhancing their military capabilities and finding solutions to social needs. In this chapter, we retrace the development of these disciplines in a very concise manner. The history of mathematics alone would require several volumes to set out in full. Here, we are only interested in identifying the key moments and discoveries for the importance they had in the development of astronomy.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_2

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The regions where all this originated are Egypt and the so-called Fertile Crescent, between the Tigris and the Euphrates rivers. These were regions rich in water that allowed for agricultural development, an essential condition for the formation of a structured society with a ruling class, including priests, figures endowed with the ability and knowledge to formalize a cosmological worldview. Historic Periods of Antiquity  Before exploring the history of the development of mathematics1 and astronomy, let us review the subdivision of the periods in the ancient history of Greece, Egypt, and Mesopotamia up to the time of the rule of the Roman Empire, and then the development of Islamic civilization and the Caliphate, for the role it played in the transmission of Hellenistic knowledge. This subdivision will serve to place more easily in time scientists, discoveries, and inventions. As far as Greece is concerned, the first period, which ranges from −60002 to −3100, is the Neolithic, in which traces of the first human settlements with early forms of social organization are recorded. The Bronze Age is dated from −3100 to −1100 (Mynoican Age), while the Iron Age, which runs from −1100 to around −700, is considered, in the terminal period, the era in which geometry emerges. The Archaic period from −700 to −562 is characterized by the structuring of the social body with the formulation of the laws of Draco and Solon. From −561 to −508 tyranny develops, followed, from −508 to −480, by the form of government of the cities (the Polis). It is during this period that the first Persian wars take place, ending in −466 in the middle of the period called classical, during which Greek civilization flourishes and prospers, especially under the rule of Pericles (−461–429). The Classical period, during which democratic forms of government alternated with tyranny, culminated with the conquest campaigns of Alexander the Great (−356–323). The Hellenistic period starts with the death of Alexander, whose empire is divided between various generals, notably Ptolemy I (−357–282) whose dynasty will rule Egypt, and Seleucus (−358–281) whose descendants will rule the Middle East. It is customary to end the Hellenistic period with the year −146 (Russo, 2022), when Greece becomes a vassal of Roma, which becomes first an ally and then completely subjugated in −30 with the death of Cleopatra VII, the  For an exposition of the origins of mathematics, see Gheverghese (2011), Kline (1996).  I will use a mathematical symbolism − and + to denote dates and periods instead of ‘BC’ and ‘AD’. Where it is obvious the dates of the modern era will not be preceded by a sign. Sometimes I will use B.C.E an C.E. instead. 1 2

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last queen of Ptolemy dynasty. As we shall see, Greek mathematics developed particularly in the Classical and Hellenistic periods. Egyptian civilization is characterized by a predynastic period lasting from −40,000 to −3150 after which a series of dynasties followed until the XXV dynasty, which ended in −672. From −672 to −332 (death of Alexander), we reach the XXX dynasty, while the period from −332 to −30 is dominated by the Ptolemy dynasty, whose last king is deposed in −30, and the Roman period begins, lasting until +642 (birth of Islamic civilization). Egyptian writing is found on papyri, subject to slow but inexorable decay, and on engravings in temples and tombs. The decipherment of Egyptian writing, thanks to Jean-­ François Champollion (1790–1832), allowed us to understand the level of mathematical and astronomical knowledge. The Egyptians’ numbering system was in base 10, and in the Fig. 2.1. left, we see the symbols used. The major source of Egyptian mathematics is in two papyri, one at the British Museum, the Ahmes papyrus written around −1650, the other in Moscow dated around −1850. They contain 112 problems with solutions (Gheverghese, 2011, p. 82). In Mesopotamia, a Sumerian period developed roughly from −3500 to around −2400, when the city-states were conquered by the Akkadians. A resurgence of Sumerian civilization is dated to around −2200 under the dominion of the Amorites, who ruled until −1730, when the region fell under the rule of the Babylonians. Sumerians, Akkadians, and Babylonians adopted cuneiform writing, consisting of signs engraved on clay tablets, which allowed for a very long preservation of documents and the recognition of remarkable mathematical ability. Around the eighth century, the Chaldean people, of Semitic origin, mixed with the Mesopotamian peoples and introduced

Fig. 2.1  Left: Egyptian numbering system. Right: Cuneiform numbering

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alphabetic symbolism into writing. Babylonian civilization continued until −550 when it fell under the rule of Cyrus the Great of Persia (−590–530). Other kingdoms alternated during this long interval. It was a period of numerous wars also against the Egyptians. Given its geographical position, the region was a hinge point in trade with Persia, India, and the Far East up to China. It was undoubtedly the Mesopotamian civilization that conducted the most important and systematic astronomical observations. The Chaldeans, connoisseurs of astrology, are credited with initiating astronomical studies, and the name of this people was once used synonymously with Babylonian.

 ncient Mathematics and Astronomy Until A the End of the Roman Empire What idea of the sky dominated the thinking of these ancient scholars? And from where did it originate? And how did the mathematical tools necessary for a deeper knowledge develop? To find an answer we have to go back to the oldest civilizations that developed in the Middle East – today’s Turkey, Syria, Iraq and, in Near Asia, Afghanistan Turkmenistan, Kyrgyzstan, Pakistan – all hinge regions between the civilizations of India and China and those of the Mediterranean. Peter Frankopan (2015) recalls that the heart of the world, at the beginning of Rome’s great power, was in Persia, at the center between the Chinese Far East and the European West. These were the hub of trade, fostering the exchange of knowledge, first and foremost, the knowledge needed to calculate – based on arithmetic and geometry – the quantities of goods and commodities traded by merchants traveling the ancient world. At the time of the Persian Wars that took place between −500 and −479, described by Herodotus of Halicarnassus (−484–425) in his Historiae, the Greek polis came into competition with this power, which was defeated by Alexander the Great in −331, whose empire extended from the Indus River in the east to Egypt in the west. With the emergence of Roman power, attention to Egypt increased: perhaps the most prosperous and rich region due to the known power of the flooding of the Nile to feed agriculture, providing very abundant harvests. Between −50 and − 30, Rome’s attention on Egypt was so strong that it resulted in a series of wars against the Ptolemy dynasty and in alliances, even amorous ones, with Cleopatra by Caius Julius Caesar (−100–44) and Marcus Antonius (−83–3), until the final defeat by Gaius Octavius Thurinus (who as emperor

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took the name Augustus) (−63 + 14), who crowned the conquest of those lands. This opened an era of great world trade (a very ancient globalization) that would enrich and contribute to the development of Western Europe. Valuable minerals and stones, grain and cereals from the west were traded with silk from China and other goods such as minerals and precious stones from all over the east. Archaeologists have found tangible evidence of these trades in the cuneiform tablets of the Assyrian and Babylonian societies and, in later times, in coins minted by the Greeks and later by Rome. But these trades did not originate with the arrival of the Roman empire: they dated back at least 2000 years earlier, with the establishment of the Assyrian, Babylonian, and Egyptian civilizations that established state organizations with laws and government structures. In Babylon between about −1800 and −1750 reigned the ruler Hammurabi, who promulgated the first code regulating life among citizens. And in those regions, cuneiform writing was invented. In order to manage and control these states and regulate trade, basic mathematics was needed, consisting of elements of geometry for calculating areas and volumes and arithmetic for calculating interest on loans, dividing up the shares of a harvest or assessing taxes. The cuneiform script included symbols to represent numbers: the very number of cuneiform characters directly indicated the digit. If we look at Fig. 2.1 right, we see the number system. The units were represented by one symbol to write numbers up to 10 and a second symbol to denote successive groups of 10 units. Numbers larger than 59 were written by multiple symbol blocks, a kind of place-value numbering. What is interesting is that 60 has many divisors, which makes it easier to calculate submultiples. This numbering system was adequate for simple arithmetic and also for dividing time into intervals, and for the first astronomical observations made in Mesopotamia, which date back to the -1700s. These were observations of events such as eclipses or planetary conjunctions that were recorded to look for periodicities and extrapolate them to predict subsequent phenomena.3 The ability to extrapolate, on the other hand, required knowledge of elements of algebra. In the Seleucid period after −300, the first tables of planetary motions had been compiled with great care, recording the speed of the Sun’s and Moon’s motions with a precision that made it possible to predict eclipses or new moons with an error of a few minutes. The length of the year, the seasons, and the sidereal year were now well  In the Appendix there are examples of simple astronomical calculus, to better understand the level of knowledge and mastery of calculus and trigonometry required to do the main observations of the motion of celestial bodies. 3

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known, and the calendar was linked to the lunar months (a calendar that was used until −45 when the Julian calendar was adopted); the constellations of the Zodiac had also been identified. The division of the circle into 360 degrees also dates back to the Babylonian civilization. A hypothesis of how they arrived at this measurement was proposed by Otto Neugebauer (1941, p. 16). The time marked by the days had an irregular pattern, in fact, the length of the days varied as the seasons changed. Around −2400, the Babylonians began to use a distance measure4 called danna, which can be translated as ‘mile’, roughly corresponding to 7 miles today. It was natural to use this measure to express the length of an entire day, as traveling 12 danna took the time of an entire day. Matching a complete revolution of the sky with 12 danna became natural. The danna in turn was divided into 30 units called uš, and it may still have been natural to divide the circumference of the sky into 360 parts = 12*30. The number 12 takes on great importance: we find it in the number of zodiacal constellations and in the number of lunations in the course of a year, albeit approximated and varying from year to year. Knowledge of arithmetic also led the Babylonians to solve the problem of division between two whole numbers with a technique that anticipated continued fractions, which we will describe extensively in the Chap. 15, on the design of a mechanical planetarium. According to Neugebauer, the development of an early form of algebra occurred in the Sumerian culture with the arrival of the Semitic peoples: The origin of mathematics in an algebraic form can be explained by a historical event, namely the complete replacement of the Sumerians by the Semitic populations, albeit in different ways. The main point is the substantial difference between the languages of the two types of population and the fact that the Semites used the Sumerian script to express their own language. Sumerian writing works with single signs for single concepts (so-called ideograms), derived from pictorial writing. The Semites used these signs in two different ways: the first in the old sense as a representation of a single concept, and the second as pure symbolic (syllabic) sounds to compose their own words phonetically. The first possibility of expression corresponds to the field of mathematics in exactly the same way as for our algebraic notation: instead of writing ‘length’ with six letters, it suffices to write l; instead of writing ‘plus’ or ‘addition’, it suffices to use the single sign + (Neugebauer, 1941, p. 23).

  The relationship of time to space is a primary concept in Einstein’s relativity, see Stephen W. Hawking (1988). 4

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The Egyptians, as mentioned, also knew the mathematics necessary for the major practical purposes, with a greater depth of certain elements of geometry, undoubtedly necessary for the determination of the boundaries of agricultural properties that could be changed by the flooding of the Nile, for collecting taxes, or for the construction of buildings. Neither the Babylonians nor the Egyptians had yet developed the conception of mathematics as a method of abstraction: mathematical objects, such as the point or the line, were conceived as real objects. The straight line was the length of a side of a field or a stick, the point was a small sphere. Seasonal periodicities led the Greeks to attempt to formulate methods of forecasting meteorological events.5 Giovanni Schiaparelli (1835–1910) notes (Schiaparelli, 1892) that systematic observations to determine with certainty the periods of the seasons and the intermediate stellar arrangements made it possible to detect discrepancies between the lunar and solar cycles: if on a certain day of the year, e.g., at the spring equinox, there is a new moon, the following year on that same day there will be no new moon. Lunar and solar calendars do not correspond exactly, marked by the seasons. The natural calendar of the succession of seasons, marked by the appearance of different stars and constellations in the sky, and by the places where the sun rises and sets, does not have a sufficiently solid basis for making predictions. It is not enough to identify the day on which Arcturus appears at sunset, or Orion is at the apex of the meridian. An idea of how the seasonal and meteorological cycle was viewed in relation to celestial configurations is provided by what Hesiod (-VIII -VII cent.) wrote in the poem Works and Day, of which I recall some verse6: He who stands lazy, does not fill the barn. The diligence of the good farmer increases the harvest, and he who puts off the work of the land overnight, must struggle with greater and greater difficulties. At the time then that almighty Jupiter brings down the autumn rains, when the burning power of the sun no longer covers us with sweat, and the skin is changed, and light is felt in the limbs, and the star of Sirius for less time of day looms over the heads of men fed for death, but for longer shines in the nocturnal silences, when the trunks of the trees grow hard to the cutting of the axe, and the leaves fall and the vegetation comes to a halt, then, remember, it is good time to cut the forest, and with the wood make your instruments.  “The Parapegma or Astro-Meteorological Calendars of the Greeks and Romans are among the most curious, if not the most important relics of ancient science; they constitute the first rational attempt to arrive at an approximate weather forecast.” (Schiaparelli 2010, p. 274). 6  Hesiodus, The Works and Days, https://books.google.it/books?id=gyN_GEQ3VCUC. Accessed May 2022. Tr. from Italian by the author. 5

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Table 2.1  Weather description in parapegmata. Selection from various authors by Schiaparelli, cit. Translation from Italian by the Author Eudoxus Caesar Philippus Democritus Democritus Caesar DOxiteus Eutemones Egyptian Hypparcus

Day 72 Day 111 Day 139 Day 156 Day 168 Day 200 Day 226 Day 295 Day 310 Day 363

Rains, thunders, and strong wind Erratic winds, rain, thunders Bad weather, storms Atmospheric disorder also on the sea Thunders and lightnings, rain or wind or both Strong southern wind: on sea thunders and light rain Nice weather, sometimes west wind Rain with hail Mistral with drizzle or southerly wind with thunder Fighting winds

Other scholars ventured into these studies, and the results of their studies led to the association of meteorological and astronomical events with the calendars that were gradually being refined, the parapegma. Table 2.1 shows fragment of parapegmata, selected by Schiaparelli from various authors and on different days (Schiaparelli, 2010, p. 321). The parapegma was published to inform the community, it was not a forecast, but indicated what kind of event could be expected at a given time in the annual calendar. Let us also recall that the notion that moonlight is produced by sunlight dates back to Parmenides of Elea, who was probably born in the mid-fourth century and died around −460. As for the cause of lunar eclipses, it probably dates back to Anaxagoras (−496–428). But these discoveries raise interesting new questions. How did these ancient astronomers manage to record the events over a period of many years, relying only on observation of the sky, the cycle of the seasons, with crude instruments to measure time? What social organization existed to allow these people to be exempted from manual labor in order to concentrate on studying the sky? Even the precession of the equinoxes had been hypothesized, but in order to be able to detect it accurately, dates and times had to be recorded for very long periods, since over 150 years, the precession is 2°! We can assume that the observation of the sky was done by priests, institutional figures whose organization often survives political upheavals, and is certainly even more stable when the central power of a royal dynasty extends over centuries. This may have been the case with the Babylonians and other ancient civilizations, such as the Chinese. In the Classical period, the presence of the Greeks spread to the eastern Mediterranean area, particularly along the coasts of present-day Turkey with the important center of Miletus, which was the terminal of trade with the Far East, and toward Italy with Crotone, which hosted the school established by

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Pythagoras, born in Samos around −580. Pythagoras is credited with the well-­ known theorem on right-angled triangles, but it may already have been known to the Babylonians. Pythagoras became interested in the relationships between numbers and made them the language to describe music, inventing musical scales from the division of the monochord into fifth (a 3:2 ratio) and octave intervals (a 2:1 ratio). The cosmological conception of the Pythagorean school did not have the characteristics of the earlier Assyrian and Babylonian or Egyptian cosmology. Rather, it was based on the harmonic principles of musical ratios, imagining fire at the center of the world around which the stars revolved, a conception that led many to imagine ‘celestial harmonies’, including, as we shall see, Kepler himself. Toward the end of the Classical period, the school of Plato (−428–348 c.) emerged, which undoubtedly gave a strong impetus to mathematical studies, especially in terms of methodology, although no specific mathematical discoveries can be attributed directly to Plato. Although they bear his name, the Platonic solids were conceived by others, as Euclid (-IV -III century) reports in the 13th book of the Elements (a fundamental treatise on Geometry, which still serves as a model for the formal mathematical method, based on axioms and demonstrations), pointing out that the cube, the pyramid and the dodecahedron can be traced back to the Pythagoreans, while the octahedron and the icosahedron are attributed to Theaetetetus (−415–369). Theaetetus is the protagonist of a dialogue by Plato, and studies on incommensurables and irrational numbers are attributed to him. These problems were further explored by a pupil of Plato’s, Eudoxus of Cnidus (−408-355c.), who dealt with incommensurables only between geometric quantities (e.g., the circle and its circumference or area), thus causing a separation between the two disciplinary fields of geometry and arithmetic. Alexander’s conquests and the subsequent reign of the Ptolemies ushered in a flourishing era in which Alexandria itself became the main center of scientific and mathematical thought of the Hellenistic period. In this city, Ptolemy II Philadelphus (−285 to 246) established a Museum and Library that became the cultural center of the Mediterranean world. In the Hellenistic period, the cosmological model was consolidated by Aristotle (−384–324), which we know from the accurate description by Claudius Ptolemy (+100+175 c.) in his Almagest (Ptolemy, 1515). Before Ptolemy, a sophisticated mathematics developed, consisting of geometry, elements of trigonometry, arithmetic and some concepts of algebra. Greek trigonometry was initially limited to the notion of the chord of the circle to represent an angle. Later, spherical trigonometry and the notion of the sine of

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Fig. 2.2  Left: The chord of an angle. Right: Stomachion

an angle was invented by Hipparchus of Nicaea (−190–120), although we know it through the writings of Ptolemy and Menelaus of Alexandria (+70+140 c.), who are credited with the development of spherical trigonometry. Let us see how the relationship between chord and angle was conceived (Fig. 2.2 left): a circle is divided into 360°, and a diameter is divided into 120 parts; each part of the circumference and diameter is further divided into 60 parts (thus we will have the arc-minute and the arc-second of an angle). In a circle, an arc AB subtends an angle 2α, Hipparchus develops a method to find the number of units of the chord AB, whose half AC corresponds to the sine of the angle α, and the segment OC is the cosine. The works of Apollonius of Perga (−262–190) on conics (ellipses, hyperbolas and parabolas) were themselves essential to the understanding and description of celestial motions. He too, like Euclid, adopted the mathematical method whereby mathematical propositions must be proved by a rigorously deductive method. Archimedes (−287–212), one of the most genial mathematicians, lived in Syracuse and part of his life is known from a biography written by Polybius (−206–118 ca.) in which his role in the Second Punic War, during which he invented and built war machines, including the famous burning mirrors, is narrated. Archimedes wrote several treatises, few of which have survived. A palimpsest tracked down in Rome in 1883 and later in Istanbul in 1906, dating back to the tenth-century and containing prayers, revealed some of Archimedes’ writings: the treatise on method, the one on floating bodies and a fragment of the Stomachion, a mathematical game consisting of composing

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a square from 3 quadrangles and 11 triangles (Fig. 2.2 right) (Netz & Noel, 2007, pos. 3447). Other surviving works are the treatise on spirals, conoids and spheroids, on the measurement of circles, on the quadrature of the parabola, and one on cattle. There are also fragments on other topics, including the reflection of mirrors and the surfaces of irregular bodies. A variety of topics cover not only geometry, but also physics problems such as that on floating bodies. On Archimedes’ contribution to the study of astronomy T.L. Heath writes (Heath, 1897, p. XXI): Another invention was that of a sphere constructed to imitate the motion of the sun, moon and the five planets in the sky. Cicero in fact saw this instrument and gave a description of it, stating that it represented the periods of the moon and the apparent motion of the sun with such accuracy that it could even show eclipses of the sun and moon (at least for a short period). Hultsch speculates that it was moved by water. We know from Pappus that Archimedes wrote a book on the construction of such a sphere (περι σφαιροποιιας), and Pappus speaks in one place of “those who understand the construction of spheres and create a model of the heavens by means of the circular motion of water”. In any case, it is certain that Archimedes dealt with astronomy…. Hipparchus says “from these observations it is clear that the differences between the years are on the whole small, but as far as the solstices are concerned, I think that both Archimedes and I have wandered the length of a quarter of a day in our observations and deductions”. It therefore emerges that Archimedes studied the problem of the length of the year… Macrobius says that he discovered the distances of the planets. Archimedes himself describes in his treatise on sand an apparatus with which he measured the apparent diameter of the sun, or the angle subtended by the eye.

I quote the description of this machine by Marcus Tullius Cicero (−106–43) from De Republica Liber I: (21) And Filo: “I will not tell you anything new or anything that I have pondered or invented; I am only reminded of C. Sulpicius Gallus, a man among the most learned, as you well know. In the days when he was said to have seen a prodigy of this kind, he happened to be with Marcellus, who had been consul with him, and ordered the sphere to be brought which his ancestor had taken, after the capture of Syracuse, from that very rich and ornate city, the only prey he had wanted to bring back home. Of this sphere of which I had heard so much, in view of the glory of Archimedes, I was at first a little disillusioned; for it was far more conspicuous and distinguished to the eye of the vulgar, that other sphere of Archimedes, which Marcellus had placed in the temple of

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Virtue; but as soon as Gallus had begun to explain to us the meaning of the work with the deepest doctrine, it seemed to me that in that Sicilian there was a genius far higher than any other human genius. In fact, Gallus told us that other sphere, solid and firm, was an earlier invention of Archimedes, and that Thales of Miletus had first given the model, which was then decorated by the Cnidius Eudoxius, a disciple of Plato, it was said, and that all this descriptive ornamentation, taken from Eudoxius, was many years later exalted in verse by Aratus, who had no astronomical training but only a certain poetic talent. However, a synthetic rotation, including the motion of the sun and moon and the five stars that are called wandering, and almost vagabond, could never, Gallus explained, have been reproduced in that primitive solid sphere, and therein lay the admirable side of Archimedes’ invention: he had found a way to reproduce in a single rotation the unparalleled motions of the stars and their varied travels. As Gallus made this sphere move, one could see the moon succeeding the sun on the earth’s horizon at each turn as it does in the sky, and the same disappearance of the sun from the sky and the same positioning of the moon in the earth’s shadow occurred as soon as the sun was on the opposite side…7

Some of contributions of Archimedes, such as concave mirrors, also formed the basis for understanding the principles of geometric optics. Overall, Archimedes’ role in the birth of modern science is primary, as Reviel Netz (Netz, 2004, pos. 311–313) comments: In a word, we can say that Archimedes' method of measurement formed the basis for the studies that led to calculus, while statics and hydrostatics formed the basis for the studies that led to mathematical physics. In this sense, Archimedes' texts are for us, quite simply, modern science.

Around the -I century, we date the origin of the so-called Alexandrian period during which Pappus of Alexandria and Ptolemy studied and perpetuated the mathematics developed in the previous centuries. Pappus wrote the Collectiones Mathematicae in which he collected a large number of theorems of geometry, and in the eighth book of the Collectiones he introduced the subject of machines. Ptolemy is the one who deepens the previous studies of astronomy and expounds Hipparchus’ theories in a structured manner by formulating the model of epicyclic motion. The positions of the planets in the course of time had been collected in tables and Ptolemy’s tables, set out in the Almagest, made it possible to predict the positions of the stars and eclipses by

 Tr. by the author.

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reporting their coordinates in the course of time: the ephemerides.8 These tables were revised in 1272 by a group of scientists at the court of Alfonso X of Castile (1221–1284), for which reason they were called the Tabulae Alphonsinae9, and became the reference for astronomical studies and navigation for a long time.10 An anecdote goes that Christopher Columbus used these tables to predict an eclipse of the Moon on 29 February 1504, gaining great respect from the Natives (Capaccioli, 2020, p. 42). The Alexandrian period ends with the dispersion of the Alexandrian School.11 This disaster opened a long period of stagnation, of oblivion, which began with the consolidation of Rome’s dominance. Many discoveries were forgotten until the end of the Middle Ages. Among the causes, as L. Russo points out (Russo, 2013, pos. 2669), with the dispersion of the Alexandrian school ‘the master-pupil chain was broken’.12 In 415, Hypatia of Alexandria (355–415), the female mathematician and head of the philosophical school, was killed as a result of the conflicts between Christians and Pagans who still followed ancient cults, leading to the final destruction of the School of Alexandria and its library.

 rab Mathematics and Astronomy during A the Middle Age It is a common idea that the Middle Ages was a dark age, in which science and technology was abandoned. This is a reductive idea, let us simply recall how mathematics, astronomy and machine building were nevertheless subjects of study continuously cultivated in medieval monasteries until Humanism and the Renaissance. Undoubtedly, the scientific method was abandoned and criticized in the Christian West, while the philosophical thought of Aristotle and Plato continued to be cultivated, neglecting mathematics as an  From Greek εφημερισ daily.  Tabulae Alphonsinae were ephemerides published around 1252 by astronomers commissioned by the King Alfonso X of Castilia e Leon, said The Saviour (el Sabio). The astronomers update Ptolemay’s tables. 10  Robert Harry van Gentt created a web page with which one can perform a simulation of the calculation of ephemerides according to the Almagest: https://webspace.science.uu.nl/~gent0113/astro/almagestephemeris.htm Accessed June 2022. 11  Founded in the -III century and progressively looted from −48 and finally completely destroyed in +642 with the Arabic conquest. 12  “The master-pupil chains, essential for the transmission of the scientific method, had been broken for centuries and this could not have failed to have devastating effects; it is therefore not surprising that Ptolemy was often unable to recover the ancient knowledge and, above all, was foreign to the method by which it had been acquired.” (Tr. by the author). 8 9

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instrument of knowledge.13 Knowledge of physics, mechanics and optics is lost, and the only conceivable astronomy is that described by Ptolemy. Mathematical and astronomical culture remains alive, however, among Arabs and Persians by virtue of the translation into Arabic of the works of Euclid, Apollonius and Archimedes. Ancient Arab civilization14 is generally divided into four phases: the first expansion with Muhammad (570–632), who in 622 started the Hegira15 by moving from Mecca to Medina; the period of the Orthodox Caliphate, until 661; the Umayyad period until 750. The next period, was the Abbasid Caliphate, whose capital was Baghdad, and during which the expansion of the empire extended as far as Spain in the west (including Sicily and southern Italy) and the Bactrian regions as far as the Indus in the east. In 1291 the Ottoman settled in Anatolia and took over the Abbasid in 1517. The role of Arab culture16 from the seventh to the fourteenth-century is crucial because it made it possible to collect a significant part of the writings of the Classical, Hellenistic and Alexandrine periods, enabling a revival of scientific studies. During these centuries, Arab culture contributed to the advancements in mathematics, gathering what came from ancient Greece and the contributions of Mesopotamian and Indian cultures. We usually consider the major contribution of Arab culture to be the translations in Arab of Greek, Mesopotamian, Persian and Syrian texts. Caliph ‘Abd al-Mâlik (685–705) decided to adopt Arabic as the official language of the empire, into which ancient works would be translated. Initially, translations and studies concerned practical problems in arithmetic and astronomy. In the Catalogue published by Ibn al-Nadîm (d. 995), we read that the first stimulus for the work of translation is due to the Abbasid Caliph al-Ma’mûn (813–833), who is said to have dreamt of Aristotle recommending him to pursue the ‘good’. This dream convinced him to send a delegation of translators to Byzantium in search of Greek scientific and philosophical manuscripts. This founding myth actually came about after ­al-­Mansûr (754–775) had already inaugurated this fundamental tradition. al-­Ma’mûn founded an academy, called the House of Wisdom, headed by  For an in-depth analysis, read Lucio Russo’s work (Russo, 2021).  I prefer to use the word “arab” instead of “islam”, since the community established during the Caliphate included many different ethnic and religious groups, like Persian and Jews to the east and southern Europe population to the west. The unifying factor was the arab language, that played the role of Latin until the end of the Renaissance among the European scholars, or English in the present days. 15  The word Hegira means migration. In Arab dating, years are measured from 622 onwards by adding the notation AH (Annus Egirae) or EH in English. 16  A concise documentation particularly focused on Arab sciences is Djebbar et al. (2005). 13 14

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Hunayn ibn Ishaq al-Ibadi (808–873), who excelled in Greek and became the most celebrated translator of the works of Plato and Aristotle and translated the works of the founders of Greek medicine: Hippocrates, Galen and Dioscorides. Linguistic and territorial unification absorbed the remaining cultural center such as Alexandria in Egypt, the city of Gundishapur in Persia, some centers in Mesopotamia and Syria in whose language there is the first partial translation of Euclid’s Elements that has come down to us. On the other hand, with the Umayyad Caliph al-Walid (705–720), the first libraries were born, which immediately became not only centers for the collection of books and manuscripts, but also meeting places for scholars and a forge for the development of new knowledge. It is also worth mentioning that during al-Mansûr’s reign the scholar Muḥammad ibn Ibrāhīm al-Fazārī from India translated astronomical tables into Sanskrit, known as Sindhind, relating to the movement of the stars, with equations established from tables that had the accuracy of half a degree, which made it possible to predict eclipses and more. This treatise became the basis of astronomical studies. During the reign of Hârûn al-Rashid (785–809), Ptolemy’s Almagest was translated from a text found in Cyprus. All these translations were frequently revised, sometimes re-translated or corrected and frequently transcribed. The territorial and administrative unity of the Arab Empire favored the development of cultural exchanges and trade in the Mediterranean, the Black Sea, the Indian Ocean, and as far as China, where Islam religion spread since the Tang Dynasty. These exchanges, also empowered by the tradition of pilgrimages, increased wealth and prosperity and financed the establishment of center of knowledge. All this constitutes fertile ground for the development of Arabic science. The theory of numbers was only partly constructed by Greek mathematicians, who favored geometry, of which trigonometry constituted a calculation tool, albeit limited to the function of the chord of the angle, to meet practical calculation needs. The knowledge of algebraic calculus was expanded with the contribution of Muhammad ibn Mūsā al-Khwārizmī (780–850), from whose name the word algorithm derives. He wrote a treatise on algebra and introduced the decimal number system, taking the concept of zero from Indian mathematics. The foundations of Greek mathematics, which was already based on the deductive method of Euclid’s Elements, were enriched with the concept of algorithm by a method of calculation that could in turn be verified. Ibn Jabir ibn Sinan al-Battani (858–929) enriched trigonometry by

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introducing the notion of tangent and cotangent, while the sine was introduced from India.17 In the field of optics, the contribution of Ibn al-Haytham, also known as al-Hazen (965–1040) was of great importance. He based his idea of optics on the concept of light rays and conducted experiments on the camera obscura. Optical studies flourished as early as the end of the ninth century with the work Kitab al_shifa (Book on Healing) by Ibn Sina (980–1037)), also known by his Latin name Avicenna, in which he dealt with the phenomena of reflection and refraction, later refined and deepened by Kamal al-Din al_Farisi (1267–1319), unifying the geometric optics of al-Haytham. We see that the role of Arab civilization cannot be considered as a mere preservation or translation activity. Gradually, an interweaving emerges between Arabic science and that of Christian countries. The moment when the legacy of ancient Greece began to be collected in Europe was around the twelfth century by virtue of the Arab presence in Sicily and Spain, with the contribution also of Spanish Jewish scientists. Previously, the relationship with European Christian civilization had been conflictual with little cultural exchange. The translation from Arabic to Latin of the Almagest by Gerard of Cremona (1114–1187) dates back to 1175, preceded by the astronomical studies of Abraham Bar Hyya (c.1070–c.1130) in Catalonia. Leonardo Pisano known as Fibonacci (1170–1240) in his Liber Abaci introduced to Europe the decimal positional number system learned from Arabic writings, which he called Modo Indorum, and described how to perform the fundamental operations. He is credited with the sequence of integers – known as the Fibonacci sequence – which he devised while studying the annual evolution of rabbit populations, and which was already known to Arabs and Hindus. The Arab contribution to astronomy is significant. Although it does not bring advances in the theory of the cosmos, it focuses on the problems of lunar motion and lunation predictions. In fact, we must remember that the Arab used a lunar calendar, and the prediction of the new moon is of fundamental importance in regulating civil and religious life, along with the determination of the hours of prayer and the ability for anyone to locate the direction of Mecca in order to correctly orient themselves for pray. The determination of the month of Ramadan, of fasting and prayer, has the same

 The etymology of sinus derives from the Sanskrit ardhajya meaning ‘half string’, a word that was translated and abbreviated in Arabic as jyb and the absence of vowel symbols led to the pronunciation jayb meaning ‘gulf ’, which in turn was translated into medieval Latin as sinus. (Gingerich, 1986, p. 77). 17

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importance as the determination of the dates of the Jewish Passover and Christian Easter. Arab astronomers organized the lunar cycle in relation to the sky by identifying 28 mansions, constellations in which the Moon is situated as the days progress, called al—manazil (moon stations). The principle of the celestial organization in lunar mansions is also found in Indian culture, called nakshatra and we will meet it again in Chinese astronomy. For these purposes, the Ptolemaic model was sufficient, but there was still the problem of measuring the position of the Sun along the ecliptic, using spherical trigonometry. That is, it was a simple matter of calculating the sides or angles of a spherical triangle whose extremes are the celestial north pole, the Zenith and the Sun. Many stars are named in Arabic, the result of systematic observations also conducted by al-Khwārizmī, who published astronomical tables, now lost, but of which a Latin translation exists. Another astronomer, al-Sinjārī (945–1020) conducted many observations on the rotation of the Earth. From data collected by Ptolemy the precession of the equinoxes was estimated to be 1° every hundred years. Five centuries after Ptolemy’s time, the shift was much greater: Arab astronomers measured a shift of 11° of the equinoctial point and another 11° of the summer solstice. These measurements led Arab astronomers to modify the method of determining the date of the solstices, which was affected by inaccuracies due to the absence of detectable shifts in the Sun’s position for several days. To make the detection of solstitial points more robust, they brought forward the date of determination not to the peak of the solstice but to the beginning of the solstitial season, when at dawn it was easier to observe the position of the Sun with respect to the fixed stars: the heliac rising method. Even more relevant are the contributions of Nasir ad-Din al-Tûsî (1201–1274) who critically addressed the Almagest by highlighting the limits of the equant hypothesis (see Sect. “Claudius Ptolemy”), anticipating by 300 years what Copernicus did. Nasir ad-Din also demonstrated a theorem known as ‘the pair of al-Tûsî’, in which he shows that the motion of a circle within a circle of double radius produces rectilinear motion. This demonstration was very successful in undermining the Aristotelian idea that the perfection of the heavens was linked to circular motions and spherical shapes, devaluing rectilinear motions unworthy of celestial perfection. Nasir ad-Din wrote a treatise in which trigonometry emerged as an autonomous mathematical discipline, and consolidated spherical trigonometry, which was essential for being able to associate a coordinate system with the heavens, by perfecting

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the methods for calculating tangent and cotangent. In Azerbaijan, he had built in 1259 an astronomical observatory, the Maragheh Observatory.

From Renaissance to XVII Century The cosmology that was universally accepted in the Christian West at the end of the Middle Ages was based on Aristotelian philosophy, permeated however with Christian thought, first and foremost by Thomas Aquinas (1225–1274), the theologian who harmonized Christian doctrine with Greek-Alexandrian philosophy. We attribute to Dante Alighieri (1265–1321) the most refined vulgarization of this cosmology in the Divina Commedia (Fig. 2.3 left). Dante was certainly influenced by the translation of a ninth century work by al-Farghani of Baghdad, whose main work Jawami or Elements, a simplified description of Ptolemy’s work, was translated by Gerard of Cremona (Gingerich, 1986). This vision of the Cosmos was still common during the Renaissance, as we see in the roof fresco by Raffaello Sanzio (1483–1520), that represents the primum mobile, the fixed star sky, put in motion by the hand of the allegorical figure of Urania (Fig. 2.3 right). It is also likely that this fresco has an

Fig. 2.3  Left: Dante’s Cosmos. Drawing by Michelangelo Caetani, 1855. Right: Prime Mover (ceiling panel), by Raffaello Sanzio (ca. 1508)

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astrological meaning, since the sky configuration would be on October 31st 1503, when Pope Giulio II was elected. During the fifteenth-century European scholars began to make new contributions to mathematics and astronomy, gathering the legacy of Arab science. Regiomontanus, pseudonym of Johannes Müller of Königsberg (1436–1476), introduced plane trigonometry. He sought to ground astrology, at the time considered the most important application of astronomical knowledge, on mathematical foundations. Regiomontanus stayed in Italy for a long time, also learning the Greek language, and returned to Germany to stay in Nüremberg where he dedicated himself to astronomical observations with instruments built by skilled local craftsmen. He also engaged in printing the writings of Georg Peuerbach (1423–1461), who had first used Arabic trigonometry to describe celestial geometry. Peuerbach and Regiomontanus had identified limitations and errors in the Ptolemaic ephemerides and undertook to correct the Tabulae Alphonsinae. In the sixteenth-century, the most important mathematical discovery was that of logarithms, by John Napier (1550–1617), who published his work in 1614. Jost Bürgi (1552–1632), whom we will meet again when examining the first astronomical machines, also invented logarithms independently, although he published his discovery 6 years after Napier in 1620. The use of logarithms greatly simplified and accelerated astronomers’ calculus, allowing multiplications and divisions to be treated as sums and differences. Naturally, tables were needed to obtain the logarithmic representation of any number. Also, in the sixteenth-century, Rafael Bombelli (1526–1572) wrote a treatise on algebra based on the translation of the writings of Diophanthus of Alexandria (III – IV) and the works of Muhammad ibn Mūsā al-Khwārizmī. Bombelli considers algebra to be the essential tool for dealing with geometry, overcoming the limits of the descriptive and constructional approach with ‘ruler and compasses’, freeing demonstrations from purely geometric reasoning. Also of great importance was the translation in 1574 of Euclid‘s Elements by Christophorus Clavius (1538–1612). It is worth mentioning that Clavius was First Mathematician of the Pope’s commission that reformed the Julian calendar, leading to the adoption in Catholic countries of the Gregorian Calendar promulgated in 1582. During the seventeenth-century René Descartes (1596–1650), a philosopher and mathematician, laid the foundations of analytic geometry, extending the algebraic methods introduced by Bombelli. Descartes’ era opens a period that is characterized by the principle of reason, the fundament of the Enlightenment and scientific progress. The methods of analytic geometry

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were applied to the study of various curves by architects such as Christopher Wren (1632–1723), who contributed to the reconstruction of London after the fire of 1666. Christiaan Huygens (1629–1695), whom we will soon meet as the builder of the first modern planetarium, proposed a wave theory of light and put the grounds of infinitesimal calculus. Pierre de Fermat (1607–1665), a magistrate by profession, was known mainly for his studies on prime numbers. Still during the seventeenth-century, Europe was shaken by the Wars of Succession and the consequences of the religious conflicts between Protestants and Catholics of the previous century. The complicated and changing politics of the kingdom of France, England, the Habsburg Empire which included Spain and Austria, and the confederated provinces of the Netherlands, made the fate of the scientists uncertain. They tried to collaborate by ignoring the borders between states, while the rulers themselves tried to invite the most knowledgeable to their courts. The Thirty Years’ War (1618–1648) caused countless deaths, devastation, plagues and the migration of entire populations. This period of conflict ended with the Peace of Westphalia (1648) from which a community of states emerged as independent entities of princes and ruling families. To the Netherlands was also granted the independence. There was still a tail end to the conflict between France and the Netherlands, partly due to rights of profession of faith and from which a new rivalry with England arose, while relationships among French and Netherlandish scientists were strained. These conflicts were also linked to the expansion of trade as a consequence of the discovery of America, which saw the emergence of powerful players such as the French, Dutch and English East India Company who laid the foundations for the colonization of the 18th and 19th centuries and who at the same time demanded ever more advanced tools and technical knowledge from scientists.

The XVIII Century The study of astronomical problems has always required a knowledge of mathematics and geometry, and the progress of astronomy goes hand in hand with the progress of increasingly rich mathematics, with trigonometry and logarithms. Isaac Newton (1643–1727) and Gottfried Wilhelm von Leibniz (1646–1716) established infinitesimal calculus. A dispute arose between Newton, who attributed the invention to himself, and Leibniz, who had independently published the book Nova Methodus pro Maximis et Minimis in the Acta Eruditorum in 1684. Apart from this dispute,

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I emphasize one aspect: Leibniz’s choice to denote the derivative operator with the symbol dy , and that of integration with the symbol ∫xdx, an elongated s dx (sum), were much more successful than the one proposed by Newton: a square symbol around y or before ▭ y for the integration and the dot over the function for the derivative, f . This highlights Leibniz’s particular focus on the importance of the choice of symbols to support mathematical reasoning. We will return to this aspect when considering the Chinese mathematics and astronomy. Leibniz was a very active mathematician and philosopher; he introduced the binary number system, with the use of the symbols 0 and 1. This idea was developed by George Boole (1815–1864) with the theory of binary logic, that forms the basis of modern computers. Leibniz had also a mind oriented to mechanics. He devised a mechanical calculator capable of carrying out the four operations. Two prototypes, unfortunately malfunctioning due to construction problems, were realized in 1694. Leibniz also ventured into the problem of the precision of clocks. Leonhard Euler (1707–1783) consolidated the theory of prime numbers and found the solution to the elliptic integral, a problem of great importance for calculating the length of planetary orbits. Euler, in collaboration with Jean d’Alambert (1717–1783), studied many problems of rational mechanics and determined exactly the motion of the Moon. Euler demonstrated that the prime numbers were infinite, and studied complex numbers, introducing the symbol i as the imaginary. He also demonstrated the relationship between trigonometric and exponential functions. These results are the main mathematical tools to the study of periodic functions in signal theory and modern electronics (Lakoff & Nuñez, 2005, pp. 467–505). Euler also systematized the theory of logarithms, introducing the constant e ≈ 2.71828, as the basis for natural or Neper logarithms. We can say that with Euler and the end of the eighteenth century, all the mathematical knowledge necessary for the understanding of planetary kinematics and thus for the construction of planetaries was completed. The use of more advanced observation instruments, such as telescopes, will allow to advance the knowledge of the solar system with the discovery of Jupiter’s and Saturn’s satellites and the planets Uranus and Neptune. In the following centuries, the foundations were laid for astrophysics, quantum physics and relativity. Indeed, in the nineteenth-century, mathematics became an academic discipline for research and higher education. The first scientific journals were born and, above all, mathematics was divided into the major branches of algebra, geometry, analysis, number theory and gradually logic and set theory.

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To give an idea of what the scientific climate was and the organization of studies in natural philosophy – as it was then called – and mathematics, I recall Johan Albrecht Bengel (1687–1752) (Bengel, 1745) who in 1745 published a book, Cyclus, on his own method of calculating planetary periods based on a reading of the Holy Scriptures. At the height of the Enlightenment, the Church still tried to reconcile astronomical discoveries with religious doctrine. Bengel claims to be able to establish the duration of planetary cycles and even the age and duration of the world, predicting its End from the prophecy of the Apocalypse. Reading the Cylus18 we can observe an example of the method underlying Bengel’s studies, who is still tied to Aristotle, and is not fully aware of the method of scientific research that has been established since Galileo. Bengel addresses, among others, two interesting topics: the notion of time – which I will discuss later – from which it emerges that time is linked to space (Bengel, 1745): § 22 … for the days to be reliably comparable, and to provide a measure of remaining time, astronomers call to their aid a measure of position in the sky, i.e. the signs of the zodiac and the degree and its parts, in which the celestial bodies flow, so that from the space of these places the duration of time can be determined

The second theme is the indifference of choice between geo-centric and helio-centric systems (Ibidem): § 3 Whether the earth alone is stationary, and the sun, moon, planets, and fixed stars, above it and around it move, or whether the sun is in the center of the universe, and whether even the fixed stars in their extreme stretches are not stationary, and the earth with the planets and the fixed stars move, is a question we need not discuss. § 4 After all, we must set aside this controversy, so that even if either the sun or the earth with that motion act, we can still contemplate them.

In essence, Bengel takes a position similar to that of Cardinal Bellarmino (1542–1621) toward Galileo and Osiander’s preface to Copernicus’ De revolutionibus. It is a position, mathematically speaking, perfectly consistent with Galilean relativity, which we will discuss further when describing Kepler’s studies of elliptical motion. In this overview we have seen mostly the contribution of Mesopotamia, Mediterranean and western countries, limiting to India and the Arab world  A translation by the author of the Cyclus is available at the web site associated to this book.

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the contribution of eastern regions. I will consider in more detail the Chinese mathematics in Chap. 13 and astronomy in Chap. 14.

Academia and University Mathematical research, even at the beginning of the seventeenth-century, was still a neglected discipline in European Universities, although the study of arithmetic and geometry were part of the curriculum. Universities had emerged in the late Middle Age: the University of Bologna was founded in 1088, in Oxford teaching started in 1096, between 1209 and 1293 were also founded the Universities of Cambridge, Salamanca, Padova, Napoli Frederick II, Coimbra, Valladolid. These institutions spread rapidly at the initiative of the Church and the Empire. They were schools to which only male students were admitted, learning the liberal arts of the trivium – grammar, rhetoric, and dialectic – and the quadrivium – music, arithmetic, geometry, and astronomy. Knowledge of Latin was essential to the teachers’ membership in religious orders. However, scientific research on mathematics was absent, and it was through the establishment of the Scientific Academies that research on mathematics flourished. We have to wait until the nineteenth century to see the emergence of the contemporary idea of the University as a place that combines teaching and research, with autonomy and freedom independent from political power. It was Wilhelm von Humboldt (1767–1835) who promoted these ideas with a writing in 1810 entitled University and Humanity, in which he argued that the purpose of the University is not only teaching but the moral formation of the individual, which must include knowledge of scientific subjects, which entails a continuous practice of research carried out in a collaborative context and not only individually. Mathematicians, who were also versed in the study of engineering, were in high demand by European Princes and Dukes, eager to devise war machines, fortifications and to develop the production capacities of their dominions. These scholars therefore carried out their research in the service of the Courts and not at university institutions, often moving across Europe, like Johannes Kepler (1571–1630) who traveled between Denmark, Poland, Austria, and the German courts. At the instigation of the most brilliant and esteemed scientists, European kings and princes set up the Scientific Academies, aware that the development of knowledge was becoming an essential instrument of power. It was through the libraries of the Academies that books, that were very expensive, became accessible, and through the Academies, the publication of essays and studies multiplied.

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The birth of the Academies began in 1603 with the establishment, in Rome by the initiative of Federico Cesi (1585–1630), of the Accademia dei Lincei, of which Galileo Galilei (1564–1642) was also a member. The Accademia dei Lincei was closed 30 years later, and was revived in modern times in 1870 by Quintino Sella (1827–1884) and is still active today. In 1657, the Accademia del Cimento was founded in Florence by Leopold de’ Medici (1617–1675) and promoted by, among others, Evangelista Torricelli (1608–1647) and his pupil Vincenzo Viviani (1622–1703), both pupils of Galileo. In France Louis XIV (1643–1715), established the Academie des Sciences in 1666 at the suggestion of King’s finance minister Jean-Baptiste Colbert (1619–1683) and the encouragement of French scientists including Blaise Pascal (1623–1662) and Pierre de Fermat (1607–1665). In England in 1662, on the initiative of King Charles II, the Royal Society was founded that had among its many fellows Robert Hooke (1635–1703), Robert Boyle (1627–1691) and John Wallis (1616–1703). In Prussia, the Berlin-Brandenburgische Akademie der Wissenschaften was founded in 1700 at the suggestion of Leibniz. In Russia, St. Petersburg, Peter the Great (1672–1725) established in 1724 the St. Petersburg Academy. Within about 100 years, all major European countries had scientific academies that rivaled the universities. The Academies soon began to publish proceedings in which the theories and results discussed among these communities of scientists could be found. Very important are the Journal des Savantes, which began publication in 1685, and the Philosophical Transactions of the Royal Society, which began in the same year. Particularly successful were the Acta Eruditorum from 1682 onwards, as it published the writings in Latin and thus achieved a very wide circulation. Some eighteenth-century mathematicians traveled to the various courts, for example Leonhard Euler was hosted in St. Petersburg, then returned to Prussia invited by Frederick the Great (1712–1786), and again returned to Russia to the court of Catherine, where he died. Almost all of these scientists were religious and adhered to the Catholic or Lutheran Church. The period in which the new mathematical and astronomical knowledge was developed coincided with a time of violent religious conflicts, between the followers of the Reformist doctrine of Martin Luther (1483–1546)19 and the Catholic Church with its Counter-Reformation. In the period between 1532 and 1593, numerous civil wars took place in France leading to the persecution of the Huguenots, who professed the  Luther’s Theses were published in 1517.

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Protestant faith. During the seventeenth-century, the Thirty Years’ War (1618–1648) took place, which began as a conflict between Catholics and Protestants and gradually turned into a struggle between France and the Habsburg Empire for domination of Europe. Louis XIV canceled the right to practice the Protestant religion in 1685 and forced Protestants to flee to Germany, Switzerland, the Netherlands, England and even North America (an estimated 200,000 to 500,000 refugees), among them many expert watchmakers who took their skills outside France.

3 Ancient Visions of the Cosmos: Orienting, Classifying and Modeling

Let us look at this history from another point of view, that of the formation of a science of astronomy, which progressively built increasingly accurate models of the sky and how this knowledge led to forms of representation that were first allegorical, then abstractly mathematical, and finally mechanical models capable of analogical calculation of celestial motions. In this excursus, we will see the transition from the first descriptive forms of the sky through the constellations to astronomical catalogs and the first geometric models of planetary motion, to the description of the kinematics of motion, and finally to the discovery of celestial dynamics. We will thus be able to gain a better understanding of the role played by the main actors in this millennial history.

Orienting in the Sky: The Constellations Until the seventeenth century, astronomy was organized as a descriptive discipline. It arose from a practical need to organize social life, and had a limited ability to predict the main astronomical phenomena. The nature of comets was not yet understood, the heavens were believed to be immutable, and the motions of the seven mobile stars (the planets, the Sun and the Moon) were an inexplicable mystery. The first step in the knowledge of the sky consisted in the identification of groups of stars that could recall figures of animals, heroes or gods, what we call constellations, and, more properly are called asterism. This was a matter of orientation, closely linked to the identification of relevant phenomena. In ancient Egypt, for example, there were three seasons: the season of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_3

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Fig. 3.1  Left: The Zodiac of Dendera (c. -50), plaster copy in the Temple of Dendera. Green arrow: constellation of the Hippopotamus with a crocodile skin on the shoulders. Red arrow: goddess Isis holding a baboon that symbolizes a solar eclipse. The date of the eclipse is March 7th – 51. The original is preserved in the Louvre Museum. Right: Goddess Nut, which represents the Milky Way, swallows the sun in the evening to give it birth the next morning, the narration of the life cycle. The goddess Nut is often painted inside the coffins of mummies. Temple of Dendera

periodic flooding of the Nile (Akhet), the season of sowing and growing agricultural products (Peret) and the harvest season (Shemu). They lasted about 4 months, and the beginning of the year corresponded to the occurrence of the flood, which took place shortly after the Ethiopian monsoon between May and August. The regularity of the event was marked by the heliacal rising of the star Sirius, called Sepdet. In the temple dedicated to the Goddess Hathor, at Dendera1, a reproduction of a complete zodiac is preserved (the original is in the Louvre Museum in Paris), in which the figures of the constellations interpreted as animals or gods are drawn (Fig. 3.1 left). Particular is the hippopotamus surmounted by a crocodile, a constellation positioned close to our constellations of the Big Dipper and Little Dipper. Even more significant is the interpretation of the Milky Way as the deity Nut, who every evening swallows the Sun and gives birth to it again at dawn, symbolizing with this the cycle of life (Fig. 3.1 right). All cultures identified constellations, for example, the asterism of Dakota, native people of North America, were naturally inspired by the animals  The final construction dates to the Ptolemaic period, -I sec.

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typical of regions (Lee, 2016). In the Greco-Latin culture, the mythology associated with the constellations was poetically described by Hyginus (first century) in his Astronomia (Chiarini & Guidorizzi, 2002), which summarizes the astronomical poetic writings of Aratus from Tarsus (c. -315–c. -240). Stars, it was believed, were the transformation of gods or animals, an event called catasterism. A belief that continued into Roman culture as Ovid’s Metamorphoses testifies. The descriptive character of the nascent science of astronomy is fully highlighted by the compilation of star catalogs. But why compile catalogs? Certainly, there is human curiosity, but accompanied by a desire to understand through some form of generalization from simple observations of the world. In the history of science, every new field of study is initially organized through some form of classification. It was Aristotle who introduced the method of hierarchical classification of concepts in the Logic. A genus is identified comparing objects, forms, phenomena. The species is identified by exploring the components of the genus. Charles Linnaeus (1707–1778) in 1735 was the first to introduce a taxonomy to classify plants and animals, which goes from the most general to the most particular class according to the progression: kingdom, class, order, genus, species. Well before Aristotle, the desire to classify the stars was present in all ancient civilizations. Examination of the sky led to a distinction between celestial bodies that changed position with the passage of days and months-the Sun, Moon and planets-from those that always maintained their position. The mere observation of the cycle of the day or the cycle of the Moon leads to the identification of the duration of the day and the month. With accurate observations, one can also identify the length of the year, referring to particular events that are easily observed such as the equinoxes. A long period, lasting centuries, was needed to perfect this knowledge, enriching simple observations with increasingly accurate measurements based on mathematical calculus. Stars and constellations played a fundamental role for the navigation, in particular the constellation close to the north pole, as we read in the Odyssey2: … and he sat and guided his raft skilfully with the steering-oar, nor did sleep fall upon his eyelids, as he watched the Pleiads, and late-setting Bootes, and the Bear, which men also call the Wain, which ever circles where it is and watches Orion, and alone has no part in the baths of Ocean.

 Homerus, Odyssey, V, 270–275. Tr. by A.T. Murray.

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Stars Catalogs The oldest star catalogs are Babylonian astronomical diaries and date back to the second millennium BCE. Alongside these catalogs are also cuneiform tablets that describe a prototype of a lunar zodiac. They are part of the library of King Ashurbanipal (−668–662). A catalog of Assyrian or Chaldean origin is called MUL.APIN meaning Plough Star, reports the rising and setting times of about 60 stars or constellations and planets. It is written in cuneiform characters and dated, on the basis of the times reported, around −1300. In order to place the contribution of the Chaldeans, Babylonians and Persians in the time scale, it is useful to remember that the presence of the Chaldean civilization is attested by artifacts from the – ninth century (Chaldeans and Babylonians are often considered to be the same people); −539 is the date when the Chaldeans were subjugated by the Persians. Chaldeans and Persians, who used them to make astrological predictions, observed the stars from the – ninth to – third century. In this same period the zodiac constellations were identified. The first to associate apparent groupings of stars with names of gods, objects or animals were, as we have seen, the Egyptians (−2500–2000). There are also very imaginative interpretations of the meaning of the constellations. Giacomo Leopardi (1798–1837) recalls the significance of the zodiac signs whose invention he attributes to the Babylonians (Leopardi, 1813, pos. 154): The invention and origin of the signs of the Zodiac also deserves special observation. Aries, expressed in ( ), shows, according to M. Pluche the robustness of the lambs, which at the beginning of spring are now ready to follow the mutton to pasture in the meadows. Taurus also figured in the sign ( ) swells the herd together with cubs, which, according to Mr Hyde’s observation, occupied the place of Gemini in the ancient Zodiac ( ). The Cancer, or Crab, which walks backwards and is obliquely marked in the ( ), expresses the retrograde and oblique motion, which the sun makes after passing this sign. The ferocity of Leo, represented in the sign ( ), symbolizes the ardor and strength of the sun's rays, as he moves towards it. The Virgin ( ), holding in her hands ears of wheat, clearly expresses the harvest. The name Erigon given to the Virgin, which meant in the East red, indicates ears of corn, which, in their perfect maturity, should be reddish, as Virgil attests: ... Libra, signified in the ( ), marking the equinox, and the poison of Scorpio ( ) to denote autumn illnesses. The hunting of wild beasts, which the ancients used to do at the approach of winter, is symbolized by Sagittarius. ( ), and the custom of the Goat to go into the mountains, climbing, in search of pasture, evidently shows the ascent that the sun makes through

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the Zodiac, after passing such a sign ( ). The Aquarius ( ) denotes the winter rains, and the sign ( ) of the Fishes, the abundant fishing, which they are used to make at the decline of the cold season.3

The observation of the sky by Arab astronomers improved the identification and naming of constellations, some of which coincided with those of the Alexandrian tradition. Particularly important is a treatise Book of Constellations and Stars (Kitab suwar al-kawakib) (Savage-Smith, 2013) written by Abd al-­ Raman al-Sufi (903–986) around 964 (353 EH) in which the constellations recognized by Bedouin communities and Greek astronomers are described and depicted. This work is not only descriptive; the stars are also identified with celestial coordinates measured at the time, an essential piece of information for the construction of astrolabes. Beginning in the fourth century, knowledge of geometry in Greece led to the formulation of a kinematic description of celestial motions. This kinematics would survive as primary astronomical knowledge until the 1600s when Kepler tackled the problem from a physical point of view, paving the way for Isaac Newton’s celestial dynamics.

The Motion and the Shape of the Cosmos Before illustrating the contribution of these ancient astronomers, we must clarify a very important astronomical phenomenon: the retrograde motion, a phenomenon that occurs in the apparent motion of every planet, more evident for the outer than the inner4 planets, that are not as easily observed due to their proximity to the Sun. The apparent motion of the planets with respect to the fixed stars is mostly west to east, but periodically it stops and reverses, so the planet’s apparent position with respect to the stars moves from east to west. This reverse motion is interrupted after some time and direct motion resumes. In the Fig. 3.2, we can see a simulation with the Stellarium program5 of Mars’ retrograde motion in the period from June 2024 to August 2025.

 Tr. by the author.  The inner planets are Venus and Mercury, closer to the Sun than the Earth. 5  https://stellarium.org/ 3 4

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Fig. 3.2  Retrograde motion of Mars, simulation with Stellarium program

Eratosthenes, Eudoxos, Callippus and Aristoteles The principal field of study of Eudoxus of Cnidus was uranography,6 his works did not survive, but his observations of the constellations were poetically collected by Hyginus. Eudoxus, who learned the principles of Egyptian astronomy in Egypt, developed a model of planetary motion based on homocentric spheres to which the planets were attached. He imagined that the Earth rotated on itself, anticipating what became clear only with Copernicus. This model is a significant example of the capacity of abstraction of mathematics, as Eudoxus’ spheres were not tangible objects, but conceptual models, geometric abstractions7 as Tumo Suntola points out (Suntola, 2018). As we shall see, this model was the inspiration for celestial machines in the sixteenth-century, The Fig. 3.3 left depicts the arrangement of the spheres for the planet Mars and shows the orientation of the various axes of rotation.8 Schiaparelli writes (Schiaparelli, 1877, pp.  7–131) that Eudoxus’ theory has been neglected by historians of astronomy of nineteenth-century: From these investigations, which did not lack any of the characteristics that constitute scientific research in the strictest sense of the term that moderns are accustomed to give it, was born the system of the homocentric spheres, for which the name of Eudoxus of Cnidus was so highly esteemed by the ancients. Although there is no complete and orderly exposition of this system left, it is still possible to reconstruct the main lines with certainty from the hints given by  Uranogaphy is the study of constellations.  See also Kline (1996, p. 182). 8  A graphic simulation of the motion of homocentric spheres can be observed on YouTube: https://www. youtube.com/watch?v=NgfMW-jcw5w. Accessed June 2022. 6 7

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Fig. 3.3  Left: Eudoxus’ scheme. The outermost sphere (1), called primum mobile, has a period of 24  hours and moves the starry sky around an axis oriented toward the celestial pole. The second sphere (2) has Mars’ sidereal period of revolution (678 days), its axis is orthogonal to the ecliptic. Sphere (3) rotates around an axis orthogonal to the axis of sphere 1. Sphere 4 rotates around an axis with an angle to axis 3 equal to the angle between axis 1 and 2. Both these axes rotate with the synodic period of 780 days, and generate retrograde motion. The direction of rotation is indicated by the arrows. The spheres of the other planets have a similar composition. Right: Hippopede of Eudoxus

Aristotle and Eudemus of Rhodes, and the peripatetic Sosigene and Simplicio. But see the power of prejudice! Eudoxus was not one of the Alexandrians, and was prior to Hipparchus; hence he was denied the status of astronomer, indeed even that of geometer “Rien ne prouve qu’Eudoxe fut géomètre”[Nothing proves that Eudoxus was a Geometer]. This enormous proposition is found in DELAMBRE, Histoire de l’Astronomie ancienne. Tome I, p. 131.” … Bailly, coming to speak of Eudoxus’ system of homocentric spheres, even calls it absurd (BAILLY, Histoire de l’Astronomie ancienne, p. 242.)9

Scholars of the time who speculated on the shape of the universe sought to ascribe to it the greatest simplicity and symmetry. This is why the orbits of celestial bodies are always conceived as perfectly circular and traveled with uniform motion. It is to this basic assumption that Eudoxus adheres, hypothesizing the geometric arrangement of the spheres concentric to the Earth. Naturally, the problem becomes very complex, since there is no possibility of translation but only that of combining rotational motions, with the great  Tr. by the author.

9

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result of maintaining an elegance compared to the much more complex constructions of Hipparchus first and Ptolemy later. In Eudoxus’ system, there is no germ of epicycles theories. The mathematical description of the motion produced by the spheres of Eudoxus is beyond the scope of this book. For those interested, we refer to the aforementioned work by Schiaparelli; however, it is interesting to recall that the trigonometric calculus to identify the trajectory of a planet lead to the description of a curve that today we call lemniscate, and that Eudoxus calls hippopede, a term used to represent the path of a horse that is trained to turn its course alternately to the right and to the left (Fig. 3.3 right). We note that the method for describing the planet’s path anticipates what Kepler was to develop for the study of Mars’ retrograde motion. Eudoxus’ model is described in the 12th book of Aristotle’s Metaphysics (ch. VIII). It was modified by Callippus of Cyzicus (fourth century BCE) to better render the motion of Mars. Aristotle, unlike Eudoxus and Callippus, tries to describe the motion of all the planets simultaneously, instead of considering them separately. Eudoxus and Callippus considering the motion of each star separately, limited the number of spheres as shown in the first column of the Table  3.1, while Aristotle, in order to describe the integrated system of all the planets, modified the primary spheres and added the reacting spheres, i.e., turning in the opposite direction. We have already mentioned that Archimedes has been credited with the construction of a planetarium that illustrated the apparent motions of the Sun, Moon and planets. It could have been a geocentric planetarium, but Cicero’s passage can be interpreted as the description of a heliocentric system. We can state that the studies of Eudoxus, Callippus and Aristotle on homocentric spheres founded the sphaeropeia that in Vitali’s Lexicon Mathematicum is thus defined: Table 3.1  Number of homocentric spheres Eudoxus Saturn Jupiter Mars Venus Mercury Sun Moon

4 4 4 4 4 3 3

Aristotle Primary

Reacting

4 4 5 5 5 5 5

3 3 4 4 4 4 0

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Subaltern branch of geometry, which deals specifically with the spherical motions of bodies around and gravitating around a center, and the natural propensity that attracts them toward it. Of which Goldinus wrote, and whose laws are described by Pappus, Collectiones Mathematices book 710 (Vitali, 1559, p. 852).

Aristarchus of Samos (−310–230) first introduced the idea that the Sun and fixed stars were motionless and that the Earth and planets revolved around the Sun. Aristarchus also estimated the distance of the Earth from the Sun and Moon and the diameters of the Sun and Moon relative to the diameter of the Earth itself. The fact that the Earth was round was well known in antiquity, and Archimedes provided a demonstration. Eratosthenes of Cyrene (−276–195) was able to estimate the earth’s radius by measuring the angle of the shadow cast at Alexandria on the day of the solstice, when the rays come perfectly orthogonal to Syene (present-day Aswan, which lies at the latitude of the Tropic of Cancer). A calculation based on two assumptions: that the Earth is spherical and that the Sun is so distant that the rays of light reach us parallel.

Hipparchus and Apollonius Hipparchus (-190 -120 c.), to whom is tributed the study of anomalies and the first catalogue, was one of the greatest astronomers of the past; numerous discoveries are attributed to him. Only one work by Hipparchus has come down to us: the Exegesis on the Phenomena of Aratus and Eudoxos11; of his other studies we have knowledge through the works of later authors. Making use of his knowledge of trigonometry and his skill as an observer, Hipparchus compiled the first truly systematic star catalog of the Hellenistic period by recording equatorial coordinates12 of 850 stars and their brightness.13 The measurements of the stars are given not only as declination and right ascension, but also as polar distance, distance to the tropics, transits to the meridian. The rising and setting of the various parts of the ecliptic are also  Tr. by the author.  A translation has been published in German by Karl Heinrich August Manitius in 1894. An Italian translation is due to Vanin and Cusinato (2022). 12  Contrary to previous scholarly opinion, Hipparchus did not compile the star catalogue in ecliptic coordinates but in equatorial coordinates. This new conception is presented in the work by Gysembergh et al. (2022). 13  In contemporary star catalogues the abbreviation HIP appears, which does not refer to the Hipparchus catalogue but to that of the stars identified by ESA Hipparcos mission from 1989 to 1993. 10 11

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indicated. The coordinates of the stars are expressed in a particular form linked to the zodiac constellations: Hipparchus writes that a star “lies on the third degree of Leo along a parallel circle” and this means that, since the equator is divided into 12 intervals of 30°, Leo would be at 120° so the star’s right ascension would be 120° + 3° (Vanin & Cusinato, 2022, p. 4). The brightness scale devised by Hipparchus, dividing the stars into six groups and comparing them with each other under new moon conditions, is still used today with minor modifications.14 Comparing his measurements with those of the Babylonians, Hipparchus realized that the time of the equinoxes had changed: it had advanced by about 20 minutes. This discovery became known as the precession of the equinoxes. At the time, the cause was unknown and was only understood with Newton’s discoveries. Hipparchus measured the length of the year as 365 days, 5 hours, 55 minutes, 12 seconds. Today the average length of the year is 365 days, 5 hours, 48 minutes and 46 seconds, a difference of less than 7 minutes. He also measured the interval between the equinoxes and solstices and thus found that the Sun’s motion had an anomaly: the hours of sunrise and sunset changed more rapidly in winter than in summer (Krafft, 2005) (Fig. 3.4 left). In Fig. 3.4 right we see the position of the Sun during a year. The curve is an analemma. It was traced on the floor of towers having a hole through which at noon the Sun projects its light. Another of Hipparchus’ discovery was related to the Moon’s orbit: he detected its inclination and eccentricity15 by studying data on lunar eclipses collected by the Babylonians, and observed that the Moon’s velocity along its orbit is not uniform, as is the case with the apparent motion of the Sun. Eudoxus was unable to grasp these phenomena, due to the limited observational capabilities of his time. He, therefore, hypothesized that the moon’s motion was a combination of the two circular motions that characterize the epicycle. The idea of Hipparchus was, therefore fully consistent with the philosophical-­geometric principle of celestial perfection and the perfection of uniform circular motion. This idea derived from Apollonius of Perga (−262–190) that the irregular motion of the Sun, Moon and planets could be described with epicycles; however, it presented difficulties for planets with orbits of high eccentricity.  Capaccioli attributes the scale of six brightness levels to Ptolemy, rather than to Hipparchus. He speculates that the method used was to record the time of dusk when new stars appeared due to the dimming of the overall brightness of the sky. In this way, a time scale would have been converted into a brightness scale (Capaccioli, 2020, pp. 35–36). 15  The eccentricity of planetary orbits is described in Appendix “Planetary Orbits”. 14

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Fig. 3.4  Left: The duration of the seasons and the intervals between solstices and equinoxes. Right: The Solar analemma captures the position of the Sun during a year. The size of the lobes is proportional to the duration of the seasons. The variable density of Sun’s position (the white dots) demonstrates speed variation along the orbit. The analemma is frequently engraved on the floor of a church, on which the sunlight through a hole is projected

Hipparchus also conducted geography studies by calculating climatic phenomena for different latitudes, described with various data such as the length of days, the shadow of a gnomon, the stars at the zenith and at the horizon. He calculated the latitude of several places, including Byzantium, Alexandria and Marseilles. For a while, he stayed in Bithynia, the northern region of present-day Turkey, where he worked on a parapegma. The question we can ask is: How did Hipparchus detect these properties of the Moon’s motions? Certainly, he made direct observations and measurements, but the results depend on a very long time series. Today historians agree that much of the data collected by Hipparchus in his catalog is of Chaldean origin. Even more interestingly, Hipparchus’ calculus of eclipses of the Sun and Moon are the result of a purely arithmetical scheme devised by Babylonian astronomers (Gingerich, 2002, p. 71).

Claudius Ptolemy Claudius Ptolemy (d. 168 c.), who lived mainly in Alexandria, studied optics, astrology and geography. Above all, he set out to base astronomical knowledge on a rigorous mathematical and geometric method. His main work is the Almagest (The Greatest) (Ptolemy, 1515), so called by the Arabs, while the original title was Syntaxis (Collection). This work was spread throughout Europe, where it had an enormous influence, with its Latin translation by

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Gerard of Cremona in the twelfth-century. Ptolemy was strongly interested in astrology and published another work of great renown on the subject: Tetrabyblos. It is a work in four volumes that attempts to ground astrology on the basis of astronomical knowledge, rather than on myths as was still the custom in his time. Giacomo Leopardi (1813, pos.3368) dismisses this work with the words: Ptolemy’s astrological works are not worth mentioning. It grieves mankind that so great a man, so versed in the science of the stars, who deserved to be called the first astronomer of antiquity, albeit to the detriment of Hipparchus, should have fallen into such palpable errors. But ease of deception has always been the heritage of man.16

Ptolemy conducted direct observations of the positions of the stars and found that, compared to the time of Hipparchus, some 250 years earlier, they had shifted by approximately 2° and 40 minutes of arc, another confirmation of the precession of the equinoxes. For his observations, Ptolemy invented an armillary sphere, described in detail in Book V, chapter 1 of the Almagest, which was based on a system of ecliptic coordinates (see Fig. 3.5). Ptolemy’s fundamental contribution to astronomy was the formulation of celestial kinematics on a geometric model, overcoming the limited descriptive capacity of the astronomical tables of the ancient Chaldeans. Ptolemy extended the purely arithmetic apparatus of his predecessors with geometry. In this way he got a more accurate method for calculating the positions of the stars and predicting celestial phenomena. The cosmological model is an evolution of the one proposed first by Eudoxus, then by Aristotle, of the homocentric spheres. For the description of these motions, Ptolemy takes up the studies of Hipparchus. Some scholars from the early nineteenth century onwards (Neugebauer, 1983, pp. 320–324) have raised the doubt that Ptolemy’s measurements are no more than those already made by Hipparchus and corrected for the precession of the equinoxes that occurred over almost three centuries and would amount to about 2.6°. This hypothesis was later disproved following the analysis of a palimpsest containing parts of Hipparchus’ catalog (Gysembergh et al., 2022). Comparing the coordinates reported there with those noted by Ptolemy shows that Ptolemy’s observations are certainly original and exposed in an ecliptic reference system. Recall that Hipparchus’s celestial coordinates  Tr. by the author.

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Fig. 3.5  Ptolemy’s armilla from Almagest

are expressed in an equatorial and not an ecliptic reference, and Ptolemy made use of spherical trigonometry techniques, unlike the simple chord method used by Hipparchus. The core of Apollonius’ and Hipparchus’ model of astronomical motions is the geometry of the epicycle and the deferent. It was indeed necessary to find a mathematical solution to the complicated apparent motion of the moving

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planets. If we observe the motion of Mars, we expect it to move relative to the fixed stars as it makes its revolution around the Earth, but at a certain point, this movement is reversed, giving rise to a retrograde motion that lasts almost 2 months (Fig.  3.2). The points at which the apparent reversal of motion occurs are called stations. To explain this behavior, we have seen that Hipparchus hypothesized that it was the result of a combined motion: the planet rotates with constant velocity around a center that in turn rotates along a larger circle, again with constant velocity, which has the Earth at its center. The circle on which the planet moves is called the epicycle, and the larger circle is the deferent. Ptolemy hypothesized that the deferent, in turn, does not have the Earth as its center but a point close to it, in order to account for the variation of the celestial body’s distance from the Earth. When the planet moves along the arc outside the deferent, it follows the motion of the deferent itself. When it moves along the inner arc the planet proceeds in the opposite direction giving rise to retrograde motion. In this way, one can explain the retrograde motion of any planet. One problem was still open: how to justify the variable velocity of celestial bodies, which Hipparchus had already identified as a solar anomaly. Ptolemy made a modification to the epicycle pattern by introducing the equant: a point placed symmetrically on the Earth with respect to the center of the deferent. The equant becomes the center around which the epicycle rotates. In this way, the apparent velocity varies between perigee and apogee. We see this situation reproduced in Fig. 3.6. Ptolemy’s idea of the epicycle diagram with equant was able to explain the retrograde motion, and the variation of the speed of motion. However, the precision of this mathematical model was still not sufficient to account for other discrepancies with the observed data. Ptolemy failed to justify the second anomaly, the precession of the Moon’s apsides.17 Nevertheless, Ptolemy’s model made it possible to predict the position of the planets with sufficient precision. He had in fact developed a method of calculation: starting from a known position, called the root, the displacement relative to the sun is determined, the displacement of the earth is then calculated, the variation due to the precession of the equinoxes is added, and finally the effect of eccentricity, i.e., the displacement from aphelion, is added or subtracted. All this was done on the basis of tables that contained values for all these components. Ptolemy’s astronomical tables were the basis for the already mentioned Tabulae Alphonsinae .  The apsides are the extreme points of the major axis of the elliptical orbit.

17

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Fig. 3.6  The trajectory of apparent motion. The deferent has its center at C, the planet P moves along the circle with its center at P1, which in turn rotates around the equant point Eq. The point Eq is opposite the Earth T so that EqC = CT. In this way, the angle  P1 , Eq , C rotates with constant velocity, but the motion of the planet P varies as can be seen from the red dots: they are denser near the Earth (at perigee) and thus faster, while they are rarer near apogee and thus slower. The figure with loops is called an epitrochoid

Ptolemy also dealt with geography, aiming at determine the extension of the ecumene18 introducing latitude and longitude to indicate the position of 8000 places in the Roman Empire. The position of his meridian 0 seems to correspond to the Canary Islands. but the measurements given by Ptolemy are conspicuously wrong. It is believed that he had underestimated the stadium measurement used by Eratosthenes to calculate the Earth’s radius, introducing a significant error (Russo, 2013) (Fig. 3.7). We will return to the description of this model; let us limit ourselves here to pointing out that before Ptolemy, astronomers were building a descriptive 18

 The ecumene is the part of the earth where exist conditions suited for the presence of human life.

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Fig. 3.7  Ecumene by Ptolemy

science, for which making predictions was difficult and imprecise, and was based on the arithmetic methods of the Babylonian era, as we mentioned when presenting the contribution of Hipparchus. Ptolemy was the first to introduce a geometric model in the description of the sky, which could then also be applied to the emerging cartography. The personality of Ptolemy is effectively described by Owen Gingerich (2002, p. 71): Claudius Ptolemy stands as one of the greatest figures of ancient science, as the first applied mathematician with an encyclopedic vision. Taking Euclid as his model Ptolemy brought mathematics to deal with cartography, astronomy, astrology, optics and harmony.

We have thus arrived at a kinematic description of planetary motions that justifies the solar and lunar anomaly, the apparent retrograde motion of the planets, still keeping the Earth at the center of the cosmos.

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Humanism and the Rebirth of the Studies In Sect. “Arab Mathematics and Astronomy During the Middle Age” we summarized the contribution of Arab culture to the development of astronomy. From the theoretical point of view, they did not make significant progress, rather they reworked and criticized the theory. It is worth noting that al-­ Haytham was very critical of the idea of the equant, which did not respect the basic perfection principle of circular uniform motion. It was with the emergence of a thought open to new approaches to scientific knowledge, with the realization that Aristotelianism had become a brake on free thought, that a new era, the Humanism, was born. Scholars sought to deepen their studies of astronomical phenomena in new ways, inventing new theories, taking up the study of ancient texts, translated from Greek into Arabic and now translated into Latin. Georg von Peuerbach (1423–1461) was an Austrian astronomer and mathematician, inventor of scientific instruments, and was Regiomontanus’ teacher. Peuerbach studied Ptolemy’s work, the Almagest, and he began its translation, but due to his early death it was completed by Regiomontanus. Peuerbach’s most interesting contribution to the development of astronomical science is the work Novae theoricae planetarum [New theories of planets], published posthumously in 1471 still by his student Regiomontanus (Peuerbach, 1562). The purpose of this theory was to demonstrate how eccentric planetary orbits (epicycles with equant) could be arranged with auxiliary spheres, called orbis19, so that the inner and outer surfaces remain concentric with the Earth, an essential requirement for a fully homocentric system of crystalline spheres. The term theorica orbium, which we could translate as theory of homocentric spheres, is used to identify not only the theoretical treatment of the problem, but also the machines built to depict the model of the spheres in three dimensions. Owen Gingerich (1986) traces Peuerbach’s theory back to the treatise On the Configuration of the World by al-Haytham in which the celestial spheres are conceived as concentric shells, tangent to each other. The inner surface of Jupiter’s shell, for example, is tangent to the outer surface of Mars’ shell. To correctly model the motion of the planets, Peuerbach argued that the existence of auxiliary spheres was necessary to ensure the concentricity of the inner and outer surfaces with the Earth, an essential requirement of the homocentric theory. Peuerbach does not refer directly to Eudoxus’ model, but to his  The term orbis should be distinguished to orbit. It was Kepler who clarified that the planets described a path called orbit. Before him the motion was considered strictly linked to spheres. 19

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7

I

Fixed stars sphere

Mars orbit

P7

P6

P5

T6

P4

P3

P2

T5 T4 T3

P1

T2 S

Earth Orbit

T7

T1

Fig. 3.8  Left: Peuerbach, the spheres of the Sun. Right: Mars retrograde motion. The Sun is in the center S, the Earth T rotates along the inner circle, and the planet P rotates in the middle circle. The outer circle represents the fixed stars, and the lines show the projection of the planet onto the fixed stars. It is evident that the retrograde motion can happen when the planet is in opposition to the Sun. Adapted from a drawing by Galileo

modification proposed by Hipparchus first and Ptolemy later, which manages to account for anomalies in the orbits. To understand how Peuerbach’s theory differs from al-Hytham’s, let us consider the three spheres required to represent the sun’s motion (see Fig. 3.8 left): the outermost sphere is concentric on its convex face to the sphere of fixed stars, but the concave face is eccentric; the innermost sphere has the concave face concentric and the convex face eccentric; and finally, the sphere intermediate between the two has both faces eccentric (Peuerbach, 1562, p. 3). The Sun has three orbis, separated from each other on all sides and also contiguous to each other. The highest [outer] of them is contiguous with the world in its convex surface, but is eccentric in its concave surface. The lowest [internal] orbis, on the other hand, is concentric in its concave surface, but eccentric in its convex one. The third orbis is intermediate to the two, and eccentric in both its concave and convex surfaces.20

 Tr. by the author.

20

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These eccentricities account for the anomaly between aphelion and perihelion. As we will see Kepler will interpret these concavities and convexities as the crassitiem (thickness) of the sphere. Peuerbach’s theory is an early attempt to provide a physical justification for the planetary motions, i.e., one capable of explaining the causes: the presence of the concentric spheres in contact each other, with their relative motion is the cause of the planets’ movements. According to Peuerbach, in fact, there was no free space between the spheres, which consequently could drag the innermost spheres in their motion from the celestial sphere, the primum mobile. We observe that Peuerbach’s idea is very naive: it is not understood how it is possible that the motion around a given axis of a sphere can generate rotation around a different axis by a simple dragging. The limitation of this idea also appears from attempts to construct armillary spheres capable of showing the motion of more than one planet at the same time. The armillary spheres that attempt to reproduce the theory of homocentric spheres present only the motion of the Moon or the Sun or a single planet, such as Mercury, with the Earth at the center.

 he Invention of the Solar System: T Nicolaus Copernicus Nicolaus Copernicus (1473–1543), in his seminal work De Revolutionibus orbium caelestium21 published shortly before his death, formulated the heliocentric model. Copernicus was born in Torun, a Polish city that became part of a Prussian confederation in the mid-1400s and was for a long time disputed between the Polish and Prussian kingdoms. Copernicus, supported by his uncle Lucas Watzenrode, conducted his early higher studies in Krakow (Gingerich, 2016). His uncle Lucas had become Bishop of the Warmian region, whose main town was Frombork (Frauenburg). In 1491 he enrolled at the Jagiellonian University, considered the best in northern Europe for the study of mathematics and astronomy, to which the young Nicolaus devoted himself. The recent invention of printing22 by Johannes Gutenberg (1400c.–1468) gave Copernicus the opportunity to acquire fundamental works, such as Euclid’s Elements and the Alphonsinae Tabulae.  On the revolutions of the orbis of the sky (Copernico, 1566).  The first printed edition of the Bible is dated 1455. This is considered the date of the invention of the print. 21 22

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The news of the discovery of the Americas by Cristoforo Colombo certainly impressed the young scholar. In 1495 Nicolaus left the University of Cracow without obtaining a degree, and his uncle Lucas put his name forward for a post at the Diocese of Warmian. To obtain this position, Copernicus had to travel to Bologna to undertake studies in canonical law. In Bologna Nicolaus shared accommodation with a young astronomy professor and active astrologer, Domenico Maria Novara (1454–1504), whom he immediately joined in astronomical observations and studies. In the summer of 1500, the 27-year-­ old Copernicus completed his legal studies in Bologna, and to celebrate the event, he traveled to Rome, where the Jubilee celebrations called by Pope Alexander VI were underway. It was the year that Martin Luther was also in Rome and was scandalized by the pomp of the celebrations and the shameful trade in indulgences. During his stay in Rome, according to the mathematician Georg Joachim von Lauten, known as Rheticus (1514–1574), Copernicus gave lectures on mathematics to a large number of students. Copernicus returned to Prussia after his Roman sojourn and was appointed Canon of Frombork Cathedral, without becoming a priest, but still taking a vow of chastity. In 1501, having promised to study medicine in order to become a physician of the community, he went back to Italy, to Padua. At the time, medicine required knowledge of astrology and of astronomy. At the end of his second year of study he had to return to Warmian, but in order not to return without a title, in 1503, at the age of 30, Nicolaus obtained the title of doctor of canonical law from the University of Ferrara. This was his last trip, however, and he did not leave Warmian again, continuing his astronomical studies and serving as Canon of the Catholic cathedral in Frombork. Astronomy at the time of Copernicus was still based on the principle of the perfection of the homocentric spheres and on the mechanism of epicycle with deferent and equant. With this model it was possible to calculate the past and future positions of the planets and other astronomical events with reasonable accuracy. But this theory considered the heavens as made if independent objects that could not be treated in a unified manner. In fact, after Peuerbach had published his translation of the Almagest, many had tried in vain to construct armillary spheres that would simultaneously show the motion of all the planets. Therefore, a central aspect of the Copernican revolution is that he considered the cosmos as a system: the Solar System. It was known to every astronomer that Mars’ retrogradation appeared when the planet was in opposition to the Sun. If we observe the diagram in Fig. 3.8 right, we can see the simplification that Copernicus introduced. The logical framework of Copernicus’ thought is based on three essential principles:

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–– any description of the apparent motion of the planets must consist of uniform circular motions; with this principle, Copernicus adheres to Eudoxus’ and Aristotle’s model of the celestial spheres. –– The only motion allowed to the spheres is a rotation around their diametral axis. –– Any motion that deviated from the circular motion would be mechanically impossible as it could not be represented by spheres. The defect that Copernicus detected in the Ptolemaic model consisted precisely in the violation of the principle of perfect circularity. In fact, to resolve the first anomaly Ptolemy had devised the equant: the epicycle rotates around a point that is symmetrical to the center of the deferent with respect to the Earth. Peuerbach had tried to perfect the model of the homocentric crystalline spheres. This violation, in the words of Copernicus, according to N. Swerdlow’s translation (Swerdlow, 2021): Nevertheless, the theories concerning these matters that have been put forth far and wide by Ptolemy and most others, although they correspond numerically [with the apparent motions], also seemed quite doubtful, for these theories were inadequate unless they also envisioned certain equant circles, on account of which it appeared that the planet never moves with uniform velocity either in its deferent sphere or with respect to its proper center. Therefore, a theory of this kind seemed neither perfect enough nor sufficiently in accordance with reason.

Copernicus was reluctant to divulge his ideas that conflicted with faith in Holy Scripture,23 he therefore tried to avoid publishing the results of his studies, except for a short booklet, the Commentariolus written between 1507 and 1512, which was lost and rediscovered in 1878. In this short work, Copernicus summarizes 7 postulates24: Therefore, when I noticed these [difficulties], I often pondered whether perhaps a more reasonable model composed of circles could be found from which every apparent irregularity would follow while everything in itself moved uniformly,  The most important biblical reference to the immobility and centrality of the Earth is in the book of Joshua X, 12,13: “On the day the Lord gave the Amorites over to Israel, Joshua said to the Lord in the presence of Israel: 23

«Sun, stand still over Gibeon, and you, moon, over the Valley of Aijalon. » So the sun stood still, and the moon stopped, till the nation avenged itself on its enemies as it is written in the Book of Jashar. The sun stopped in the middle of the sky and delayed going down about a full day.” (Bible, 2011–2022). 24  Translation of Copernicus postulates by Swerdlow (2021, p. 436).

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just as the principle of perfect motion requires. After I had attacked this exceedingly difficult and nearly insoluble problem, it at last occurred to me how it could be done with fewer and far more suitable devices than had formerly put forth if some postulates, called axioms, are granted to us, which follow in this order: 1 . There is no one center of all celestial spheres (orbium) or spheres (sphaerarum) 2. The center of the earth is not the center of the universe, but only the center toward which heavy things move and the center of the lunar sphere. 3. All spheres surround the sun as though it were in the middle of all of them, and therefore the center of the universe is near the sun. 4. The ratio of the distance between the sun and the earth to the height of the sphere of the fixed stars is so much smaller than the ratio of the semidiameter of the earth to the distance of the sun that the distance between the sun and earth is imperceptible compared to the great height of the sphere of the fixed stars. 5. Whatever motion appears in the sphere of the fixed stars belongs not to it but to the earth. Thus, the entire earth along with the nearby elements rotates with a daily motion on its fixed poles while the sphere of the fixed stars remains immovable and the outermost heaven. 6. Whatever motions appear to us to belong to the sun are not due to [motion] of the sun but [to the motion] of the earth and our sphere with which we revolve around the sun just as any other planet. And thus, the earth is carried by more than one motion. 7. The retrograde and direct motion that appears in the planets belongs not to them but to the [motion] of the earth. Thus, the motion of the earth by itself accounts for a considerable number of apparently irregular motions in the heavens.

Reading these postulates reveals Copernicus’ reasoning: how was it possible for the Earth to revolve around itself if it is so small compared to stars like the Sun, and the fixed stars are so far away from it? In the third postulate, Copernicus still considers the planets as bound to their own sphere, and in the sixth postulate he writes that the changes in the apparent motion of the sun are due to the motion of the Earth in its own sphere. In the seventh postulate he again ascribes the apparent retrograde motion of the planets to the motion of the Earth. In postulate 2 we can see a concept of a force of gravity that attracts the bodies and consequently explains the spherical shapes of the Earth itself and of the planets. Even the motion exists because it is sustained by a force of attraction.

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Table 3.2  Copernican parameters for determining the size of orbits (Gingerich, 2016, pp. 27–29) Venus Earth Mars Jupiter Saturn

Deferent radius

Epicycle radius

Orbital period

18 25 38 130.25 230.8

25 25 25 25 25

225 365 687 4333 10,756

To the objection that if the Earth rotates on itself and around the Sun then our bodies and things themselves would be ripped apart, Copernicus replies that everything on Earth participates in its motion, and we walk on it as if it were stationary. These are arguments that Galileo would develop extensively in his writings. Postulate 4 reveals that Copernicus had guessed that the size of the starry sky was enormously larger than previously thought. Copernicus then developed a method to estimate the distances between planets from the duration of their orbits. This work probably dates back to 1510, to the beginning of his studies, and in some notes, he observes that in the Almagest the size of the deferent of Mars, Jupiter or Saturn is always 60 units, since they are treated independently and are not part of a system. The epicycle conversely contains the comparative dimensions of the orbits. Copernicus vice versa sets a standard size of the epicycle instead of the deferent, assigning 25 units instead of 60, including the epicycle of the Earth. With this assumption, Copernicus obtains this list of values for the diameter of the deferent, which are relatively in good proportion to the periods of revolution (Table 3.2). Copernicus’ alternative to the equant is well described by Otto Gingerich (2016).25 In the elementary formulation of the new heliocentric model, every planetary sphere has the Sun at its center, and to describe the eccentricity of the planets, it is necessary to characterize its value and the direction in which it lies, i.e., the direction of the axis of the apsides; a problem that was completely solved by Johannes Kepler, according to whom Copernicus merely interpreted Ptolemy instead of the heavens. Copernicus kept his model still in the Aristotelian groove of the perfection of the circle, and to account for the nonuniformity of orbital motion with uniform circular motions—which Ptolemy had solved by introducing the equant—Copernicus substituted the Sun for the Earth at the center of the deferent as in the Fig. 3.9.

25

 See pp. 73–74.

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Fig. 3.9  The equant is replaced by the epicyclet, small epicycles of the same radius

In this way Copernicus was able to consolidate his conception of the solar system. Concerning the distances of the planets, we note that in the drawing published in De Revolutionibus (Fig. 3.10), the planets are placed at an almost equal distance from each other; undoubtedly to try to reproduce planetary distances to scale, the space of a page would not have been enough. In the concentric circles, which are not orbits, the positions of the planets’ spheres and their durations are indicated. The Lutheran schism having been accomplished, and even the Protestant theologians were opposed to his hypothesis. Copernicus spent some 30  years writing his work. The young Rheticus traveled to Frombork in 1539 and assisted Copernicus in his work. Rheticus had been trained in mathematics at the University of Wittenburg under Philipp Melanchthon (1497–1560), a theologian and astrologer and one of the most active supporters of the Lutheran Reformation. He was thus able to put his knowledge at the service of Copernicus and endeavored, in vain, to publish his work in Nüremberg. The work De Revolutionibus, was printed shortly before Copernicus’ death by the Protestant theologian Andrea Osiander (1498–1552) who, to avoid condemnation for heresy, wrote in the preface (Capaccioli, 2020, p.  24; Copernico, 1566): For it is the task of the astronomer to compose, by diligent and skillful observation, the history of the celestial motions, and thus to search for their causes, or, since it is impossible to grasp the true ones, to imagine and invent any ­hypotheses

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Fig. 3.10  The Copernican model

on the basis of which these motions, whether they concern the future or the past, can be calculated exactly in accordance with the principles of geometry. And the author of this work has performed these two tasks admirably … It is not necessary for these hypotheses to be true, or even plausible, but it is sufficient that they show the calculation to be in harmony with the phenomena observed.26

26

 Tr, by the author.

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Caution against Church beliefs was a must to avoid the risk of accusations of heresy or criticism from fanatical scholars, but this preface nonetheless makes Copernicus’ work clear. In brief, Copernicus’ model placed the Sun at the center and all the rotating planets, always fixed to the homocentric spheres, around it and the Moon around the Earth. This simple model still did not solve other problems, including the variation of declination, which was later identified in the inclination of the ecliptic, and the anomaly of its velocity varying between aphelion and perihelion. To overcome this second limitation, Copernicus retained the principle of epicycles, and to overcome the variation in declination he imputed to the Earth the trepidation, which consisted of an oscillatory motion combined with a change in the obliquity of the ecliptic and the precession of the equinoxes; a hypothesis that was abandoned by Kepler.

4 A Lucky Astral Conjunction

In the second half of the sixteenth century, the availability and protection of grand dukes and princes, their interest in science, and the collaborative involvement of the greatest scientists created favourable conditions for a great leap forward that would soon lead to the foundation of modern astronomy. The first key figure was Landgrave Wilhelm IV of Kassel (1532–1592) (Fig. 4.1 left), known as the Wise, who studied in Strasbourg in his youth and visited its astronomical clock, meeting Conrad Dasypodius (1532–1600), with whom he became friends and whom he frequented throughout his life. His interest in astronomy led him to build an astronomical observatory in Kassel, where he personally conducted observations that led to the writing of a catalogue of a hundred stars. He called Jost Bürgi to Kassel, whose contribution we will examine later on. Other fine mechanics collaborated with the Landgrave, such as Eberhard Baldewein (1525–1593). The Landgrave was indeed determined to have mechanisms built that could represent the cosmos. The second key figure was Rudolph II of Habsburg (1552–1612) (Fig. 4.1 centre), who offered his protection and financed the studies of Tycho Brahe and Johannes Kepler, the other two scientists who are the protagonists here. To these noble benefactors, we must add Frederick II of Denmark and Norway (1534–1588) (Fig. 4.1 right), who protected Tycho Brahe. We should not, however, think that these protections and this interest in the advancement of science on the part of some Dukes and Kings was an easy matter. The Courts were often hostile to these scientists who had too much influence distracting the prince from the duties of State, and often upon their death

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_4

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Fig. 4.1  Left: Wilhelm IV of Kassel. Centre: Rudolph II of Habsburg. Right: Frederik II of Denmark

funding ceased, and the scholars were forced, as we shall see, to leave the country that hosted them and find other benefactors and protectors. The most important centre of this concurrence of events was certainly Praha, where the university school founded by the Jesuits, invited to Praha by the Habsburgs in 1556, was established. They built the Collegium Maximum, a university school, in the remains of a Dominican convent near the church of St. Clement, destroyed during earlier religious conflicts, to which they gave the name Clementinum (Šima, 2006). Tadeaš Hàyek (1525–1600), astronomer and personal physician to Rudolph II, convinced the emperor to invite Tycho Brahe. The period of Rudolph II’s reign is considered a golden age for science, culture, and art. The emperor collected valuable works, now preserved in the Kunstkammer of the Kunsthistorisches Museum in Wien, and allowed great freedom of thought. He had made Praha a kind of Promised Land for European scientists and artists.

In Search of Precision: Tycho Brahe His actual name was Tyge Ottesen Brahe (1546–1601), and he was born in 1546 on the Scania peninsula at Knutstorp Castle, then Danish. He was the first of 12 children, of whom only eight survived, from a noble Danish family. At the age of two, he was entrusted to his uncle Jørgen Brahe, who had no children and obtained permission from his brother to raise him until the age of 18. Between the ages of 6 and 12, Tycho attended a school where he learnt Latin. Later, in 1559, he enrolled at the University of

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Copenhagen, where he initially studied law as his uncle wished in order to launch him into a career in the government. He soon became interested in astronomy and physics, learning the principles according to the Aristotelian tradition.1 On 21 August 1560, he had the opportunity to observe a solar eclipse (in the Danish regions the occultation was partial) that had been predicted by astronomers, albeit a day in advance. The inaccuracy aroused the interest of Tycho, who realized the need for more precise observations. He invested two thalers to purchase a copy of Ptolemy’s work, published in Basel in 1551. The careful study of this treatise—preserved in the University Library in Copenhagen—is evidenced by the numerous notes in the margins written by Tycho’s hand. He stayed in Copenhagen for three years; when his uncle Jørgen judged it was time to send him to other schools, he sent him to Leipzig with a tutor a few years older, the 19-year-old Ander Vedel. They arrived in Leipzig and matriculated at the Lutheran University of Leipzig in March 1562. Despite Vedel’s efforts, Tycho continued to cultivate his astronomical studies, buying books and equipping himself with a small celestial globe as a guide for observing the constellations. The following year, between 17 and 24 August, he observed a conjunction of the planets Jupiter and Saturn, constructed a rudimentary compass to measure the angular distance between the two stars, and again found that the Tabulae Alphonsinae had an error of a whole month, and were therefore not precise enough to predict such an astronomical event. The Tabulae Prutenicae2, published in 1551 by Erasums Reinhold (1511–1553), based partly on the book De revolutionibus orbium caelestium of Copernicus, were also inadequate, although they reduced the difference between the predicted and actual date of the conjunction to a few days. In his mind, he became even more convinced that astronomy was a discipline that required the highest and most rigorous precision in observations and measurements, using the most accurate and precise instruments available at the time, so he began to devise new ones. Tycho combined the study of astronomy with that of astrology, as was the custom at the time, a knowledge that enabled him to prepare horoscopes for various personalities. In a letter written in 1588 to the mathematician Caspar Peucer (1525–1602), son-in-law of the mathematician Philipp Melanchthon, he predicted that Peucer would have great misfortune by suffering exile or  For a biography and works of Tycho see Dreyer (1890).  Compiled in 1551 by the astronomer Erasums Reinhold are ephemerides based on the Copernican model. They take the position of Saturn in 1490 as reference data. They are so named in honour of the Duke of Prussia Albert I. 1 2

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imprisonment, from which he would be freed at the age of 60. The prediction turned out to be correct! When the two young men returned from Leipzig in 1565, Denmark was at war with Sweden, and their uncle Jørgen, vice-admiral of the Danish fleet, had won a major battle by sinking the warship Mars. However, he was later defeated and shortly afterwards died of a fever, possibly due to pneumonia contracted from jumping into the water after a night of drinking with Denmark’s King Frederick II to save him. After his uncle’s death, his wife Inger Oxe inherited his estate and continued to support Tycho like a son. In 1566 during his stay at the University of Rostock, where he studied Medicine, the 21-year-old Tycho challenged Manderup Parsberg (1546–1625) to a duel to argue his superiority in mathematics and lost his nose, which was severed with a sword blow. From then on, he always wore a prosthesis, sometimes made of gold or silver but more often of bronze. In 2012, an investigation of Tycho Brahe’s remains using modern analytical techniques showed that bronze residues remained on the bone.3 In April 1567, having completed his studies, Tycho decided to devote himself to astronomy. He went to Augsburg, where he built a large quadrant with a radius of about 19 feet (just under 6 metres), with which he could measure angles of one minute of arc. The instrument was destroyed by a storm in December 1574, but could be used for observations of the stella nova. In 1568 he returned to Knutstorp Castle because his father was ill and died in May 1571. In 1570 he left Augsburg and returned to Denmark, where another uncle, Steen Bille, invited him to settle in Herrevad and helped him build the first observatory in one of the monasteries. At first, Tycho devoted himself to other activities including the construction of a paper mill, the first one built in Denmark. In 1572, he resumed his astronomical studies with the observation of a supernova in the constellation Cassiopeia. He wrote in 1573 a short treatise on the subject,4 in which the supernova is depicted among the constellation’s main stars. Tycho waited some time to publish the study, considering it inappropriate for a nobleman to publish books, but was finally persuaded by other astronomers, also to counter the much nonsense that was being written on the subject. The scientific question that inflamed the debate among astronomers was the origin of the phenomenon. The Aristotelians, convinced of the immutability of the sky, claimed it was a phenomenon related to the Earth or at most  https://phys.org/news/2012-11-mercury-poisoning-tycho-brahe-death.html. Accessed November 2021.  De nova et nullius aevi memoria prius visa stella, iam pridem anno a nato Christo 1572, mense Novembri primum conspecta, contemplatio mathematica [A mathematical investigation of the new star never remembered in the past, first seen in the year of our Lord 1572, in the month of November]. 3 4

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to the Moon. Tycho, noting that a parallax could not be measured and that the star did not move across the sky like the other planets, concluded that it was a much more distant entity than the Moon, the Sun or the planets. He called it a “new star”, criticizing those who did not recognize the facts in the preface to his work “O dull minds. O blind spectators of heaven”. In this work, Tycho also exposed his cosmological theory, according to which the planets revolved around the Sun, which in turn revolved around the earth like the moon. The fame that this discovery earned him, and other publications of detailed observations made at Herrevad, put him in contact with other scientists. Tycho would have liked to go abroad for some time but was held back by an illness, probably malaria, and new family affections. In fact, Tycho became united with Christine, a young woman of humble origins, perhaps unsuitable for marriage to a nobleman. She was de facto his wife and, although not formally married, bore him several children, of whom Magdalene, Tyge, Jørgen and three other daughters survived. When problems arose over inheritance, Tycho’s sister Sophia wrote a letter certifying the full legitimacy of the sons of Tycho Brahe. After his illness, Tycho resumed his astronomical observations with the help of his sister Sophia, who had had an excellent education and accompanied him for a long time, sharing with him her interest in astrology and astronomy. In 1575, Tycho was finally able to set off, leaving his family behind and travelling first of all to Kassel to visit the observatory that the Landgrave Wilhelm IV had built in 1561, from which the Landgrave personally made frequent observations. The Landgrave brought to Tycho’s attention the phenomenon of the apparent slowing of the sun at sunset, which al-Haytham had attributed to the refraction of the atmosphere. Tycho travelled on to Frankfurt, Basel and Venice where he acted as agent for the King of Denmark, inviting various artists to decorate his new palace in Elsinore. During these journeys, he had several instruments built to conduct astronomical observations. He sent the instruments to Denmark and obtained a copy of Copernicus’ Commentariolus, which he helped disseminate to other scholars. Tycho believed that Copernicus had not measured the parallax of the planets, and thus rejected the Copernican model, and undertook precise measurements to obtain the distances between the planets, finding values very close to those of Copernicus (Šima, 2006, p. 100). On his return to Denmark, King Frederick II offered Tycho prestigious positions and property, but Tycho was more interested in pursuing his astronomical studies and planned to return to Basel to join the vibrant local scientific and academic life. The King, having learned of Tycho’s plans, offered him

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the island of Hven, until then the exclusive property of the Crown, to establish an astronomical observatory there. Work on the construction of the observatory began in 1576. Tycho immediately made use of his feudal rights by urging the inhabitants to increase their food production and recruiting them for the construction work of the observatory, decisions that provoked protests that were resolved after negotiations under royal patronage. The observatory designed by Tycho was intended to be dedicated to science and the arts. Tycho gave it the name Uraniborg referring to Urania, the muse of Astronomy. The architecture was loosely inspired by the Italian Renaissance, in particular the buildings of Andrea Palladio (1508–1580), whose works Tycho had seen during his trip to Venice (Fig. 4.2 left). It included several observation towers that often proved to be too exposed to the winds, so in 1584 Tycho built a second centre, Stjerneborg, this time underground and consisting of hemispherical crypts covered with movable hemispherical domes. Quadrants and armillary spheres were mounted on steps to allow the observer to get closer to the instruments, thus taking more accurate measurements. In this way, he could also accommodate a larger number of collaborators. Uraniborg soon became a full-fledged research centre that housed up to 100 students. It was equipped with an alchemical laboratory with ovens for

Fig. 4.2  Left: Uraniburg model. Right: The large globe on which the positions of the stars were marked

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conducting experiments in distillation and alchemy and a press centre that published research results internally. Tycho Brahe was extremely jealous of the results of his studies and, in order to prevent confidential information from spreading without his knowledge, he set up a papermaking laboratory that imprinted a personal watermark on the sheets. A large globe with a diameter of 5 feet, which he had built in Augsburg, was mounted in the library (Fig. 4.2 right). On it were mounted the bronze rings for the zodiac, equator, and graduated meridian. The positions of the stars were marked on the globe as their celestial coordinates were determined, a task that was completed over the course of 20 years. The other very important instrument was the large quadrant mounted on a wall (Fig. 4.3). The bronze arc had a radius of about 2 metres and a width of 12 cm, sub-divided into 5° intervals, which in turn were sub-divided into arc minutes; between the minute marks were diagonal dotted lines that made it possible to determine the arc seconds with a maximum error of 3″; this subdivision was called transverse subdivision and was used until the invention of the nonius (an example can be seen on the rim of the sextant by Bürgi in Fig. 6.6 right). The quadrant was equipped with two aiming points that could run along the arc; in the geometric centre of the arc, on the south wall, there was a hole through which the star was sighted with the sights and angular measurements were thus taken. Basically, it was used as an instrument to observe the meridian transit. Tycho decorated the wall on which the quadrant was mounted with a painting showing him pointing through the observation hole; in the background is the laboratory with the various instruments, and the large globe can be seen in the centre between the two intermediate arches; a bookcase is depicted behind the astronomer. On the wall, below the dial, two clocks can be seen to record the times of the transit. Two assistants collaborate, transcribing the notes and telling the time. At the top a cartouche reads: ‘EFFIGIES TYCHONI BRAHE O.F. AEDIFICII ET INSTRUMENTORUM ASTRONOMICORUM STRUCTORIS ANNO DOMINI 1587, AETATIS SUAE 40’. The date of construction was 1582, and the painting was the work of Tobias Gemperling of Augsburg, dated 1587. This drawing is included in Tycho Brahe’s (1598) work Astronomiae Instauratae Mechanica, dedicated to Rudolf II. It is a book of great beauty for its plates and of great interest for understanding the use of astronomical instruments of the time. Each instrument is accompanied by a description of its structure and how to use it. The main instruments that were used at Uraniborg were, in addition to the large wall quadrant and the large globe, the triquetrum, the crossbow, various

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Fig. 4.3  The large wall quadrant

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armillary spheres, azimuth dials, and sextants (on the characteristics of these instruments, see Sect. “Instruments to Guide the Observation”). The largest of these instruments were fixed, others, smaller, were portable. In order to be able to build instruments with the necessary precision Tycho had to have substantial funds at his disposal, and he was paid a regular salary from 1576 onwards; in addition, the King had granted him the income from an estate in Kullagaard on the western end of Scania, where a lighthouse for navigation was in operation. Tycho had to commit himself to keep the lighthouse burning in order to be able to continue to use the estate for the timber needed to heat Uraniborg. With this service, Tycho tried to earn more money by demanding an annual contribution from the ship owners, but the king objected, considering the already agreed remuneration sufficient. Tycho received further financial aid from King Frederick II, and even after the king’s death in 1588, he asked for a large payment for his accumulated debts, which was granted to continue his studies. Among the main collaborators who worked at Uraniborg under Tycho’s direction were Christian Sørensen (Longomontanus) (1562–1647), the instrument maker and goldsmith Hans Crol (d. 1591),5 Peder Flemløse (1554–1598) who also worked on meteorology and medicine, Paul Wittich (1546–1586), a mathematician who stayed very little in Hven and a few years later informed the Landgrave in Kassel about some of Tycho’s measuring instruments that were immediately built and improved by Jost Bürgi. Other pupils of Tycho spent many years in Hven. Elias Olsen Cimber Morsing (1540–1590) was sent on a mission by Tycho to Frauenburg to verify the difference in the measurement of the inclination of the ecliptic calculated by Copernicus, who had probably neglected refraction. On his return, Olsen brought Tycho the 8-foot-long triquetrum, which Copernicus had personally constructed. A peculiar person lived in Uraniborg: a dwarf called Jeppe or Jep who chattered incessantly and, according to Longomontanus, was gifted with second sight. One day during a meal they were waiting for the return of two attendants, when the dwarf said to Tycho “look how your people are washing up in the sea”, hearing this and fearing that they had been shipwrecked Tycho sent a person to watch from a tower of the building who indeed saw a boat capsized in the sea and two soaking wet men who had just come ashore. Jep was also sent on watch to see Tycho’s return from a voyage in advance and warn  Tycho wrote words of great esteem for Crol: ‘my goldsmith Hans Crol, who lived with me and looked after my instruments, many of which he made with his own hands, and with diligence and fidelity served me for many years’. 5

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the students in time. Jep, when consulted about someone’s health, predicted their death or recovery and was always right. Tycho was in constant correspondence with numerous scholars, including English ones. One of the most valued interlocutors was the Landgrave Wilhelm IV of Kassel who had the best instruments for measuring time, built for him, as we shall see, by Jost Bürgi. The landgrave and the astronomer from Kassel, Christoph Rothmann (c. 1540–1601), mathematician and astronomer, exchanged information with Tycho on observation techniques, results concerning particular astronomical events and constantly updated ephemeris tables. Despite the help and trust continually granted to Tycho, King Frederick II never visited Hven and the observatory. Instead, Tycho hosted Queen Sophia for a few days. On several occasions he hosted also nobles and dignitaries from various countries, who were interested not only in astronomy but also in astrology and alchemical studies. On the other hand, Tycho frequently expressed his gratitude to the King by compiling horoscopes and astrological forecasts. In 1590, King James VI (1566–1625) of Scotland visited Uraniborg, having spent several months in Norway to join his promised bride, Elizabeth, daughter of Frederick II, who had been stranded to Hven by bad weather on her way to Scotland. The wedding ceremony between James VI and Elizabeth then took place in Oslo.6 Also in 1590, the astronomer Rothman came from Kassel to discuss the cosmological models of Copernicus and Ptolemy. Rothman never returned to Kassel, where the Landgrave awaited an account of the instruments used in Uraniburg. After a short time, Tycho had a detailed description of all 28 instruments in the observatory compiled and sent the aforementioned work Astronomiae Instauratae Mechanica to the Landgrave. In 1594, Tycho conducted further important observations of Mars, Jupiter and Saturn. In 1595, he resumed his work to determine the position of other fixed stars, 777 of which had been determined in previous years. In 1602 the work Astronomiae Instauratae Progymnasmata listed in total 1000 stars, with their positions reported on the great globe. It was published posthumously by Kepler. In 1596 Christian IV (1577–1648) was crowned king of Denmark; courtesans hostile to Tycho assumed greater power at court, and the king withdrew some of his privileges and sources of income, and denied his commitment to maintain funding for the observatory. Tycho’s situation in Denmark was also deteriorating due to conflicts over economic issues with his pupils. Because of  Some scholars have suggested that the episode is the basis for Shakespeare’s Tempest, which was actually performed at James VI’s wedding. The figure of Prospero would be none other than Tycho. 6

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gossip about his marriage and the legitimacy of his children, Tycho was having difficulty to get his daughter Magdalena married. Tycho was now determined to leave Hven, but new obstacles made his choice difficult, due to an accusation by the inhabitants of Hven that he had lived outside the sacraments and with a concubine for 18 years. In April 1597, Tycho and his family left Hven and moved to Copenhagen. After a few months, Tycho realized that he would not be able to accomplish anything in Copenhagen and embarked for Rostock, where he had already studied in his youth, with about 20 family members and students, among them Frans Tengnagel (1576–1622) who soon became his son-in-law. The plague appeared in Rostock and Tycho thought he had made a mistake in leaving Denmark. He wrote to the King asking again for his benevolence, but the reply he received was very harsh. The King found it insulting that Tycho had not asked his permission to leave Hven and now wanted to retrace his steps. Tycho therefore accepted the invitation of the intellectual Heinrich Rantzov (1526–1598) to travel to one of his castles, and Tycho chose Warndsbeck, near Hamburg, where he believed he could rebuild his observatory. He arrived with his family and pupils at Warndsbeck in October 1597, and there he met Georg Ludwig Frobenius (1566–1645) a mathematician, history scholar and bookseller, who had visited Hven years earlier. The cultural milieu of Warndsbeck allowed Tycho to resume his studies, but he still tried, in vain, to return to Denmark with the help of various intermediaries, until, in 1599, Emperor Rudolf II invited him to Praha. The Emperor’s respect and admiration for Tycho allowed him to finally find a stable, comfortable and suitable accommodation for his work: Rudolph II offered him a choice of several castles and Tycho chose Benatky as his new Uraniburg. Observation instruments, books, writings, and notebooks arrived in Praha in 1600, the same year when Kepler came to Praha after his time in Graz. We will discuss Kepler’s contribution to Tycho’s work and the initial difficult relations between the two in the next paragraph. Here it will suffice to recall that the absence of Longomontanus—who had completed his study of the motion of the Moon—had deprived Tycho of a valuable mathematician, so he saw in Kepler the best person to assist him. In fact, he assigned him a task of great importance: the study of the retrograde motion of Mars. With the study of the motion of Mercury and Venus Kepler had begun to doubt Tycho’s cosmological model (Fig. 4.4). Committed to consolidating the family’s move to Bohemia as well, Tycho slowed down his astronomical studies a little, and on 13 October 1601, he was invited to a dinner at Baron Rosenberg’s, during which he suffered a

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Fig. 4.4  The Tychonic System

bladder or kidney infection possibly produced by alchemy experiments with dangerous substances. Instead of leaving the banquet, Tycho stayed until the end, aggravating himself, and after a painful 11-day suffering he passed away on 24 October after advising Kepler to complete the astronomical tables that were to become the Tabulae Rudolphinae. Tycho tried to reconcile the geometric benefits of the Copernican theory with the philosophical benefits of the Ptolemaic system. His astronomical model correctly placed the Moon orbiting the Earth and the planets orbiting the Sun, but he still believed that the Sun orbited the Earth. Moreover, he considered Copernican model not sufficiently accurate, since the predictions based on this model deviated from the accurate measurements made by Brahe. His astronomical observations carried out without the telescope, which had not yet been invented, were based on a scientific methodology that relied on empirical facts, and on extremely accurate observation instruments. One instrument that does appear in the drawing of the large wall quadrant, the

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clock, does not appear to have been used systematically by Tycho. The clocks he had at his disposal were very irregular, and Tycho preferred to use clepsydrae to measure short intervals, and observation of the position of the stars to determine the time to an accuracy of about 1 minute. Tycho extended the range of astronomical instruments, which before him were limited to astrolabes or zodiacal armillary spheres. He introduced equatorial spheres and built a very large one. Among the 28 instruments described in the Astronomiae Instauratae Mechanica are numerous movable quadrants. Tycho’s contribution to the advancement of astronomy consists, as mentioned, in the exceptional precision of his measurements. This is due not only to the precision of its instruments but also to the original method of measurement adopted. To determine right ascension and declination of a celestial body, Tycho initially uses the data measured at the transit, but the determination of the time of transit7 was uncertain due to the imprecision of the clocks at his disposal. Therefore, the only valid data at the transit is the declination of the celestial body. To determine the right ascension, Tycho uses an indirect method. Starting from the very accurate data of the lunar and solar motion model, he derives from them the AR (right ascension) of Venus chosen as the reference body for its luminosity. Finally, he derives the hour angle of the star as a difference from that of Venus.8

Johannes Kepler: A Journey in the Space Let’s resume the state of the art. Ptolemy’s geocentric model sought to explain the apparent retrograde motion of Mars, Jupiter, and Saturn. Recall that before Ptolemy it was assumed that these planets moved along a perfect circumference, the deferent, with the Earth at its centre and at the same time rotated around the circle of the epicycle, whose centre moves along the deferent. But the variation of Mars’ apparent velocity showed the inaccuracy of this hypothesis. Ptolemy hypothesized that the deferent circle was not centred on the Earth and that the motion of the epicycle revolved around the equant. Thus, opposite angles of equal amplitude were travelled at different velocities. In Ptolemy’s theory, the planets, the sun, and the fixed stars are moved by spheres rotating around axes with different inclinations. The epicycles in turn revolve around these spheres. The problem of the decentralization of the spheres in the equant raised problems and doubts in many scholars. Copernicus  See Appendix “Positional Astronomy”.  An in-depth analysis of Tycho’s method was presented by Carman (2022).

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in his postulates eliminates the equant, which is no longer necessary once the Sun is placed in the centre. Today we know that the orbits of planets and satellites are elliptical and have different eccentricities, but how could Kepler realize that orbits cannot be circular, thus freeing himself from the cumbersome equant hypothesis? Was he imaging a journey in the space? We must therefore devote a few pages to illustrating the life and personality of this scientist, to whom we owe decisive steps in the development of knowledge of the cosmos, in an era devastated by religious conflicts. The Early Years 1571–15949  Johannes Kepler (1571–1630) was born in Weil der Stadt in Württemberg on 27 December 1571 into a family of modest nobility from Nüremberg. Johannes always used the coat of arms the family as a seal. His grandfather, Sebald, was a firm believer of the Lutheran Reformation, even though Weil der Stadt was a centre where Lutheranism failed to gain a majority. Educated according to Reformation principles, Johannes developed a very solid faith. His father’s morality, on the other hand, was weak. Litigious, attracted to military life, he treats his wife, in turn a chatty and unpleasant person, with brutality. Family life is therefore not the happiest one and this perhaps contributes to Johannes’ sensitive and irascible character at the same time. In 1574, his father enlists to fight in the Netherlands and abandons the family. Little Johannes is brought up by his grandparents with little affection. In 1576 his father returned and moved the family to the nearby town of Leonberg, where Johannes enrolled in school at the age of 7. The schools in the Grand Duchy of Württemberg were of a high standard and very well organized: they took the education of children away from the monasteries and taught Latin and arithmetic in order to form a class of civil servants of the state. Already the first years of school revealed Johannes’ extraordinary abilities and he won the admiration of the teachers. It was during these years that his passion for astronomy was born, when in 1577 his mother took him to a mountain to observe a comet.10 In 1580 it was his father’s turn to make him observe an eclipse of the Moon.11 In 1584, having passed his state exams, Johannes enrolled in the boarding school in Adelberg. After two years he moved to the higher seminary in the

 A biography of Johannes Kepler has been written by Caspar (1993).  Now classified as C/1577 V1. 11  In 1580 there were two total eclipses of the Moon, the first on 31 January, the second on 26 July. 9

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Cistercian monastery in Maulbronn.12 The religious climate in these centres tends to foster doubts about the principles of the Catholic faith, but Johannes’ Lutheran faith is strengthened, making him adhere to the principles of the Augusburg Confession.13 In 1589 at the age of 18, Johannes passed the entrance exams to the University of Tübingen, the most important centre of Lutheran inspiration in Württemberg. The subjects studied in the first 2 years were Greek, Hebrew, dialectic, rhetoric, ethics, astronomy, and physics. Theology was studied only from the third year onwards. Right from the start Kepler was awarded a scholarship. In 1591 he passed his exams and enrolled in higher studies. Among the authors that most contributed to his education was certainly Platonic and neo-Platonic philosophy, as well as the school of Pythagoras. Fundamental was the tutoring of Michael Mästlin (1550–1630), a distinguished mathematician and astronomer, who became a mentor with whom Kepler maintained relations for many years. It was Mästlin who first introduced Kepler to Copernicus and his cosmological model. In those years, Mästlin supported the Ptolemaic theory in public seminars, but privately he was now convinced of the validity of Copernicus’ ideas. In this regard Kepler wrote (Caspar, 1993, p. 46): In Tübingen, when I was attentively following the lectures of the famous Master Michael Mästlin, I realized how cumbersome the widespread notion of the structure of the universe was in many respects. I was then delighted to discover Copernicus, whom my teacher often quoted in his lectures, to the extent that I not only often recalled his view in candidate disputes, but also conducted in-depth disputes regarding the thesis that the first mobile (the revolution of the heaven of the fixed stars) derived from the earth’s rotation. I had already set to work on attributing the motion of the sun to the earth, basing it on physical grounds, or, if one prefers, on metaphysical grounds, as Copernicus does on mathematical grounds. For this purpose, I gradually identified the mathematical advantages Copernicus has over Ptolemy, partly drawing this from Mästlin’s lectures and partly from my own studies.

We must not overlook the fact that Kepler here recalls his youthful experience when he was not yet fully acquainted with Copernicus’s thought, but we can see in these words his desire to interpret the cosmos with mathematical perfection.  At the time, interreligious conflict in the German-speaking and northern regions was relatively contained and religious schools also offered hospitality to young people of different beliefs. This changed radically from the Thirty Years’ War (1618–1648) onwards. 13  Document published in 1530 summarizing the fundamentals of the Lutheran faith. See https://en. wikipedia.org/wiki/Augsburg_Confession. Accessed May 2022. 12

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Kepler was not only concerned with astronomy in Tübingen: theological studies were prevalent. From his astrological knowledge, he was able to earn money for the horoscopes he was commissioned to make. His theological studies were due to end in 1594, but even before that, Kepler was invited to Graz to occupy the chair of mathematics. Kepler was surprised and, having wished to dedicate his life to the Church, the proposal aroused strong doubts. In the end he accepted, asking for and being granted the privilege of being able to return home to Württemberg and take up religious life. Graz 1594–1600  From 1594 to 1600, Kepler moved to Graz, part of the Holy Roman Empire, then ruled by Rudolf II from 1560 to 1612 of the House of Habsburg, where Lutheranism was fought according to the principle cuius regio eius religio14 agreed upon in Augsburg in 1555. The school at which he taught was still of the Protestant faith and in competition with the Jesuit schools, and was intended to educate the sons of the local bourgeoisie, who apparently had little interest in mathematics, which is why Kepler had very few students. This fact, which did not prevent the Rector and other professors from holding Kepler in the highest esteem, allowed him to devote himself completely to his astronomical and mathematical studies. It was during his stay in Graz that Kepler published his first work Mysterium Cosmographicum in 1596 (Kepler, 1596), the first step towards his great discoveries. Following the principles of Copernicus, Kepler pursued the search for a mathematical order linking the motions of the planets and the earth around the sun, with the belief that the perfection of divine creation could be found in mathematical perfection. After all, the principle of perfectly circular motions was also inspired by an Aristotelian idea of perfection, but Kepler went further. In the preface to the Mysterium Kepler observes that the conjunctions15 between Saturn and Jupiter occur successively after eight zodiacal positions, and drawing a diagram (Fig. 4.5) showing the sequence of conjunctions reveals a family of equilateral triangles whose envelope16 is a perfect circle, which inspires him to find geometric figures that can be related to the orbis of the planets.17

 Whose realm their religion.  For the concept of conjunction see the Appendix. 16  The envelope of a family of triangles is a curve that is tangent at one point to the three sides of all triangles. 17  In this writing Kepler still uses the term orbis to denote the path of a planet, which he calls via anticipating the future designation of orbit. The notion of orbit will become fully clear with the publication of the Astronomia nova and fully defined in the Epitome Astronomiae Copernicanae. We will return later to clarify the concepts of orbis, sphere, and orbit. 14 15

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Fig. 4.5  Left: The conjunctions of Jupiter and Saturn. Mysterium Cosmographicum. Right: Kepler’s drawing of the Platonic solids enclosing the planetary orbits. Mysterium Cosmographicum

I thought: what, according to Copernicus, is the size and proportion of the six heavens? Only five figures remain among the infinite ones we can find, which have certain specific properties. And what plane figures are there among the filled orbis? More suitable they would be solids. And here, reader, is the discovery and content of this little work. Five regular bodies that surround orbis or are contained in them …

Using data from the Tabulae Prutenicae for measuring the distance of the planets from the Sun as Copernicus had estimated it, Kepler theorized that the different orbis could be enclosed within Platonic solids, deriving the foundations of his hypotheses from geometry. Let us circumscribe a regular solid of 12 faces around the orbis of the Earth; the sphere circumscribing a dodecahedron will be the orbis of Mars. Let us now circumscribe the orbis of Mars with a solid of four faces and the sphere that circumscribes it will be the orbis of Jupiter. Let us now circumscribe the orbis of Jupiter with a cube: the sphere circumscribing it will be the orbis of Saturn. Now, inscribing a solid of 20 faces within the orbis of the earth; the sphere inscribed in turn in the icosahedron will be the orbis of Venus. Finally, we inscribe an octahedron in the orbis of Venus, and the sphere inscribed in it will be the orbis of Mercury (Fig. 4.5 right). The five solids are the symbol of geometric perfection, they are the only regular solid figures, i.e., with equal faces, angles and edges, unlike regular

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plane polygons that can have any number of equal sides. The number 5 fits perfectly to justify the number of planets then known, placed between them. They were considered in the number 6, including the earth, and 6 is the number of spheres that can circumscribe and inscribe the six spherical orbits. This finding convinced Kepler even more that he had to describe the universe with mathematical tools, and that he had to find ever more accurate data on the motions of the planets in order to perfect his method and demonstrate that planetary periods were linked to their distance from the Sun.18 The result of his study was imbued with an idea of the harmony of the cosmos, which was expressed through the geometric properties he had identified. Kepler extended this geometric harmony and music associated to the planets distances intervals—according to the Pythagorean principles that musical intervals were ratios between whole numbers19—and wrote pleasing melodies for each planet (Kepler, 1619). Between 1596 and 1599, Kepler tried to make a model of his invention, which would be discussed later, asking the Duke of Württemberg Friederich I for financial help. These were difficult years, however, in the constant search for secure financial means, a need that haunted him throughout his life. Kepler sent his study to Galileo but did not receive any comments, while Tycho Brahe was positively impressed by it and the mathematical knowledge it revealed. The relationship with Tycho Brahe was of great importance and soon developed into a close collaboration. In 1600, religious conflicts in Graz grew to the point where Lutherans were forced to leave the city or convert. Kepler, who was deeply rooted in the Lutheran faith, decided to leave Graz and tried to return to Tübingen, but he also desired to work with Tycho Brahe, and eventually, he moved to Praha, where Tycho had established a study centre with many collaborators, under the patronage of Emperor Rudolf II. Praha 1600–1612  The relationship between Kepler and Brahe began with difficulty due to misunderstandings that arose after Kepler’s involvement in Ursus’ shameful behaviour20 and to Kepler’s irritable temperament. Kepler  In 1776, Johan Daniel Titius (1729–1796) discovered a mathematical relationship between interplanetary distances that was formally published in 1772 by Johan Elert Bode (1747–1826), called the Titius-­ Bode law. The semi major axis of the orbit of a planet expressed in Astronomical Units is: a = 0.4 + 0.3 ∗ 2i; i =  −  ∞ , 0, 1, 2, … . 19  These are the well-known ratios 2:1 of octave, 2:3 of fifth, 3:4 of fourth, etc. Plato himself in the Timaeus, stated that the ratios of octave, fifth and fourth generated consonant musical chords. 20  We must devote a few words to a figure who intersected the scientific path of Tycho and Kepler: Nicolaus Reimar Bär, also known as Reimarus Ursus, briefly Ursus (1551–1600). Ursus, who had had no training in his youth, managed to become a self-taught mathematician and astronomer until he was appointed Imperial Mathematician at the court of Rudolph II. In 1584 he had the opportunity to visit 18

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sought security for himself and his family (he had in fact married in 1597). Kepler’s wife had a lot of property in Graz and her father was reluctant for them to move away. However, they were eventually forced by events and religious conflicts and Kepler obtained to go to Prague to work with Tycho.  The period in Praha lasted from 1600 to 1612. Brahe was known as the astronomer capable of the most precise observations and one of Kepler’s goals was to gain access to that data to develop his mathematical theory of the Universe. For the first few months Kepler still hoped to return to Tübingen, but he was soon discouraged and had to resign himself to staying in Bohemia. His Lutheran faith was fortunately better tolerated than in Graz, but he suffered financially as he was entirely dependent on Tycho Brahe, who tended to see Kepler’s compensation as mere benevolence rather than due recognition for the work he was doing. Tycho soon entrusted him with the long task of studying the retrograde motion of Mars, which Christen Sørensen Longomontanus had been working on without much success. Kepler returned to Graz after his father-in-law’s death to try to sell his wife’s property and obtain a large sum of money, but the result was modest. Fortunately, on his return to Praha, Tycho Brahe finally introduced him to the Emperor Rudolph to whom he had promised to dedicate the astronomical tables he was compiling. A few days after this meeting Brahe fell seriously ill and shortly before his death, on 24 October 1601, he left his scientific papers and instruments to Kepler with the commitment that he would complete his work following his own cosmological model and not that of Copernicus. A few days after Brahe’s funeral, a messenger from Emperor Rudolf II informed Kepler that he would be entrusted with the continuation of Tycho Brahe’s work, that all instruments would be transferred to him and that he would receive an adequate salary. In practice, he was appointed successor to the scientist who had previously helped him. The entire European scientific community was convinced that there was no better person to receive Tycho Brahe’s legacy than Kepler. However, the astronomers who had worked with Uranjborg, and 2 years later he informed Landgrave Wilhelm IV of Kassel that he had devised a new system of the world that perfectly matched the Tychonic system. Tycho immediately accused Ursus of plagiarizing his theory and initiated legal action against him that continued even after Ursus’ death in October 1600. In 1595 the young and unknown Kepler, eager to get in touch with prestigious mathematicians and astronomers, wrote a letter to Ursus full of praise, acknowledging him as having been a master of mathematics, and asking his opinion on a trigonometric calculation concerning the apogee of the planets. In 1597 Ursus, who had read Kepler’s writing Mysterium Cosmographicum, replied to this letter, in turn praising Kepler. Ursus then published his model in the work Fondamentum Astronomicum in 1588, enclosing Kepler’s adulatory letter as support of a scholar of recognized value, without even asking Kepler’s permission. Kepler’s rudeness was excused by Tycho after some time, finally opening up a collaboration that was fundamental to the progress of astronomy.

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Brahe and Frans Tengnagel, Brahe’s son-in-law, obstructed Kepler’s work for a long time. Kepler was able to use and complete Brahe’s tables only after promising Tengnagel to allow him to write a preface to his work, Astronomia Nova, which he published in 1609. We will return to the results of this work later on. Let us now recall other works by Kepler and the last phase of his life as a scientist, constantly travelling through Europe. While studying planetary motions, Kepler was also interested in the exact determination of the phenomena of eclipses of the sun and moon, which had different durations in the countless reports. He also tried to explain the phenomenon of what we now call the Sun’s corona, the appearance of a strong luminosity during the total occultation, or the reddish colour of eclipses of the Moon. All this led him to study the phenomena of refraction, and to study what had been written in 1270 by Erazmus Witelo (1230 c.–1300 c.) known as Vitello, who in turn reported on al-Haytham‘s studies of optics. The result of Kepler’s investigations was published in Ad Vitellionem Paralipomena, quibus Astronomia Pars Optica, briefly called Astronomiae Pars Optica, completed in 1603 and published in 1604. This treatise is considered the founding moment of modern optics, in which Kepler analysed the anatomical structure of the eye, formulated the reasoning that would lead Snell to identify the law of refraction, and linked the principles of perspective to those of optical projection. The religious conflicts in Bohemia were not completely resolved, the different faiths coexisted with difficulty, but Kepler, who had been expelled from Graz as a Lutheran, could rely on the protection of Emperor Rudolf II in Praha, who was more interested in science and the works of artists than in their religion. Praha had thus become an important centre for the arts and science. It was in this climate that the observation of the stella nova in 1604 took place: on 17 October, Kepler was able to observe a star as bright as Jupiter in the constellation Ophiuchus (also called Serpens-Snake) in the vicinity of the three planets in conjunction Jupiter, Saturn and Mars. His observations led to the publication of De Stella Nova in pede serpentarii in Praha in 1606. In those years Kepler considered himself to be in financial straits, and certainly what was paid to him from the Imperial Treasury was not enough to compensate for permanent collaborators who could help him with the laborious calculus and studies. On the other hand, Rudolph II spent enormous sums on both his collections and to finance his army. 1611 was an unlucky year for Kepler, the conflict between Bohemia and the neighbouring duchies resumed, family life suffered due to his wife Barbara’s illness and the death of one of his youngest sons, his last attempt to return to Tübingen failed and his

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wife herself died shortly before the autumn. A new season opened: the Linz period. Linz 1612–1626  Kepler arrives in Linz in 1612, a small town where his knowledge stands out but is so high that it is poorly understood by the community. Kepler was employed as a teacher of mathematics and had to depend on bureaucratic figures of whom he had low esteem. Moreover, his move to Linz took place with the complete approval of Rudolf II’s successor, Matthias, who had also renewed his appointment as imperial mathematician. However, Kepler preferred to move closer to his wife’s birthplace. Over time he had acquired a reputation as a Calvinist, which had disqualified him from returning to Tübingen, and even in Linz this suspicion of unreliability, of being a ‘lost sheep’, was reinforced to the point of excluding him from communion. In October 1613, Kepler married the young Susanna Reutinger, an orphan who had been adopted and protected by Baroness Elizabeth von Starhemberg. It proved to be a good marriage and Kepler was able to keep the two young children from his previous marriage with him, and he had six more children by Susanna, three of whom died in infancy. The Witchcraft Trial of Katharina Kepler  It seemed to be a happy time, having resumed his studies, but between 1615 and 1616 in Leonberg, where his mother Katharina Kepler lived, six women were condemned for the crime of witchcraft, and in his neighbouring hometown of Weil der Stadt between 1616 and 1629, 38 women were also condemned for this crime. The trial of Kepler’s mother lasted 6 years, during which she was imprisoned on false and confusing charges.  Kepler finally obtained the support of the duke of Württemberg, Frederich I, and on September 4th 1620, the trial began and a legal battle between the defense and the prosecution began, which produced pages of documents. A decision was quickly reached by the Duke on September 10th: the accusations were not sufficiently substantiated so the woman would have to be brought before the instruments of torture and forced to confess. On September 28 1621, Katharine Kepler was brought into the torture chamber, shown the instruments that would be used if she did not confess. The woman knelt down, affirmed her innocence and recited the Lord’s Prayer, trusting in the Lord’s will. The Duke, informed, declared that the woman had invalidated the circumstances of the accusation and should therefore be acquitted, which was

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done on October 4 1621, when she was finally freed. She did not enjoy her freedom for long because she died on April 13, 1622. During this painful period, Kepler did not cease his studies: he completed another book Harmonices Mundi libri V, which he had begun to conceive when he was still living in Graz and of which some traits are present in Mysterium Cosmographicum. By 1618 he had formulated the third law of planetary motion. The mathematical structure erected to understand the universe was based entirely on numbers, which represent the pure harmony that stimulates the thought, whereas musical harmony stimulates the senses alone. Geometry, music and astronomy come together through harmony. In this work, Kepler investigates the study of regular solids and relates their proportions to musical ratios on the one hand, and to planetary distances on the other. But it is above all a study of musical harmony. Returning to Linz in 1621, he resumed his mathematical studies. He had heard about Napier’s invention of logarithms and decided to write his own work to confirm the validity of a mathematical method that greatly simplified calculus. He published a treatise on logarithms, Chilias Logarithmorum, in 1623. During 1626 Kepler was busy printing the Tabulae Rudolphinae after a long dispute with the heirs of Tycho Brahe, but the work was interrupted by the war. Ulm, Sagan, Regensburg 1626–1630  In October 1626, Kepler obtained permission to leave Linz and travel to Ulm to escape the risks of religious persecution and the dangers of war, taking with him his family, books, manuscripts and all the printed materials of the Tables. Leaving his family in Regensburg, he continued his journey to Ulm where he resumed the printing of his work. Kepler had the printing material with him, including the special typefaces he had cast for the astronomical symbols. At the time, the printing of an astronomical treatise was a very complex process that required the direct participation of the author to correct the proofs and help the printer with the typesetting. Kepler devoted himself to it with daily effort, all the more onerous because of the nature of the work: countless pages of neatly aligned numbers. T   o conclude the work, one more step was needed: the writing of the dedication, which had to be agreed upon with Tycho Brahe’s heirs. Kepler received the text proposed by them and added a few changes to better explain the reason for the long duration of the work. The cost of printing the work had

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been entirely borne by Kepler who had been granted by the Emperor, in agreement with Brahe’s heirs, the privilege of selling the copies by collecting the full amount of the expenses. Once the expenses had been offset, he was to share the profits with Brahe’s heirs. To organize the sale of the book, Kepler travelled to Frankfurt in September, while the autumn fair was taking place, and tried to agree on a fair price. Unfortunately, this took longer and was concluded after the end of the fair, leaving Kepler, however, with the hope of selling numerous copies at the following spring fair.21 The frontispiece of the work (Fig. 4.6) was designed by Kepler himself and engraved by Georg Celer (1599–1638). It depicts several great astronomers who paved the way for Kepler himself. In the background are four columns in wooden trunks, representing the ancient rudimentary astronomy with a Chaldean with a Phrygean cap. On the rough brick columns are the names of Meton, Aratus, Hipparchus, Ptolemy. The figures of Brahe—wearing a rich gown similar to that of the portrait on the great quadrant—and Copernicus have also the names written in two solid and clean columns in centre. Hipparchus with his catalogue is on the left and Ptolemy, sitting on the right with his book, complete the scene. On the roof of the small temple are six allegorical representations of physics, mathematics and astronomy, on the top of the temple stands the Imperial eagle. Kepler himself is represented on the left face of the basement studying at his desk by candlelight, in the central square is the island of Hven where Brahe conducted his observations. The printing process of the Tabulae is depicted on the two rightmost panels and in the leftmost there is one of the Tycho heirs.22 Back in Ulm after his trip to Frankfurt Kepler again tried to find a city where he could move to and teach and continue his studies. His numerous contacts with friends and governors in the different centre were fruitless, hampered by the religious and political conflicts that raged across Europe. He could have returned to Praha, where the Emperor held him in high esteem,  The history of the Frankfurt Book Fair began in the fifteenth century when Johannes Gutenberg, in Mainz (near Frankfurt), invented movable letter printing, which would revolutionize the world of books. Until the seventeenth century, the Frankfurt Fair remained the most important in Europe. Then, political and economic changes, but above all the strict imperial censorship introduced with the Catholic Counter-­ Reformation caused many publishers to move to Leipzig, which gradually became the new center of the book and publishing industry. After the Second World War, since Leipzig was in the DDR, the Fair went back to Frankfurt. 22  A study of this frontispiece is Gattei (2014). Brahe’s heirs rejected a first drawing—probably made by Kepler’s friend Shickhard—because Tycho was not wearing the collar and badge of the Order of the Elephant awarded in 1585. The second and final version was accompanied by a poem that celebrated the Temple of Urania; the frontispiece is an allegory of the history of astronomy. 21

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Fig. 4.6  The frontispiece of Tabulae Rudolphinae

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but court life disgusted him, and hospitality in the Habsburg kingdom meant converting to Catholicism, which Kepler did not intend to do. Kepler was offered to move to Sagan, a fief in western Poland assigned by the Emperor to Albrecht von Wallenstein (1583–1634),23 for whom he had written a horoscope (Kepler, 2009a, p. 445). In May 1628, Kepler left for Regensburg to join his family and returned to Linz to obtain permission to finish his work as a mathematician, which was finally granted allowing him to finally move to his new centre, Sagan, where he arrived in August. In reality, Kepler did not intend to stay there for long, so much so that he had left many of his possessions in Regensburg in the hands of trusted friends. Unlike in Praha, there were no scientists or scholars of his level living in Sagan, and Kepler suffered as he felt isolated. In addition, he was still waiting to receive important compensation from the imperial treasury, repeatedly promised, for his many commitments; even with Wallenstein’s help, steps were taken to give him property from which he could earn an income. Wallenstein also promised to allocate funds for the establishment of a printing press in the city of Sagan, something Kepler relied heavily on. In fact, a printing press was set up in Kepler’s own house in 1629, and he engaged himself in the work of a printer,24 primarily for the printing of ephemerides long awaited by all the astronomers of Europe, as a result of the publication of the Tables. This work went into production in 1630, and during the breaks Kepler devoted himself to another work: Somnium, seu opus posthumum de astronomia lunari [Dream or posthumous work on lunar astronomy]. During those weeks, a young man, Jakob Bartsch (1600–1633), offered himself as a collaborator to Kepler who welcomed him with great joy: he could finally have someone to help him compile the ephemerides and also assist him in drawing up the astrological calendars that were requested. As was often the case in those years, the pupil Jakob married Susanna, the master’s daughter, in March 1630. The ceremony took place in Strasburg, but Kepler had to stay in Sagan with his wife who gave birth to a daughter, Anna Maria, shortly afterwards.

 Wallenstein was the supreme commander of the Imperial Army of the Holy Roman Emperor Ferdinand II of Habsburg during Thirty Years’ War. 24  The first opera printed by the new machine was the ‘Commentatiuncula’, the answer to a letter from the Jesuit Terrentius (Sect. “The Arrival of the Jesuits and Western Astronomy”). Two months later, the Duke let Kepler know that he was satisfied with his work, that he honored him and his printers with 250 guilders and that he expected the ephemeris work that was currently being printed be dedicate to him. 23

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The conflicts of the war resumed even in Sagan, once again causing Kepler difficulties with the local religious communities. These reasons and the desire to collect the credits accumulated in Linz prompted Kepler to set off again, first for Leipzig, then for Regensburg where he arrived ill on November 2nd, and within a few days he died on November 15th 1630. panis quadregesimalis  How did Kepler come to conceive the new laws of planetary motion? The path of the planets according to Aristotelian principles had to be a circle a perfect as the perfection that befitted the heavens. At the same time, a purely circular motion could not justify the retrograde path of the planets and therefore epicycles were used. However, this was not sufficient because it did not explain the anomalies of the orbits, i.e. the difference in velocity between perigee and apogee, and Ptolemy had introduced the equant, making the description of planetary motions increasingly cumbersome, complicated and in any case unsatisfactory. Kepler was the first astronomer who imagined ‘hovering’ in the sky, to observe the motion of the planets from outside the Earth, positioning himself at a distant point perpendicular to the plane of the ecliptic. Thus, using data from the Tabulae Rudolphinae (Gingerich, 2004, p. 49), performed long and complex calculus to make an orthographic projection and plot the apparent path of Mars around the centre of the deferent over a period of about 16 years from 1580 to 1596. Imagine for a moment that you only have a long table of numbers describing the celestial coordinates of Mars’ position as seen from Earth. The problem now becomes to transform these numbers to represent the coordinates of Mars’ position on a plane. Thus, Kepler ideally chose the viewpoint of an astronaut flying above the planets. He had to repeat these calculus 40 times, as he himself writes, and without any aids. An astonishing plot emerged that Kepler called panis quadregesimalis—Lenten bread as it recalls the shape of the pretzel, which was a typical Lenten period cake. If we look at Fig. 4.7, we notice loops representing the path during apparent retrograde motion. The letter a designates the position of the Earth, while the letter b is the centre of the deferent. We have already seen that the curve generated by an epicyclic motion is an epitrochoid, but a perfect epicycle should generate an epitrochoid in which the rings are all equal, but this is not the case. The loops are at varying distances from the Earth, and they have different sizes.

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Fig. 4.7  The drawing of Mars motion, so-called panis quadregesimalis

All this strongly impressed Kepler, who became convinced that circular motion with the equant shift to account for the irregularities of motion did not stand up precisely because of the absence of perfect symmetry and regularity of the curves, thus contradicting the Aristotelian principles. The caption of the drawing is: If one groups all these things together, and believes at the same time, that the sun really moves through the zodiac in the course of a year, as Ptolemy and Tycho Braheus believed; then it is necessary to admit that the circuits of the three higher planets through ethereal space, as they are composed of many motions, are really spirals; not as before, in the manner of a cluster of threads, the spirals arranged next to each other; but more truly in the shape of a quadregesimal bread, in about this manner

The Three Laws  From this investigation, Kepler arrived at the conception of the elliptical motion of the planets: the principle that planetary motions were perfectly circular had to be abandoned. Kepler, daring to go against a

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thousand-­year-old tradition, hypothesized that the planets’ motion took place along elliptical orbits, placing the Sun at one of the foci of the ellipse, thus eliminating the complications of epicycles and equants. Kepler’s theory was not just based on mathematical considerations but assumed an underlying physical principle. Until Tycho Brahe, the scientific world was convinced that the physical cause of motion lay in the celestial spheres on which the planets were rigidly located. Now Kepler hypothesized that the sun exerted a form of magnetic attraction (as Kepler called it) that drew the planet along its orbit to its closest point, perihelion, and then away from it again to aphelion and resumed the cycle again.  Kepler had now come within a step of Newton, and had also founded astronomy on physical principles. Remember that the full title of Astronomia nova is: Astronomia Nova ΑΙΤΙOΛΟΓΗΤΟΣ seu Physica Coelestis, tradita commentariis de Motibus Stellae Martis ex Observationibus G.V. Tychonis Brahe.25 These assumptions led him to formulate his famous three laws of planetary motion: 1. The orbit described by a planet is an ellipse, of which the Sun occupies one of the two foci. 2. The segment (vector) joining the centre of the Sun with the centre of the planet describes equal areas at equal times. 3. The squares of the times the planets take to travel their orbits are proportional to the cube of the semi-major axis. The first law gets rid of the Aristotelian perfection of circular motions, and accounts for the variation of the planets’ apparent distance between aphelion and perihelion. The second law solves the planetary anomaly problem and accounts for the variation of the apparent velocity of the planets along their orbit. Finally, the third law links the distance of the planets from the sun to the period of their orbits and mathematically justifies the planetary distances. It is interesting to note that the first discovered law is the second, as in letters written in 1602, while the first law is described in a letter of 1605 to David Fabricius (1564–1617) and the third law was discovered after leaving Praha.

 New Astronomy justified, or Celestial Physics, Treated by Means of Commentaries on the Motion of the Star Mars, from Observations of Tycho Brahe, Gent. Tr. by the author. 25

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Spheres and Orbits  Goldstein and Hon (2005) shed light on another aspect of the Keplerian scientific revolution. For Eudoxus the planets were anchored to spheres, but this hypothesis was a pure mathematical abstraction. Ptolemy’s emphasis on the hypothesis of the epicycles and the equant overshadowed the theory of the spheres, which, however, re-emerged in the Middle Ages, when the celestial spheres were seen as concrete and real entities, filling the entire space between the planets. And above all, they offer the solution to the physical cause of the celestial motions: the configuration of the homocentric spheres incorporated into each other. The outermost of the fixed stars drags them all along and is therefore called primum mobile, or immobile motor.26  The work of Peuerbach Theoricae novae planetarum of 1472 became the reference work for Ptolemaic theories during the sixteenth century. In it, Peuerbach names the celestial spheres orbis. Kepler in the Mysterium Cosmographicum still makes use of the term orbis, but rejects the notion of concreteness assumed by Peuerbach. He still has to solve two problems: is there something between the planets’ orbis? What are the distances between the planets that are not made explicit in the previous theories? It is in relation to this second problem that Kepler makes use of the Platonic solids, whose presence between two planets determines the different interplanetary distances. The concept of orbis for Kepler has only the meaning of a geometric constraint, losing its value as a real object. This is most clearly seen in the description of a planetary machine project that we will examine in Sect. “Paving the Way to the New Cosmos: Kepler’s Planetary Machine”. Kepler takes a further step in the Astronomia nova, where he adopts the new term orbit, which has the current meaning of the path of a celestial body, but there is already a sign of this meaning in his earlier work Mysterium, when he writes, in Chap. XIV: Igitur ut ad principale propositum veniamus: notum est, vias planetarum esse eccentricas et proinde recepta physicis sententia, quod obtineant orbes tantam crassitiem quanta ad demonstrandas motuum varietate requiritur

In other words: accepted that the paths of the planets (vias planetarum) are eccentric, therefore one accepts the physicists’ view that the fatness (crassitiem) of the spheres must be sufficient to contain the path of the star in its irregular motion between equant and deferent, as we see in Fig. 4.8.  Dante’s cosmology is entirely founded on the principle of the primum mobile whose origin Dante, in the Convivio, attributes to Ptolemy. 26

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Fig. 4.8  Kepler’s scheme of the thickness of an orbis. A is the orbis of Saturn, B is the orbis of Jupiter

Here Kepler’s path departs from that of his predecessors: Peuerbach held that there was no empty space between the spheres, that each orbis was tangent to the previous and the next. Kepler delved into the structure of the orbis as shells within which the planets move. The introduction of Platonic solids within this empty space was intended to determine the distances between what he soon called orbits. Eudoxus also moved along the same line of thought, when he recognized the retrograde path as the hippopede, which is similar to the panis quadragesimalis diagram. Until Kepler, celestial observations were a series of pairs of numbers: time and position, there was no graph of positions as a function of time, and this is precisely what Kepler does by plotting the path of Mars (the Lenten bread) over the course of 16 years. Kepler’s journey into the space results in the depiction of a trajectory of Mars, as seen from the Earth, projected on the plane of the ecliptic. Although no direct contact between Kepler and Descartes is documented, the Cartesian idea of analytic geometry was in the air. After all, the very title of the work, Mysterium Cosmographicum, bring a sense of graphical perfection. We note that at the time the term graphice has a connotation related to drawing, as in the relevant entry in Vitali’s Lexicon Mathematicum (Vitali, 1559, p. 311): GRAPHICÉ, means, in Greek, overhang, decorative element. It mainly relates to Optics, and is a skill that teaches the way in which what is erect can be deformed in the plane, e.g., in what measure and form shadows and lines should be projected. In particular it is called Analemmatographica when it consists in

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examining the circles and parallels of the celestial sphere, whose image falls on a horizontal or vertical plane, or on a wall inclined in any way. It belongs to Perspective when other things are to be transported and delineated on a plane at a long or short distance. For this reason, Vitruvius notes in the beginning of his book de Architectura, it is a requirement of a good architect to possess the science of using a pencil with which he can more easily modify drawn examples of a constructed model27

Already at the time of this first work, Kepler realized that he was on his way to plotting the path of the stars. Referring to the Mysterium, Lucio Russo observes (Russo, 2013, p. 169): The crystalline sphere of fixed stars had lost its usefulness when it was realized that the rigid diurnal motion of the constellations is an illusion due to the earth's rotation and was therefore abolished by Heraclides Ponticus in the fourth-century BC, leaving in its place a theoretical spherical surface, used as a mathematical model to locate the stars. After the cultural collapse, the sphere of fixed stars regained its heavy corporeity, in which Kepler still believed in the seventeenth-century.28

Even so, Kepler soon gets rid of this concrete conception, although in the planetarium project that we will examine later, homocentric spheres are still present. Later, abandoning the conception of the sphere (orbis) as a physical entity, he manages to conceive of the motion of the planets as a path (iter /via) that he will eventually call an orbit. However, it is now necessary to consider the physical cause of the motion of the planets, which is no longer traced back to the homocentric spheres. And Kepler identifies it in the Sun, which exerts a magnetic force at a distance on the planets, thus paving the way for Newton. A Science Fiction Novel: Somnium  The book Somnium, seu opus posthumum de astronomia lunari (Kepler, 1993, pp. 319–331) was published by Kepler’s heirs in 1634, immediately after the printing of the Tabulae Rudolphinae. It was conceived by Kepler in October 1605. A handwritten copy of the short text circulated in Württemberg after 1609 and contributed to the escalation of the case against Kepler’s mother. Indeed, there is a passage in the text of the Somnium in which Kepler speaks of “scrawny hags” (dürren

27 28

 Tr. By the author.  Tr. By the author.

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Vetteln), capable of travelling long distances in the night on forks (furcae) and thus particularly suitable for lunar flight. Kepler makes use of the literary genre of the dream narrative, which allows the greatest possible freedom. With this writing he wants to offer an argument in favour of the annual movement of the earth against the objections, through the example of the moon. After a foreword Kepler writes that he fell into a deep sleep in which he evokes the figure of Duracotus. Duracotus is a native of Iceland whose fate is closely linked to Fiolxhilde,29 in whom we can recognize Kepler’s mother, Katharina. Fiolxhilde knows magic, trades and works with herbs and magic remedies and also initiates her son Duracotus into this art. Let us read a short excerpt30 of Kepler’s words. My name is Duracotum, my homeland Iceland, which the ancients called Thule. My mother was Fiolxhilde. Now deceased, I can write about her without fear of slander or insult. ... On the feast day of St. John, when the Sun stays in the sky for 24 hours, she used to take me to Mount Haecle. She gathered many herbs and cook them with much ceremony. With goat skins she used to form bags to sell to sailors. When one day I opened a bag that my mother was selling, fearing I wanted to deprive this small gain from her, she angrily left me to the sailor. I was only 14 years old, and on arriving at the island of Hven I was taught by Tycho Brahe, who ushered me into astronomy. I watched the moon, about which my mother was telling to me and used to converse with. I learned the science of divination, which prepared me for greater things. After a few years I was seized with the desire to return home, said farewell to Brahe and returned to my homeland. Great was my happiness to find my mother still alive, and she too was happy to have her son whom she feared lost. The year was turning to autumn, my mother was not working, she did not want to be parted from me, and she let me tell her about the lands I had visited and the sky I learned. She would compare what she knew with my tales and exclaimed she was ready to die because her son had now her legacy of wisdom. By nature most curious, I asked about her arts and who were her masters. She told me of the spirits who conversed with her, nine of them the most important and one of them very familiar to her, the most harmless and mildest who is summoned by twenty-one spirits. I wanted her to tell me the wonderful things said of Levania.31  In the notes Kepler explains that he was inspired by the name Flox seen on a map of Iceland.  Free translation from Latin by the author. 31  Kepler specifies in the notes that in Hebrew the Moon is called levanah (‫)לבנה‬. The choice of this name is justified by Kepler who ascribes knowledge of magic to Jews. 29 30

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It was already spring, the Moon was rising with Saturn in the Taurus; my mother in a few words, pronounced her summons; and when the ceremony was over, she held up the right hand, ordering silence, and sat. Just wrapped the head in the robe and the creature appears with an uncertain, deaf voice, speaking in the Icelandic idiom.

The Demon describes the journey to the Moon. In the deep sky isolated is Levania, five hundred thousand Germanic miles away. The journey from here can rarely be made and is easy for us, but for humans is arduous and dangerous. They risk their lives from air rarefaction. Men accustomed to long journeys are chosen, not the weak, nor the sedentary nor those who are too stout. The journey is done in the shadow cone of the Moon within an eclipse and lasts four hours. So together with great effort we hurl the traveler up in the sky. At first, he is shaken, as if thrown by a bombard. We drug him to sleep with opium and ­narcotics to prevent violence from dismembering him. Cold and inability to breathe are the other difficulties, so we place a sponge imbued with water near his nostrils. Once the first part is completed, the journey gets easier. The body enfolds itself like a spider, that we easily lead with the force of gravity to get it to the intended place. But the critical moment is important, as the motion accelerates and we have to be ahead of the body avoiding the harsh impact with the Moon. Upon awakening on the moon, the human being feels great fatigue and dizziness.

Somnium story continues as the Demon describes lunar astronomy, which forms the main part of the entire work. Demon describes in detail, from a selenostatic point of view, all celestial phenomena known to Earth such as the movements of the Sun and mobile stars, orbital anomalies, eclipses, the length of day and night on the Moon, etc. In Levania no seasons exist, but the torrid zone has a temperature of 10 degrees while the others are frigid. For lunar geography, the Demon uses the terms volva (the side facing the earth) and subvolva (the opposite side). The lunar surface constitutes only a quarter of the earth’s surface, and Demon describes it as crowned by high mountains and crossed by deep, long valleys. The hollows offer the moon’s inhabitants protection against the scorching, unbearable heat of the sun during the lunar day, which lasts 15 Earth days. Clouds and heavy rains provide relief from the heat. The Demon describes all the celestial phenomena as seen from the Moon, that are explained by Kepler in the notes to the narrative. This novel reveal Kepler’s desire to travel in the space to observe the solar system for understanding deeply its structure and motions and to explain it.

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It is the same desire that inspired his planetary machine. He figures also the difficulties of travelling in the space, by glimpsing the idea of gravitation and vacuum. I drive the attention of the reader to the description of the journey described by the Demon, where we can see how the notion of gravity was already clear to Kepler. He is aware of its action at a distance and the interaction between Earth and Moon gravities. There is a critical point where the forces are in equilibrium. In fact, in the words of Demon there is a critical moment (Kepler writes ροπή in Greek) where the attraction inverts and the traveller’s body accelerate. Somnium testifies how Kepler was fully immersed in the culture of his time. He chooses a fantasy novel as an allegory to explain the Copernican theory and criticize the erroneous arguments against it, with the many notes, integral part of the book. In the Netherlands the Renaissance had seen the emergence of artists such as Hieronymus Bosch (1450–1516) (Fig. 4.9), who depicted the fantastic in painting. The dream of a journey in the afterlife was born in the Middle Age. It was published in 1474 in the famous book Visio Tnugdali [The vision of Tnudgalus], translated in 15 languages. In Italy Ludovico Ariosto (1474–1533) dealt with the theme of the journey to the Moon to recover Orlando’s intellect. Many artists like Albrecht Dürer (1471–1528) or Jan Brügel the Elder (1568–1625), took up Bosch’s fantastical theme, whose bestiary could depict otherworldly beings. Works of these artists were part of the Habsburg artistic collection and were well known in all the European Courts. Nobles of the 1600s collected also objects and animals from the New World by gathering and displaying them in Wunderkammer. It is not improper, therefore, to think that Kepler’s Somnium belongs to the literary field of this genre.

Galileo: Looking into the Deep Sky Galileo Galilei (1564–1642) was born in Pisa on 15 February 1564, to the Florentine Vincenzo Galilei and Giulia degli Ammannati. I limit here to recall the main dates and events in the life of Galileo, of whom there are countless biographies.32 In 1574, the family left Pisa and moved to Florence. In 1581, Galileo enrolled at the University of Pisa to study medicine, following his father’s wish. During his studies, he became interested in physics and in 1583  A synthetic and comprehensive biography of Galileo can be found on the Britannica Encyclopedia: https://www.britannica.com/biography/Galileo-Galilei. Accessed June 2022. 32

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Fig. 4.9  Hieronymus Bosch 1550 c., central panel of the Last Judgement

formulated the hypothesis of isochronism of the oscillation of the pendulum. In 1585 he returned to Florence without having completed his studies, and began to devote himself to physics and mathematics, also giving private lessons. In 1586 he invented the hydrostatic balance.

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In 1588 he obtained a chair in mathematics at the University of Pisa, which he held until 1592. It was during this period that he became interested in the fall of bodies and wrote De Motu. In 1592, Galileo obtained a chair in mathematics (geometry and astronomy) at the University of Padua, where he remained until 1610. In 1599 he married Marina Gamba, who gave him three children: Maria Celeste, Arcangela and Vincenzio. In 1602, he conducted some experiments on the pendulum during a study on accelerated motion. In 1604, Galileo observed a supernova that appeared in the sky during autumn. In 1606, he invented the thermoscope, a primitive thermometer. In the years to follow, he devoted himself to studies of hydrostatics and the resistance of materials, built his hydrostatic balance and discovered the parabolic motion of projectiles. When Kepler published Astronomia nova in 1609, Galileo began to take an interest in a new instrument, built in Holland: the telescope. Until then, astronomical observations had been made with the naked eye. After making improvements to the telescope, Galileo presented a specimen to the Venetian Senate, to which he gave the name perspicillum. With the new instrument, Galileo carried out a series of observations of the Moon in Padua and on 7 January 1610 observed bright ‘small stars’ near Jupiter. In March 1610, he revealed in Sidereus Nuncius that these were four satellites of Jupiter, which he named Astri Medicei in honour of Cosimo II de’ Medici, Grand Duke of Tuscany. Only later, at Kepler’s suggestion, would the satellites take the names by which they are known today: Europa, Io, Ganymede and Callisto. Galileo continued his studies of the planets through the telescope and observed Saturn. Since he could not distinguish its rings with his instrument, he believed it to be composed of three distinct celestial bodies, and therefore gave the planet the name Saturn tricorporeal. In 1611 Galileo returned to Florence and observed the phases of Venus and sunspots; also, in this year he was welcomed into the Accademia dei Lincei. Galileo’s interpretation of his planetary observations is contrary to the Ptolemaic theory and instead confirms the Copernican theory. The Holy Inquisition declares this theory heretical and formally forbids Galileo to support it. Copernicus‘book De Revolutionibus Orbium Coelestium is also put on the Index librorum prohibitorum. In 1630, Galileo concluded writing the Dialogo sopra i due massimi sistemi del mondo (Dialogue on the Two Chief World Systems) in which he compared the Copernican and Ptolemaic theories; he later agreed on a number of changes in order to have the work printed, and it was published in Florence in 1632. The Dialogue aroused criticism and denunciations of heresy, which led Pope Urban VIII, previously in favour of Galileo, to ban its distribution

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and the Inquisition opened a trial against Galileo. In 1633 Galileo was summoned to Rome for trial and explicitly asked to abandon the Copernican theory; he was imprisoned and threatened with torture even though he was now ill. Forced to make a public abjuration, he was sentenced to life imprisonment, which he was allowed to serve at his villa in Arcetri. He died in 1642. Countless are the writings on Galileo.33 Ludovico Geymonat (1957) captures the most interesting insights into his life and scientific contribution. Geymonat extends Galileo’s role, which is not only limited to the scientific field. He was perhaps the first scientist who also took on a political role. In his works, one can see two intentions: firstly, to spread Copernicus’ new cosmic model and secondly, to urge the Catholic Church to accept this scientific revolution rather than oppose it. Geymonat writes that Galileo, for the first of these purposes, was undoubtedly successful, while the second was a failure certified by the condemnation of the Holy Office. Part of the success of the first objective was the use of the vernacular language,34 instead of Latin, thus offering the new knowledge also to less educated social classes. Perhaps the second objective was not a total failure, in fact, after his house arrest to Arcetri, other scientists and friends took care to spread his writings in Europe. One in particular, Elia Diodati (1576, 1661), who was born in Lucca and lived in Switzerland for a long time, edited Galileo’s books in Europe, earning the author’s gratitude. We can say that Galileo’s condemnation had a side effect that compensated for the partial failure of the second goal. The book, included in the index of prohibited books, aroused the interest of all scholars, and its dissemination was accelerated by multiple reprints. These events highlight the differences in the religious climate in Europe at the time, which allowed scholars from northern Europe a relatively greater freedom in dealing with subjects that were sensitive in terms of faith. We have already explained the two fundamental contributions that Galileo made to astronomy through two instruments: the telescope for observations and the pendulum for measuring time. But his scientific contribution goes beyond this, as he developed in mathematical form, i.e., through demonstrations according to the method of Euclid‘s Elements, some fundamental aspects of mechanics. Let us remember that after being condemned by the Sant’Uffizio, Galileo exiled himself to Arcetri where he enthusiastically resumed his mathematical studies, refraining from the controversies that had done him great harm. In 1638, he published Discorsi e dimostrazioni matematiche intorno a  See e.g. Camerota (2004), Heilbronn (2013).  The vernacular of Galileo is a splendid example of Renaissance Italian, that is studied at the high school as an example of elegant and refined writing. 33 34

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due nuove scienze [Mathematical discourses and demonstrations on two new sciences] (Galilei, 1638), pertaining to mechanics and local motion, for the publisher Elsevier of Leiden. In this work, the Galilean method is systematically used: the basis of scientific knowledge is the experiment from which laws are derived that must be formulated and demonstrated mathematically. The printer writes in the preface to the work: … But the grace bestowed on him by God and nature (though through much labor and vigilance) is much more evident in the present work, in which he is seen to have been the discoverer of two entire new sciences, and from their first principles and foundations conclusively, that is, geometrically, demonstrated: One of the two sciences, which must make this work more marvellous, is concerned with an eternal principle of nature, which has been speculated on by all the great philosophers, and on which there are many volumes written; I speak of local motion, a subject of infinite admirable accidents, none of which has so far been found or demonstrated by anyone. The other science, which has also been demonstrated from its principles, is about the resistance of solid bodies to being violently broken; this is of great use, especially in the sciences and arts of mechanics, and it is also full of accidents and propositions that have not been observed up to now.35

Newton had not yet been born, and Galileo was already expounding the new paradigm that also mechanics must adopt: experimentation and mathematical demonstration; it can therefore be said that the concept of rational mechanics was already introduced by Galileo. The value of experimentation is perfectly illustrated in the astronomical observation conducted with the telescope, which leads to contradicting theories and hypotheses formulated through imaginative speculation alone. Again, in the words of the printer, the synthesis of astronomical discoveries: … and also for having, by means of the telescope (which first came into use in these parts, but was then reduced by it too much greater perfection), discovered and given, first of all, the news of the four satellite stars of Jupiter, of the true and certain demonstration of the Milky Way, of sunspots, of the roughness and nebulous parts of the Moon, of Saturn three-bodied, Venus scythe-shaped, of the quality and arrangement of comets; all these things never known to astronomers nor to ancient philosophers …

 All excerpts from Galileo writing are translated from Italian by the author.

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It is fascinating to discover how Galileo’s reasoning proceeds. He chooses a specific theme, linked to the practical experience of ship builders of the Venice Arsenal, a topic of great importance in model building. The point is that moving from a small size machine to a larger size does not guarantee that the same properties are retained and in particular that the solidity of the small machine remains the same in the large machine. This in physics is the problem of scale invariance, and Galileo in discussing it considers different situations: the size of a vessel, ropes or animals. The ingenious point is that the invariance of scale does not exist unless all the variables involved are considered: if you enlarge a vessel while leaving gravity constant it weakens, but let us read Salviati’s words (Galilei, 1638, Giornata Seconda [Second Day])36: Salviati …. Now let them see how, from the things that have been openly demonstrated thus far, the impossibility of the ability not only of art, but of nature itself, to increase its machines to immense vastness is evident: so that it would be impossible to build large ships, palaces or temples, whose oars, masts, beams, iron chains, and all the other parts of them consisted; just as it would be impossible for nature to make trees of immense size, since their branches, weighed down by their own weight, would finally collapse; similarly, it would be impossible to make structures of bones for men, horses or other animals, which could subsist and perform their duties proportionately, while these animals would have to grow to immense heights, if we did not remove much harder and more resistant matter than usual, or deform these bones, disproportionately enlarging them, so that the appearance of the animal would be monstrously large … Simplicio But if this is so, great occasion gives me to doubt the immense moles that we see in fishes; for such a whale, as far as I can see, will be the size of ten elephants; and yet they support themselves. Salviati. Your doubt, Mr. Simplicio, makes me realise a condition that I did not notice before, which is still powerful enough to make giants and other vast animals consist and stir no less than the smaller ones: and this would follow when one not only added strength to the bones and other parts, the function of which is to support one's own and the weight above it; but, leaving the structure of the bones in the same proportions, the same constructions would consist in the same way, or rather more easily, when the gravity of the material of the bones themselves, and that of the flesh or other material that rests on the bones, were diminished in that proportion. And nature has prevailed with this second artifice in the manufacture of fish, making their bones and flesh not only very light, but without any gravity.  This dialogue is taken from Discorsi, where Galileo uses the same three characters of the Dialogo sopra i due massimi sistemi. Simplicio who plays the role of an advocate of Aristotelian and Ptolemaic doctrine, Sagredo is an impartial interlocutor who animates the discussion and Salviati is Galileo himself, defender of the Copernican doctrine. 36

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The study of gravity had led Galileo to try to measure the speed at which bodies fall. At the time, all scholars still relied on Aristotle, according to whom a constant force causes uniform motion, the speed of which is proportional to the force. But falling bodies do not fall with constant velocity, so Galileo devised the necessary experiments to disprove this theory and formulate a new one. Since he did not have suitable instruments to measure the speed of the fall directly, he used an inclined plane along which the fall is slower. It was not until 1664 that we learn that Robert Hooke invented a machine to measure the falling speed of bodies and had conducted experiments that confirmed Galileo’s law of uniformly accelerated motion: the space travelled is proportional to the square of the velocity, which implies that acceleration is constant, and acceleration will be associated with the concept of force by Newton, finally disproving Aristotle’s theory. The law found by Galileo is based on a principle of simplicity of natural laws, expressed by William of Ockham in the fourteenth century and known as “Ockham’s razor”. Another example of this principle is in the laws of reflection and refraction of light, by Pierre de Fermat, who assumes that a light ray travels the shortest path. Of course, it did not escape Galileo’s notice that a ball of lead falls faster than a grain of sand and he realized that this is due to the presence of air that produces a force that opposes movement. At the time, there were no instruments to create a vacuum; his pupil Evangelista Torricelli succeeded in forming a vacuum using mercury in 1643, a year after Galileo’s death, and the vacuum pump was invented by Otto von Guericke (1602–1686) who published his studies on the vacuum in 1663 with his work Experimenta Nova Magdeburgica de Vacuo Spatio. Fall experiments carried out under vacuum conditions once again fully confirm Galileo’s results. Observing the motion of the pendulum, Galileo realizes that it is a form of a fall of a grave not free, but bound to a wire that leads him to complete an arc trajectory of a circle. The friction of the air dampens until the oscillations stop but this happens slowly, allowing measurements to be carried out. But let Salviati speak (Galilei, 1638, Giornata Prima [First Day]): Salviati … and at last I took two balls, one of lead and one of cork, the one well over a hundred times more heavy than this one, and each of them I attached to two thin equal strings, four or five arms long, tied at the top; having then removed one and the other ball from the perpendicular state, I set them off at the same time, and they, descending through the circumferences of the circles described by the equal strings, passed beyond the perpendicular, and then went back along the same paths; and by repeating a hundred times for themselves the going and the returning, they have sensibly shown how the grave goes so far

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below the time of the lighter, that neither in a hundred vibrations, nor in a thousand, does it anticipate the time of the slightest moment, but walks with very equal pace. The operation of the medium can also be discerned, which, by creating some impediment to the motion, much more diminishes the vibrations of the cork than those of the lead, but not, however, that it makes them more or less frequent; on the contrary, when the arcs passed by the cork were no more than five or six degrees, and those of the lead fifty or sixty, they passed under the same times.

and Simplicio’s objection: Simplicio. If this is so, how then will not the velocity of the lead be greater than the velocity of the cork, making that sixty degrees of travel in time that this one barely spends six?

to which Salviati replies: Salviati. But what would you say, Mr. Simplicio, if both sent their journeys at the same time, while the cork, having moved away from the perpendicular point by thirty degrees, passed through an arc of sixty, and the lead, having moved away from the same point by only two degrees, passed through an arc of four? Would not the cork then be just as much faster? Observe, however, that when the lead pendulum is enlarged by, say, fifty degrees from the perpendicular and is left at liberty from there, it flows, and passing beyond the perpendicular almost another fifty degrees, describes an arc of almost one hundred degrees, and returning by itself, it describes another slightly smaller arc, and continuing its vibrations, after a great number of them it finally comes to rest. Each of these vibrations is made at equal times, as much that of ninety degrees as that of fifty, of twenty, of ten and of four; so that, in consequence, the velocity of the mobile is always languishing, since at equal times it goes through successively smaller and smaller arcs. A similar, or rather the same, effect is produced by the cork hanging from an equally long wire, except that in fewer vibrations it is brought to stillness, as it is less able, by its lightness, to overcome the obstacle of the air; whereby all the vibrations, great and small, are made in equal times, and equal again to the times of the vibrations of the lead. Hence it is true that if, while the lead passes an arc of fifty degrees, the cork passes one of ten, the cork will then be slower than the lead; but it will still happen, on the other hand, that the cork passes the arc of fifty, when the lead passes that of ten or six; and thus, at different times, now the lead will be faster and now the cork. But if the same pieces of furniture still pass, under the same times, equal arcs, then certainly their velocities may be said to be equal.

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In addition to the proof of the independence of the speed of fall from mass, here we see clearly expressed the principle that the pendulum’s oscillation is isochronous and independent of mass. The legend that this idea came to Galileo while observing a lamp oscillating in the Pisa cathedral is perhaps just such a legend, even though Vincenzo Viviani himself recounts it. In any case, Galileo arrived at this fundamental property experimentally and derived three laws from it: 1. The oscillations of the pendulum are isochronous; their duration is independent of amplitude. 2. The period is independent of the weight of the oscillating body. 3. The periods of two pendulums are related to each other as the square roots of their lengths. The validity of the first law is limited to the case of small oscillations. When the amplitude of oscillation increases from 1° to 90° the period increases by 18%, so two pendulums with amplitudes 1° and 90° will have a difference in frequency of about six oscillations. Galileo did not realise this; probably, following the generalization principle of falling bodies in the absence of air, he might have assumed that in vacuum isochronism was respected. But Galileo also identifies the resonance phenomenon of an oscillation (Ibidem): Salviati. Before anything else, it should be noted that each pendulum has the time of its vibrations so limited and prefixed that it is impossible to make it move under any other period than its natural one. Let anyone who wants to take the rope to which the weight is attached into his hand and try as he pleases to increase or decrease the frequency of its vibrations; it will be effort wasted in vain: But when we meet a pendulum, even if it is heavy and at rest, by blowing into it we will give it motion, and a great deal of motion by reiterating the blows, but under the time that is that of its vibrations; If we remove the first blow from the perpendicular half a finger, and add the second after it has returned towards us, the second vibration will begin, we will give it new motion, and so successively with other breaths, but given at a given time, and not when the pendulum comes towards us (which would prevent it from moving and would not help it); and following this, with many impulses we will give it such impetus that we will need more force than that of a single breath to stop it.

The repeated blow has the role of reinforcing the oscillation, and the way the oscillations are maintained in the pendulum of a clock is by the impulse

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of the escape wheel.37 By counting the number of oscillations, Galileo recalculated the diameter of certain stars, finding, according to him, more accurate values than with other methods. In this way, he gave birth to the revolution in clock construction to be used in astronomical observations. Another of Galileo’s discoveries is of particular importance: relativity. The principle is discussed by imagining a ship in motion and observers on the inside who are unaware of the motion, or on the outside who may believe that by observing the motion of another ship they are observing that it is moving and theirs is stationary. If two reference systems are inertial, i.e., they are in uniform rectilinear motion with respect to each other, acceleration is also independent of the chosen reference system, and we can then express it in either reference with the Galilean transformations. We can conclude here the brief description of Galileo’s contribution to the development of astronomy by noting that the millennial discussion on the centrality of the Earth or the Sun is completely indifferent to our observation of celestial phenomena; the laws of motion will have to take into account the relationship between possible reference systems. The revolution in astronomy bears the name of Copernicus, but the real revolutionary change is due to Kepler. Indeed, Copernicus merely changes the place of the Earth, but retains an entirely earthly view of the cosmos and does not understand the physical basis of planetary motion, which he continues to think is caused by divine action on the primum mobile. Kepler, on the other hand, assumes an extra-terrestrial point of view and this allows him to free himself entirely from the constraints of the celestial spheres and epicycles and, above all, allows him to see the planets moving along their orbits. The releasing of the spheres also brings out the need to understand the physical basis of the planetary motions, freeing them from the drive of the divine will. Finally, Galileo, by observing Jupiter’s satellites with the telescope got rid of the Aristotelian principle of the absolute perfection and immobility of the sky: coupe de grâce to the old astronomy. We are now in the middle of the seventeenth century, since 1473 there have been four scientists who have revolutionized astronomical thinking built on the ancient Babylonians and Chaldeans, and described mathematically by Hipparchus and Ptolemy: Nicolaus Copernicus, from 1473 to 1543, Tycho Brahe, from 1546 to 1601, Galileo Galilei, from 1564 to 1642 and Johannes Kepler, from 1571 to 1630. Towards the end of this fantastic period, Isaac Newton was born.

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 The mechanism of escapement will be discussed in Sect. “Measuring the Time”.

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The Cause of Celestial Motion: Isaac Newton Isaac Newton (1642–1727) was born in Woolsthorpe, County Lincolnshire, in 1642 into a small middle-class farming family. He was orphaned at only 3 years of age and was left by his mother in the care of his grandmother, perhaps giving rise to a reserved character. An anecdote reported in the introductory biography to an edition of the Principia published in the United States in 1863, says that as a child he delighted in building kites, carefully choosing the points where to connect the strands, and also paper lanterns that he and his playmates attached to the kites at night to frighten the peasants with the fear that they were comets. Other small inventions of the young Newton are also recounted, perhaps with a slightly hagiographic tone. After his stepfather’s death, the inheritance left enabled his mother to lead a wealthy life, but his son Isaac had no interest in managing her estate and, after attending high school in Grantham, he was admitted to Trinity College, Cambridge, in 1661, where he began his mathematical studies under Isaac Barrow (1630–1677). In 1665 he obtained the academic title of bachelor, but due to the plague he was forced to return to his native village. In 1665–1666, he developed a new method of calculation, which he called the method of fluxions, with which he intended to describe change in the velocity of quantities that he called fluctuating, i.e., that can change, such as temperature, lengths, areas and volumes. To do this he used sums of infinite elements of smaller and smaller magnitudes, anticipating infinitesimal calculus. He returned to Cambridge in 1667, and two years later, he was given the chair of mathematics by the concession of King Charles II, despite not having been ordained an Anglican cleric. Ironically, the school that received him was Trinity College, although Newton was anti-Trinitarian for theological reasons. Newton observed the phenomenon of colour splitting in rainbows and hypothesized that white light was composed of a mixture of all colours, publishing a paper at the Royal Society in 1672. He also took an interest in Alchemy, which he considered an exact science, perhaps trying to anticipate chemistry as a scientific discipline. These studies led to an accumulation of mercury in his body that was detected after his exhumation. In 1704, he published his studies of optics in the treatise Opticks. But far more important was the publication in 1687 of the treatise Philosophiae Naturalis Principia Mathematica, in which he presented the theory of gravitation. During a conversation with Christopher Wren and Edmund Halley, Newton heard Halley state that he knew the law governing the fall of bodies, but being unable to formulate a mathematical equation, he asked Newton for help. Newton, in

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turn, declared that he had already found the solution but had lost his notes. So, he had to rewrite the reasoning that he delivered to Halley in 1684, entitled De motu corporum in gyro, or of the motion of rotating bodies. Then he extended his work to write the three books that made up the treatise, published at Halley’s own expense. The reputation this work gave him led Newton to take an active part in political life, and he was elected Member of Parliament. In 1696 he was appointed director of the Royal Mint in London, where he remained until 1701 and helped to reform the English monetary system. He was appointed president of the Royal Society in 1703 and knighted in 1705, becoming one of the most influential scientific figures. He died in 1727 and was buried in Westminster Abbey. Newton’s contribution to the understanding of the cosmos was fundamental, not only in relation to the theory of gravitation, but also in the field of optics, which enabled the construction of more powerful telescopes than those that Galileo used. Newton in his treatise on optics (Newton, 1704, reprint 1979) set out a theory on the nature of light, which consists of corpuscles that move in a straight line and are deflected by reflections or when passing through a transparent medium. The white light of the sun, in turn, is composed of lights of different colours, which explains the phenomenon of chromatic aberration that made observations unreliable in telescopes and microscopes. Newton applied a typically Galilean method in this research, as a letter to Henry Oldenburg on September 21st 1672 illustrates (Mamiani, 2002): In fulfilment of your invitation […] I compiled a series of such experiments with the purpose of reducing the theory of colors to propositions and to prove each proposition with one or more of these experiments with the help of common notions drawn up in the form of definitions and axioms in imitation of the method mathematicians are used to prove their doctrines.

In Book I of the Treatise on Optics, he sets out the geometrical properties of the propagation of light through a prism, formulating the theory of geometrical optics. In Book III, he expounds the corpuscular theory of light as opposed to Huygens’ wave theory, and studies the phenomenon of diffraction, discovered and so-called by Francesco Maria Grimaldi (1618, 1663), described in his treatise Physico-Mathesis de Lumine, Coloribus et Iride. Studying the reflection of parabolic mirrors, he demonstrates that light coming from a parallel direction is concentrated in the focus of the parabola and

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to observe the reflected image he interposes a small mirror supported by thin foils along the axis of the parabola: the Newtonian telescope. In the preface to his fundamental work, the Principia, Newton presents a philosophical work that combines natural philosophy, geometry, mathematics and mechanics, thus a different way of philosophizing than the ancients. Newton formulates the definitions (Newton, 1846): Of mass: « The quantity of matter is the measure of the same, arising from its density and hulk conjunctly»; of quantity of motion: «the quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly»; of inertia: «The vis insita or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to preserve in its present state, whether it be of rest, or of moving uniformly forward in a right line.»; of the imprinted force: «An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.»

He then sets out axiomatically a series of laws, the first: every body perseveres in its state of quiet or uniform motion in a straight line, unless it is forced to change its state by impressed forces. The second law: the change in motion is proportional to the impressed driving force, and occurs in the direction of the straight line in which that force is impressed. The third law: an action always corresponds to an equal and opposite reaction: that is, the reciprocal actions of two bodies are always equal and directed in the opposite direction. From all this he derives the law F = m.a which links force, mass and acceleration, the relationship between weight and mass through the acceleration of gravity g: weight = g.m. The step to generalizing the relationship between mass and force with the idea of universal gravitation was a short one, the idea that gravity acted between distant bodies was widespread among the scientists of the time, it only lacked to be written down in mathematical form by identifying the relationship that links force to the inverse of the square of distance, which Newton did, allowing him to justify the elliptical motions discovered by Kepler with dynamic properties. Newton can demonstrate the precession of the apsides with these laws, in propositions XLIII, XLIV and XLV of Book I. The Principia conclude with a Scolium Generale (Newton, 1687, reprint 1846, p. 506–507), in which Newton states that the regular motions of the cosmos do not originate from mechanical causes and therefore could not have come into being without divine intervention. But even the laws he identified are not sufficient to understand the causes of other phenomena, and here again Newton’s powerful vision stands out, capable of identifying the

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scientific programme of the following centuries, from physics with its studies on electromagnetism, to biology, neurology and modern neuro-science. Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very center of the sun and planets, without suffering the least diminution of its force; that operates not according to the quantity of the surfaces of the particles upon which it acts (as mechanical causes use to do), but according to the quantity of the solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always in the duplicate proportion of the distances. Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body is composed; and in receding from the sun decreases ­accurately in the duplicate proportion of the distances as far as the orb of Saturn, as evidently appears from the quiescence of the aphelions of the planets; nay, and even to the remotest aphelions of the comets, if those aphelions are also quiescent. But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypothesis; for whatever is not deduced from the phenomena is to be called and hypothesis; and hypothesis, whether metaphysical or physical, whether of occult qualities or mechanical have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterward rendered general by induction. Thus it was the impenetrability, the mobility, and the impulsive force of bodies, and the laws of motion and of gravitation, were discovered. And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea. And now we might add something concerning a certain most subtle Spirit which pervades and lies hid in all gross bodies; by the force and action of which Spirit the particles of bodies mutually attract one another at near distances, and cohere, if contiguous; and electric bodies operate to greater distances, as well repelling as attracting the neighboring corpuscles; and light is emitted, reflected, inflects and heats bodies; and all sensations is excited, and the members of animal bodies move as the command of the will, namely, by the vibrations of this Spirit, mutually propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles. But these are things that cannot be explained in few words, nor are we furnished with that sufficiency of experiments which is required to an accurate determination and demonstration of the laws by which this electric and elastic Spirit operates.

5 The New Vision of the Cosmos

The view of the Cosmos had become incompatible with the new observations. Aristotle, on the one hand, conceived the Cosmos organized in a geocentric view, and on the other, he divided celestial phenomena between sublunar and those that took place beyond the sphere of the moon. The Cosmos was seen as an invariable entity in which the sun, moon, and planets moved with regularity, and nothing about it ever changed. The sublunar phenomena concerned everything that took place on the earth and in the atmosphere, and to them, according to the Aristotelians, belonged meteors and comets, but in-­ depth study of comets and new stars had demonstrated the inconsistency of these theories. To understand the reason for the complex motions of Sun, Moon, and planets it took centuries of study and observation, with increasingly precise measurements and above all, the formulation of Newton’s theory of gravitation, which forms the basis of planetary dynamics. The equations needed to describe and predict the Moon’s motion were not completed until the mid-nineteenth century. Newton’s equations can be solved by calculation when only two bodies are involved, but the Moon’s motion is also influenced by the presence of the Sun as well as the other planets. In this case, approximate methods are used, in which a large number of periodic components (expressed in the form of sinusoidal functions) that describe the motion of the other celestial bodies come into play. With Newton’s discoveries, moreover, knowledge of the solar planetary system reaches its zenith. What happens next is linked to observations through instruments such as the telescope, which also leads to the discovery of new planets and, above all, to the refinement of data collected over more than two millennia. It © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_5

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is therefore time to summarize the conclusive data that eighteenth-century scientific and technical thought had arrived at.

Shape and Motion of the Earth The annual duration of the revolution around the Sun is measured in different ways. The tropical year, or solar year, is the time it takes the Sun to return to the spring equinox. Due to the precession motion, this position is reached approximately 20 minutes earlier each year. The sidereal year, on the other hand, is the time it takes the Earth to see the Sun in the same position relative to the fixed stars,1 and is, therefore, about 20′ longer than the tropical year. During the course of the year, the Earth reaches pivotal positions that mark the changing of the seasons: the solstices and equinoxes. Solstices correspond to the positions where the Sun reaches its maximum or minimum elevation above the horizon (summer solstice or winter solstice). Equinoxes are the positions where the duration of day and night is the same. The dates of many religious and civil celebrations were determined by the dates of the equinoxes. Because of the precession motion, calendars no longer corresponded to astronomical positions. The dates of many religious and civil celebrations were determined by the dates of the equinoxes. Because of the precession motion, calendars no longer corresponded to astronomical positions. The Earth makes one complete revolution around the Sun in approximately 365.25 days. At the same time, it revolves around its own axis, taking 23.93447 hours, which we conventionally round up to 24 hours. In the course of the revolution the Earth’s speed varies, greater at perihelion (in the winter period) and lesser at aphelion (in the summer period). So, the time between two passages between equinoctial or solstitial points varies, and consequently, the length of the year varies depending on the time chosen to measure the time, e.g., 365.2427 if you start at the winter solstice, and 365.2416 if you start at the summer solstice. An average value of 365.2422 is therefore adopted today. This is the duration of the mean tropical year. Today, time is measured in different ways: GMT (Greenwich Mean Time) time indicates the time referred to the Greenwich meridian and is based on the rotation time of the earth. UTC (Coordinated Universal Time) is based on atomic clocks, and while the hours and days are equal to those measured according to GMT time, the minutes undergo corrections of the order of a  To observe the Sun in relation to fixed stars, it is used to record the moment when a particular star appears just before sunrise, the heliacal rising. 1

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second to synchronize the two times. Through this change in minute length, the delay between UTC time and ‘earth time’ is kept within 0.9 seconds. The GPS (Global Positioning System) time is that measured on the network of positioning satellites. It is corrected for relativistic time-variation; the count starts from 0 at midnight on January 6 1980 (UTC time), so adding up the value generated by the on-board atomic clocks gives the current date and time. For example, the local time 11:08:322 UTC + 2 on August 11, 2021 (summer time in Italy) corresponds to 09:08:32 UTC and 09:08:50 GPS time. We will see the relationship between these measurements and the International Atomic Time adopted in contemporary astronomy when we discuss the atomic oscillator (see Sect. “Measuring the Time”). The Earth’s rotational motion is not simple, and it moves like a whirligig. The Earth axis rotates in about 25,000 years to complete. Consequently, the Earth’s axis, which today points toward the Polaris star (α Ursa Minor), will, in about 12,000 years, point toward the star Vega (α Lyrae), and this orientation, although variable, identifies the North Celestial pole. But it does not end there: the whirligig motion is perturbed by nutation, i.e., an oscillation along the precession trajectory. According to the Babylonian calendar, the tropical year had a duration of 360 days, whereas the Egyptians noticed a deviation of 5 days, as the flooding of the Nile was synchronized with the appearance of the star Sirius just before sunrise. Egyptians considered the duration of the tropical year to be 365 days. Since the average length of a tropical year is 365.2422 days, after about 4 years the calendar would err by one day, which in 360 years would cause a true reversal of the seasons, exchanging solstices for equinoxes. The calendar was, therefore, periodically reformed but without a precise rule. In -46 Julius Caesar, on the advice of the Egyptian astronomer Sosigene (-I century), altered Rome’s civil calendar by inserting, after each cycle of 3 years with a duration of 365 days, a year with an extra day, called a leap year. In the Julian calendar, the extra day is February 24. This reform corresponded to considering the duration of the tropical year to be 365.25 days, slightly longer than the actual duration. But the consequence of this approximation was that at the Council of Nicaea in +325, it was found that 3 days had accumulated in advance and it was decided to change the date of the spring equinox to March 21. The final calendar reform was promulgated by Gregory XIII (1501–1585), who ordered that October 4, 1582 be followed by October 15, 1582, and reformed the calculation of leap years by removing the centuries unless they were multiples of 400 (Capaccioli, 2020). Another problem, related to the shape of the Earth, was the determination of longitude. Imagining the globe divided into 24 segments corresponding to

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the 24 hours of the day, the angular distance between them amounts to 15°, and this difference corresponds to an interval of one hour. The method proposed by cartographer Johannes Werner (1468–1522) in 1514 for determining the longitude was based on the observation of eclipses of the Moon (Sobel & Andrews, 1998, p. 28). By having ephemerides of the eclipses with associated start and end times, it was possible to observe them at different locations and calculate the time difference. This apparently simple method presented several difficulties. During the time interval between the time recorded on the ephemeris and the time of observation, the Moon had moved along its orbit, but the Moon’s motion was not yet well known, and the error could amount up to 20°, exceeding one hour. In addition, the position of the stars had not yet been determined with sufficient precision. That the earth was a kind of sphere had been well known for thousands of years, and even its diameter had been determined to a fairly good approximation by Eratosthenes. What was still little known was the exact shape of the earth, the length of the meridian, which, once measured, made it clear that the shape is slightly flattened at the poles. Still partly unknown were the continents, America still little explored, and even less known was Australia, which was discovered for the European world in 1770, and the polar regions, particularly Antarctica, whose existence was confirmed in 1820 by Russian explorers. We can therefore say that knowledge of the earth in the seventeenth century was still partial.

The Satellites of Jupiter Pointing his telescope at the planet Jupiter, Galileo discovered four satellites taking note of their position during time (Fig. 5.1). They got the names of nymphs of Greek mythology – Io, Ganymede, Europa and Callisto. The revolution periods of the four satellites are: Io = 1.77 days; Europa = 3.55 days; Ganymede = 7.15 days; Callisto = 16.69 days. These times can be accurately observed by measuring the times of occultation by the planet Jupiter. Galileo Galilei perfected a method for measuring the longitude at the ground based on occultations occurring with perfect regularity. Galileo used a graduated scale to compare the distances between the satellites and Jupiter, and for use at sea he had devised the celatone, a sort of helmet on which a telescope was mounted, thus trying to simplify pointing and measuring (Stefani, 2004). Nevertheless, the method was unsuccessful, as even the beating of the heart could put Jupiter out of the telescope’s field of view at the

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Fig. 5.1  A sheet of Galileo’s observations of the Medicean satellites. The circles denote Jupiter, the dots are the position of the satellites

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time. Galileo also constructed an instrument, probably a volvelle2 for calculating local longitude, the jovilabium. The idea of jovilabium was taken up by other authors, as we will see in Sect. “Ole Rømer: Jovilabium and Planispherium”. More modern instruments made it possible to use Jupiter’s satellites occultation to measure the longitude on Earth, which aroused great interest among seventeenth-century cartographers. Giovanni Domenico Cassini (1625–1712) published highly accurate ephemerides of observations of Jupiter’s satellites in 1668, a work that brought him fame in France. Jean-Baptiste Colbert commissioned Cassini to build the astronomical observatory in Paris. Among the many cartographers who drew maps and globes thanks to these accurate measurements was Vincenzo Coronelli (1650–1718), a Franciscan monk who founded a geographical society and built some of the finest globes for seventeenth-­century kings and countless terrestrial and celestial maps. We will see in Sect. “A Challenge Between French and English: Measuring the Meridian and the Time in Navigation” that to determine longitude with the accuracy required for accurate chart construction and safe navigation, it will be necessary to construct marine chronometers for timekeeping.

The Motion of the Moon The length of the lunar month and the multi-year cycles of the Moon’s motion were well known to both the Babylonians and the Egyptians, who adopted a lunar calendar. The Moon’s motion marks the division of the year into months, although there is no exact correspondence to the duration of the calendar months. Depending on how the Moon’s reoccurrence is observed, different months and different monthly cycles are identified over the years. The Moon’s orbit is inclined about 5° to the Earth’s orbit, the ecliptic. The ideal points of intersection between the Moon’s orbit and the ecliptic are called nodes, and a distinction is made between the ascending node, when the Moon is rising toward the full Moon, and the descending node when the Moon is proceeding from the full Moon to the new Moon. The Moon’s orbit also performs a precession motion whereby the nodes move with a period of about 18.6 years. Different months and lunar cycles are considered on the basis of these complex motions.  Instrument on which the mean motions of the four satellites are marked, consisting of two discs of different diameters. The disks rotate and with a moving rod indicate the positions of the satellites observed from Earth, identifying moments of occultation. 2

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Sidereal Month  is defined as the time taken by the Moon to be in the same position in relation to the fixed stars, and lasts 27 days, 7 h:43′:11.6″ equal to 27.3217 days. Tropical Month  is the time between passages of the Moon through the same celestial longitude. Its length is 27 days 7 h:43′:52), equal to 27.321582 days, only 7 seconds shorter than the sidereal month. Synodic Month  is the time it takes the Moon in the same position relative to the Sun. During a revolution of the Moon, the Earth also moves relative to the Sun, so we need to add about 2 days, so the length of the synodic month is 29 days, 12:44′:2.9″ equal to 29.530589 days. Anomalistic Month  is the time it takes the Moon to return to perigee, the closest point to the Earth along its orbit. The line joining perigee and apogee is called the line of the apsides, and it rotates around the focus of the ellipse of the orbit in 8.85 years: it is the precession of the apsides. The anomalistic month is 27 days 13:18′:33.2″ or 27.55455 days and is therefore longer than the sidereal month. Draconic Month  This is the time it takes the Moon to return to the ascending node. The Draconic month is 27 days 5:5′:38.865″ or 27.212221 days. Cycle of Meton  If we observe that at the spring equinox the Moon is in the new Moon phase, at the equinox the following year it will not be in the same situation. In order to find a new Moon at the spring equinox, it is necessary to wait about 19 years. The discovery of this periodicity, although already known to Babylonian astronomers, is attributed to Meton of Athens, who observed that 235 synodic months correspond to 19 tropical years. The Metonic cycle is then divided into 12 years consisting of 12 lunar months and 7 years consisting of 13 months. To achieve synchronization with the 365-day period of the year, however, it is necessary to add a month, called embolismic every two or three years. The Jewish and Arab lunar and luni-solar calendars are also based on this subdivision. The cycle is 6939.691 days long, a value that we will find in the description of the Antikythera machine. Cycle of Callippus  Callippus of Cyzicus reviewed the Metonic cycle by proposing to remove one day for every four Metonic cycles, thus Callippus’ cycle has a duration of 940 lunar months, equal to 76 years. The duration of the

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synodic lunar month of 29.5306 days was correctly related to the duration of the tropical year of 365.2422 days with an error over the 76 years of 4′36″. The total days of the Callippus cycle are therefore: 27,758.764. Saros Cycle  Saros cycle3 was identified by Babylonian astronomers, as documented in some cuneiform tablets, and has a duration of 223 1/3 synodic months, or 18.03 years. Eclipses of the Sun and Moon occur on exactly the same days and times each cycle, during which there are 71 eclipses, of which 43 are solar and 28 lunars. Exeligmos  Another cycle linked to the Saros is the Exeligmos (the Greek word means turn of the wheel), which corresponds to three Saros cycles and corresponds to 669 synodic months. By associating the Exeligmos cycle with the Saros cycle, an eclipse of the Sun or Moon will have the same type and duration as the previous cycle.

The Tides Almost all the scholars we have mentioned have also sought an explanation for the tidal phenomenon, Lucio Russo (2020) recalls in great detail the succession of hypotheses. We limit ourselves here to mention that already the hypothesis that the tides were the result of the influence of the moon dates back to the Hellenistic period, and it was hypothesized that the tides were the effect of sympathy between the earth and the moon, as the vibration of strings by consonance. It is only at the beginning of the Renaissance that the problem is again addressed. We want to mention here in particular the theory of Jacopo Dondi4 , who in 1355 published a manuscript in which he asserted that tides are the joint effect of the Sun and Moon, and that they occur with greater or lesser amplitude depending on the relative positions of the two stars with respect to the earth: maximums in conjunctions and oppositions, when Moon and Sun act by attracting the waters of the seas in the same direction, and minimums in quadrature. A spring tide is a tide of maximal range, near the time of new

 Saros is an accadic word that means a large quantity or a quantity of 3600 in Babylonia.  The manuscript transcribed by Paolo Revelli in 1912 is available online: http://www.mat.uniroma2. it/~simca/Testi/Dondi.pdf 3 4

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and full moon when the Sun and Moon are in syzygy5—i.e., aligned with the Earth, either in conjunction or in opposition. This interpretation is definitively confirmed only by Newton, based on his theory of gravitation. We note that Galileo also tried his hand at the subject, his kinetic theory being based on experimentation with the motion of water in vessels placed in rotation; once again Galileo adopts his experimental method, but the results prove erroneous due to a still incomplete understanding of planetary mechanisms that are not describable by kinematics alone.

The Speed of Light The study of light and optics, already initiated by Huygens and Newton, also deepened during this period. The determination of the speed of light, which was believed to be infinite, i.e., to arrive instantaneously at any illuminated place, completes the picture of knowledge of the Cosmos in the seventeenth century and shows with great clarity the methods of investigation of astronomers and how collaborations between the various scientists were carried out. In 1671 (Bobis & Lequeux, 2008), the French astronomer Jean Picard (1620–1682) was sent to Denmark to measure the difference in longitude between the Paris observatory and Uraniborg, Tycho Brahe’s observatory. The measurement of longitude was done by synchronizing time with the occultations of Jupiter’s satellites, as proposed by Galileo, in particular Io, which had a very short period. In this work, Picard was assisted by the young Ole Rømer (1644–1710). On his return to Paris, Picard took Rømer to work with Cassini, helping to construct ephemerides of the Medicean satellites to synchronize the clocks. Rømer and Cassini’s observations showed that successive occultations of the same satellite were irregular and varied according to the position of the Earth: shorter when the Earth is close to Jupiter, longer the further away from it. Cassini rejected the idea that these differences were due to the speed of light because the values observed for the other satellites did not correspond. Rømer calculated a difference of 22 minutes for the satellite Io due to the time it took light to travel the distance equal to the diameter of the Earth’s orbit, and on the occasion of Io’s eclipse on November 9 1676, he predicted an advance of 10 minutes which turned out to be very close to the observed value. Huygens learned of Rømer’s theory of the finiteness of the speed of light from an English translation published in the Journal des Savantes in 1677. This hypothesis was necessary for Huygens to justify the reflection  A syzygy is when three or more celestial bodies are aligned.

5

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Table 5.1  Planetary periods by different scholars as collected by J.A. Bengel Mercury Venus Hypparcus J. Sturm M. Mästlin J. Kepler J.J. Heinlin A. Reyher P. de La Hire J. Keill J.J.Scheuchzer A. Bengel NASA/JPL

Moon Synodic

Moon sidereal

Mars

Jupiter

Saturn

29.531

87.968

224.708 224.708 224.740

27.322

10759.241 686.930 4330.627 10759.207 4331.340

29.531 87.972 87.969 87.967 87.968 87.971

224.688 224.708 29.531 224.705 29.5306

686.977 4332.514 10759.275 27.322 27.3217

686.975 4332.467 10759.379 686.992 4332.896 10755.887

and refraction phenomena of his wave theory. Huygens calculated the speed from Cassini and Rømer’s data and observed that it was more than 600,000 times the speed of sound, which with contemporary values would amount to 230,000 km/sec, a relatively small difference from the current estimate of 299,792 km/sec.

The Planetary Periods After Copernicus many scholars studied planetary periods of sidereal revolution of the different planets, the duration of the tropical year, and the synodic and sidereal month. Table 5.1 shows the values proposed by the major scholars6 collected by A. Bengel (1745). The unit of measurement of the periods is the day, expressed in decimal form instead of sexagesimal to facilitate comparison. We can see that the estimates of synodic lunar motion have been close to a perfect match since the time of Hipparchus, for the inner planets, there are very small differences of less than 2 minutes, while for the outer planets they are much more significant, on the order of days. In particular, the difference in period estimates for Jupiter makes all studies of the Medicean satellites particularly critical, as we shall see later.

 Johannes Sturm, Germany (1507, 1589); Andreas Reyher, Germany (1601, 1673); Johann Jacob Scheuchzer, Switzerland (1672, 1733); Johan Jakob Heinlin, Germany (1588, 1660); John Keill, Scotland (1671–1721); Philippe de la Hire, France (1640–1718)). 6

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The Astrological Vision of the Cosmos The astronomical data we have summarized here constitute the fundamentals of the conception of the Universe that scholars had arrived at in the eighteenth century. We will see later how this translated into the construction of instruments to disseminate the new astronomical knowledge. However, a vision of the Cosmos that links the astronomical configurations of the planets, sun and moon to human nature, to the character and individual destiny of the person survives and will survive to this day: astrology.7 Astrology originated with the observation of relevant celestial phenomena such as eclipses or the passage of comets, interpreted as heralding disaster or illness. The correspondence between the length of the fertility cycle and the lunar cycle leads to the belief that the stars influence people’s character. Astronomers become astrologers, diviners who scrutinize the heavens to predict events and examine the dispositions of the planets at the birth of kings trying to grasp their destiny. The horoscope is depicted with diagrams highlighting the succession of the signs of the zodiac, as we see in the picture of a horoscope designed by Gerolamo Cardano (1663) (Fig. 5.2 left). Cardano (1501–1576) was a mathematician and physician. He is also known as the inventor of a mechanical joint that bears his name. In the diagram in Fig. 5.2 left, one can recognize in the triangles the sequence of zodiac signs and the symbols of the planets with the degrees of their longitude. In the center is the date and time of birth of the person for whom the horoscope was studied. The different zodiacal zones are called mansions. The planetary opposition or conjunction is also given importance in determining which element of the character should prevail. In Fig. 5.2 right, we see the method proposed by Campano8 to analyze a horoscope using an astrolabe.9 Astrologer think that the planets have different influence on characters, e.g., the influence of Saturn is described by Kepler in Wallenstein‘s horoscope as (Kepler, 2009a, p. 453): Saturn in the ascendant makes thinking deep, melancholic, unruly, brings inclination to Alchemy, magic, witchcraft, communion with spirits, contempt and  An interesting collection of studies on Astrology in sixteenth century see Hoppmann (1997).  Campano da Novara (c. 1220–1296) was a mathematician, astronomer and physician at the Curia Vaticana. He published a treatise on astrology. 9  For a modern presentation of astrology see: Boxer (2020). Alexander Boxer has also implemented a web site where horoscopes can be computed and rendered on a virtual astrological astrolabe: https://alexboxer. com. Accessed September 2022. 7 8

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Fig. 5.2  Left: Outline of the Horoscope. Right: Subdivision of the zodiacal mansions in a computational astrolabe. Green: Campanus uniform subdivision. Purple: Regiomontanus nonuniform subdivision

disrespect for human commandments and customs, even for all religions, make everything suspect and suspicious, what God or men do, as if it were all a lie and something else is then believed.10

The planets are also related to gender: Mars, Jupiter, and Saturn to masculine, Moon and Venus to Feminine. A relationship to gender is also associated with the zodiac signs. Kepler writes about astrology: The astrologers devised the division of the 12 mansions given at the beginning for this very reason, in order to be able to give a different answer to everything that man wishes to know, but I consider this knowledge impossible, superstitious, scaramantic and an appendage of Arabian sortilege, since one answers yes or no to any question that is put to man at that hour, even without knowing his hour of birth, and one thus wishes to make astrology an oracle, relying accordingly on the inspiration of the (innumerable) celestial spirits. Since I do not need to go through all the mansions and listen to particular questions, I should not worry about this unless I leave it with good reflection.11

Consequently, Kepler makes the study of character the predominant element in the formulation of a horoscope, thus considering astrology a discipline based on mathematics.  Horoskope für Wallenstein, Horoskope Sammlung.  Ibidem.

10 11

6 The Instruments

The study of astronomy was made possible by the instruments built by the other actors in this intricate and long history: mechanics. In the Hellenistic period, mechanical activity, although scarcely appreciated by philosophers, was most probably practiced by mathematicians. Exemplary is the case of Archimedes, who invented many machines and also offered his studies for military purposes. However, the figure of the mechanic and even mechanics has never enjoyed a good reputation among historians.

Mechanics What is Mechanics? The question seems trivial; mechanics became the means of the technological development of civilization. It is a scientific discipline; it is part of the academic curriculum in Engineering and Physics. If we investigate the question from the point of view of the history of science, questions of great interest emerge that are closely related to astronomical machines. Pappus of Alexandria, who collected the mathematical knowledge of the ancients in an eight-volume work, compares mechanics with mathematical and astronomical knowledge and qualifies mechanics as an inferior discipline compared to astronomical-mathematical disciplines, even though he recognizes among its practitioners’ abilities that are difficult to equal. He goes so far as to state that no one can be fully educated and versed in both disciplines (Pappus, 1878, pp. 306–308):

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_6

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The mechanics followers of Heron believe that one part of mechanics consists in mathematical demonstration and one part in manual activity; and the part they call rational consists in geometry, arithmetic, astronomy and physics, while the part that requires manual activity consists in the art of manufacture (or working with bronze), architecture, carpentry, painting and their manual practice. And it is said that he who from childhood has devoted himself to the aforementioned sciences and acquired experience in the aforementioned arts, and moreover has an agile mind, will be an excellent inventor of mechanical works and architect. But since it is not possible for the same person to excel in so many disciplines and learn the aforesaid arts at the same time, they suggest that those who wish to deal with mechanical works should avail themselves of the proper arts required for each thing. Of all the arts relating to mechanics, those most necessary to the uses of life are the arts of those who make and steer machines, which are also called mechanics according to the ancients (for they lift great weights, which are immovable by nature, by moving them with less force), then the arts of those who make war machines, which are also called mechanics (for projectiles of stone, iron and the like are launched a long distance by the catapults they make); in addition to these, the art of those who properly call themselves machine-­ builders (for from a great depth water is more easily lifted aloft by the instruments they make for drawing water).1

A new vision of Mechanics is proposed by Newton, who states in the preface to the Principia (Newton, 1846, p. lxvii): Since the ancients (as we are told by Pappus), made great account of the science of mechanics in the investigation of natural thing: and the moderns, laying aside substantial forms and occult qualities, have endeavored to subject the phenomena of nature to the laws of mathematics. I have in this treatise cultivated mathematics so far as it regards philosophy. The ancients considered mechanics in a twofold respect: as rational, which proceeds accurately by demonstration, and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical, what is less so, is called mechanical. … Geometry does not teach us to draw lines, but requires them to be drawn for it requires that the learner should first be taught; to describe these accurately, before he enters upon geometry; then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics; and by geometry the use of them, when so solved, is shown; and it is the glory of geometry that from those few principles, brought from without, it is  Tr. from Greek by Giuliana Boirivant.

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able to produce so many things. Therefore, geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that geometry is commonly referred to their magnitudes, and mechanics to their motion. In this sense rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated.

This passage is enlightening and resolves the dichotomy made explicit by Pappus that separates practical mechanics from geometry and mathematics. Newton clearly describes a deep relationship between the two fields and resolves the separation with the notion of rational mechanics. In the study of astronomical machines, the distinction between these two fields is evident in the collaboration between scientists and skilled craftsmen and clockmakers. The scientist submits problems and hypotheses to mechanics, and mechanics implements the desired tools. A collaborative exchange is established that requires a full understanding by both of them of the basic principles of both disciplines. The relationship between mechanics and science in the Enlightenment period becomes increasingly close, but the roles are clearer. For example, we will see that the Sphére Mouvent by Passemant brings his name, but the constructor was Louis Dauthiau (see Sect. “Claude Simeon Passemant ”), while Passemant was the designer. On the other hand, Antide Janvier or Philip Mathäus Hahn were designers and builders at the same time.

Instruments to Guide the Observation The most basic astronomical observing aid is a fixed reference point from which one can look at the sky from the same position at all times. The most important observable astronomical events are the rising or setting of the Sun or the Moon and the meridian transit of the Sun, Moon, or a star. The meridian is the maximum circle passing through the north pole in a given place, and the transit is the instant of time when the Sun is at mid-day or a star is at the peak of its apparent motion, called culmination. To make these observations with any accuracy, a fixed reference point must be identified. The observer is forced into a fixed position and can compare the transit of other stars or the same one on different days. It is very likely that the site of Stonehenge, England, is precisely an observatory for the rising or

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setting of the Sun, consisting of stones arranged according to the logic of a fixed reference system. Another fine example is the astronomical observatory in Jaipur, India, founded in 1724 by Maharaja Sawai Jay Singh II (1688–1743), called Jantar Mantan, a word derived from Sanskrit meaning a calculating instrument. Similar locations existed in ancient Mesopotamia or Persia and in the Far East in China and allowed to record accurately the positions of the stars. These observatories are probably much older; the site of Göbekli Tepe near Lake Van in Turkey, which may date back 12,000 years, is considered a religious site and a primitive astronomical observatory. However, much more precise instruments are needed to make more accurate observations. Armillary Spheres The armillary sphere2 is an instrument of enormous importance in the history of astronomy (Fig. 6.1).  In general, the armilla is made up of graduated metal rings representing the ecliptic, the meridians, the equator and parallel to it, the Tropic of Cancer and the Tropic of Capricorn; lastly, it may include two more maximal circles: the equinoctial and the solstitial colure. It is therefore, a three-dimensional model of the celestial coordinate system, which includes the, circles of greatest importance for identifying the positions of the stars. Some armillae contain the globe in the center, and by varying the orientation of the terrestrial globe according to time and season, the astronomical coordinates are related to the positions of the zodiac. The zodiacal or ecliptic armilla is made up of a maximum circle that is the meridian that can rotate around the axis of the ecliptic that points roughly in the constellation of the Dragon. Within this, a second ring pivots on the axis orthogonal to the ecliptic. In the equatorial armilla the maximum circle is parallel to the equator and its axis is oriented toward the celestial north pole. The armillary spheres were probably invented by Eratosthenes around the -II century. Numerous armillary spheres were constructed from the fifth-century onwards. Let us examine an armillary sphere in detail using the illustration (Fig. 6.2) in Jerome Lalande’s astronomy textbook (Lalande, 1795). The horizontal ring supported by the four feet represents the horizon, orthogonally to this ring and embedded in two notches, there is another ring representing the meridian of the place, i.e., the arc of maximum culmination of the sun. The circle of the horizon is fixed on the support, that of the meridian can rotate by sliding in  From Latin armilla: ring.

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Fig. 6.1  Armillary Sphere, seventeenth-century

the notches change the orientation of the axis depending on the latitude of observation. Inside is a kind of cage made up of moving circles that rotate around PR axis. There are four maximum circles: the equator, the ecliptic, and the colures (Colur des S[olstice] and Colur des Eq[uinox]). The colure of the solstices passes by the poles and the points of the solstices and is a meridian that serves to measure the obliquity of the ecliptic. All the stars on this meridian have a right ascension of 90° or 270°. The colure of the equinoxes always passes through the poles, and the equinoctial points, are perpendicular to the former; all the stars on this colure have the right ascension of 0° or 180°.

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Fig. 6.2  Armillary sphere. Lalande Pl. II. Cit

Four other smaller circles are the tropical circles (HM and DI) and the polar circles (SO and XV). In between these, there is the Equator – They are not used for astronomical observations but for geography to indicate temperate and polar regions. The Zodiac is a celestial belt or zone of the ecliptic with an extension of 17° (a spherical zone), 8°30′ on each side, and indicates the

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space occupied by the orbits of the planets whose inclinations with respect to the ecliptic are within this interval. The earth is represented by a small globe at the center of the sphere. A ring KL is positioned around pole P on which the 24 hours are marked; an indicator integral with the axis PR shows the time as the sphere rotates. An arm V is fixed to it; it carries the Sun with one arc and the Moon with a second, smaller arc. The two mobiles can thus be rotated around the Earth. Finally, at the base, there is a compass to orient the PR axis to the north and a level bubble to place the object perfectly horizontal. There are two types of armillae: zodiacal armilla and the equatorial armilla. They are associated to two different celestial coordinate systems (Appendix “Positional Astronomy”). We will see that this difference has relevant side effects in the study and interpretation of the sky (Chap. 14). Armillary spheres were made both as aids to astronomers and as decorative and prestigious objects for the nobility. The armillary sphere in artistic iconography became the symbol of the scientist, of knowledge, of the cosmos. Cross Staff  The cross staff, known in Italian as balestriglia, in French as arbalestrille and in Spanish as balestilla (Fig. 6.3 left), was an astronomical instrument first described in 1328 by the Catalan Jewish astronomer Levy Ben Gerson (1288–1344), so much so that it was also called Jacob’s staff by Christians. It was initially used in the observation of celestial phenomena: the German astronomer Regiomontanus used it to measure the diameter of a comet that appeared in 1472, which 210  years later would be known as Halley’s Comet. Introduced on ships in the first half of the sixteenth century by the Portuguese, who called it tavoletas da Índia, a name that suggests its oriental origin, it consisted of a 1.5–1.8  m long wooden rod. The rod has a

Fig. 6.3  Left: Cross staff. Right: Drawing of Copernicus’ Triquetrum

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graduation; a crossbar, the hammer, can slide along the rod. The upper end of the hammer is used to gaze at a celestial body while the lower end is aligned to the horizon. The measurement taken along the rod is proportional to the tangent of the subtending angle, which can be converted to an angular value with the use of trigonometric tables. To measure the altitude of the sun, the shadow of the upper end of cross staff is projected onto a smaller hammer attached to the lower end. A smoked glass can also be used to observe the sun. Like any measuring instrument, the crossbar is affected by errors, including one known in physics as the parallax3 error. Another error depended on the need to obtain the alignment of the sun and the horizon at the same time. Triquetrum The triquetrum (Fig. 6.3 right) is a tool for marking the position of the stars by measuring angles. The structure is very simple: two rods of equal length form an isosceles triangle. The rod BD is oriented toward a star, the angle in B is the altitude to the zenith of the star, its value can be computed from its chord CA. This is a very ancient instrument, already in use in the classical period in Greece. It was probably invented by Ptolemy, and has been accurately described in 1593 by Antonio Santucci in his manuscript Trattato di diversi instrumenti matematici [Treatise on several mathematical instruments]. The instrument is used to measure the height of the Sun on the horizon, the distances between celestial bodies, and terrestrial distances, as well as for surveying. Tycho Brahe came into possession of a large instrument personally built by Copernicus. Astrolab  The invention of the astrolabe (Fig. 6.4 left) dates back to the -IV century probably invented by Eudoxus of Cnidus. Hipparchus of Nicaea also used astrolabe for his studies. During the Alexandrine period Ptolemy, Theon of Alexandria (335–405), and Iohannes Philiponus (490c.–570c.) improved the astrolabe, that were further built and refined by Arab scientist, like alFarghani (850c.) and al-Zarqali (1029–1087). Other Arab astronomers built innumerable astrolabes, many of which, richly decorated. Arab astronomers measured the shift in celestial longitude (precession of the equinoxes) on the order of a degree over a period of about 70 years, so astrolabes were often reconstructed to be up-to-date.4

 See Appendix “Positional Astronomy”.  Readers interested in learning more about the history and the major builders will find a concise exposition in Darin Hayton’s work (Hayton, 2012). 3 4

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Fig. 6.4  Left: Astrolabe. Right: Drawing of a quadrant

The astrolabe (Fig. 6.5) consists of a main disc, called the mater, with a ring (the throne) to hold it vertically with one finger; the edge of the mater is raised and is called the limb. On the limb it is engraved a scale, graduated from 0° to 360° and divided into 24 sectors, corresponding to the hours, 12 daytime and 12 nighttime; sometime also the zodiacal signs are engraved. On the back of the mater are various engravings, such as altitude tables, calendars, etc. Inside the mater there is a disc, called the climates or tympanum, and an open-work disc called the rete. At the throne it is marked the local meridian, on the tympanum disc is also marked the tropics, the equator, the horizon and the celestial meridians, plotted with the stereographic projection5 on the local horizon plane, from the South Pole for northern sky and vice versa. On the tympanum are projected the meridian circles and the parallel for a given latitude. Since the astrolabe uses the alt-azimuth coordinate system, the projected meridian circles are the azimuthal circles centered on the local zenith. They are called almucantarat and are used to determine the current hour. The projection plane for the climates of the astrolabe is the horizontal plane at the latitude of the observer, so if the instrument is used on different latitudes the tympanum has to be changed. The open-work rete represents the projection of the celestial sphere with a geo-centric perspective, the tips of the small flames identify the position of the brightest stars, in the most complex astrolabes there are up to 33 stars. By  For a description of stereographic projection see the Appendix.

5

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Fig. 6.5  Components of an astrolabe

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rotating the rete to the current hour reference the tips of the rete will point to the correct celestial coordinates. All the components are held together by a pivot on which the alidade is hinged. The alidade is a movable indicator that has a hole at one end and a sort of fin at the other. While the astrolabe is held suspended, the alidade is rotated until the hole is pointed toward the sun so that the light projected on the opposite fin is collimated. The height of the sun is read on the mother’s graduated scale. There were also astrolabes in which the indication of hours took into account the different length of day between summer and winter (unequal hours). Curves were engraved for this purpose, showing the different length of day as the seasons changed. The simplest astrolabes were constructed to determine latitude and time in navigation and can be considered the antecedent of the sextant. The more complete astrolabes can be used for many calculations: measuring the height of a building, determining the time based on the position of the sun or stars, determining sunrise or sunset, determining the position of the planets on a certain date for horoscopes, determining the time of specific prayers or religious festivities. Sextant  It is the instrument to measure the angular distance between two stars. Tycho Brahe had several sextants and quadrants built by Jost Bürgi and Erasmus Habermel (c. 1530–1606). The sextant is composed of two steel rods mounted at an angle of 60°, an intermediate rod rotating around the center called the alidade, which allows to read the angle engraved in the brass arch, called the limb (see Fig. 6.6 left). This instrument was mounted on a rigid stand and suspended by the center of gravity so that it was perfectly balanced. After rotating it around the vertical axis to align it with the zenith, the altitude of a star could be measured moving the alidade and observing the angles engraved on the limb. To measure the angular distance between two stars it was rotated to align the plane of the instrument with the plane identified by the two stars and the center of observation. The sextant built for navigation allowed to align the observation to the Sun by means of mirrors to filter the sun light and provide a measure to determine the longitude, given an accurate measure of time. You can see in Fig. 6.6 right the transverse subdivision for measuring fractions of a degree.

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Fig. 6.6  Left: Sextant by Habermel, 1600. Right: detail of Bürgi’s sextant, 1600. Technique National Museum, Prague

Astrological Astrolab  Astrological study was one of the main uses of the astrolabe, which in this case contained indications for determining the ­position of the planets, sun or moon in the different zodiacal mansions. For the division of the zodiac into the 12 mansions, 12 curves were engraved. The curves correspond to the projection of great circles drawn from the observer’s north and south horizons to 12 points on the celestial equator. The method devised by Campano (Fig. 5.2 right, green lines) to draw the curve is based on a uniform subdivision of the zodiacal mansions into 30°intervals, and it supports fractional mansion coordinates. This subdivision, however, does not take into account the different extension of the constellations of the zodiac. For this purpose, Regiomontanus devised the unequal sizes subdivision of the zodiac, trying to include between the pairs of curves the set of stars of each constellation (Fig. 5.2 right, purple lines). Equatorium, Nocturnal, Quadrants The equatorium is a mechanical instrument for locating the position of the Sun, Moon, and 5 planets relative to the zodiac circle, without having to resort to calculations and ephemeris tables. The main use of this instrument is to locate the positions of the moving stars for the preparation of astrological horoscopes. The equatorium derives from the astrolabe, it includes movable pointers to indicate the position of the planets relative to the zodiac and may also be without mechanisms. (Fig. 6.7 left). The nocturnal or nocturlab, was used to determination time during the night; this instrument derives from the astrolabe. By observing the polar star, the alidade is aligned with a star of the Ursa Major, and the current time can be read from the ephemeris (Fig. 6.7 right).

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Fig. 6.7  Left: Equatorium, Veneranda Biblioteca Ambrosiana. Right: Nocturnal. Technique National Museum, Prague

Another instrument, probably invented by the Chaldeans or Babylonians, is the astronomical quadrant (Fig. 6.4 right), used to measure the height of a star relative to the horizon. Many quadrants from the Arab era allow to read on the back also the value of the sine and cosine of an angle measured with the instrument. By knowing the latitude and measuring the declination, it is possible to tell the local time. Increasingly accurate measurement requirements lead to the construction of large quadrants, in 995 Abu al-Wafa al Buzjani (940–998) built a quadrant with a radius of about 670 cm, and Hamid ibn al Khidr al-Khojandi’s (940–1000) sextant measured about 17 meters. Before the invention of the telescope, graduated quadrants were the main instrument, and we have seen their use by Tycho Brahe at the Uraniborg observatory. These instruments were also used for a long time by Arab and Persian astronomers. Their function, as we have seen, was essentially to measure vertical angles in order to determine celestial coordinates, aiming at the stars with various tricks. It was only with the invention of the telescope and in particular of the equatorial mount, that it finally became easy to orient the observation toward the desired star and to read the coordinates on graduated scales rigidly attached to the instrument. Transit Instruments  Ole Rømer built the first instrument to observe the transit, a telescope placed in a fixed position on the meridian of the place that could only move in elevation, which allowed him to determine the c­ oordinates of numerous stars with high accuracy. Accompanied by a precision clock, the

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Fig. 6.8  Left: Ole Rømer and his transit instrument. Right: Transit instrument. Insert in top left: a detail of the Illuminator of the collimator

instrument made it possible to record the instant of transit observed through the telescope. In addition, a reticle visible in the eyepiece allowed to center the observed object. In the Fig. 6.8 (left) several clocks can be seen, one of which is a pendulum clock at the bottom that makes a very wide swing. With this instrument, Rømer achieved a measurement accuracy that allowed him to estimate the speed of light. After the invention of the telescope the transit instrument became more complex. In the middle of the tube, there has a collimation grid that can be adjusted to measure distance between nearby stars. The collimator is illuminated from the sides by a candle (Fig. 6.8 right). Torquetum  These measuring instruments provided celestial coordinates in the three reference systems: the alt-azimuth, the ecliptic, and the equatorial (see Appendix “Positional Astronomy”). The conversion of coordinates between different reference systems was a long and complex calculation with frequent errors. The torquetum is a kind of analog calculator consisting of graduated scales engraved on two aligned planes such as the ecliptic and the celestial equator and a vertical plane on which an alidade rotates (Fig. 6.9). Reading the equatorial or ecliptic coordinates is immediate on either plane or the other. Early torquetum was built in thirteenth century and continued until the sixteenth century. Telescopes  For an in-depth study of the history of the telescope, see Henry C. King (1955); his work is based on an accurate reading of ancient texts, which also highlights the role of technicians and mechanics who contributed

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Fig. 6.9  Left: Drawing of a Torquetum. Right: A Torquetum, late sixteenth century. Technique National Museum, Prague

to the development of an increasingly precise instrument. I will limit here to recalling a few dates and aspects that are fundamental to understanding the role of the telescope among observation instruments during the seventeenth and eighteenth centuries. Roger Bacon (1220–1292) is believed to be the first to exploit the refractive power of glass to construct convex lenses, presenting the first laws of geometric optics in his treatise Opus Majus. To see the application of lenses to astronomy, however, we have to wait until 1608, when the Dutchman Hans Lippershey (1570–1619) built the first telescope, perfected by Galileo in the following years. The discovery of Jupiter’s satellites with the telescope strongly consolidated the Copernican hypothesis, disrupting the idea of a perfect sky in which nothing changes: it was a hard blow to prove that satellites also revolved around Jupiter. There are two main kinds of telescope: refracting and reflecting. The refracting telescope used by Galileo, and later greatly perfected, consists of two lenses: a plane-convex objective lens that causes a convergence of the light beam at a focal point within a tube, and a plane-concave ocular lens that enlarges the previous image and can be observed by the eye at a close distance. Lenses, however, have various forms of aberration. Spherical aberration is due to the spherical shape of lens surfaces, resulting in a blurred image at the edges. Chromatic aberration is due to the different deviation of light at different wavelengths. The solution to chromatic aberration was the use of two or three lenses in the eyepiece that compensated for the deviation based on the different wavelengths. Huygens devised an eyepiece whose composition is still used today due to its low cost, and retains the name Huygens eyepiece.

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Fig. 6.10  Left: Reflective telescope by Gregory. Right: The refractive telescope by Hevelius (1611–1687), 60 feet long

Ingenious solutions to these shortcomings were also found by Giuseppe Campani (1635–1715) (whom we shall meet again when talking about clocks), who started a business in Rome as a telescope and microscope maker, using his great skill in grinding lenses and the supply of glass from Venetian manufacturers. The Scotsman James Gregory (1638–1675) designed the reflecting telescope (Fig. 6.10 left) before Newton (which was later built by Robert Hooke) and, as a mathematician, made important contributions by laying the foundations of the notion of integral later developed by Bernhard Riemann (1826–1866) (Mamiani, 2002). By the second half of the seventeenth-­ century, therefore, optical telescopes had become the main instrument for observing the sky. All that remained was to solve the problem of measuring time. Capaccioli (2020) recalls that the lenses for Galileo’s telescopes were probably made in Venice. Later, professional glassmaking and polishing came to Flanders, also developing a craft of the cutting and polishing of diamonds. The optics of transparent media was still little known; the law of refraction was proposed by Willebrord Snell (1580–1626). He proved that the deviation of light when traversing two media of different refraction index, such as air-­ water or air-glass, is proportional to the ratio of the sines of the angles of the incident and refracted rays.6 Some studies in optics had been conducted by the Arab al-Haytham, who imagined a kind of corpuscular theory of light and  Given the index of refraction of the two media n1, n2 and θi, θr the angles to the normal to the separating sin θi n2 surfaces of incident and refracted rays holds the ratio . = sin θr n1 6

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observed the effects of the reversal of the image passing through a very small hole: the principle of the camera obscura. However, he did not produce any relevant results or hypotheses on refraction. The reflective telescope, on the other hand, was based on properties of light reflection that had long been known and could be expressed geometrically. The assumption that light rays coming from outside propagated in a parallel manner made it possible to geometrically resolve their reflection and refraction on concave mirrors or lenses. The construction of lenses and mirrors was, however, a very complex and expensive business, both because of the quality of the material, which had to be free of impurities and have a perfectly symmetrical curvature. For this reason, in general, lenses and mirrors were small.7 The description of Galileo’s telescope in the catalog of the Museo Galileo in Florence8 gives the main measures: the plano-convex objective, with the convex side outwards, measures 37 mm in diameter, has an aperture of 15 mm, focal length of 980 mm and a thickness at the center of 2.0 mm. The original eyepiece is lost and was replaced in the nineteenth-century by a biconcave eyepiece with a diameter of 22 mm, thickness at the center of 1.8 mm, focal length of −47.5 mm (the negative focal length indicates that it is a diverging lens). The instrument can magnify objects 21 times and has a field of view of 15’. It is clearly an instrument with very little luminosity, a problem that perhaps was less important at the time than today, given the total absence of light pollution. The name telescope was given to this instrument by Federico Cesi, a name derived from the Greek τελοσ - far, σκοπεο - to see. In order to achieve significant magnifications with refracting telescopes, however, it was necessary to build very long telescopes, which were, therefore, unstable and difficult to maneuver. However, they had very low luminosity to retain small but accurate lenses and were therefore suitable for observing planets and the Moon and not for observing the deep sky: galaxies, nebulae, and star clusters. Even in the eighteenth century, long telescopes were maneuvered by systems of ropes and pulleys (Fig. 6.10 right), and aiming also proved to be very complex. A very long telescope by Giuseppe Campani was installed at the Paris Observatory, at the time directed by Giovanni Domenico Cassini (1625–1712), who discovered four of Saturn’s satellites and recognized the  The magnifying power of a telescope depends on two parameters: the focal distance F and the brightness F where f, the aperture, D is the same dimensional unit found in photographic lenses. 8  catalogo.museogalileo.it/oggetto/CannocchialeGalileo.html 7

f, which are linked through the diameter of the lens D by the relationship: f =

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band structure of Saturn’s rings, whose main subdivision bears his name. Cassini also discovered Jupiter’s red spot and observed the differential rotation of Jupiter’s atmosphere. A major problem concerned the tracking of a star during observation, which, due to the Earth’s rotation, quickly left the field of view. To overcome this limitation, geared mounts were invented. Specifically, the equatorial mount has an axis of rotation parallel to the earth’s axis, allowing a celestial body to be tracked by means of a single gear (Fig. 6.11). Reflecting telescopes could have a mirror arrangement that, with multiple reflections (but at the expense of brightness) allowed long focal lengths while keeping the instrument very compact. For observing planets, magnification was undoubtedly more important, which is generally achieved by extending the focal length and thus the length of the tube. For deep sky observations, on the other hand, brightness is the most important parameter. Since the amount of light captured by a telescope depends on the diameter of the lens or mirror, the construction of telescopes with larger and larger apertures has been constant over time. This led to the construction of very large mirror telescopes, the most famous being the telescope at Mount Palomar, California, which has a 5 meters diameter mirror. Beyond this size, it was no longer possible to go for reasons of cost, weight, and machining precision. Since the 1970s, adaptive optics technology has been developed that allows the construction of reflecting telescopes with small composite mirrors that can be maneuvered to correct any irregularities in image formation. This also made it possible to build compact, lightweight telescopes that could be put into orbit outside the Earth’s atmosphere.

Measuring the Time In Babylon and Egypt, the day was divided into 12 hours during the day and 12  hours at night. This division remained constant until the Middle Ages when the measurement of time was related more to religious rituals than to civil or scientific requirements. The beginning or end of the day was not uniform throughout Europe. Until the late eighteenth century, Italic time was in force in Italy, as opposed to the transmontane time of France. In Italy, 24th hour began with the setting of the Sun, thus varying by about 3 hours between summer and winter. Conversely, transmontane time identified noon with the

Fig. 6.11  Left: Schematic of an equatorial mount. Right: equatorial mount of a reflecting telescope

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transit of the Sun and midnight with the twelfth hour thereafter. The dials of some ancient Italian bell towers still bear the indication of Italic time. The earliest clocks9 of the late Middle Ages certainly did not have sufficient precision to improve astronomical measurements. Until the sixteenth century, the verge escapement with foliot, which we will describe later, did not exceed an accuracy of more or less one hour, which dropped to half an hour in the mid-1500s when some minor improvements were introduced. The clocks built by Jost Bürgi after 1586 brought the accuracy down to ±30″. The Huygens pendulum is slightly more accurate ±20″ but very irregular over long intervals of time. The introduction of the anchor escapement toward the end of the seventeenth-century brought the error to ±10″, and in 1710, the famous English clockmaker George Graham (1674–1751) introduced the deadbeat escapement, bringing the error below ±3″ (von Bertele, 1953, p. 800). We should not think that the reason for perfecting the measurement of time was only speculative, to increase astronomical knowledge. Between the sixteenth and seventeenth centuries, the most important explorations of the Earth took place, most of which were carried out by sailing the oceans. The problem of knowing the exact position of a ship and plotting it on maps, which were the main tool for describing the shape of the Earth, was therefore present and dramatic. Astronomical observatories became the place where the measurement of time took on an official character, and all European countries gradually established observatories for both astronomical studies and the official measurement of time. Preceded by the Arab observatories of Baghdad in 828, Maragheh in 1259, and Ulugh Beg in 1420 (all now gone), the oldest in the modern era is Tycho Brahe’s Uraniborg observatory, then in 1633, the Leiden observatory in the Netherlands. The Copenhagen observatory was established in 1642, the Paris observatory in 1667 under the direction of Cassini, and the Greenwich observatory in 1675 directed by John Flamsteed (1646–1719).10 European kings established monetary prizes to solve the problem of measuring time in navigation, which encouraged the study of innumerable solutions. The evolution of clocks as instruments for measuring time is in turn part of the development of mechanics. The nineteenth century was the era of the triumph of the most refined mechanical technique for the construction of clocks. Ferdinand Berthoud (1727–1807), a prominent French watchmaker, published a history of time measurement in 1802 (Berthoud 1802a), in which he states:  For a general history of clocks see (Turner, 2022)  The Observatory was founded by King Charles II in 1675, after the Great Fire of 1666, and designed by Christopher Wren. 9

10

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Among the immense and marvellous productions of Mechanics, the Art of measuring time by clocks is the one that holds the first rank, as much by its usefulness as by the varied extent of its inventions, by the subtlety of its effects, by the genius and depth of its conceptions, and by the extreme delicacy of the parts that compose it.

Ferdinand Berthoud lists four main uses of clocks: (1) a civil and public use in tower clocks and personal clocks with pendulums and pocket watches; (2) a use for astronomy; (3) a use for navigation and geography and (4) a use to imitate the movement of the stars. We will not follow this classification; it is more interesting to highlight, on the one hand, the complicated and multifaceted nature of time measurement and, on the other, the major innovations up to the end of the eighteenth century. Both the ingenuity of the inventors and the scientific basis on which these innovations were based emerge from this narrative. After this overview of the development of clocks, I will return to astronomical clocks and the imitation of the motion of the stars. What to Measure?  In this chapter I choose to change the way of telling the story of time measurement by introducing a strictly physical concept of time based on the identification of a measurable quantity. I came to regard the counting of oscillations as the dominant principle. The notion of time is as difficult as ever to explain in both scientific and philosophical terms, and I certainly do not intend to engage in philosophical discussions.11 One question to which one can try to find an answer is: how is time measured? The first step in measuring time emerges from the scanning of days, months and years. We already recalled a hypothesis by Otto Neugebauer (1941, reprint 2015, p. 16), of how this came about (see Sect. “Ancient Mathematics and Astronomy Until the End of the Roman Empire”). In this chapter, I would like to change my point of view and answer a different question: what should we measure in order to obtain an accurate and reliable measurement of time? I consider measurement as a function that associates a number with some observable. The philosophical difficulty of defining time can be circumvented. I do not consider time as an observable in itself. With this assumption, the answer to my question becomes simpler: what we can measure must be a different quantity that is related to this abstract entity that is time. The first measurement of time, as we have seen, can be traced back to the counting of 11

 See also Eco (2020).

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days and months derived from observing the day/night cycle or lunar cycles, but what we mean by the measurement of time is something more precise, something that allows us to divide the time of the day into hours and then into minutes and seconds. We know that the earliest instruments for measuring time were sundials and water or sand hourglasses. The oldest Egyptian sundials date from the Middle Kingdom (-2000-1630). Water clocks, also called clepsydra, are probably equally ancient. The shadow is certainly an observable object in a physical sense, and its change in length and direction can be measured. The change in length of the shadow at the same position of the Sun on different days can be related to the cycle of the seasons. The measurement of the change in direction over the course of a day can be related to the passage of hours. The measurement of time obtained by measuring a projected shadow is certainly accurate, but it is only possible during the day and not when the clouds mask the Sun. It has also the limit of providing a time scan bounded to the location where it is measured. In the case of the clepsydra, what is measured is a quantity of matter, water or sand, that is consumed or moved in a controlled manner, and this count is related to the passing of hours. The accuracy of this method is subject to significant irregularities due to the nature of the matter that varies with temperature conditions, such as the density of the water or the weight of the sand. The very name has a Greek etymon κλεψυδρα meaning water stealer, although the term is used to denote sand instruments too. There are two types of clepsydrae, those that measure the time it takes water to leave a container (outflow) and those that measure the time it takes to fill it (inflow). Babylonian clepsydrae are mostly outflow, while Egyptian ones are both but mostly outflow. In the Alexandrian period, Ctesibius (−285–222) devised a solution with a floating indicator attached to a graduated cylinder and moved with gears, in an inflow clepsydra (Fig.  6.12 left). In Sect. ”Astronomical Instruments and Machines” we will see a different and more complex clepsydra invented, around the year 1000, in ancient China. The limitations and inaccuracy of clepsydrae and sand hourglasses were overcome by inventing a different quantity to measure: the number of oscillations. Usually, watchmaking histories tend to present the technical evolution of pendulum clocks and pocket or wristwatches on the basis of escapement-­ related inventions. However, this principle cannot be fully understood if one does not grasp that the pendulum or balance are mechanical oscillators, which, left free, dampen their motion to a stop and must therefore be maintained in a state of forced oscillation. The physical theory of oscillators can be

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Fig. 6.12  The clepsydra by Ctesibius, according to Vitruvius. Right: Huygens’ cycloidal pendulum

of help in understanding the characteristics and limitations of instruments that counts oscillations. The Oscillator  The oscillator is a device capable of oscillating with constant frequency and continuity. A clock with this device, instead of grains of sand or drops of water, counts oscillations. Three problems must be solved: (1) constant frequency or isochronism, (2) continuity of oscillations, and (3) a mechanism for counting oscillations: the escapement. Galileo studied the oscillation of the pendulum and discovered its isochronism for small oscillations. A second important property is that the frequency depends on the length of the pendulum. Galileo did not build a pendulum clock, although a drawing due to Viviani and his son Vincenzo was found in 1855 and is kept at the Museo Galileo in Florence. The ideal pendulum is a perfect harmonic oscillator. It consists of a mass m suspended from an inextensible wire. When the angle is small, the oscillations are isochronous, with a period that depends only on the length and not on the

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mass.12 The hypothesis of small oscillations is very difficult to guarantee in practice, if, for example, we have a pendulum that oscillates with an angle of 14° the error is 1%, and ancient clocks have oscillation angles much larger. Huygens demonstrated that the property of isochronism is always valid if the oscillating mass follow a cycloidal path instead of a circular path. Huygens made prototypes with a cycloidal pendulum suspended from a wire that changes its profile by resting on two cycloidal foils, which are called cheeks as in the (Fig.  6.12 right). Unfortunately, this solution introduces significant friction. The period can change with the pendulum length. In the ideal pendulum the mass has no influence on the period. To overcome the problem of the length, clockmakers made many experiments, devising many different solutions. The gridiron pendulum (see Fig. 6.19 left), invented by John Harrison (1693–1776), consists of a series of parallel metal plates with different thermal dilatation coefficients that compensate for each other. Wooden pendulums are less sensitive to elongation due to temperature changes, but are sensitive to humidity. Mercury pendulum (invented by George Graham) compensate temperature change by the expansion and reduction of its mass that contrast the elongation of the suspending bar. When the pendulum is used on a moving system, such as a ship, the plane of oscillation does not remain constant, which causes loss of isochronism and, in the case of strong movements also the stopping of the clock. This problem was, for a long time, a major obstacle in solving the measurement of time in navigation. Hooke‘s studies on elasticity led to the invention of the circular oscillator, the balance. The balance oscillates in a plane orthogonal to the suspension axis (Fig. 6.13 left). In this case, the periodic rotation occurs around the suspension axis, but the torsion foil is shaped as a very thin steel spiral that forces the

 The period T varies according to the law: T = 2π

12

l where g is the acceleration of gravity and l is the g

length. In the cycloidal pendulum, the period is given by: T = 2π

4a where a is the radius of the g

generating circle of the cycloid. In practice the pendulum does not correspond to the ideal scheme but consists of a body of mass m made up of the wire and the suspended object. In this case, again for small oscillations, the period of the I where I is the rotational inertia (which depends on length and shape), m pendulum is: T = 2π mgd the suspended mass and d the distance of the suspension point from the center of mass of the pendulum.

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Fig. 6.13  Balance oscillator with spiral. Center: limit cycle of recoil pendulum. Right: limit cycle of deadbeat pendulum

balance to oscillate in its own plane. The balance oscillator has now become the most accurate solution for mechanical watches.13 All these different types of mechanical oscillators in free conditions are subject to damping due to friction with air and friction generated by the type of suspension. The theory of harmonic oscillations considers damped oscillations and forced oscillations. A free pendulum is an oscillator damped by friction and, as mentioned, it stops after a while. A forced oscillation system has the same frequency as the free oscillator but is phase-delayed as π, by means of the escapement. Under these conditions, the oscillator reaches a resonance frequency very close to the frequency of free oscillation. The first mathematical analysis of pendulum dynamics is due to Georg B. Airy (1801–1892) who was Director of the Greenwich Observatory, and in 1827 published an in-depth study of pendulum dynamics examining the effect of the escapement (Airy, 1830). Mark Denny (2002), using modern mathematical methods, draws the so-called limit cycle, i.e., the diagram of the amplitude (the horizontal axis in the diagrams of Fig. 6.13 of the oscillations with respect to their speed (the vertical axis in the diagrams). In the case of a free oscillation, the limit cycle is a spiral14 converging toward the origin (Fig. 6.13 right), whereas in a forced pendulum with escapement, the limit cycle, after a complete period during which the speed decreases, resumes with the initial maximum speed. In Fig. 6.13 (center) we see the limit cycle with a recoil escapement and on the right the deadbeat escapement.

13

 The period of a circular oscillator is: T = 2π

I where R is the radius of the balance wheel and I is mgR

the rotational inertia.  The spiral is not shown; it can be recognized in the left diagram, where the amplitude reduces until a force is applied. 14

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The theory of mechanical oscillators makes it possible to describe periodic motion under conditions of forced oscillation. In such a case, the oscillator reaches a resonant condition, when its own frequency corresponds to the frequency of the force that maintains the oscillation (Vermot et al., 2011). The measurement of oscillation in clocks and watches is counted in pulsation, while the frequency is counted in Herz, one complete oscillation is, therefore, two pulsations (the tic-tac of the pendulum). The Escapement  The escapement15 maintains the oscillation and allows to count the number of oscillations. It is not known who invented the escapement, von Bertele reports that the verge escapement was invented by St. Gerbert, who became Pope Sylvester II (970, 1003) (von Bertele, 1953, p. 802). There are, however, documents describing a mechanical clock made by Pacifico (776–846), Archdeacon of Verona (Berthoud 1802a, p. 47). Consider a crown wheel16 and an arbor with two pallets protruding, one at each end set at a little more than a right angle to each other. As the wheel revolves its teeth engage the arbor alternately, one side a pallet and the opposite side the other. In this way, the crown wheel is stopped and released, and when it is released, it gives an impulse to the arbor to change its rotation direction. The foliot is a bar fitted to the arbor, to its two arms, two weights are suspended. By varying the distance of the weights, the rotation speed changes: faster when near the pivot, and slower when farther from the pivot (Fig. 6.14). This device is not yet an oscillator: it allows to count some ticks and keep the system going but is not isochronous. From a physical point of view, we are dealing with a mass with central symmetry that is put into rotation, stopped, and reversed by the pallett mechanism. The first clocks with the foliot mechanism probably date back to the fourth-­ century, although clockmaking historians have suggested that foliot clocks had already been built by the end of 900. But this hypothesis is unconvincing: it would not explain how another 300 years passed without an invention of this utility being widespread. The verge escapement is present in the oldest public tower clocks, such as the one in Chioggia, and in early Renaissance monastic clocks, typically used to determine the hours of prayer.  A very detailed discussion of the main types of escapements, including design guidelines is Gazeley (1956). 16  The crown wheel has the teeth orthogonal to the wheel plane. It is also called the Catherine wheel because it recalls the shape of the cogwheel depicted in the iconography of St Catherine’s martyrdom. 15

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Fig. 6.14  Left: Verge escapement with foliot. Right: Verge and foliot (blocked by the wooden piece), XIV century

About five centuries had to pass before a more effective mechanism was devised. With the introduction of the pendulum, the escapement took the form of an anchor, which in early versions had a flaw: the residual thrust of the pendulum caused the gear wheel to rotate in the opposite direction with a loss of energy (recoil escapement Fig. 6.15 left). George Graham introduced the deadbeat escapement (Fig. 6.15 center), in which the gearwheel remains stationary and when the pendulum reverses its motion, it provides the thrust to maintain the oscillation. The anchor has two polished surfaces, curved in the recoil escapement and flat in the deadbeat escapement, which alternately stop the wheel and receive an impulse that maintains the oscillation.17 Since the beginning of the eighteenth century, continuous research has been conducted to improve the escapement mechanism with the aim of reducing the friction between the teeth and the anchor planes and ensuring the isochronism of the oscillations as much as possible. A deadbeat escapement is also the pin-wheel escapement adopted in many tower clocks, which pre-dates the anchor escapement (Fig. 6.15 right). The Driving Force  A driving force is required to operate the machine. The weight of a stone can transmit, by means of ropes, the driving force to the axle of a wheel, which, by means of other gears, is transmitted to the escapement that count oscillations and maintains motion. The ratios of reduction or multiplication of the motion generated by the driving force make it possible to

17

 The diagrams of the escapements are taken from Headrick (1997).

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Fig. 6.15  Anchor escapement. Left with recoil. Center deadbeat by Graham. Right: pin wheel escapement

divide the time interval so that an indicator, attached to the axis of the driving wheel, completes an entire revolution in 12 or 24 hours. In early clocks, the weights were very large and heavy and ran on wheels of very limited quality, so the friction and wear on gears, pinions, and pivots was considerable and led to measurement errors and frequent maintenance. Because of this, clocks were only used to measure short time intervals. For example, the astronomer Michael Mästlin estimated the apparent diameter of the Sun at 34′ 13″ (Berthoud 1802a, p. 59). Using a clock that had a frequency of 1264 oscillations per hour, equal to 2528 pulsations, he counted 146 pulsations taken by the Sun for the transit, (the measurement today is between 31’ 29” at perihelion and 32′ 33″ at aphelion). Only a more accurate manufacturing technique made it possible to reduce the weights, wear and friction of the movements. The driving force can also come from a spring, a steel foil wound around an axle, but in this case the force exerted is not constant: as the spring unwinds, it becomes progressively weaker (Fig.  6.16 left). To overcome this, several solutions were invented to try to keep the force constant, the first solution being the stackfreed18 (Fig.  6.16 right), a spring running along a cam that rotates with the main spring, thus varying the force generated: a waste of energy! Later, the conoid was used,19 around which a rope or thin chain is wound in order to vary the torque applied to the gears by the spring (Fig. 6.17  It consists of an arc spring that presses on an eccentric disc integral with the spring barrel; as the radius with which the spring presses on the disc varies, there is a slowing effect that is greater when the spring is loaded and gradually decreases when the spring is unloaded. 19  The driving spring does not act directly on the gear train, but by means of a very thin chain wound on a cone-shaped barrel with a gradually increasing cross-section. The fusée-and-chain transmission works like an infinitely variable gearbox. It equalises the waning force of the mainspring and makes sure that the movement always receives a constant amount of energy. This keeps the watch running at an exact rate. 18

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Fig. 6.16  Left: The gradual torque reduction of an unwinding spring. Right: The stackfreed with the cam. The spring A has a roller B that slides along D with variable pressure

left). Modern technology has made it possible to construct springs with appropriate elastic constants. Robert Hooke was the first to study elastic forces by formulating the law that links the force of a spring to its length.20 Leibniz also ventured into finding a way to make the motion of a watch more regular, particularly during the winding phase (“Tardy” 1969, p. 16). In 1675 he published in the Journal des Savantes a project, never realized, in which two springs are alternately wound by a mainspring, so that one unwinds while the other is wound again, when the winding of the latter is complete and the former is fully unwound the two functions are interchanged. As each spring unwinds, it sets in motion a gear wheel attached to a pinion connected to a flywheel whose moment of inertia governs the duration of the spring’s unwinding (Fig. 6.17 right). Gears and Motion Transmission  Historical evidence concerning Archimedes’ planetary machine reveals the knowledge and use of gear wheels, and the Antikythera mechanism is a clear proof of their use in the -I century. Heron of Alexandria (+first-century) describes machines for lifting great weights. Earlier civilizations in Mesopotamia and Egypt also had the knowledge to transmit movement, if not with gear wheels certainly with ropes and winches. In more recent times, water clocks with gear wheels were built by Cassiodorus (+485, +580). Berthoud reports the description by Father Gaubil (1689–1759) – Jesuit astronomer and missionary to China - of the machine built by the Chinese astronomer Y-Hang.21 20 21

 F = − kE·Δl·x where F is the applied force, kE the elastic constant, Δl the elongation and x its direction.  Probably the name of the astronomer is I-Hsing. (Berthoud 1802b, p. 178). See Chap. 14.

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Fig. 6.17  Left: Conoid and barrel linked with a string or chain. Right: Leibniz’s project to regulate a clock Water moved several wheels, by which the proper and common motion of the Sun, Moon and the five planets; conjunctions, oppositions, eclipses of the Sun and Moon; and occultations of the stars and other planets were represented. The length of days and nights for Si-Gan-fou,22 the stars visible and not visible on its horizon were shown. Two needles marked the day and night, kè, the hundredth part of the day, and the hours. When the stylus or needle was on the kè, a small wooden statue suddenly appeared and rang a bell and then disappeared. When the stylus was on the hour, a wooden statue would appear on the stage and strike a bell; when the stroke was given, it would retreat.

Haroun el Rashid, in +809, built a clepsydra with gears that he sent to Charlemagne.23 Thus, there are records showing that the use of gear wheels for time measurement goes back a long way. The earliest gears were built with pegs fixed between two discs that meshed with other pegs placed orthogonally to the circle of the driven gear, or with gear wheels. The Romans used a chariot with wooden gear wheels to measure distances, invented by Heron, the odometer. Leonardo da Vinci (1452–1519) designed machines based on wooden gears. Studies on gears are also attributed to Gerolamo Cardano (1501–1576), who was in correspondence with Leonardo. The shape of the teeth in metal wheels was originally triangular with significant friction. It was Philippe de La Hire who, in 1694, invented the involute profile by which the teeth do not slide but rotate tangent to each other, reducing friction. Currently, the tooth profile meets international standards, which in watchmaking refer either to the English BHS standard or the Swiss standard.24 Electronic and Atomic Oscillator  In the second half of the nineteenth century, a new type of oscillator was invented: the electronic oscillator. Electronic  The capital of the province of Shen, at the Han time called Kwan-Chung.  Ibidem 24  Gear theory will be briefly described in Chap. 15. 22 23

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harmonic oscillators can be built with resistors, inductors, and capacitors; their frequency depends on the physical characteristics of the circuit. Mr. and Mrs. Curie in 1880 discovered piezoelectricity, a property of quartz crystal to generate an electrical charge when deformed or subjected to an electric field. A resonance effect produces an oscillating frequency that is constant over a long period of time. Small changes in the Earth’s rotation were discovered in 1934 by means of a quartz clock. Cesium atoms have been chosen as the frequency generator of today’s atomic clocks. The unit of time, the second, is now defined as 9,192,631,770 oscillations of a Cesium clock. Today, the international standard for astronomical observations adopts International Atomic Time (TAI), obtained as the average of the measurements of several Cesium clocks distributed in various places on Earth. UTC time25 corresponds to TAI minus a correction factor that is introduced after a few years; the last update of 34 seconds was on 31 December 2008, so UTC = TAI - 34 sec (Sutton, 2012). The Evolution of Clock Making for the Development of Science The development of watchmaking can also be considered from a trading and production point of view. We have seen that the first clocks were born in the most important Renaissance centers: Italy, France, and the Netherlands. Watchmaking technology spread to the rest of Europe following the emigration from France of Protestants persecuted in religious conflicts: the Huguenots. From France, many fled to England, to Germany, particularly to the southern states of Bavaria and Württemberg, where the Black Forest, known for its wooden cuckoo clocks, was located, and to Switzerland, to the canton of Neuchâtel, on the border with Franche-Comté to the north, where Morbier or Comtoise clocks were made. The great Swiss watchmaking industry then took shape in the Swiss Jura valley. Geneva, too, became an important watchmaking center, starting in 1540 when it hosted Huguenot refugees from Catholic France. In 1541, the Calvinist Reformation banned wearing jewelry, and many goldsmiths turned to make watches, which masked their natural status as luxury items with utility. Others fled to England, fleeing religious persecution, bringing watchmaking technology that soon took hold. And it was England that long competed with France for the production of luxury and precision watches. Religion has also played a role in the history of horology in Italy. The power of the Catholic Church persecuted Galileo; Giordano Bruno was burned at the stake in 1600, and Copernicus’ heliocentric theory was placed on the 25

 Coordinated Universal Time.

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index of prohibited books. Giordano Bruno strongly supported the Copernican theory in his work De l’infinito, universo e mondi [On infinite universe and worlds] published in Venice in 1584 (Giordano Bruno, 1584)26: For the resolution of what you seek, you must first see that, since the universe is infinite and immovable, you must not seek the motor of it. Secondly, since the worlds contained in the universe are infinite, such as the earths, the fires, and other kinds of bodies called stars, all of them move from their internal principle, which is their own soul, as we have proved elsewhere; and therefore, it is vain to search for their extrinsic motor. Thirdly, that these worldly bodies move in the ethereal region, no more attached or nailed to any body than this earth, which is one of them, is attached; which, however, we prove that from the internal animal instinct it circles its own center, in several ways, and the sun. Having set forth these instructions according to our principles, we are not obliged to demonstrate either active or passive motion of an infinite power intensively; for the movable and the motile is infinite, and the moving soul and the moving body concur in a finite subject; in each, I say, of the said worldly stars. So much so, that the first principle is not the one that moves; but, quiet and motionless, it gives the power to move to infinite and innumerable worlds, great and small animals placed in the most ample region of the universe, of which each, according to the condition of its own virtue, has the reason for mobility, motion, and other accidents.27

These facts discouraged the fugitive Huguenots from coming to Italy, taking away the opportunity to participate in the development of the new technique, with one important exception: the Campani brothers, who worked in Rome during the seventeenth-century. Pier Tommaso Campani (1630–1700) started a watchmaker’s business and, with the collaboration of his brother Giuseppe made a night watch for Pope Alexander VII (Fabio Chigi), who suffered from insomnia. In addition to an original system of backlighting a disc indicating the night hours and quarter hours, they adopted a mercury escapement (Secchi & Proja, 1860). This invention was a huge success at European courts, and numerous examples were produced by Campani and others, decorated in the Baroque style with paintings by important artists (Zanetti, 2020). Giuseppe Campani visited the Medici court and learned about the invention of the Huygens pendulum clock in 1657. Campani’s clock was however recognized as an instrument of great quality, so much so that a patent was granted by the Grand Duke of Tuscany. These clocks soon became luxury items for princes, and the Pope himself often made gifts of them to Ambassadors and Sovereigns (Fig. 6.18). Another of the Campani brothers, Matteo (1620–1678), published a booklet in which he set out a design for a watch suitable for navigation at sea (Campani degli Alimeni, 1673). Matteo Campani identified several  https://www.pensierofilosofico.it/ebooks_file/deinfinitouniverso13588679021.pdf. September 2022. 27  Tr. by the author. 26

Accessed

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Fig. 6.18  Night Clock by Giuseppe Campani. Musei Capitolini, Roma

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problems: the effect of moisture, unequal spring force, gear wear, and the need to keep the machine upright. The solution devised was, therefore, to enclose the movement in a carefully sealed vacuum container and to make a support that would keep it upright at all times. To guarantee the regularity of motion, he also proposed a double pendulum. In support of his invention, he wrote (Campani degli Alimeni, 1673, p. 15): Because it happened to me above to speak of my other invention of making closed Orioli [clocks], it seemed well to me, to relate here the same figure of it, with which from the beginning, I expressed the easiest manner of it, to be understood by artisans even of mediocre ingenuity. Cautioning the benign reader, however, that it expresses nothing more than the principles of invention, and the very easy possibility of many and various far better ways of making such Orioli, so well enclosed in glass, or elsewhere, that there remains neither even the slightest glimmer of being able to leak air into it.28

Although subject to the political domination of the Church, scientific research activities continued apace in Italy, consolidating ties with other European scholars. In central European countries, the first watchmakers were mainly blacksmiths who soon organized themselves into guilds (Britten, 1982). In London, the Worshipful Company of Clockmakers was established in 1631 by Royal decree. The first guild in the Germanic countries was founded in Anneburg, Saxony, in 1543, in Nüremberg much later, in 1656. In France, the guilds were established in Paris in 1544 and in Blois (another important French manufacturing center) in 1597. The guild of Genevan watchmakers was born in 1601.

 Challenge Between French and English: A Measuring the Meridian and the Time in Navigation The determination of longitude on land had been solved by the end of the seventeenth century. The problem of longitude in navigation, however, still remained open. An in-depth description of the events of this long search was published by Dava Sobel and William D.J.  Andrews (Sobel & Andrews, 1998). The book by Rupert T. Gould (1989) is a more technical work. Clocks built from the 1300s to the first half of the 1500s had an error between 8 and 15 minutes corresponding to an angular error between 2° and 3.75°. Toward the end of the 1500s the accuracy of the clocks increased with  Tr. by the author.

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an error of 2′ 30″ equal to an angular error of 0.625°. With Bürgi clocks, the error in timekeeping still drops to 0′ 15″ corresponding to an angular error in the order of 3.75 arcmin (Matther & Sánchez-Barrios, 2017). If we imagine to move along the earth’s equator, 1 arcmin corresponds to a distance of about 1.85 km. and at latitude 45° the distance is about 1.31 km. These are ranges that over long ocean distances could lead to catastrophic course errors. Moreover, the period of a pendulum varies, albeit by very little, as a function of latitude, given the shape of the earth and its effect on the acceleration of gravity. Various proposals were experimented: (1) eclipses of Jupiter’s satellites, (2) occultations of a star by the Moon, (3) eclipses or other phenomena such as planetary conjunctions, (4) Moon’s transit, (5) lunar distances and finally (6) the use of a clock. In 1714, the English Parliament awarded a prize of £20,000 for anyone who managed to build a clock that could measure the time with sufficient accuracy to determine the longitude at sea. There were numerous English watchmakers working from the eighteenth-century onwards. The most active were: George Graham, John Harrison, Larcum Kendall (1721–1790), Thomas Mudge (1715–1794), John Arnold (1736–1799), and Thomas Earnshow (1749–1829). From their work, horology arrived at the perfection of the marine chronometer. George Graham  Although he did not take part in the search for a solution for measuring time at sea, George Graham’s role cannot be forgotten, for two reasons: as a fine mechanic and scientific instrument maker and for his help to John Harrison in his quest for perfection. Graham worked with Thomas Thompion (1639–1713) until he inherited his business. We have already briefly illustrated his contribution to the technical progress of horology, but his activity did not stop there. He played a great role in the creation of scientific machines, as remembered by marquis de Maupertuis (1698, 1759), who led an expedition to the Polar Circle in Lapland to determine the shape of the Earth and to carry out longitude measurements. During these studies, the scientists made use of some of Graham’s instruments: one sector he built is described as follows (Maupertuis, 1739, p. 36): The instrument [a Sector] was made in London, under the eyes of Mr. Graham of the Royal Society of England. This skillful mechanic had applied himself to provide it with all the advantages and conveniences that we might need: finally, he had divided the limb himself.

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A special Graham pendulum was also used in this expedition to assess weight changes with latitude (Ibidem p. 185–186): The Pendulum is composed of a heavy Lenticule which holds on to a flat Verge of copper. This Rod is finished at the top by a piece of steel which is perpendicular to it, & whose ends are two Knives, which, instead of being received between two inclined planes, or between cylinders, bear on two flat steel tablets, which are both in the same horizontal plane.

On another occasion he provided a pendulum clock with which the effect of temperature changes was observed in London. The expedition to Lapland allowed de Maupertuis to confirm Newton‘s theory that the Earth was shaped like a rotating ellipsoid, slightly flattened at the poles, which was disputed by Cassini who claimed it was elongated at the poles. John Harrison  John Harrison was the son of a carpenter and followed his father’s profession, developing a passion for mechanisms from a very young age. He had no formal education but studied independently with the assistance of a parish clergyman, who provided him with books on physics and mechanics. With this study and early practical experience, he soon became an expert in repairing clocks and built his first pendulum in 1715. This work immediately highlighted the problem of regularity as a function of temperature, the effect of which on simple pendulums is to produce elongations or shortenings that change the period of oscillation. We have already mentioned his invention of the gridiron pendulum (Fig. 6.19 left), and following the success of his invention, Harrison became convinced that it was necessary to minimize escapement friction and devised the grasshopper escapement (Fig. 6.19 right). By 1726 he had completed two regulators29 with his pendulum and the grasshopper escapement, which did not vary more than one second per month over the course of 14 years. In 1728 he traveled to London to propose himself to the Royal astronomer Edmond Halley (1656–1742)30 for the awarded prize, but Halley advised him to meet Graham and make a prototype before submitting his application to the Longitude Commission, of which Halley was the chairman. George Graham was remembered as ‘the good guy’, and in  The regulator is a high-quality pendulum clock whose sole purpose is to measure time with the utmost precision. It often has dial faces to indicate hours, minutes and seconds and was present in watchmaking workshops before the spread of electronic clocks. Every astronomical observatory had regulators. 30  By examining comet sightings in the years 1456, 1531, 1607, and 1683, he concluded that it was the same comet that was to reoccur in 1758. Prediction fulfilled and the comet was given the name Halley. 29

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Fig. 6.19  Left: gridiron pendulum, made with rods of steel and brass having different thermal expansion coefficients. Right: Grasshopper escapement

fact, he supported John Harrison’s research from the outset by funding with £200 the construction of the first marine clock, which was not a pendulum because of the difficulty of keeping its oscillating plane in navigation. This first clock, H1, had two balances connected so as to maintain an opposite oscillation, which helped to compensate for the oscillations of a ship. Equipped with the grasshopper escapement, it proved to be a very accurate watch, but Harrison still did not win the prize. Harrison devoted himself to the problem for almost 40 years and built a total of four prototypes, the fourth of which, called H4, was a large pocket watch with a balance oscillator with a balance spring of a special steel, and a modified escapement with ruby pallets. This clock was not mounted on a universal joint to keep the machine horizontal, as was done with earlier prototypes, it was kept in a box. The first test trip was to Jamaica from 1761 to 1762, and the second trip was to Barbados in 1764. In the course of the evaluations, the Royal Astronomer Nevil Maskelyne (1732–1811), who was an ardent supporter of the method of lunar observations, interfered in many ways to disqualify Harrison’s work, who failed to get proper recognition for his work. Nevertheless, Harrison was awarded a prize of £10,000 to compensate for the costs incurred and to enable the details of the project to be published. All four chronometers were handed over to the Astronomical Observatory.

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In 1775 he published a book A Description concerning such Mechanism as will afford a nice, or true Mensuration of Time; together with some Account of the attempts for the discovery of the Longitude by the Moon and also an Account of the Discovery of the Scale of Musik.31 Larcum Kendall, Thomas Mudge, John Arnold, and Thomas Earnshow  Larcum Kendall, in 1767, was commissioned to build a marine chronometer copy of Harrison’s H4 model, which was completed in 1770. Kendall was a trained watchmaker, and the chronograph he built is of the highest quality. The purpose of the Longitude Commission was both to verify the validity of Harrison’s design and to assess whether it could be built at a limited cost. Captain Cook, on his last expedition in 1776, used the K1 chronometer. The Longitude Commission asked Kendall to instruct other workers to build more copies, and Kendall suggested changes that could bring the cost down to around £200. For this purpose, he removed some nonessential parts and built the K2 and K3 models. Although less accurate than the K1 model, they were used frequently, and in particular, the K2 model was used by Captain Bligh of the famous HMS Bounty. John Arnold invented the spring detent escapement (Fig. 6.20), which did not interfere with the regularity of the movement of the escape wheel, a fundamental requirement to ensure regularity of measurement. However, the first models that were embarked on Cook’s ship together with the K1, revealed significant irregularities due to insufficient compensation of the balance spring as the temperature changed. Arnold, therefore, invented a new type of balance spring with a cylindrical shape, in which the change in length due to temperature did not affect the oscillation frequency. He also adopted a variant of the escapement: pivoted detent, which required no special lubrication. Delivered to Greenwich for evaluation, it was used in the pocket for 13 months and proved to be of excellent accuracy. Later he made other chronometers in a box suspended with a Cardan joint. Thomas Mudge was the inventor of the lever escapement, and participated in the expert commissions invited to evaluate Harrison’s H4 chronometer. Mudge thereafter engaged in the construction of marine chronometers, and in 1774 he delivered a chronometer to the Commission to be evaluated at the Greenwich Observatory. Its characteristics are spring-driven; eight-day movement; powered by two separate mainsprings housed in a single barrel; fusee with Harrison’s maintaining power; Mudge’s constant force escapement; temperature compensation achieved by the use of two bimetallic strips; separate  A reprint has been published by the Antiquarian Horological Society in 2018.

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Fig. 6.20  Arnold’s spring detent Escapement. The balance wheel has been removed to see the spring and its connection to the escapement wheel

enamel dials showing separately minutes of time and equivalent in degrees, hours and degrees, seconds.32 Maskelyne’s judgment found that in 109 days it had lost only 1′ 19″, an exceptional value that made it the most accurate watch of the time. Harrison and Kendall took about 4 years to build their chronometers H4 and K1, while the K2 and K3 models took 3 years. Mudge also took 3 years for the first chronometer and 2 years to build the next two models called Blue and Green. This was an excessive amount of time and was reflected in the price. The transformation to industrial production and the consequent reduction in cost was due to Arnold and Earnshow. Thomas Earnshow made the definitive improvements that constituted the English-built marine chronometers until the advent of electronic timing. Earnshaw perfected the pivoted detent escapement and invented a temperature-compensated balance. Every mechanical watch still made today has variants of these inventions. Pierre Le Roy  While in England, the studies and experiments were going on, France was also busy solving the problem of determining the longitude. It was a competition that went hand in hand with the political and commercial rivalry between the two countries. Two watchmakers were particularly dedicated to this work, Pierre Le Roy and Fedinand Berthoud. There was bad blood between Berthoud and Le Roy, and on numerous occasions, Berthoud 32

 From British Museum catalogue.

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denounced Le Roy’s attempts to appropriate his inventions and vice versa. Berthoud published a pamphlet in which he contested Le Roy’s attempts to appropriate his invention and discussed the results of experiments (Berthoud, 1773). Pierre Le Roy (1717–1785) was the son of the most prestigious French watchmaker of the time, Julien (1686–1759), one of whose pendulums was used by de Maupertuis for studies of the effects of temperature changes on clocks (Maupertuis, 1739, pp. 195–196). Pierre Le Roy devoted himself in particular to perfecting two problems: the compensation of temperature variations and the “detachment” escapement. It is difficult to recognize the priority of the inventions, both in England and in France, the watchmakers who excelled were all working on these problems. However, it is important to remember that Le Roy independently adopted a compensating balance and a detachment escapement similar to those of Arnold and Earnshaw. The earliest marine chronometers made by Pierre Le Roy were always mounted on a gyroscopic mount; one example, completed in 1761, is currently in the Musée d’Arts et Mestieres in Paris (Fig. 6.21 left). Ferdinand Berthoud (1727, 1807) was born in Neuchâtel in Switzerland but worked mainly in Paris. He was trained as a watchmaker in Julien Le Roy’s workshop and worked alongside his son Pierre Le Roy. Berthoud had theoretical skills and an education that led Denis Diderot (1713–1784) to entrust him with writing a section of the Encyclopédie. Berthoud wrote very important works on watchmaking theory and history.

Fig. 6.21  Left: Marine Chronometer by Pierre Le Roy. Right: Ferdinand Berthoud Marine chronometer

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Berthoud traveled to London several times to meet Harrison and understand the working principles of his chronometers in vain. Anyway, Berthoud met Mudge, who naively described to him in detail the H4 chronometer. Berthoud completed his first marine chronometer in 1763, which was not particularly accurate. However, he built countless variants until he arrived at a solution very similar to that of Arnold and Earnshaw (Fig. 6.21 right). It was mounted on a suspension to swing freely aboard the ship. Concluding this brief exposition of the construction of marine chronometers, it can be observed that all the best watchmakers concentrated on three problems: (1) how to overcome the difficulties of maintaining pendulum or balance oscillations, (2) how to guarantee isochronism when temperature varies and (3) how to minimize irregularities due to escapement friction. And the solutions were similar: instead of the pendulum, the balance was adopted, and temperature compensation was solved with a variant of the gridiron, namely the bimetallic, deformable balance with variable geometry and the reduction of escapement friction with detachment mechanisms. Their focus reveals a deep understanding of the physical and mathematical principles that underpin the construction of high-quality mechanisms; we are dealing with figures that it is certainly limiting to describe as simply mechanics or craftsmen.

7 The First Astronomical Machine: Antikythera

The history of astronomical machines does not begin with the clockmakers of the Enlightenment: it dates back to the Hellenistic period, as evidenced by Archimedes‘sphaera, mentioned by Cicero, and even more so by the Antikythera machine. From the Renaissance onwards, the use of machines to represent the cosmos in its continuous mutation became more and more frequent. There are few written records of ancient complex mechanisms. The physician Ctesibius of Alexandria invented many mechanical instruments and automata moved by water and steam, war machines, and, according to Pliny the Elder (d. +79), even a hydraulic organ (Ctesibius’ works are known to us indirectly, mainly through Marcus Vitruvius Pollio (−80, −15)). The first machine I analyze is that of Antikythera, an extremely complex machine that has come down to us not through written records but as a material object. I want to devote a section to this machine because of its fascination and the confirmation it offers about the astronomical and technical knowledge of the Hellenistic period, which here takes the form of an extraordinary artifact.1

 An extended version of this chapter has been published in (Marini, 2022a).

1

Supplementary Information The online version contains supplementary material available at https:// doi.org/10.1007/978-­3-­031-­30944-­1_7. The videos can be accessed individually by clicking the DOI link in the accompanying figure caption or by scanning this link with the SN More Media App.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_7

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Fig. 7.1  Left: Antikythera Island. Right: A short video of the main fragments of the mechanism: fragment A in the center, B on the left and C on the right. (▶ https://doi. org/10.1007/000-­a96)

The Discovery In 1900 a group of sponge gatherers found the remains of an ancient shipwreck near the island of Antikythera (Fig. 7.1 left).2 The recovery of the findings took about 2 years. Initially, the archaeologists’ attention was drawn mainly to fragments and statues of bronze and marble, ceramics, jewelry and coins. The discovery caused quite a sensation because among the findings were numerous valuable objects, including a philosopher’s head and a life-size bronze statue of an ephebe, an absolute rarity. Today, all these findings are preserved in the National Museum of Archaeology in Athens. Among the artifacts was a bunch of corroded bronze that had been left in storage. But those fragments turned out to be even more interesting than the other findings, after archaeologist and museum director Valerio Stais noticed that the fragments, totaling 35, revealed the presence of gears with inscriptions (Fig.  7.1 right links to a short video of the major fragments see also Fig. 7.2 right). I will not dwell on the adventurous tale of the recovery of what was found and the initial research to understand the nature of this artifact.3 I will therefore consider only a few aspects: the dating, functions and structure of the machine with its gear trains, the interpretative hypotheses for the ­cosmological model of lunar motion, the question of whether it is only an astronomical calendar or is also a planetarium, and the question of the  Also called Cerigotto, just to the north is the island of Kithera (Cerigo). It is located in a very dangerous sea area. On its coast sank the ship that was carrying to England the Parthenon marbles stolen by lord Elgin. A novelistic description of this affair was told by Marta Boneschi (2022). 3  See Jones (2017), Marchant (2009), Kaltsas et al. (2013). 2

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Fig. 7.2  Left: Reconstruction in 2010 by Dionysios Kriaris. Right: side view of the major fragments. National Archeological Museum, Athens

construction techniques available at the time, which allowed for the creation of a very sophisticated and highly accurate instrument.

The Dating Initially, it was assumed that the object dated to the second century of the current era. Historians recalled a description by Pausanias of the existence of artistic works (bronzes and marbles) of great value in Argos at that time. A more thorough analysis of the objects proved that the bronzes were dated between the -third and -second centuries, fragments of marble statues between the -second and -first centuries, and glass, jewelry and ceramics from the -first century. Some objects, including parts of human skeletons, were found in later expeditions in 1976, and in the period 2012–2016. The coins found were dated, the most recent, around the year -70. This led scholars to assume that the shipwreck occurred in the period between the years -70 and -60. It remains to determine the dating of the machine itself. In 1955 Derek de Solla Price (1959), historian of science and technology, began a systematic study and, in particular, analyzed, with the help of a paleographer, the inscriptions on the fragments that led him to hypothesize that it was an astronomical machine. Other paleographers analyzed a series of inscriptions on other fragments belonging to the back layer, identifying a parapegma. An even more recent paleographic study has led to the conclusion that the first of the 235 cells of the Saros cycle dial (see below) correspond to

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a probable eclipse of 27 April -205. Traces of several calendars have been recognized among the fragments, which we will examine later on. Paul Iverson and Alexander Jones (2019) have also pointed out that the calendar identified in the machine is typical of the Corinthian family of calendars. In fact, each Greek city adopted its own calendar. The different calendars were synchronized by considering the Panhellenic games, which were held every 4 years in different places and in honor of different deities. Tony Freeth identifies, through the inscriptions on the Saros dial, an epoch that would have begun in -205, shortly after the death of Archimedes (Freeth, 2014). The initial configuration of the calendar does not necessarily indicate the date of construction. It has been put forward hypothesis that the machine was made in Rhodes since the indication of the Olympic calendar relates to the Panhellenic games that took place in Rhodes. The paleographic study, noting small differences between the alphabetical signs, also revealed the possibility that at least two different hands worked on the machine. All clues and studies agree that the construction and the sinking should have taken place in the first quarter of the -first century. A possible constructor of this machine could have been Posidonius of Rhodes (-135-51), who was living in Rhodes. Hipparchus stayed and worked on that same island, making Rhodes a center of astronomical studies and the place where the machine may have been built. On the other hand, Corinth could also be the place of origin, considering the use of the Corinthian calendar.

Structure and Functions of the Mechanism To get a reference of the general shape of the machine, let us look at the reconstruction based on the most recent hypothesis made in 2010 by Dionysios Kriaris (Fig. 7.2 left). The size is roughly that of a dictionary. There are two panels: inside the front one there are engravings that look like instructions for interpreting the data displayed by the machine. On the back panel is the inscription of the parapegma. This fragment, for example, was interpreted (Paizis, 2016, p. 6): Π Διδυμοι άρχονται ἑπιτ[ελλειν] [P The rise of Gemini begins] Ρ Ἀεηοτξ ἐπιτέλλει ἑσπε[ριος] [R The eagle rises at twilight] Σ Ἀρκτοῦρος δύνει ἑ[ῶιομς] [S Arcturus rises at dawn]

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The front face of the machine contains the annual calendar with a hand that locates the position of the Sun in relation to the zodiac and the Moon with its phase. The rear face contains two vertically arranged spiral dials; the upper one indicates the month according to the cycle of Meton. Inside this dial there are two smaller ones: on the left for the cycle of Callippus and on the right for the year relative to the Panhellenic games. The lower dial, also a spiral, contains the indication of possible lunar or solar eclipses according to the Saros cycle and a smaller dial to indicates the exeligmos (Fig. 7.3). A knob

Fig. 7.3  The back dials. Reconstruction by Karakalos, National Archeological Museum, Athens

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on the side of the machine rotates a crown wheel and operates the entire mechanism. The first research was carried out by Derek de Solla Price in the 70s. He identified a series of gears and proposed the hypothesis that it was a kind of astronomical calculator that could indicate the synodic month, the lunar year, and the positions of the Sun and the Moon. In 1970, the first radiographs were taken by C. Karakalos of the ‘Demokritos’ Scientific Research Center, and published by de Solla Price in 1974 (de Solla Price, 1974). In the 1980s and 1990s, further studies were carried out by Alan G. Bromley (1990) and M.T. Wrigh (2002), with new low-resolution tomographic X-rays. Bromley’s advanced expertise in horological technology led him to refine de Solla Price’s hypothesis. He postulated that the machine’s front dial not only indicated the sidereal month and year but also provided information on upcoming eclipses. This updated theory represented a significant contribution to the field of archeoastronomy. The rear dial would show the synodic month and lunar year with the draconic month and the cycles of Meton and Callippus. Tom Malzbender, a researcher at Hewlett Packard, contributed to these studies by applying an image processing technique, called PTM (Polynomial Texture Mapping), which greatly improved the legibility of the engravings (Malzbender et al., 2001). In 2005, the Antikythera Mechanism Research Project (AMRP) was established, which, continuing the collaboration with the Archaeological Museum and Thessaloniki University, involved Cardiff University and scientists from other universities.4 AMRP’s first effort was high-resolution X-ray analysis to more accurately identify the internal structure of the main fragment. We reproduce (Fig. 7.4) some high-resolution images of the scan of fragment A, taken from the volumetric A6 JPEG stack dataset, consisting of 544 images with a total of 653 MB, representing the reconstructed volume with 25 projections, with a distance between sections of 100 microns and a spatial resolution of 46 microns.5 The maximum thickness of the fragment is 54.4 mm. Other details became evident of the b2 wheel with 64 teeth and the l1 and c1 wheels both with 38 teeth.6

 http://www.antikythera-mechanism.gr/project/overview  X-ray Data courtesy X-Tek Systems (now owned by Nikon Metrology), 2005. Equipment loaned by X-Tek Systems Ltd. (now owned by Nikon Metrology) was used to collect the data. https://dataverse. harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/UCXZWU Accessed May 2021. 6  For the numbering of wheels see Fig. 7.6 The scheme proposed by the AMRP research group. The wheels are identified with numbered letters and the number of teeth. 4 5

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Fig. 7.4  Left. The wheels of the pin&slot mechanism. Right Rendering with the MIP method by the author. Pin&slot wheels are highlighted

Fig. 7.5  Wheel cross structure b1. Left: front side. Right: rear side. Rendering with MIP method by the author

Another picture (Fig. 7.5 right) shows the two 32-tooth wheels b3 and e1, and finally, the fragments of the 50-tooth wheels e6 and k1, the main components of the lunar epicyclic motion that we will examine in more detail later on, and implemented with a mechanism called pin&slot (Fig. 7.4 right). An examination of the X-ray images reveals the complexity of the mechanism. The first images consist of the individual ‘slices’ of the 3D reconstruction of the X-ray scan. In the central wheel, one can see the presence of a large hole that allows the passage of a large hollow axle suitable for accommodating

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Fig. 7.6  A short animation of the rendering of fragment A. The pin&slot mechanism. (▶ https://doi.org/10.1007/000-­a95)

other concentric hollow axles for the front dial hands (Fig. 7.5 left). A brief animation is in Fig. 7.6. These wheels can be better seen when the CT is depicted using the rendering method that simulates the conventional radiography effect, called MIP (Maximum Intensity Projection). Less dense materials are rendered in darker gray, and denser material in light gray, with a transparency effect. In this way, one can better recognize the in-depth arrangement of the gears and the presence of other components such as the support plates of the larger wheels. The results of the AMRP project group’s work were presented at a conference in Athens in 2006.7 They led researchers to share the belief that the machine was an astronomical calendar that could predict the following phenomena and events –– the synodic lunar cycle –– the lunar cycle of Meton and Callippus –– the Saros cycle and the Exeligmos –– the sidereal month and year –– The four-year cycle of the Panhellenic games: Olympic Games at Olympia, the Pythian Games at Delphi, the Nemean Games at Nemea and the Isthmian Games at Corinth. Probably were also included the Naia games at Dodona and Halieia at Rhodes.  Decoding the Antikythera Mechanism, Intern. Conf., 2006, Athens.

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Tony Freeth et al. (2008) published a very accurate reconstruction of the dials of the machine. The front face of the machine (fragment C) has a circular graduated scale with 365 subdivisions corresponding to the length of the Egyptian year and an outer circle with 360 subdivisions on which it is indicated the Zodiac. The hypothesis is that two hands, for the Sun and the Moon, mounted on concentric hollow axes (like the hour and minute hands in modern clocks) indicate the day and the age of the Moon, which is moved by epicyclic gears. The Moon’s hand rotates on itself to highlight the phase (the synodic month). The back face (see Fig. 7.3) shows two spiral scales of five turns, the upper one has 235 subdivisions (47 subdivisions for each turn) corresponding to the Metonic cycle of 19 years; within this spiral there are two smaller dials both subdivided into four parts to indicate, on the left, the cycle of Callippus and on the right the Panhellenic games. The lower spiral of the back face is composed of four turns divided into 223 parts, corresponding to the duration of a Saros cycle of about 18 years. In the subdivisions, eclipses of the Sun or Moon are indicated with the symbols H (Ηλιος, the Sun) or Σ (Σεληνη, the Moon), respectively. Within this spiral, there is a small dial divided into three parts showing the Exeligmos, i.e., one of the three cycles of Saros for a total period of 3*18 = 54 complete cycles for the exact repetition of eclipses. The Gear Trains  The general structure of the gear train (see Fig. 7.7) starts with wheel b1, which is connected to the Sun hand (missing), so one complete rotation of this wheel corresponds to one calendar year. It can be moved directly by the dial via the crown wheel a1 of 48 teeth. In order to understand all the other gears, it is simpler to start with the dials and their hands, and for

Fig. 7.7  The scheme proposed by the AMRP research group. The wheels are identified with numbered letters and the number of teeth

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each of them calculate the reduction ratios and verify that each gear realizes the desired periods. The hands on the dials are integral with the axles of the wheels: (a) n1 for the Metonic cycle (b) q1 for the Callippus cycle (c) p2 for the Olympian cycle (d) g1 for the Saros cycle (e) i1 for the Exeligmos (a) The gear train for the Metonic cycle therefore runs from b2 to n1 and consists of the wheels b2, l1, l2, m1, m2, n1. The reduction factor of this 38 96 53 gear train is:    3, 8 which multiplied by the five turns of the 64 53 15 spiral gives 19, the number of years in the Metonic cycle. (b) For the Callippus cycle, the gear train continues from wheel n3, p1, p2, 60 60 q1. The reduction factor is:   20 which multiplied by 3.8 (the 15 12 factor of the Metonic cycle) gives 76, the number of years of the Callippus cycle. (c) The cycle of the Olympics, or rather the Panhellenic games, is realized with the gear pair n2, o1, which has a reduction ratio (to be multiplied by the factor of the Meton cycle) that is exactly the interval 60 between games:  3.8  4. 57 (d) The Saros cycle is realized by the train: b2, l1, l2, m1, m3, e3, e4, f1, f2, 38 96 223 53 54 g1. The reduction factor is:      4.507 which 54 53 27 188 30 multiplied by 365.25 (number of days in a tropical year) results in 1646.18 days, which corresponds to one of the four turns of the spiral for a total of 6584.72 days, an excellent approximation to the Saros cycle. (e) For the Exeligmos, the wheels g2, h1, h2, i1, realize a reduction 60 60   12, that divided by the four turns along the spiral gives three 20 15 Saros cycles, thus equal to the cycle of the Exeligmos. All these periods are summarized in Table A.7 in Appendix A. The front dial (fragment C) indicates the day and the position of the Sun in the zodiac. The wheels above b1 are missing, so a question remains: what other information could have been indicated? Probably some wheels were

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used for the variability of the lunar revolution and the precession of the lunar apsides - also known as the lunar or Hipparchus anomaly.

 he Missing Parts: The Motion of the Moon, T the Sun, and the Planetarium An analysis of the high-resolution X-ray sections shows the presence of a pin on wheel k1 running in a slot along a diameter of wheel k2 (see Fig. 7.8). In addition, k2 would be mounted on an eccentric axis relative to that of k1, so that the pin slides back and forth along the slot of k2 (pin-and-slot coupling), varying the speed of rotation. This wheel, in turn, meshes with e6. So, the train would proceed with e5 driving k1, k1 drives k2 by varying its radius, and thus its speed of rotation and k2 drives e6, which is finally on the same axis as e1. This mechanism implements an epicyclic motion, which corresponds to the model of the precession of moon apsides. Sidereal Month and Year and Lunar Anomaly  The complete sequence of gears comprising the pin&slot epicycle is: b2, c1, c2, d1, d2, e2, e5, k1, k2, e6, e1, b2, b0, q1. The wheels e5, k1, k2, e6, e1 and b2, being only transmis64 48 127 sion, do not contribute, so the reduction factor is:    13.3684 In 38 24 32 other words, one complete rotation of the wheel b1 corresponds to one year, during which there are 13.3685 sidereal months (365.25/27.3217). The front dial shows the calendar and the position of the Sun along the zodiac; the hand with the sphere of the Moon makes 13.3682 rotations, close to a sidereal month. Wheel k1 is fixed on e3, e5 has a hollow axis fixed on e3 so it transmits the rotation of e6, but at the same time generates the period of the lunar anomaly

Fig. 7.8  Scheme of the pin&slot system. Detail of Fig. 7.6

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on e3. In fact, the ratio 88.85 is realized, to a good approximation, by the 38 96 223 wheel sequence:    88.826. 64 53 27 The synodic month would be shown by the rotation of the Moon’s own sphere with a period of 29.53 days, by means of a crown coupling between b0 and m1 with a number of teeth 27 and 25, that are in the ratio 1.08 close to the ratio 29.53/27.3217 = 1.08083.

The Planetarium The hypothesis that the machine also included a planetarium has been considered by all who have been able to examine it. Wright, in 2002, proposed a reconstruction (Fig. 7.9 left) (Wright, 2002). Freeth (Freeth 2012) observes that next to the large wheel b1 with four arms, there are pillars that most likely supported another plate or acted as axes for other wheels; furthermore, the same wheel is responsible for the movement of the Sun. Therefore, the motion of the inner planets, Mercury and Venus, could have been controlled by the large wheel with 224 teeth (see also Diolatzis 2019a).

Construction Technology and Materials We can ask whether metalworking techniques during the second or first century BCE were so sophisticated as to allow the construction of gears with the required precision, and whether knowledge of mechanics was sufficiently

Fig. 7.9  Left: planetarium hypothesis According to M.T. Wright (2002). Right: Astronomical watch by Hublot Company, mimicking Antikythera mechanism

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advanced to conceive of a gear train capable of calculating periods of tens of years. Hellenistic Gear Mechanics Pappus, in Book VIII of the Collectiones (Pappus, 1878), Vitruvius (−first century) and Heron (+first century) described in their works a great variety of machines mainly used for the movement of large weights or water and often based on gears. Unfortunately, few writings, translated into Arabic, survive of Heron, but tradition holds that he was a mathematician with engineering skills. Among the works attributed to Vitruvius there is the odometer. It measures distances along roads. A series of gears transmit the movement of a wheel that counts the number of rotations which, multiplied by the ­circumference of the wheel, gives the distance traveled. A machine, described by Pappus, is the barulcus (Commandino, Liber VIII 1640, p. 461) which he attributed to Heron, a kind of winch for lifting large weights by means of a gear train that greatly reduces the required torque (Fig. 7.10). Construction Methods and Materials Voulgaris et al. (2018) identified the materials used: bronze, copper and tin. They suggest that it is a unique copy, almost a prototype, which was adapted

Fig. 7.10  Barulcus of Pappus

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Fig. 7.11  Use of a vertical drill. Bas-relief of the tomb of Petosiris. Tuna el Gebel, Egypt

probably to solve problems of regularity of movement by introducing washers to better align some gears. The authors also analyzed the chemical composition of the materials. In particular, they found washers between the e5-e6 and k1-k2 wheels (the pin&slot mechanism), probably necessary to minimize friction. This is a problem that arises in mechanisms where the wheels and pinions mesh in different planes: alignment errors of a few tenths of millimeters can block or make the movement irregular.8 The authors using chemical analysis conclude that most of the machine’s components are made in bronze. The washers could have been made of an alloy of tin and lead, and the various pins identified could have been made of iron. Voulgaris et al., in a different paper, examined construction technologies (Voulgaris et al., 2019). The authors analyzed in particular the process of drilling wood and metal using a tool operated by a bow. They point out that even in the Homeric poems, there are references to tools for the construction of objects. Moreover, the use of lathe is reported in several ancient works. The use of a vertical lathe is depicted in a bas-relief from the tomb of Petosiris, an Egyptian priest from the -third century (Fig. 7.11). The teeth of the Antikythera mechanism have a triangular shape, whereas modern technology requires that gear teeth have a cycloidal shape that allows friction-free contact. A major problem in wheels with triangular teeth gears concerns the regularity of the mechanism. Poor meshing or insufficient force to overcome internal resistance can jam the machine.  In Chap. 15, we will address this topic in more detail.

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Manos Roumeliotis points out that the reconstructions carried out to date present problems (Roumeliotis, 2018). All hypotheses and reconstructions assume that the driving gear a1 of 48 teeth, driven by a crank, engages with the Sun wheel b1 of 223/224 teeth. The authors calculate the driving torque required to transmit motion to the various gear trains from a torque of 10 N mm (Newtons per millimeter) equal to 0.102 kg cm. The assumption that the driving wheel is b1 driven by a1 leads to maximum torque values in the order of 6900 N mm, equal to 70.36 kg cm, a rather high but still acceptable value.9 A particularly interesting study is also that of Efstathiou et al. (2012) who, in order to design other mechanical reconstructions, examine from a strictly mechanical point of view the geometry and size of the teeth to identify the primitive diameter that guarantees the best meshing without the risk of the gear getting stuck. Understanding these parameters allowed the authors to hypothesize a construction procedure similar to that used in contemporary technology. However, other questions remain unanswered regarding the skill and construction technique involved in working the bronze to obtain hollow shafts. In the reconstruction hypotheses, all scholars agree that on the front side, there should have been a dial with an indication of the annual calendar and the position of the Sun in relation to the zodiac, and an indication of the lunar month. In order to construct such a mechanism, one would have to construct an axis that moves the date hand within which the axis of the Sun hand rotates, and within the latter, the third axis with the Moon hand rotates. Even more complex would be the construction of a succession of hollow axes that rotate one inside the other and move the planet hands. In the hypothesis of a complete planetarium, there are five tubes to move the planets, plus the Sun, the Moon, and the hour, thus arriving at a total of eight hollow axes. With present-day technology, we can make tubes with even very small diameters by means of extrusion techniques or by bending and welding sheet metal. It is possible that in the -first century, to obtain the thin tube without welding, casting was used. With the lost-wax technique, it is possible to prepare a cylinder of any diameter in the wax that will become the tube cavity during casting. An alternative solution could have been the technique of rolling a thin bronze foil. Finally, another possibility is that the tube would be constructed by drilling with hardened iron bits, working on the lathe to ensure perfect centering: the most difficult method.  To get an idea of these torque values, I recall that a stepper motor in use in small robotic systems has a maximum torque of 6.8 kg cm. 9

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Fig. 7.12  Left: Parts of the reconstruction in 1930 by Ioannis Theophanidis. Right: reconstruction in 2009 by M.T. Wright

Copies and Simulations One of the earliest reconstructions is by Ioannis Theophanidis in 1930 (Fig. 7.12 left). Many others followed as research continued. Derek de Solla Price and M.T. Wright also made the reconstruction of the first interpretation of the machine structure (Fig. 7.12 right). Diomidis Spinellis (Spinellis 2008) created an application for educational purposes with the ETOYS software,10 based on the ARMP reconstruction, which describes the mechanism in great detail. The interaction allows to examine separately the gear trains of the different functions. Another multimedia simulation has been realized by I.S. Diolatzis and G. Pavlogeorgatos (2019b); it includes a 3D reconstruction that facilitates the exploration of the gear system and includes the hypothesis of reconstructing solar and lunar motion with the visualization of eclipses. Diolatzis et al. (Diolatzis, 2012) make systematic use of computational simulation techniques for the geometry of epicyclic motions, in particular with the GeoGebra software.11 They test the hypothesis of epicyclic motion for Venus and Mercury, in order to validate the hypothesis of the presence of a planetarium in the machine. A wristwatch inspired by the Antikythera machine was built by the Hublot company (Fig. 7.9 right), which contributed to funding submarine archaeological missions.

 https://www.spinellis.gr/sw/ameso/. Accessed May 2021.  https://www.geogebra.org. Accessed May 2021.

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Until a few years ago, even the most qualified scholars believed that the only instruments for measuring the time that the ancients had developed were sundials or clepsydrae. Today, we can say that the Antikythera machine shatters this image of the ancients’ knowledge and know-how. It is an absolute unicum: a mechanism created more than 2000 years ago that makes it possible to calculate calendars and predict eclipses. The Antikythera machine, however, is not only this. It is also a model of the cosmos: a computational model. It demonstrates the vision of the motion of celestial bodies that great thinkers like Hipparchus had formulated without having the technical instruments that we have today. It is a vision aware of the irregularity of the motion of the Moon and the Sun, of the different durations of months and years. With Ptolemy, these irregularities were described geometrically with epicycles because circular motion was considered the perfect motion for an unchanging sky. In the study of the Antikythera machine there are traces of the epicyclic motions of the Sun and Moon, which were certainly already known at the time of its construction. Perhaps the Antikythera machine also included the inner planets, Mercury and Venus, and the outer planets, Mars, Jupiter and Saturn; the relevant gears are destroyed, but the reconstruction hypotheses seem plausible. We can therefore imagine with a fair degree of probability that the Antikythera machine was a mechanical planetarium and an analog calculator, the results of which are displayed by moving the hands to different dials. We do not know the relationship that existed between the builder(s) of Antikythera and the great mathematicians and astronomers of the Hellenistic period, but the complexity and richness of the machine reveal that they had access to the most advanced astronomical knowledge of the time, and the solutions for the arrangement of the gears are those of skilled craftsmen.

8 Astronomical Machines and Clocks from the Arab Times to the Renaissance

The period following the destruction of the library of Alexandria led to centuries during which mathematical and astronomical studies came to a halt in the West; however, as we have seen, from the seventh century onwards, the legacy of science and technology was partly preserved by Arabs.

Astronomical Instruments of the Caliphate Arab scholars designed and built time-measuring water-powered machines. Only drawings of these machines survive, some very imaginative and equipped with mechanisms for moving automata. Mario G. Losano (1990) observes that mechanics also continued its history like the other Alexandrian sciences, by virtue of Arab civilization. Mūsā ibn-Shākir, an astronomer from Baghdad in the eleventh century, whose sons, collectively called Banu Musa (sons of Musa), collected and created libraries with Greek writings and perfected machines by inventing new ones, especially those driven by air pressure. They invented a conical valve so famous that Leonardo da Vinci drew a sketch of it that became known throughout Europe. Banu Musa published ingenious mechanical designs in the Book of Mechanisms with a special interest in fluids. These studies were later taken up by al-Jazarī (1136–1206), a mathematician, astronomer, inventor and engineer born in Diyarbakir, Turkey, who wrote an important treatise ‘Book of the Knowledge of Ingenious Mechanisms’ in which he describes some mechanisms still in use today, such as cams,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_8

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crankshafts and different types of gears. In another work, he describes a mechanical clock known as the Castle Clock (Fig. 8.1 left), of which Salim Al-Hassani made a computer reconstruction (Al-Hassani, 2008). This clock was a clepsydra: water flows from a cylinder on which floats a weight connected by ropes to pulleys that set the entire mechanism in motion. The flow regulator maintains the correct pressure in the water reservoir and can be modified to adapt to the different lengths of the days over the course of the year. At the output of this regulator, there is a system of valves that open at the sixth, ninth and twelfth hours and activate the movement of a series of chime figures, also letting out a stream of air for the flute player. Other connections activate the sun, the zodiac and the moon, which appears through a crescent-­ shaped window. We note that this machine, of which only the detailed description survives, has no exact mechanism for counting the hours and requires daily adjustment. It is more of an automaton than a real astronomical clock. Al-Jazari designed several automata, with the intention of creating objects of entertainment and prestige rather than scientific machines. These include

Fig. 8.1  Left: Castel clock by Al-Jazari. From the Book of Knowledge. Right: engraving of the elephant clock

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the Elephant Clock (Fig. 8.1 right) and the Candle Clock, an example of combustion clock. In Fez are the remains of a large water clock completed in 1357 on commission from the Sultan Al-Mutawakkil Abu ‘Inan. The clock has 13 small windows and as many supports with bronze bowls into which a metal ball fell every hour, opening a small window and creating a chime. The mechanism was based on a hydraulic principle that dragged the balls and sequentially filled the water bowls that opened the windows. (Fig. 8.2 left). This mechanism is described in the 1203 treatise written by Ridwan bin al Saati entitled On the construction of clocks and their use, where one can see a drawing very similar to the clock in Fez (Fig. 8.2 right). The water clock of Jayrun built by Muhammad al-Sa’ati was located in Damascus at the exit of the Umayyad Mosque in the twelfth century. These works arouse admiration for their imaginative decorations and technical solutions, but their greatest limitation is evident: they are mechanisms without automatisms, which must be continuously adjusted and synchronized with astronomical events. This is probably a consequence of the fact that Arab engineers did not fully understand the mathematical modeling of the Cosmos constructed by Greek thought. Many documents describe armillary spheres designed by the astronomer of the Caliphate, but no one survives (Savage-Smith & Belloli, 1985). Dia’ al-­ Din Muhammad, to show the position of the major stars, built numerous celestial globes as the one exemplified in Fig. 8.3 center. On the other hand, many Arab astrolabes still exist (Fig. 8.3 left), as well as instruments to help in finding the direction of Mecca for prayer (Fig. 8.3 right).

Fig. 8.2  Left: Water Clock of Fez (Morocco) after restoration. Right: Damascus water clock Drawing of the thirteenth-century

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Fig. 8.3  Left: Astrolabe fifteenth century; Center: Celestial globe 1070 Hegira/1659, Dia’ al-Din Muhammad; Right: Instrument for finding the direction of Mecca, 1200 Hegira. Museum of Islamic Art, Cairo

Renaissance In continental Europe, and particularly in Italy, began to appear mechanisms for measuring time moved by springs or weights. These were clocks with a foliot and verge escapement, built to determine the hours of prayer in Convents, or for public use in civil towers or church bell towers. The oldest tower clock is recorded in Italy and is the clock in the bell tower next to the church of S. Andrea in Chioggia (Fig. 8.4 left). A study conducted by Marisa Addomine et al. (2006) found official documents showing that this clock was built before 1386. In the Communal archives of Chioggia, it is recorded the expenses incurred to pay the temprator:1 Make available to the municipal treasurers the sum to pay the costs of the clock and to keep it in order and in working order February 26th 1386

It is not an astronomical clock, and its pendulum oscillator with anchor escapement has been modified many years after its construction. In England, Richard of Wallingford (1292–1336) wrote a Tractatus Horologii Astronomicii and started the construction of a clock, probably completed about 30 years after his death,2 which also displayed only the hours. In Strasbourg Cathedral,  Archivio Comunale di Chioggia: Archivio Antico “Dino Renier”, Statuti e Consigli 1381-1390, volume XXV - c. 33 r - anno 1386, 26 febbraio. 2  Marisa Addomine, personal communication. 1

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Fig. 8.4  Left: The clock of the S. Andrea Tower in Chioggia. Right: Astronomical Clock of Strasbourg

the first clock was built in 1354. Known astronomical clocks on civic towers or Churches date one century later. The Astronomical Clock of Strasbourg Cathedral It was built between 1352 and 1354 (Oestmann, 2020). Defossez makes the hypothesis that it was constructed by Heinrich Halder (Defossez, 1946, p. 52), a Basel-born watchmaker. It was a rather simple clock, with a calendar, an astrolabe and an automaton, which at each hour activated the Three Wise Men, who knelt in front of the Virgin Mary and set off a chime.3 The brothers Isaac and Josué Habrecht, sons of the watchmaker Joachim Habrecht (c. 1500–1567), built a new machine between 1572 and 1574. The calculus for this machine was particularly complex, and the Habrecht brothers relied on the Strasbourg mathematician and astronomer Chretien Herlin (d. 1562) and, upon his death, his pupil Conrad Dasypodius (1532–1600). Jost Bürgi also collaborated on this work during his apprenticeship. The clock included complicated automata that were animated in given hours.  Another description of the history and characteristics of this work is (Ludwigh, 2022).

3

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The present clock (Fig. 8.4 right) is the result of the restoration carried out by Jean Baptiste Schwiligué (1786–1856) starting in 1838 and completed in 1842. The case and automatons have remained original, while the mechanism was extensively rebuilt. The clock is very tall, about 17 m., the dial shows official and local time, and at each quarter several mechanical figures are set in motion. At 12:30 p.m., the betrayal of Judas and the Passion of Jesus is re-enacted with the procession of the Apostles and the triple crowing of the rooster. The time is symbolically represented by other mythological figures. The painted celestial globe at the bottom shows around 5000 stars; it completes one rotation in a day and represents a Ptolemaic depiction of celestial motions, indicating the motion of the Moon and possible eclipses. Günther Oestmann observes that the astrolabe and the globe are a representation of a geocentric theory, and argues that Dasypdius was not in favor of the Copernican heliocentric theory: the presence of Copernicus’ portrait among the clock’s decorations could therefore be a simple acknowledgment of his qualities as an astronomer and mathematician, but probably not of his cosmological views (Oestmann, 2020, pp. 95–111). Oestmann describes also the gear system and identifies the gear trains for: quarters and hours, a carillon with 10 bells and 6 drums that rang every 4 h, and finally the mechanism of the train of time. The astrolabe shows the positions of the planets, the tympanum is engraved for the latitude of Strasbourg 48.5°. The astrolabe has a mechanism for the planets motion; Venus and Mercury are linked to the motion of the sun indicator, while the moon, and the external planets have their own gears. Oestmann (Ibidem pp. 169–177) has computed the periods from the gear trains and compared to the known ones calculating the error (see Table A.9). Giovanni Dondi Giovanni Dondi (1330–1388) known as Dondi dell’Orologio was born in Chioggia to the physician Jacopo Dondi (1293–1359). Jacopo Dondi is credited with the construction in 1344 of an astronomical clock in Padua. Giovanni Dondi studied medicine, later taught at the University of Padua, and wrote about astronomy, philosophy and logic. The Astrarium he built, completed in 1364 in Padua, is described in detail in the Tractatus Astrarii. This machine computes the motion of the Moon and planets, movable feasts, and the calendar on the basis of Claudius Ptolemy’s astronomical model, by means of a complex system of gears (about 107). The Astrarium was purchased in 1381 by Gian Galeazzo Visconti and kept in the library of the Castello Sforzesco in Pavia; it was then forgotten, as no

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one was able to repair the damage that time had inflicted on this complex machine. The Astrarium by Giovanni Dondi’s can be considered a model of planet motion, where the movement of the planets, Moon and Sun can be observed on its dials. Twelve manuscripts exist of the Tractatus Astrarii which report three different versions of the machine, probably due to transcriptions by different hands. Three different descriptions have been identified, denoted as Version A, B or C (Dresti & Mosello, 2016). The reconstruction by Luigi Pippa can be seen in Fig. 8.5. Dresti and Mosello list 15 reconstruction of this machine. Bedini and Maddison (1966) published an important study of this machine. The machine has seven dials, five of which are for the planets, whose position is shown according to the epicycles model, one for the Sun and one for the Moon whose motion takes into account the Hipparchus anomaly. Two opposite dials at the bottom indicate the hours of the day, and the possible eclipses. The days of the year with Saints and religious festivities are displayed in the middle ring. One of the features of this machine is the device to solve the problem of the motion of Mercury and the Moon, to simulates the epicyclic and the retrograde motion of the planets. The planets and the Moon are positioned along a slot of an arm that rotates around a center. This is a pin-slot pattern similar to that found in the Antikythera machine. In this machine Giovanni Dondi creates also an entirely original solution: motion is transmitted to the planet, which rotates around the epicycle, by a pair of oval gears controlled by the deferent circle.

Fig. 8.5  Left: Dondi’s Astrarium. Reconstruction by Luigi Pippa, 1963. Right: The astrolabe dial

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The manuscript preserved in Venice, a Version C, describe an Astrarium with three special additions: an accessory for determining the dates of the movable festivities, a device for indicating the time of sunrise and sunset in the city of Chioggia, and a mechanism for reducing the friction of the annual wheel. This device is also present in the reconstructions preserved at the Smithsonian Institution. Dresti and Mosello constructed a model based on manuscript version A in 2010, and in 2016 undertook the construction of a model from manuscript version C, which includes the calculation of moving festivities. The main movable festivity is Easter, which is closely linked to the lunar cycle. A distinction must be made between the Christian and the Jewish Passover. The former celebrates the flight from Egypt, the name Pesach (‫)פסח‬ means passage (Passover) and denotes the crossing of the Red Sea, the date is the 14th of Nissan (‫)ניסו‬. Christian Easter celebrates the Crucifixion of Christ, which according to tradition takes place during the celebration of Passover. During the Council of Nicaea in 325 it was decided to separate the two festivities. Before the Council, Easter was celebrated on different dates among Christian communities: some celebrated it on the same date as the Jewish Pesach, others on the Sunday immediately following, and still others on the Sunday after the spring equinox. The Council resolved that Easter should be celebrated on the same day in all Christian communities, but the exact definition of the day was not specified, except that it should be on the first Sunday following the spring equinox, which at the time fell on 21 March. In order to determine the date according to the Julian calendar (at Dondi’s time this was the calendar in use), it is necessary a complex calculus that I describe in Appendix A.3. The mechanical solution devised by Dondi consisted in identifying the vernal equinox, then finding the full Moon and then finding the first Sunday after it. Apart from the date of Easter other movable festivities related to Easter are Septuagesima (9 weeks before Easter), Lent (6 weeks before Easter), Rogation (5 weeks after Easter) and Pentecost (7 weeks after Easter). In their work, Dresdi and Mosello describe the mechanism for calculating the movable festivities devised by Dondi, which is based on three chain mechanisms that realize the flow of years, the lunar cycle, and a third chain for indiction, unrelated to the Easter cycle but used to draw up contracts and pay taxes. The solar cycle chain consists of 28 links, as many as the years of the solar cycle—at the end of which the days of the week recur on the same

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dates—each link is engraved with the number of the cycle and the corresponding Littera Dominicalis (dominical letter). The chain of the lunar cycle has 19 links—the Metonic cycle—in each link are engraved: the golden number, the epact, the five movable festivities with their dates, including Easter. The chain of indiction has 15 links engraved with a progressive number. The chain of movable festivities is moved by a double-wheel train that generates the reduction ratios from the annual calendar wheel to the two five-­ faced sprockets of the chain. Dondi’s Astrarium is an extremely complex machine in relation to the period in which it was made. It reveals a deep knowledge of astronomy and technical and machining skills worthy of the best watchmakers of the nineteenth century, knowledge and techniques reminiscent of those of the Antikythera machine. Models of the Homocentric Spheres: Theoricae Orbium  During the fifteenth and sixteenth centuries, Ptolemy’s planetary model was still widely accepted by scholars. Georg von Peuerbach described a new model of homocentric spheres to construct the mechanisms of planetary motions. Johann Schöner in the Opera mathematica of 1551 (Gingerich, 1977, p. 39) described the construction of a theorica orbium in three-dimensions. Gerolamo Della Volpaia (c. 1530–1613), a maker of astronomical instrument, made various machines to illustrate the motion of Mercury and other planets. Some of his works are preserved in the Museo Galileo in Florence, others in the Biblioteca Laurenziana in Rome. The model of the Moon motion in Fig. 8.6 left has three orbis: the deferent, the epicycle and the fixed stars. On the horizontal ring is engraved the zodiac with the division into degrees. The moon is identified by a small sphere on the vertical disc. The model of Mercury’s motion (Fig. 8.6 right) highlights the difficulty of making a mechanical model according to the theory of homocentric spheres. Indeed, the various spheres (represented with disks) are pivoted eccentrically from one layer to the other in order to represent the precession of Mercury’s apsides. The typical model is a three-orb-system, and is clearly described in relation to the Sun as we have seen in section “Humanism and the Rebirth of the Studies” in Chap. 3. Elly Dekker (2004) considers the works of Gerolamo della Volpaia and other authors as the materialization of Peuerbach’s planetary theory (Peuerbach 1562). Of the 13 theoricae analyzed, the author identifies two groups. The

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Fig. 8.6  Left: Theorica Lunae. Right: Theorica Mercurii, 1557. Musei Vaticani, Roma

first group has the orb system attached to an outer sphere fixed on a stand with a horizontal ring. This outer sphere represents the primum mobile, and the model looks like an armillary sphere. The second group includes instruments in which the orb-system is mounted in an outer sphere which is oriented as the ecliptic, that presumably represents the fixed stars. In this model, the primum mobile and the horizon ring, typical of armillary spheres, are absent. Otto Gingerich proposed to call these models orbarium (Gingerich 1977). Peuerbach’s orbis are thought of as shells and are of two types: of uniform or nonuniform thickness, so that the planet can move eccentrically. It was impossible to make a mechanism that showed the movement of four concentric spheres, each revolving around its own axis. These instruments were limited, at best, to being moved by hand, thus depicting the Earth, the Sun and a single planet or the Moon. All attempts to understand how motion transfers to the entire Cosmos were unsuccessful.

 phères Mouvant and Globes of Sixteenth Century. S Eberhard Baldewein and Oronce Finé In Europe, especially in Germany and France, instruments similar to the astronomical clock, often called moving spheres, became widespread. They were equipped with several dials to mark the time, the calendar, the moon and

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often also the position of the planets, solving the problem of the first anomaly with an original mechanism. These clocks were works of sublime quality and luxury items highly prized by cardinals and dukes. It is known that Emperor Charles V had a machine made by Geminello Torriani (c. 1500–1585), automata builder and engineer born in Cremona. The Cardinal Charles de Lorrain had a machine built by Oronce Finé, Philipp Imsser built in 1555 a clock for the Elector Ottheinrich of Palatinate. The Landgrave Wilhelm IV of Hessen and Elector August I of Saxony had both a machine built by Eberhard Baldewein. Gessner et al. (2020) recall that of these machines, only four survive. Philipp Imsser’s astronomical clock is preserved at the Technisches Museum für Industrie und Gewerbe in Wien. Two astronomical clocks were built by Eberhard Baldewein between 1563 and 1568, one at the Astronomisch-­ Physikalisches Kabinett in Kassel (Baldewein I) and the second at the Mathematisch-Physikalischer Salon in Dresden (Baldewein II). The last machine, by the mathematician Oronce Finé, is preserved at the Biblioteque de Saint Génevieve in Paris. Calling these machines moving spheres takes us back to the time of Archimedes and Cicero, that in the Latin of the time the study of these machines was called sphaeropeia. It is therefore in this meaning that we must interpret the expression moving spheres. It is a way to represent the Cosmos, that persists in the Renaissance, imitating the motion of planets. Eberhard Baldewein (1525–1593) was a master builder servicing the Landgrave Ludwig IV of Kassel since 1567. From 1569 until 1579 he continued with the successor Wilhelm IV, a renowned astronomer, for whom he built from 1563 to 1568 two astronomical clock, one shown in Fig. 8.7. The Landgrave Wilhelm collaborated with Baldewein by calculating the gears. Baldewein was assisted for the construction by the goldsmith Hermann Diepel. The machine has separate dials, two on each side. Six dials are for the planets (Mercury, Venus, Mars, Jupiter, Saturn and the Moon), a dial has an astrolabe and the last is a calendar. The power of the clock is provided by a spring. The Sun appears in a celestial globe on the top of the machine, showing the Sun’s position in the sky of the fixed stars. The characteristics of this clock are the same as the one preserved in Kassel. Anomaly is achieved in this machine without eccentric or epicyclic gears, but with an original solution (Gessner et al., 2020, p.219). There is a single central gear with 360 teeth that drives the entire mechanism. The teeth of this wheel are unequally spaced. The separation of the teeth averages 1° but is less near the apogee and greater at the perigee. This large gear is driven by a worm

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Fig. 8.7  Baldewein astronomical clock. Right: detail of Mars’ dial

screw that advances a single tooth with each revolution. The precision of this mechanism is quite high, considering that the teeth were sawn and filed by hand. In Fig. 8.7 right, a detail of the Venus dial shows enameled zodiac symbols; the hand has a slit that mimics the planet’s retrograde motion. It is a large machine, height 1180 mm and width 625 mm. Oronce Finé  (a.k.a. Fineo) (1494—1555) was born in Briançon in 1494 and studied mathematics in Paris at the Collège de Navarre, which rivaled the Sorbonne. In 1517 François I of Valois (1494–1547), king of France, ended a long dispute over the power to appoint bishops by signing a Concordat with Pope Leo X in Bologna. He thus obtained the right to control dioceses and archdioceses. Finé opposed this and was imprisoned until 1524. Later, however, he was appointed by the same sovereign to the chair of mathematics at the Collège Royal (1532). He wrote mathematical and astronomical treatises, La Théorie des Planetes published in 1557, and was also the inventor and maker of mathematical instruments and sundials, described in Quadrans Astrolabicus (Paris, 1527) and De Solaribus Horologijs et Quadrantibus (Paris, 1531). He was also a cartographer. In 1552 publishes in French Le Sphere du Monde (Finé 1525), a work of cosmology, dedicated to King of France Henry II, in which he

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describes the nature of the Universe from the composition of matter to the constitution of the earth and sky. In this work, he summarizes the entire cosmological vision of his era. Finé describes the system of astronomical circles that lead to the construction of the armillary spheres, summarizes the arrangement of the planets according to the geocentric model, describes the projections of the sphere on the plane for cartography, and examines the measurement of time. The mobile sphere of Oronce Finé (Fig. 8.8) is described by Berthoud (Berthoud, 1802b, pp. 180–188) and by Dubois (1849, pp. 155–160), and both reproduce the same description of the work that they claim was

Fig. 8.8  Left: Center: Sphère mouvant by Oronce Finè. Right: detail of the internal mechanism, made of steel

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made for the Cardinal of Lorraine and presented by Finé in 1553, a description preserved in the same Biblioteque. The dials display the same information as the Baldewein clocks. The description reports the periods of all moving objects. The clock is 3 feet (98 cm.) tall, on a basement of the same height. Dubois (Ibidem p. 161) expresses admiration for this work but at the same time writes that he would have preferred Finé to make it even more perfect, as he would not have calculated the motion of the planets accurately enough. In particular, for the motion of the sun, he notes that it has a period of 365 days; instead of 365.25, 3  48  180  48  146 5  73 Finé’s gears are in fact:   365 . Curiously, 12  36  48  24 1 Dubois compares Finé’s gears with that of Dom Jacques Alexandre (1653–1734) who lived 200 years later, two epochs that are not at all comparable. Henry King and Millburn (1978, pp. 67–68) are doubtful about the attribution of this work to Finé, who probably intervened to modify an instrument made in Germany. King also notes discrepancies between the mean sidereal period of the moon from the Tabulae Alphonsinae, 27.315 and the periods calculated by counting the teeth of the wheels 28.1409. A similar discrepancy is the period of Jupiter: in Tabulae Alphonsinae the period is 4330.7 while from the teeth count it is 4368.0. In a communication to the Académie des Inscription et Belles-Lettres, Emmanuel Poulle (1974) reviews the gears and the periods, that differ from those by Dubois and Berthoud. Moreover, Poulle computed the periods of the epicycles. We may remember that these machines are still based on the Ptolemy’s theory, and the epicycle shall display the retrograde motion of the planet. The scheme of the epicycle implemented by the clockmaker of Finè sphere is illustrated in Fig. 8.9. The periods of this machine are summarized in Table A.8. Celestial Globes  The purpose of celestial globes is to show the positions of stars on a sphere. They are fairly simple instruments that can be rotated to observe the positions at different times of the stars and constellations. Some globes have an internal mechanism that moves the globe and the Sun along the ecliptic line, allowing its position relative to the zodiac to be observed at the time displayed above the Earth’s axis. Some scholars also include in this category of machines the armillary spheres of the Renaissance and late Renaissance period, which quickly became

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H A m

Î O T

G

E

D

A

T

D

E

Fig. 8.9  Left.: Deferent and epicycle. Right: axis for setting the center of rotation at the equant. T is the position of the Earth, D is the center of rotation of the deferent, E is the center of the equant. The planet P rotatates around O, the center of the epicycle

both instruments of scientific popularization and an astronomer’s working tool. The greatest difficulty in making celestial globes is the precise placement of the stars. The stars represented were usually those with a high brightness, up to magnitude 3. The symbolic representation of the main constellation was engraved or enameled on metal globes, painted on cardboard ones. The Doppelpokal4 preserved in Wien was built by Abraham Gessner (1552–1613) around 1590. It is a terrestrial globe made of two goblets that forms the sphere, and an armilla on top (Fig. 8.10 left). The globe was modeled on a map by the cosmographer Gerard Mercator, created in 1569 and adapted in 1587 with the Earth’ axis oriented vertically. Gessner was a goldsmith who made a similar globe preserved in Zurich, Landesmuseum, that has a celestial globe on top instead of the armilla. These were decorative objects, which often became cups to hold wine and spirits called Kredenzpokal,5 and were appreciated by the aristocracy of the time. It was a visual artform to disseminate the new knowledge about the shape of the Earth, the discovery of new continents, the complexity of the sky. Gerhard Emmoser (1556–1584) constructed a celestial globe in 1579 for Rudolf II (Fig. 8.10 center), has an internal mechanism for the movement of the celestial sphere and the Sun runs along the division of the two

 Double goblet.  Credenza cup.

4 5

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Fig. 8.10  Left: Terrestrial globe with armillary sphere ‘Doppelpokal’. Center: Celestial Globe by Gerhard Emmoser. Right: Eberhard Baldewein, Rechenglobus

hemispheres that corresponds to the ecliptic; it is a masterpiece of jewelry as well as mechanics that is only 27.3 cm high. Much larger is the Rechenglobus6 by Baldewein (Fig. 8.10 right), 115 cm high and with a diameter of 89 cm. The use of the Rechenglobus is completely different. On its surface, stars are precisely marked, and their celestial coordinates can be read. By rotating the globe until a star is placed on the meridian, its altitude on the meridian arc and azimuth along the horizon ring can be read. Wilhelm IV, around 1560, commissioned Baldewein to make a simple copper sphere devoid of any ornaments. In 1577, the copper globe stood in the room behind the observatory, as can be seen in a painting of Landgrave Wilhelm IV and his wife Sabina von Württemberg in front of astronomical measuring instruments (Fig. 4.1 left). The engraver Lennep was commissioned by the Landgrave Carl of Hesse-Kassel (1677–1730) to decorate it by inscribing the constellations. The image of a Dutch globe by Willem Blaeu served as a model. The double calendar on the horizon ring, with Julian and Gregorian indications, also dates to this time. Other mechanical globus built Georg Roll (1546–1592) and Johannes Reinhold (1550–1592) in 1584 is preserved in the collection of Rudlph II, at the Kunstihistorisches Museum, Wien.

 Can be translated as “Calculus globus”.

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J ost Bürgi: Mathematician, Mechanic, Clock Maker, Astronomer Jost Bürgi (1552–1631) is one of the most interesting figures in the history of technology, astronomy and mathematics.7 Bürgi was not only a watchmaker of the highest level, he was also an astronomer who conducted observations and noted measurements. He was also a mathematician, and his mathematical skill was essential in designing his clocks and astronomical machines. Bürgi’s contribution to astronomy has been acknowledged by giving its name to a group of Moon craters. The Life  Jost Bürgi was born in Lichtensteig in the canton of St. Gallen. Very little is known of his youth. Bürgi collaborated with Isaak Habrecht (1544–1620) to the construction of the second astronomical clock in Strasbourg, under the direction of the mathematician Conrad Dasypodius. In 1579, he was called to the court of the Landgrave of Hesse-Kassel Wilhelm IV to assist him in astronomical observations. The observatory was equipped with various clocks and had many collaborators included Willebrord Snell, who invented the triangulation technique to determine the distance between points on the ground, and Christoph Rothmann. The Landgrave greatly appreciated the collaboration with Bürgi as we read in a letter signed by the Landgrave to Tycho Brahe on 14 April 1586 in which he writes: (von Bertele 1953) «Recently, Orion’s longitude has been observed with great precision with the Bürgi clock indicating minutes and seconds, with such accuracy that between two culminations there is a deviation of less than 1 minute.» The quality of the Bürgi clocks in use in Kassel is still confirmed by Rothmann who in 1585 writes: As far as our clocks are concerned, we have three for observation. But it would be tedious and take too much space to describe them. However, one thing must be mentioned, and that is that the first of these clocks has three hands, which indicate not only the hours and minutes but also the individual seconds. The duration of a second is not very short, but it roughly equals the duration of the shortest note of a moderately slow melody. The escapement does not operate in the usual way. On the contrary, it is implemented using a newly invented method so that each movement corresponds to one second.  The sources about mathematical and astronomical discoveries and watchmaking inventions are (Defossez, 1946, pp. 57–70; von Bertele, 1953; Brusa, 2005). 7

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Tycho Brahe, during his stay in Kassel, discussed with Rothmann problems of astronomy, methods and instruments of observation. After his return to Uraniborg, he remained in constant contact with the Landgrave and Rothmann, who went to Uraniborg in 1590, never to return to Kassel. The observations carried out in Kassel by the Landgrave, led to the compilation of a catalog of around 400 fixed stars. The method used was to measure the altitude of the star at transit, and for this work it was necessary to have very precise time-measuring instruments, which Bürgi was able to build. Bürgi traveled to Praha in 1592 to the court of Rudolf II, the Landgrave’s grandson, bringing with him a celestial globe that he had made and for which Rudolf II paid the sum of 300 thalers. The same year the Landgrave Wilhelm died and his successor Mauritius renewed the contract with Bürgi. In 1604, the Landgrave Mauritius again sent his watchmaker to Praha with a sextant (Fig. 6.6 right) and a triangular instrument invented by Bürgi himself. Bürgi was appointed Watchmaker to the Imperial Chamber in Praha and took on several apprentices who later contributed to the spread of instruments making and watchmaking technology in German-speaking countries. At the imperial court, Bürgi gained more and more prestige and became a renowned watchmaker, but he continued to conduct astronomical observations, as Kepler himself attests when speaking of the Serpent’s New Star: «…At vespers I watched it with Justus Byrgius, the Emperor’s mechanic », reconfirming the experience in a letter to his Master Michael Mästlin in 1605: «… and here is also Justus Byrgius, mechanic of the Landgrave, methodical and conscientious observer of the fixed stars…». Bürgi was granted a noble title, as recorded in the Imperial Seal Register kept at the Court Library in Vienna, dated February 3, 1611: «Nobilitatio und Wappen für Jobsten Burgi, Kammeruhrmacher».8 He continued to work as imperial watchmaker for Rudolf II’s successors, Emperors Matthias and Ferdinand II, until his death in Prague in 1631. Bürgi the astronomer  began his first observations as early as 1584, in 1590 he replaced Rothmann who was ill; after the Landgrave’s death in 1592 he took over as director of the observatory, as reported by Defossez (1946). Willebrord Snell in 1618 under the title ‘Coeli et siderum in eo errantium observationes Hessiacae’[Hessian observation of sky and bodies moving in it], published the observations done in Hessen by Wilhelm IV and his ­collaborators. Bürgi’s observations made from 1590 to 1597 are from page 15  Noble title and coat of arms for Jost Bürgi, Chamber Watchmaker.

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to 69. We read: «Observations of the planets made by means of the sextant at the Kassel Observatory by Jost Bürgi, instrument-maker of the illustrious princes Wilhelm and Mauritius, Landgraves of Hesse». Further another note from pages 109 to 113 regarding observations of the Sun: «Meridian observations of the Sun by Jost Bürgi of the Kassel Observatory, by which we determine the apogee». Bürgi’s mathematical skills  and knowledge can be seen in his mastery of geometry; in fact, he invented two instruments: the triangular instrument and the proportional compass, an instrument invented also by Galileo. The triangular instrument works on the same principle of operation as the modern rangefinder. Jost Bürgi invented the logarithms independently of Napier.9 Kepler used them to compile the Tabulae Rudolphinae. He writes in the introduction: … thanks to Jost Burgi's logarithms predating Napier s publication by several years. And indeed, this man, a temporizer and guardian of his secrets, who keeps them in embryo, and does not consign them to public use ….

Bürgi’s logarithms are most likely in a base that approximates the Euler constant e. His half-brother Benjamin Bremer published Bürgis’ tables of logarithms in 1620. Bürgi was not versed in scientific writing and also his geometry studies were published by Bremer in 1618. Bürgi’s mathematical studies were closely linked to the demands of observation. His experience as a watchmaker had made him realize that the time spent building an instrument was crucial to the accuracy of the results of observations. He applied the same principle by building himself the mathematical and mechanical instruments that allowed him to perform astronomical calculations as accurately as possible. Bürgi as a Clockmaker  contributed to the perfection of clocks. Some claim that Bürgi invented the pendulum clock, since some of his instruments are equipped with this oscillator. The issue is controversial: some may be pendulums added during later restorations; on the other hand, if scientists such as Kepler and Tycho Brahe had noticed Bürgi’s use of the pendulum, they would have immediately recognized its superior qualities to the foliot. However, Bürgi introduced an original oscillator that enabled him to improve the precision of his clocks: the cross-beat oscillator analyzed by Chandler and Vincent (1981) (see Fig. A.10). Its aspect may be misleading and can be confused with a double pendulum.  Napier invented the logarithms in 1614, publishing a book Mirifici logarithmorum canonis description [Description of the wonderful table of logarithm] Defossez (1946, p. 59). 9

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Another ingenious invention of Bürgi’s is the winding mechanism called a remontoire or constant-force device, consisting of two springs, one that supplies the driving force directly to the escapement and a second that provides the movement of the gear train. In this way, the force acting on the escapement is decoupled from the gear train, thus eliminating irregularities due to motion faults, external disturbances, and reduced torque when the train spring is discharged. The escapement spring is periodically recharged by the train spring, and although it does not generate thrust on the escapement during this interval, the escapement continues to advance smoothly. Bürgi introduced the independent seconds hand with a dedicated dial. He also invented a mechanism for the equation of time. All these innovations appear to be closely aimed at the goal of providing astronomical observation with a reliable and accurate instrument for measuring time. This reveals a mind capable of blending scientific knowledge, mechanical inventiveness and manual skill. Bürgi’s astronomical clocks and machines that exist today can be found in several museums: a celestial globe at the Musée des Arts et Métieres in Paris; two celestial globes and a square horizontal table clock at the Hessisches Landesmuseum in Kassel; an astronomical table clock contained in rock crystal; and an astronomical clock called the Wiener Planetenuhr at the Kunsthistorisches Museum in Vienna, built in 1604/05. The rock Crystal clock, built in 1622/23, has a celestial globe on the top (Fig.  8.11), probably engraved by Ottavio Miseroni (1567–1624), an outstanding artist at the court of Rudolph II. It has a cross-beat escapement and a remontoire, its dials indicate hour, minutes and seconds, and the strikes the

Fig. 8.11  Bürgi, Rock crystal clock. Note the small globus in the right figure. Kunstakammer, Kunsthistorisches Museum Wien

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hours and quarters. The seconds are shown in a separate dial for easy observation; it is probably the oldest surviving clock with a second hand. The clock mechanism moves the globe and the Sun along the zodiac. The clock indicates the month, moon age and phase. The clock is small, 185 mm high and the diameter of the globe is 56 mm. The Wiener Planetenuhr (Fig. 8.12), described by Staudacher (2018), on top has a heliocentric planetary system with the Sun in center. In the two sides there are two small silver statues of Hermes (Mercury) and Apollo made by Jan Vermeyen. On the other two sides are the dials. In total, this clock can provide information on 10 different astronomic functions, including calendar, eclipse, precession of lunar nodes, position of the planets and the Sun relative to the zodiac. The celestial globe preserved at the Schweizerisches Landesmuseum in Zurich is of exceptional beauty, made by Jost Bürgi for Emperor Rudolf II of Habsburg in 1594 (Fig. 8.13). In this work, Bürgi summarizes the

Fig. 8.12 Wiener Museum Wien

Planetenuhr

by

Bürgi.

Kunstakammer,

Kunsthistorisches

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Fig. 8.13  Celestial globe by Bürgi. National Museum, Zürich

astronomical knowledge of the time in relation to the passage of the seasons and the viewing of the starry sky.10 This is a work of great artistic value; the materials are brass, iron, and steel, with gilding and silvering, and the mounts show a Mannerist style. There are chiseled wise men’s heads and feline heads  A detailed analysis of this work was published by L. Oechslin (2000).

10

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and paws, a style that anticipates the Baroque despite the fact that this is a fully Renaissance work.11 The sphere is in the shape of a small globe-a diameter of 14.2 cm and a total height of 15.5 cm-on which is engraved the map of the sky with the positions of the stars according to the Tabulae Rudolphinae, with the constellations and their symbols, the graduated equator, the parallels of the tropics, and the meridians. Along the cut separating the two halves of the globe moves the small disk of the sun. A dial placed on the earth’s axis indicates the hours and minutes with the 12 digits repeated for a 24-h cycle. The celestial globe rotates around this axis, which is tilted 50°, the latitude of Prague. A ring placed at the height of the horizon indicates the days and months of the year, the dates and major holidays. Orthogonally to it a semicircle indicates the Meridian, so that the transit of the engraved stars can be observed. The globe rotates thanks to a mechanism inside it, which also drives the clock. The driving force of the movement is a spring, stabilized by a chain and a conoid. The date displayed along the ring is moved by a gear train inside the ring, which is driven by the movement through a gearbox. Ludwig Oechslin (2000) adeptly portrays the seamless blend of technique and culture evident in Bürgi’s work, which is characterized by meticulous attention to both function and aesthetics. As a purpose-built scientific instrument, it not only serves as a valuable tool for researchers but also holds significant prestige for its owner and serves as a means of scholarly publication for scientists. Furthermore, the instrument doubles as an excellent teaching aid for educators. However, it is limited to a representation of the sky and the motion of the Sun as it can be observed from Earth, which therefore does not allow us to grasp the relative positions in space of the planets. The stars engraved on the Globus are the result of astronomical observations made by Bürgi and Rothmann in Kassel. Bürgi, before the Globus in Zurich, constructed other celestial globes, two of which between 1580 and 1582 are now in Kassel and Weimar, and two others between 1582 and 1587 are now in Paris and Kassel.

11

 The chiseling is probably due to Anton Eisenhoit (1554–1603) from Warburg (Brusa, 2005, p. 9).

9 Toward Planetary Machines

In the previous chapters, we examined planetary machines that depicted the cosmos using dials with indicators to identify the positions of planets based on celestial coordinates or the zodiac. The Antikythera machine even had a planetarium to display the relative positions of planets. These machines were clever inventions designed to aid the study of the heavens, popularize astronomical knowledge, and serve as luxury items. However, due to mechanical imprecision and a limited understanding of celestial motions, these machines had limitations in solving computations of the motion of stars. One notable exception was Galileo’s jovilabium, which was a true calculating instrument based on a simple model of Jupiter’s satellites’ apparent motion. Astrolabes had been appearing on astronomical clocks in towers and cathedrals for at least two centuries. However, instead of serving as tools for understanding the configuration of the sky, they were more astrological instruments for specialists. A new need was emerging, which Kepler was the first to recognize: the need to form a mental representation of the solar system. This could only arise from simplifying the models that had been followed for centuries. Kepler was the first to realize that by freeing the planets from the encumbrance of the homocentric spheres, one could imagine the planets moving freely through space.

Supplementary Information The online version contains supplementary material available at https:// doi.org/10.1007/978-­3-­031-­30944-­1_9. The videos can be accessed individually by clicking the DOI link in the accompanying figure caption or by scanning this link with the SN More Media App.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_9

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He not only took flight to represent the motion of Mars but also paved the way for a new era of astronomical understanding.1

 aving the Way to the New Cosmos: Kepler’s P Planetary Machine Kepler realized that to describe his findings it was necessary to represent them with material objects; descriptions and drawings were inadequate. In the Epitome Astronomiae Copernicanae, he emphasizes the usefulness of the ‘Automata Coelestia’ as a teaching aid, supporting calculation and reasoning about celestial motions, even when overcast skies make direct observation impossible.2 In 1596 he described the first concept to explain what he had written in Mysterium cosmographicum: the order of the planets around the sun was governed by the geometric perfection of the Platonic solids. He had already drawn a cup representing the sphere of fixed stars within which the spheres of the planets were bounded by the regular solids (Fig. 4.5 right). So, he decided to honor the duke Friederich I von Württemberg (1557–1608) with a luxury gift: a silver cup. The First Concept—1596  On 17 February 1596, Kepler wrote a letter to the Duke in which he proposed to make a Kredenzpokal to show the arrangement of the planets and the dimensions of their orbits. He also attaches a drawing (see Fig. 9.1) that clarifies the use of the cup. Each sphere will contain liquors, beer and wine. In the drawing, the different liquors are assigned to each sphere: Sun “Aquavite”, Mercury “Brantwein [distilled wine or brandy]”, Venus “Meth” [?], Mars “Vermouth”, Jupiter “White wine”, Saturn “Other wine or beer”. It will be a luxury and decorative object to be kept in the Duke’s Wunderkammer. This object was never built because the outer sphere would have been too large to have a reasonably sized inner sphere.  To give a clearer idea of the Kredenzpokal I have made a three-dimensional virtual reconstruction of Kepler’s concept. To make this reconstruction I had to model the series of the platonic solids. Kepler in Epitome Astronomiae Copernicanae3 notes the relative dimensions of the solids by expressing them as ratios of the radii of the  A more extended description of Kepler’s project has been presented to XL Symposium of Scientific Instruments Commission, Athens September, 2022 https://doi.org/10.48550/arXiv.2208.04815 (Marini, 2022b). 2  Kepler (1618, p. 26). 3  Ibidem, p. 273 (p. 468 of the manuscript). 1

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Fig. 9.1  Drawing of the Kredenzpokal by Kepler in Letter 17th February 1596 to the Duke. Along the border are seven spigots to spill liquor from each sphere

different spheres that are to circumscribe the Platonic solids. I used the same ratios normalized to a unitary size of the edge of the octahedron. In Fig. 9.2 the three-dimensional reconstruction of Kredenzpokal with the spheres and platonic solids. The pokal can be opened and liquor poured into the hemispheres and poured from the spigots. The Second Concept: A Planetarium—1598  After giving up the project of the Kredenzpokal Kepler had the new idea to make a moving object, a mechanical representation of the moving planets. In fact, Kepler formed the idea that the sky was a machine.4 In a mechanical model of the Cosmos each celestial  … Caelestem machinam dicam non esse instar divinj animalis, sed instar horologij … [The sky machine is not like a divine animal, but it is similar to a clock]. Letter to Herwart von Hohenburg, February 10th 1605 KGW XV p. 146. 4

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Fig. 9.2  Top: Rendering of the reconstruction with the cup open (left) end closed (right), with the tubes to spill the liquors. Bottom: Details of the Platonic Solids

body had to move independently, under the action of a single motor but with clearly separated gears.  The project of the astronomical machine is described with details by Kepler in the letter to Mästlin (Kepler, 1945, p.162–186) January sixth 1598. He introduces the subject5: Moreover, to continue about this issue, I am writing to you for an advice about the whole construction. I was considering to build an instrument to demonstrate my invention and at the same time the primum and second mobiles. To get this already difficult result, I had to solve further difficulties. Firstly, all the orbis and the five bodies [planets] must appear raised and standing out. The orbis will be split in half not to obstruct each other. Regarding the bodies I had to devise a solution and arrange them so that their supports (σκελος sive

 Translation of Kepler’s letters by the author.

5

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latus6[leg]) are not broken or split. In order to achieve an artistic work, I also made sure that they do not intertwine and that they can be dismantled like clockwork. If this proves too difficult, all the orbis and bodies (from the smallest to the largest) will be fastened together with rivets. The entire opus is therefore divided in the middle into two parts, so that the flat circles representing the paths [viam planetariam] are inserted (in order to be able to examine my work) or removed in order to use them to show the Theory. Furthermore, to justify the name (Theoria primi et secundorum mobilium), I had to arrange the inside of the entire work transparently so that one can look through and see them dragged through the middle of the zodiac signs supported by four supports. In this way one can mark the position in degrees of the Sun or the Earth on the circle of the Earth (constructed in the same way and engraved regularly as the circle of Saturn) and make the machine march until it reaches the same sign in degrees in the middle of the starry globe.7

From this short description, we can figure out a general view of the machine: a globe divided into two parts, inside which the planets and orbs can move and show their motion. The machine, if possible, could be disassembled. It can be used to examine his theory by inserting circles that display the planetary paths, or to show his theory, by removing the circles. The globe will be engraved with the position of the stars; along the planets and zodiac circles are marked and engraved divisions into degrees. Kepler draws seven sketches to illustrate his project (Fig. 9.3). Construction work is progressing very slowly. During the construction, the goldsmith made a huge (immanem) error: the Saturn circle was divided into 396° instead of 360! He thinks he can rectify the error, but he understands that the goldsmith is not a professional engraver. He considers the possibility to give back the silver to the Duke, to apologize and ask to buy new silver, but the work already cost 1.000 florin. If manufactured in Augsburg, Nüremberg or Antorff (Anvers) the cost would be ten times less. Kepler considers to move the work to another town with more professional artisans, and hope to get the authorization of the Duke. In the meantime, he examines three options: to complete the opus with the defect, or to give up and return the silver, and finally, to stop the current work and start again from scratch, risking new problems. Kepler would choose the third option, but the silver given to the goldsmith has lost the fineness of its alloy, for multiple melting, and this could cause litigations with the goldsmith. He will need more silver and support other costs. He is

 In Greek in the original.  Caspar (1993, p. 173).

6 7

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Fig. 9.3  Kepler’s sketches

also very concerned about associating his name with a failure. On 12 April 1599, Kepler informed Mästlin of his decision to abandon the project. The Virtual Reconstruction  Let us recall the requirements as written by Kepler to Mästlin on June 1598. The main requirement is the ability to move the machine so that we can observe its evolution over the course of a year, with the planets sliding by one after another and being able to compare their speed. The machine should move in the same way we see bodies move in the sky, either following normal time (like a clock) or speeding up to run hundreds of years or more, so as to highlight the precession of the equinoxes. Kepler realized that this machine would be very expensive, not least because it would have to run for a thousand years! Another requirement is the representation of Venus and Mercury anomalies, latitude variations, and mean motion. He understands how difficult it is and curbs his enthusiasm by emphasizing that he will have to limit himself to imitating nature: ‘natura imitari quantum sufficit’. About the moon, he writes in the project description: «Nothing to say about the moon, since its motion is uniform» and he did not describe the implementation of moon’s motion. However, the work will include the meridian circle and the horizon, avoiding excessive ornamentation precisely in imitation of nature. Kepler wants to move each planet with only two wheels; the clock mechanism will have to be built by a master watchmaker, and the gears will be placed under a supporting base.  Kepler’s design is different from the celestial globes constructed at the time, and the planets will appear as standing out, rotating at their proper distance inside their orbs. While the general

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c­ oncept is clear, its detailed description is disorganized, and the sketches are not clean technical drawings; when the description becomes confusing, the author writes: «non possum pingere, tu intellige» [I can’t draw, you must understand]. On the other hand, sketch 5 in Fig. 9.3 gives an immediate understanding of the general setup. This sketch shows a series of concentric tubes and their connection to the driving wheels and to the curved arms that support the planets. The inscription “cubic line” confirms that he used his theory of Platonic solids to correctly define planetary distances and sphere sizes. The planet (in Fig. 9.3 sketch 5, it is Saturn) moves between two surfaces, in chap. XIV of Mysterium Cosmographicum Kepler writes: «Igitur ut ad principale propositum veniamus: notum est, vias planetarum esse eccentricas et proinde recepta physicis sententia, quod obtineant orbes tantam crassitiem quanta ad demonstrandas motuum varietate requiritur» (Kepler, 1596, p.  47) [Once accepted that the path of planets (vias planetarum) is eccentric it is also accepted the opinion of physicists that the thickness (crassitiem8) of the spheres must be sufficient to contain change of the planet’s motion (as required by the equant and deferent)]. With this observation, Kepler modifies the idea of his predecessors: physicists used to think that the space between the orbis was not empty, Kepler considers each orbis as two hulls within which the planets move. The space between the various orbis is empty and its extension is determined by the eccentricity, while the size of the external hull is determined by the size of the platonic solid. The exterior of the whole globe is made of silver with engraved stars. The zodiac is supported by four pillars with the four zodiac symbols. The orientation of the globus is not described, but we can assume that the motion axes are vertical and oriented toward the celestial pole; therefore, the ecliptic circle will be inclined 23.5°. The globe may also have armillary circles and at least the zodiac ring engraved with signs and degrees, possibly with subdivision to measure arc minutes. Periods and Gears  Kepler calculated the number of teeth of the gears that move the machine. He writes that he has many numbers and has chosen the ones that are best for his purpose. We do not have any information about the mathematical method used to calculate these numbers. We know that the first to use the method of continued fractions to approximate a rational number was Christian Huygens, one century later. In his letter, Kepler also calculates the accuracy of his approximation in terms of degrees after 10.000 years. I  We already discussed this issue in section “Johannes Kepler: A Journey in the Space” in Chap. 4.

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have computed the periods from the wheelwork, comparing it with present-­ day periods (Table A.10).  First of all, Kepler writes (see Fig. 9.3 sketch 2): « … the wheels A will be fixed to their axis and the wheels E will reverse rotate free [on their tubes] (in this way wheels in E have reverse rotation with respect to wheels A) ». This means that there is a single axis on which all the driving wheels (A) are fixed, while the driven wheels (E) are mounted on a series of concentric tubes (see Fig.  9.3 sketch 5). From a construction viewpoint, a decision is necessary: whether the wheels are assembled in two parallel directions or along an orthogonal direction. Sketch 2  in Fig.  9.3 and Kepler’s description suggest first an orthogonal assembly; in fact, he writes, « NB the wheels in A are not vertical, on the contrary they are aligned horizontally and perpendicular to CE ». Later, during the description of the motor, it is understood that the axes are parallel and vertical. He decides to have just two wheels for each planet, so that the final assembly will look like the double cone, as will be done later by Ole Rømer. To fix the measures for a feasible construction, I computed the diameter of the wheels so that they match a given distance between the two axes. So, I had to select a different module9 for each couple of wheels, keeping a constant distance between their centers. The result of my computation for the module is in Table 9.1. The teeth number computed by Kepler has many problems; for instance, a wheel with 191 teeth driving 46 teeth requires a relatively strong torque; the couple 60:60 needs a very high module number: 2.79. To cut the teeth is a difficult technical problem, in general, large teeth number are difficult to cut and to divide a circle into 395 parts, for instance, is not an easy geometrical problem. Table 9.1  Module of each couple of gears, center distance 167.50 mm Parameter of the wheels Primitive diameter Wheel work

Planet

Module

Driving

Driven

11:324 344:29 79:42 60:60 243:395 46:191

Saturn Jupiter Mars Earth Venus Mercury

1.00 0.90 2.76 2.79 0.53 1.41

11.00 26.10 116.00 167.40 127.58 269.31

324.00 309.60 218.20 167.40 207.38 64.86

 Elements of gear theory are summarized in section “Background” in Chap. 15.

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Kepler is not fully aware of the difficulties of this construction; he only notes the difficulties related to gears with a small radius. Anyway, Schickard and Kretzmeyer, who were technical experts, raised the issue with Mästlin and the Duke. The Spheres  The discussion on the weight of the spheres reveals the adherence to the theory of homocentric spheres, as already said, but it is unclear how the spheres should be positioned and how would be solved the problem of supporting their weight.  Kepler writes that The load is on E [Fig. 9.3 sketch 2 and 7] as much as you want, nevertheless it weights but separately, as we will see. And if the weight is mutually on each other, nevertheless they will be light once removed the bodies and the orbes.». Further « I have already written that the opus C, suspended in its center to the external part of the fixed stars globe, with, try to understand, balanced supports, maybe difficult to build. C is driven by E, that rotates (reversed) around B, preserving the fixed parts. Similarly, BE in A is bend so that while AE and EB match, A makes B rotate concentrically and E eccentrically I said that the hole for B shall not be in the zodiac pole, but at a distance large as the size of AE. This distance determines the size of the work, how to build the wheels as small as is possible.

Later he writes that the orbis (opus C in Fig. 9.3 sketch 1, 2 and 3) are solids and divided in half, not pierced. We can conclude that Kepler speaks of hemispheres, that allows observing the interior and the motion; therefore, the upper hemisphere could be transparent or removed. Moreover, the whole opus rotates around B, not around the zodiac pole. A question arises: how the planets are positioned and how do they move with respect to the spheres? First of all, Kepler writes: « Part C is driven so that the earth remains at the center of the orbis of fixed stars». This means that his idea was not to show the motion of the planets around the Sun, rather to show how the motion of the planets around the Sun as seen from the Earth. He continues: It is evident that if the motion in C follows a path HIK, it is necessary an opposite motion KIH that keeps it in the center [Fig. 9.3 sketch 4]. Because if the earth moves correctly also all other planets will move accordingly in the sky, and together with the earth.

This point is further analyzed in the last part about the primum mobile. Kepler writes:

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Now let’s come to C. It will be made with orbis divided in half and without holes. They will be fixed to the bodies so that no orbis will be movable,

and from this, we could conclude that the spheres will stay fixed and the planets will move inside them. But fixed to what bodies, «corpora»? Kepler draws sketch 6 and writes: « I presume that it will not be too difficult if the machine is extended beyond the half globe and the planets will pass, like under a yoke, under three or four small rods. » In my interpretation, this will be solved with the bars that sustain the spheres (Fig. 9.4 left) that are modified so that the planets can pass under them. For the variation from aphelion to perihelion, Kepler has a simple and clever solution: the external surface of the orbs is deformed into a convexity and an opposite concavity, while the arm of the planet will be flexible and elastic to slide on the surface and be pushed toward the center at the aphelion and pulled out at the perihelion. The deformation should be set coherently to the direction of the apsidal line. I have modeled this deformation only for the planet Saturn with oval shaped hemispheres. The middle hemisphere (Fig. 9.4 right) is the circumscribed sphere of the cube, the platonic solid of Saturn. The other two spheres delimit the thickness for the Saturn orbit and are eccentric to the central axis. Kepler describes how he would implement the eccentricity of the orbits and writes an obscure description: «Ich will mich drüber setzen und raitten (quandoquidem ipsum meum inventum, quod par est, in opere exprimi, calculum anteà mutat) ob es vil austrage, wan ich ubique 1/10 diametrj pro eccentricitate näme » [I want to fix and calculate (since my finding, which is convenient to express in the work, changes the previous calculation) whether it is wrong to take eccentricity everywhere as 1/10th of the diameter].

Fig. 9.4  Left: Bars to support the spheres. Right: Thickness and ovalization of Saturn orbis

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Now we have two solutions for the eccentricity of the orbits: first the ovalization of the spheres, second their off-center positioning of 1/10th of the diameter. In my reconstruction I have considered the first solution, since the notion of eccentricity used by Kepler is not mathematically correct (see Appendix A.5). To implement the change in latitude Kepler suggests to put under the hull of Saturn a surface inclined as its orbital plane. I could not figure out how to implement this solution, given the obstacle of hemispheres. The solution adopted in planetary machines constructed from the eighteenth-century had a tilted orbital surface, over which an axis slides varying the latitude of the planet; the description by Kepler looks similar to this solution. Primum Motum  This is the core of Kepler’s concept, drawn in Fig.  9.3 sketch 7. Kepler writes a long description for this device that he calls primum motum and later ground motion. In his view it reproduces the Copernican system as seen from the earth.  Let’s recall what Kepler writes, while observing the scheme in Fig. 9.5. … in point B let be an axis BA that rotates around the fixed-point A and keeping CD always parallel to itself. The point B on the bar CD describes a circle G with center A, while others have different centers, e.g., E describes a circle with center F». By this description the bar CD rotates parallel to itself so that points B and E describe circles centered in A and F.

The motion of this mechanism is described by Kepler: Thus, the circle around F led by E is equal to the orbis of the Earth and it is as if F were led along a circle around E. Now imagine 6 motor circles pivoted in B, and passing through a hole in the rod CD. In E there are 6 free wheels adapted to the above. Now let B rotate towards G, then E will rotate towards A and H will rotate on the axis towards G (NB: it would be better to reverse the whole motion from right to left.) Let there be a circle around E led by the adjacent circle in the opposite direction towards I and D. Whatever the motion of the lower wheel, such will be that of the upper planet which is represented in KF,10 and that is the earth. In this situation F is the Earth, E the position of the Sun. An intermediate wheel L reverses the motion so that the Sun in E moves towards A, H moves towards C, point I towards B and F towards D. The Earth is and remains in F. This is also the case for the other planets. They, because of the different pairs of wheels in B and in E, will move with different velocities. Thus, I 10

 Letter K does not exist, read as HF.

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Fig. 9.5  Kepler’s scheme, with the driver (the ring with internal teeth) the Sun, the Earth and Mars. Mars distance respects Kepler’s platonic solid theory. In yellow the Ab, EF wheel H and I after rotation of 120° clockwise

will move 6 planets, 12 wheels and 6 actors. The length of BA can also be adjusted at the extremes C or D, so that there is enough space for wheels of many teeth.

In this description there is a problem: the direction of rotation must be anti-clockwise, so that the rotation of I moves correctly the Sun around the Earth. But the presence of the intermediate wheel L forces H to rotate in the same direction, as well as the driving wheels fixed on the B axis, therefore the wheels pivoted on E will drive the planets clockwise. As a result, wheels I and H must rotate in the opposite direction, and wheel L must be removed. Moreover, the wheels that move the Earth in the concentric-­tubes assembly should also be removed; otherwise, the Earth would rotate anti-clockwise around E moving it from the desired position F. The solution I propose is shown in Fig. 9.6 right, where the wheel L and the driving and driven wheels of the Earth are removed, so that the Earth is subject only to the rotation around A generated by the ring wheel. To activate the

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Fig. 9.6  Left: Kepler’s description. Right: feasible implementation. Blue dashed circles indicate circles with centers A and F

rotation of wheel I, a second identical wheel is fixed to its same axis and below the bar CD; this wheel is moved by another wheel, that could be shaped as a ring with internal teeth. Now assume that the Earth is positioned along the F axis, assume moreover that the ring wheel is fixed in center F; therefore, its anti-clockwise rotation has the effect to move wheel I in the same direction and the bar CD parallel to itself, so that the whole machine, inside the sphere of the fixed star, rotates around the center F, and, as said by Kepler, the Earth remains on F, and the Sun rotates anti-clockwise around it. Moreover, wheel I transfers an inverted rotation to the wheel H and to the axis to which all the driving wheels of the planets are fixed, and each of them transfers the motion clockwise to the planets, excluding the Earth. But in this case the Earth is subject to two motions: one around F for the rotation of the ring wheel and one around its own axis for the rotation of its driven wheel. Therefore, the Earth will not keep the central position on F. To avoid this, Kepler considers the effect of the H, I and L wheel sequence that rotates clockwise the Earth, keeping it fixe in center F. This motion, produced by the rotation of the bar CD parallel to itself, makes the whole solar system to rotate around the center F, where Kepler keeps the Earth fixed. We get finally a configuration of the motion that emulates a Tychonic model, allowing to show the motion of the mobile stars as seen from the Earth. The fixed star sphere will be much larger to host this complex mechanism, and will be fixed and centered on the Earth, while the orbs rotate with the ring wheel and bar CD, and the planets’ motion will be the combination of this motion and their own rotation around their common axes centered on the Sun.

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Fig. 9.7  Final rendering of the hypothetical reconstruction with and without the platonic solids and Spheres. The red line in the right figure indicates the center of the opus, which coincides with the earth

In Fig. 9.7 the final rendering of the reconstruction. I have also inserted the colures, the zodiac and the equator to clarify Earth axis orientation. The globe is divided into two parts, the top one can be removed to observe the interior of the planetarium. In Fig. 9.8 we see how the planets rotate around the Sun while the Earth remains in the center of the whole system. The Way to the New Cosmos  When Kepler proposed the first concept, the Kredenzpokal, he did not have the ample vision that he gradually developed. The difficulties of the construction were related to practical problems with goldsmiths or mechanics experts rather than to the astronomical concept. With the second concept, Kepler poses a problem that is itself astronomical. It includes not only the invention of the Platonic solids to define the distances of the planets, but especially the idea of visualizing with a machine, for the first time, the apparent motion of the planets as caused by a single motor. Also this project was a failure, but Kepler traced a pathway for future mechanical inventions for the construction of planetary machines.  The project of Kepler is an anticipation of his major innovative approach to the study of the sky: to move outside of the Earth, flying with his imagination high in the space to observe the motion of the planets and to describe their paths, what he did while computing Mars retrograde motion and plotting the drawing of the panis quadragesimalis. An effective way to explain this concept can be done today with computational methods. An example is the simulation of Kepler’s planetary

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Fig. 9.8  The Earth remains fixed in the center of the system; the planets rotate around the Sun and altogether rotate around the Earth

Fig. 9.9  Computational simulation. Viewpoint: left heliocentric; right geocentric. In both cases the motion is heliocentric

configuration implemented by Paolo Maraner11 using Mathematica™ (Fig. 9.9), where the path of the Earth, the Sun and Mars are traced as observed from the Sun or from the Earth. Kepler’s theory of platonic solids to determine planetary distances gives an estimate of 1.259 astronomical units (AU) of Mars—Sun distance, while the correct value should be 1.52 AU. To evaluate the effect on retrograde motion as the planet’s distance from the Sun varies, the result of a 15-year simulation, computed with Matlab™, can be seen in Fig. 9.10. For the first time, the solar system is conceived as planets moving independently around the Sun, under the effect of a unique force, the primum mobile, that later Kepler identifies as a kind of magnetic force, as noted by Krafft 11

 Ptolemaic to Copernican World System Continuum http://demonstrations.wolfram.com/PtolemaicToCopernicanWorldSystemContinuum/ Wolfram Demonstrations Project, Published: July 13, 2017.

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Fig. 9.10  Difference of retrograde motion as Mars distance varies from its actual value (left) and the value computed from Platonic Solid theory (right). Note the different size of the loops

(2005) acting from the Sun to all the planets. His model does not contain the motion of the Moon and neither the annual motion, therefore, it is not an analog astronomical computer like (hypothetically) the Antikythera machine. It is an educational instrument for the lay people and for the scholar to better understand the Copernican theory and the cinematics of planet apparent motion. Kepler’s theory of primum mobile, that he properly calls ground motion, is clearly represented in the virtual reconstruction: the planets are moved by a gear system, possibly with a single clock as motor. There is no need of complex artifacts as required for the nested homocentric spheres. Kepler initially declared that he wants to implement the eccentricity of the motion of the planets by using epicycles. Apart from this sentence at the beginning of the project description, no other details are provided about epicycle implementation. Indeed, the eccentricity of planet motion will be solved by a trick: the deformation of the sphere (orbs) that contains the planet orbit into an oval, that will force the flexible support of the planet’s body to bend and move closer or farther to the Sun. Kepler conceived and designed such a machine, but scientists and clockmakers of his time did not consider his project. Kepler wrote only letters and did not publish anything else about this subject. One attempt to develop Kepler’s conception is attributed to Wilhelm Schickard (1592–1635), a mathematician, astronomer and mechanic (he invented a machine for arithmetic calculus), who discussed with Kepler between 1617 and 1628.

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Fig. 9.11  Left: Portrait of Wilhelm Schickard with a tellurium in hand. Right: Schickard’s sketch of Kepler’s project

Frank D. Prager (1971) reproduced a sketch by Schickard that outlines the shape of Kepler’s machine (Fig. 9.11 right). It would take almost a century for Huygens to succeed in constructing a planetary machine. Throughout the eighteenth-century and into the mid-­ nineteenth-­century these machines would reach heights of great precision, right up to the projection planetariums of the early twentieth-century.

Ole Rømer: Jovilabium and Planispherium Interest in the motion of Jupiter’s satellites was dictated by practical needs: determining the longitude of a place and determining longitude in navigation. Galileo’s volvelle, the jovilabium, provided undoubted help, but even more useful seemed to be a machine that reproduced the motion of the satellites around the planet, a mechanical version of the astrolabe. We have already mentioned Rømer for his contribution to the observation of transit, but his scientific activity as an astronomer and instrument designer is very significant. Born in Århus in 1644 and died in Copenhagen in 1710, he traveled extensively among European scientific centers. In 1672 he went to Paris where he spent 9 years at the Royal Observatory directed by Gian Domenico Cassini, who had him study Galileo’s idea of using the occultations of Jupiter’s moons as a universal clock for navigation. In 1679 Rømer went on a scientific mission to England and met Isaac Newton, John Flamsteed (the director of Greenwich Observatory) and Edmond Halley.

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On his return in 1681 he was given the chair of Astronomy at the University of Copenhagen. Rømer’s contributions are not limited to astronomy, he was the first to propose the epicyclical profile of gear teeth. Defossez (1946, pp. 275–276) recalls that s note presented in 1674 to the Académie Royal ‘Sur la forme à donner aux dents des roues pour qu’elles s’engrènent le mieux possible sans sauts ni accotements’ was, however, not published. 20  years later, the astronomer Philippe La Hire (1640–1718) claimed this invention by publishing a Traité des épicicloïdes; an uncertain attribution due to the fact that Rømer and La Hire collaborated for a long time at the Paris Observatory. Studies of Jupiter’s satellite occultations led Rømer to design a jovilabium in 167712 in the form of a planetarium. Table A.11 summarizes the reduction ratios and corresponding periods for the wheel work. Probably built by Isaac Thuret (1630, 1706), Louis XIV’s renowned watchmaker and later clock and machine builder for Christiaan Huygens, it turned out to be inaccurate. The watchmaker Thuret seems to have built a Rømer’s saturnilabium, which, however, is not described in any documents, but was supposedly presented to the Académie Royal in 1678. On that occasion, Rømer is said to have presented a conical gear scheme to reproduce the irregularity of the satellites’ rotation from aphelion to perihelion by studying oval gears in conical arrangement. Rømer presented to the Académie Royal in 1680 a planetarium (whose periods are summarized in Table A.12) built by the watchmaker Thuret, and was awarded by Minister Colbert 3000 pounds, a very substantial sum. He also designed and commissioned an eclipsarium for determining the dates of eclipses. Both machines are preserved at the Biblioteque National in Paris (Fig. 9.12). Thuret made a copy of these two machines for King Christian V of Denmark, preserved today at Rosenborg Castle. In the planetarium13 or better Planispherium as Rømer himself called this instrument, on one side is depicted the starry sky and two small windows with the date. On the other side, the planets are represented on a plane and rotate with off-center orbits along slots under which are the arms connected to the gear system. The planets Mercury, Mars and Jupiter move along their orbits respecting the velocity anomaly by varying the length of the guiding arms, which is minimum at perihelion and maximum at aphelion. The motion of the Moon around the Earth is obtained by connecting an 8-tooth gear to the 99-tooth gear that generates the annual motion, resulting in a reduction of 12.375, which approximates the number of annual  King e Millburn (1978, pp. 107–108).  Ibidem, pp. 109–110.

12 13

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Fig. 9.12  Left: Roemer’s Planispherium. Right: Roemer’s Eclipsarium

lunations with an average duration of 29.5 days. The precision of the planets’ motions can be seen in the table of reduction ratios Table A.12. In 1681, Rømer returned to Copenhagen and continued his work on astronomy. He initially thought of installing an observatory in Copenhagen’s Round Tower where Christian Longomontanus, who was also Tycho Brahe’s assistant, had worked, but it was too exposed to the wind, so he chose his own home, from where he continued his observations until his death. The King Louis XIV wished to make Paris the world center of astronomical observations, therefore in 1669 commissioned another astronomer, the Frenchman Jean Picard (1620, 1682), to travel to Denmark and visit Tycho Brahe’s Uraniborg observatory to determine its longitude and thus be able to perform the calculus necessary to express celestial and terrestrial coordinates according to the Paris meridian. The observatory at Uraniborg and the Astronomical Tower in Copenhagen were within sight, and through multiple lightings of fires Picard was able to

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accurately determine the distance between the two places and subsequently the correct longitude of Tycho Brahe’s observatory. In 1672, on his return, Picard published data that Tycho was unable to publish in Denmark; he was also accompanied by Ole Rømer, who from then on started his collaboration with Cassini and the French astronomers. The importance of the Copenhagen Observatory continued even after Rømer’s death. In the observatory, Thuret produced another planisphere based on Rømer’s design, which was, however, destroyed in a fire that also burnt a large part of Rømer’s writings. This planisphere was reconstructed in 1740 and can still be seen in the Copenhagen Tower (Fig.  9.13 left). In Fig. 9.13 right we can observe the double cone mechanism devised by Rømer.

Christiaan Huygens Huygens was born in 1629 into a prominent family, his father Constantin was secretary to the Prince of Orange, and was first minister of accounts to King William III of England. Christiaan studied law and mathematics at the University of Leiden. At 17 years, he published a study on the problem of the quadrature of conic curves that attracted Descartes’ attention. After finishing his law studies in France, he devoted himself exclusively to mathematics, finding solutions to the measurement of time for astronomy and the determination of longitude in navigation. With his father, he also devoted himself to

Fig. 9.13  Left: The Planetarium of the Copenhagen Tower, reconstructed in 1740. Right: the double cone mechanism used by Rømer

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lens construction, an experience that aroused in him a great interest in the nature of light and the phenomena of refraction and reflection. Huygens supported the wave theory of light and published the Traité de la lumière in 1690. He is credited with a law of propagation of light, according to which each point on the wavefront is the origin of a new wavefront, a principle that makes it possible to justify the phenomena of slit diffraction. He built a telescope with which he made astronomical observations of the planets, the Orion Nebula and the Moon, and discovered Saturn’s rings. These observations were published in Systema Saturnium (The Hague, 1659), dedicated to Prince Leopold de’ Medici. He maintained a correspondence with the scientific community of the Accademia del Cimento, discussing in particular the nature of Saturn’s rings. His studies on the measurement of time led him to experiment with oscillations constrained by a cycloidal curve (Fig.  9.14, right), and he demonstrated that in this way the oscillations were isochronous regardless of their amplitude. There was a controversy with Vincenzo Viviani who claimed Galileo’s application of the clock to count oscillations. In 1665, Huygens received an invitation from Minister Colbert to visit Paris and contribute to the development of science, following the deaths of renowned intellectuals Descartes and Pascal. He was subsequently appointed by the King to lead the organization of the Royal Academy, reflecting his esteemed status in the scientific community. However, due to ongoing religious and political conflicts between France and the Netherlands, Huygens was compelled to return to The Hague in 1681, possibly also due to health concerns. The first pendulum clock designed by Huygens was built in 1657 by Salomon Coster (1620–1659), which is now preserved in the Museum Boerhaave in Leiden (Fig.  9.14). He published a treatise, Horologium Oscillatorium sive de Motu Pendolorum ad Horologia aptato demonstrationes geometricae (Huygens, 1673) in which he demonstrated the properties of the cycloidal pendulum and put forward hypotheses for a solution to the stability of oscillations in navigation. Perfecting the cycloidal oscillator involved numerous experiments with Coster, until Huygens was able to mathematically calculate the configuration of the curved foils. During his stay in Paris, Huygens designed a planetarium of the solar system according to the Copernican model, and obtained funding for its construction from Colbert. In 1682, after returning to the Netherlands, he completed the construction of the machine and published a short treatise Automata Planetarii (Huygens, 1703) which described its structure and explained the calculus of the gears.

Fig. 9.14  Huygens clock 1657

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The planetarium has an octagonal shape and can be turned to observe its internal structure (see Fig. 9.15). Frontally, it has a gold-plated brass surface, protected by glass, on which run the planets, set in circular grooves, with the Sun in the center. At the bottom, two small openings indicate the month, year and date. Ascending and descending nodes are marked on the orbits. The orbits are circular but slightly off-center in proportion to the eccentricity of the elliptical orbit, this, as we shall see, to simulate the planetary anomaly. Saturn and Jupiter’s satellites are simply etched on a small disk that rotates with the planet. The outermost circle indicates the Zodiac and is divided into 360° to read the longitude. At the bottom there are two small windows: the upper one indicates the month, the lower one the current year. These indicators run by virtue of two circles, one of 360° and the other corresponding to a 300-year  cycle. The semicircular aperture at the top indicates the time and is moved, together with the lower indicators, by a balance oscillator clock equipped with a winding spring. The machine, built by Johannes van Ceulen (1629–1695), is operated by a crank. The Gear Trains  A very important aspect of Huygens’ project concerns the calculation of the gears. For the first time, to compute the reduction ratios for clocks or planetary machines, he makes use of the mathematical method of continued fractions, which he describes in detail in the treatise (for a brief history and mathematical description of the method see section “Gear Computation” in Chap. 15).  Huygens starts with the periods of revolution of the planets, the Earth and the Moon, and from these he derives a ratio, which

Fig. 9.15  Huygens Planetarium. Right: interior

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being a rational number requires an approximation as a ratio of integers: the number of gear teeth. To illustrate the method, he uses the case of Saturn. Huygens gets Saturn’s period from the date of Giovanni Riccioli (1598–1671), a Jesuit who, in 1665, published an update of the ephemeris tables in his work Astronomiae reformatae, in which he attempted to refute the Copernican model with his own model, partially inspired by that of Tycho Brahe. The motion of Saturn is implemented from the rotational motion of the Earth. Each year Saturn progresses 12° 13′ 34.18″ equal to 2,640,858 ms, the Earth progresses each year by 359° 45′ 40.31″ equal to 77,708,431 ms. The reduction ratio is therefore 77,708,431/2,640,858 = 29.42545. It is now a question of finding an approximation to this value that allows a gear pair with a reasonable number of teeth. Now Huygens applies the method of continued fractions. The approximation that would bring the error to 0 is given by the ratio 10,803/367, but the numerator can only be broken down as 3 × 3601, which is too large for a gear. To get reasonable gears Huygens accepts a limited error in the approximation and takes the ratio 206/7  =  29.42857 with an error of 0.0074. The Accuracy  The machine created by Huygens was primarily intended for study and teaching purposes. Although Huygens believed that it could provide reasonably accurate indications to determine the positions of the planets, he never claimed that it could predict eclipses. The accuracy of the machine can be evaluated from different perspectives, such as the still inaccurate determination of planetary periods in the seventeenth century, the precision of the gearwork, and calibration of the instrument. Moreover, telescope and meridian transit observing instrument were still rarely used. So, if we compare the periods produced by the gear work with those known at the time from Riccioli’s tables, we get the data in Table A.13.  It is also interesting to measure the error with respect to the periods calculated by Huygens, with the nonzero approximation. If the ratio is not a rational number, the approximation will be affected by an error. But even in the case of a rational number, one may be forced to choose an approximation because better precision cannot be obtained with reasonable gears. The cumulated error with the chosen gears is analyzed by Huygens who writes: Actually, as this example shows, the wheel of 7 teeth mounted on the common axis leads Saturn with a wheel of 206 teeth. It is therefore necessary that in the

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course of 206 years Saturn completes 7 complete periods. Since the ratio of the motion of the Earth with respect to Saturn is 77708431 to 2640858 it will be found that in 206 years Saturn will not precisely complete 7 periods but about 7 times 1/1346. Therefore in 206 years Saturn delays in this machine by 1/1346, and in every single year it will delay its motion by a small fraction of a tooth so in the course of 1346 years it will delay its motion by a whole tooth, therefore after this time Saturn must be moved forward. This same wheel of 206 teeth is a complete circle of 360°, therefore one of its teeth corresponds to an angle of 105′ and so Saturn must be moved forward after 206  years, and even after 20 years there will be a delay of 1′:34″ The same thing happens for all the others.

The error is high for Saturn and Jupiter compared to Riccioli’s tables, but also for the other planets and especially the Earth. This makes the machine unsuitable for an ephemeris calculation. Henry King and Millburn14 observe that if Huygens had implemented annual rotation in 365.242 instead of 365  days, the periods of all planets would have been the same as those of Riccioli. Construction In Automata Planetarii, Huygens described the construction of the planetarium in great detail, including four plates showing both the front view and the interior of the mechanism, excluding the clock. Huygens planetarium has been studied by Antide Janvier (Janvier, 1812), who made a French translation15 of the description of the instrument, that is also reproduced by Pierre Dubois (1849). Recent studies on the use of continued fractions by Huygens have been conducted by Dutch researchers (Amin, 2008; van den Bosch, 2018). The Anomaly To resolve the anomalies of the motion from perihelion to aphelion Huygens arranged the circles of the planetary orbits off-center from the Sun by an amount proportional to the eccentricity of the elliptical orbit. The planet is moved by a long pinion that allows the driven wheel to slide by varying the radius and thus the speed of rotation according to the eccentricity (Fig. 9.16 left). This machine reveals, on the one hand, Huygens’ great knowledge of mathematics and astronomy, and on the other hand, his modest knowledge of the construction methods for precision mechanics, limitations that Janvier duly observed. The machine was not particularly successful, perhaps because  King e Millburn (1978, p. 115).  The translation by the author of Janvier’s treatise is in the web site of this book (“Huygens.pdf ” in supplement). 14 15

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Fig. 9.16  Left: Huygens, Anomaly simulation. Right: Planetarium of Desaguliers

Huygens himself kept it for his own use. However, it did constitute a demonstration of the possibility of describing planetary motions according to Kepler‘s theory, avoiding the complications of epicyclic rotations while also simulating the anomaly. This problem was later addressed by others, in particular by John T. Desaguliers (1683–1744), who inaugurated a discipline of Experimental Philosophy inspired by Isaac Newton, giving numerous lectures on astronomy, hydraulics and optics. In the context of research into planetary machines, Desaguliers devised pairs of elliptical gears (Fig. 9.16 right), with which he built a mechanical planetarium in 1734, which he used for his lectures on popularizing science (Desaguliers, 1745). Huygens’ use of continued fractions has aroused the interest of numerous scholars who have taken up his method in order to deepen their understanding of its mathematical foundations or to apply it to other problems of gear train determinati.

10 Orreries and Astronomical Clocks

The Planetary Machine by Thompion and Graham The skill and know-how of the best English watchmakers finally realized Kepler’s dream of a planetary machine capable of showing planets floating in space. Thomas Thompion and George Graham built a mechanical planetarium, placed on top of a clock. The instrument was built around 1719, probably the first Orrery 1 ever made; it illustrates the movement of Earth and Moon around the Sun. There are two series of markers showing the dates on the calendar, spaced 11  days for the Julian and Gregorian calendars. 2 The history of the change of ownership of this work was studied by Silvio A. Bedini. 3 Prince Eugene of Savoy acquired the instrument between 1705 and 1710 before his visit to England or during the 2 months he spent in London. It would have been used in one of his palaces in Wien. After Prince Eugene’s death in 1736, it was sold with his other properties to Emperor Charles VI. It was then inherited by the Emperor’s daughter Maria Theresa. It returned to the Imperial collection in 1753, but in some of the subsequent frequent transfers, it disappeared. It seems to have been owned by the Salzburg monastery of Erzabtei St. Peter, who sold it in 1930, after which it returned

 For a general treatment of orreries see (Buick, 2014).  The Gregorian calendar was adopted in England in 1752. 3  Silvio A. Bedini, In pursuit of provenance, the George Graham proto-orreries, 1994, 54–77). 1 2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_10

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to London through an Austrian dealer. After being in the hands of three dealers, it was finally bought by the History of Science Museum in Oxford at an auction in 1948. The first machine bears the signature of Geo. Graham, although he was still employed by Thompion at the time, and was most likely built as a prototype. A second machine preserved at the Adler Planetarium in Chicago is signed Tho. Thompion – Geo. Graham. The octagon containing the machine, made of ebony wood, is about 20 cm high and has a maximum width of 32 cm. The brass top has a central area painted green, outside of which are three graduated circles indicating the days of the month, months of the year, and symbols of the zodiac. The mechanism is not moved by the clock but by a crank inserted in the center pin of the dial, which moves the top plane with the zodiac indicator. The sun in the center is gilded and has a diameter of 19 mm. A “ray” emerges from the sun, pointing toward the earth and indicating the position of the meridian. The sphere of the earth is inclined to its axis and is engraved with circles of latitude and longitude. The moon moves with a rod attached to a ring around the earth. The internal mechanism of the machine preserved in Chicago has a 365-tooth main wheel that generates the rotation of the upper plane. This wheel also moves the gears that cause the rotation of the sun and the constant direction of the earth’s axis northward. From an astronomical point of view, this machine is more of a Tellurium 4 than a planetarium, and it is still very simple, lacking in fact a representation of lunar cycles due to the inclination of the Moon’s orbit. These approximations indicate that Graham’s astronomical knowledge was not deep enough for the construction of a complex machine. Unfortunately, the machine, while built with Graham’s typical care, is not accurate. This work is undoubtedly the first evidence of the creativity of English manufacture, which gave rise to a veritable fashion that lasted throughout the eighteenth century. Every university, nobleman or academy wished to own an Orrery.

 King and Millburn (1978, p. 154) uses this nomenclature: tellurion is a machine where the motion of the earth is simulated, lunarium is a machine where the motion of the moon is fully modelled, planetarium is a machine where the rotation of the earth is absent, as in Huygens’ model. I prefer to call Tellurium any machine that limits the planetary simulation to the Earth, Moon and Sun, while planetarium is any machine that includes some planets. 4

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 he Dissemination of the New Astronomy T in England Numerous astronomical machines illustrating the motion of the planets for educational purposes, created during the eighteenth and nineteenth centuries, are named Orrery. The name comes from Charles Boyle (1674–1731), fourth Earl of Orrery, who commissioned John Rowley to build the first complete machine in England. Charles Boyle was related to the eminent scientist Robert Boyle, a distinguished chemist and physicist; he studied at Oxford, and his kinship with Robert Boyle must certainly have influenced his education. The family library numbered about 10,000 volumes and was considered one of the most important, with books in Latin and Greek, as well as French and Italian. Charles developed a great interest in the physical sciences and astronomy. As early as 1706 he had amassed a large collection of instruments such as microscopes, telescopes, spheres, quadrants, globes and planetary. Charles Boyle’s library, donated after his death to Christ Church College Library, remained long forgotten until its “rediscovery” in 1930. It is considered one of the most extensive collections of scientific works from the eighteenth century (Smith, 1994). The first of these instruments were the work of John Rowley (1668–1728) a London clock and instrument maker, commissioned directly by the Earl. For the planetary machine Rowley was inspired by that of Graham and Thompion. Built between 1712 and 1713, it is a mechanism that demonstrates the motion of the Earth and Moon around the Sun and for this reason, like Graham’s, is more properly a Tellurium (Fig.  10.1). Rowley later built several machines with planets. Its fame grew rapidly, especially after the publication of Isaac Newton’s treatise. Rowley’s machine was much admired in his day and was given the name Orrery in honor of its owner, probably also

Fig. 10.1  The first Orrery made by Rowley. Right: The mechanism

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playing on the word horary. In any case, the name Orrery became common in the United Kingdom to describe planetary machines. Isaac Newton’s studies had great resonance in England. With Desaguliers, a new profession was born for scholars, which we now call scientific popularization or popular science. This new profession was closely linked to the construction of planetary machines, which were the key teaching tool for illustrating the new view of the universe. George Graham had shown how a planetary machine could be built, and, inspired by this early model, a real commercial product was born, which was built in the many workshops of English clockmakers. Clockmakers publicized their machines with advertisements and commercial postcards, which also listed selling prices.

George Adams George Adams (1750–1795) was a scientist, an optician who invented illumination systems for microscopes and a microscopic image projector to facilitate the reproduction of the object observed. He wrote several essays on elementary mathematics, optics and microscopy. He devoted himself to the manufacture of several Grand Orrery, which became the reference for more sophisticated planetary machines (Fig. 10.2 left). The one on display at the Musée d’Histoire des Sciences in Gèneve is enclosed in an armillary sphere oriented like the earth with its ecliptic, and this configuration is characteristic of the grand orrery. On the outside of the

Fig. 10.2  Left: Grand Orrery by George Adams Right: Philosophical Table by George Adams

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armillary sphere are the outer planets, Mars, Jupiter and Saturn with their satellites, which protrude from the machine. George Adams supplied numerous scientific machines to King George III (1738–1820), who had a passion for science and set his policy of developing the nation on Enlightenment principles. Between 1761 and 1762, Adams built for the king a Philosophical Table (now on display at the Science Museum in London), a mechanical device for demonstrating various phenomena, including pendulum isochronism, collisions, and the effects of central forces (Fig. 10.2 right).

Thomas Wright, Thomas Heath, Benjamin Martin Thomas Wright, Rowley’s assistant and probably Rowley’s successor and George Graham’s collaborator, built a large Orrery now preserved at the Adler Planetarium in Chicago. Wright was very prolific in his production of Orrery, not just the grand version. Another maker of large Orrery was Thomas Heath (1719–1773); one of his machines is preserved at the Adler Planetarium, very similar to Adams’ (Fig. 10.3 left). Benjamin Martin (1704–1782) wrote in 1756 that:

Fig. 10.3  Left: Grand Orrery by Thomas Heath. Right: Benjamin Martin, Tellurium, c. 1770

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the high cost of the Orrery was the reason why this useful machine is not as common as one would wish, considering the service of easily disseminating an adequate idea of the true form of the system of the World and the first principles of Astronomy and Geography, such essential parts of the education our English youth should receive. (Millburn, 1973).

Martin was a self-taught man who published numerous popular treatises, maps and catalogs. He set up a business as an optician and instrument maker. In Fig. 10.3 right we see a Tellurium, preserved at Stuttgart Landesmuseum Württemberg. The arrow protruding from the earth denotes the meridian transit. Martin directed a school and taught subjects ranging from writing to astronomy. In his lectures on experimental philosophy, he used demonstration instruments and apparatus, probably built by Heath and two by Rowley. One of these had been commissioned by the East India Company around 1725 but was never shipped. Martin strove to try to simplify the Orrery so that it could be produced at a lower cost. He criticized and got rid of the armillary sphere which he felt was totally inappropriate. He then published a series of lectures in 1747, entitled Philosophia Britannica, in which he described a «simple and transportable mechanism that any gentleman could construct at little expense». The core of this machine consisted of a series of coaxial tubes, one for each of the planets, around a fixed rod for the Sun. Each tube had a gear at one end and an arm at the other to support the spheres of the planets. The proposed scheme matched Rømer’s (and Kepler’s) idea: a series of pairs of gears for each planet, arranged in a double-cone as Kepler’s design. The result in terms of accuracy of motion was very limited, but considering the purely illustrative purpose of the structure of the solar system it was certainly adequate (see Table A.16). In Fig. 10.4 top we see an orrery by Martin, that has a label with the periods of the planets (Fig. 10.4 bottom). In 1756 he started a trade in planetary machines with prices ranging from £2.12 up to a maximum of £22.1. In 1771 he improved the complexity of his machines and published a catalog offering large Orrery with prices ranging from 12 guineas for the simpler models to 150 guineas for the most complete. The machines designed and built were mostly hand-operated and could be adapted to demonstrate various phenomena, such as the retrogradation of the planets, the change of the seasons throughout the year, eclipses (but the moon’s orbit was not inclined), and the comparison between the Ptolemaic and Copernican systems by exchanging the earth for the sun. Harvard

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Fig. 10.4  Top: An Orrery by Martin. Bottom: Periods of the planets

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University ordered several Orreries from Martin during 1764 for which he received a fee of £90 against an order worth £50. 5 A commercial competition took place between Adams, Martin, Wright and other manufacturers. Martin died in bankruptcy and of the 644 lots offered for sale there was one complete manual Orrery. The commercial history of Orrery production in eighteenth-century England is an original example of how an important scientific outreach activity was implemented. It was also based on the study and development of variants of planetary machines. By the late eighteenth century there were two standard structures: the simpler one based on double-cone gears and the more accurate one that illustrated more celestial phenomena. In the more complex machines only the outer planets were moved in a double-cone assembly, while the Earth, Moon and inner planets were mounted on an arm with their own tellurium-like gearing system. It is important to emphasize again that all of these machines were created for the purpose of illustrating the arrangement of the planets in the solar system and describing major astronomical phenomena, but they are in no way suitable for scientific functions, such as predicting eclipses or the arrangement of celestial bodies. The planetary periods are very approximate and are of purely indicative value.

James Ferguson Born into a modest family, James Ferguson (1710–1776) learnt from his father to read and write as well as woodturning. At the age of 18 years, he built a world map on which he drew continents, meridians and parallels with indications of degrees. James Ferguson is an interesting, self-taught scientist who acquired sufficient mathematical, mechanical and astronomical knowledge to design and build planetariums and improve their characteristics. He invented a system of revolving cards that he called astronomical rotula, a kind of paper astrolabe (also called volvelle) that Ferguson was able to sell easily for a few shillings. This device aroused the interest of a famous Scottish mathematician, Colin Maclaurin (1698–1746). A copy printed in 1819 is kept at the History of Science Museum in Oxford. Ferguson also undertook to practice the experimental philosophy inaugurated by Desaguliers, and to this end he devoted himself to the construction of various planetary instruments to describe individual phenomena, such as  The order of magnitude of the value of a pound in the eighteenth-century to today is about 1:100, so a £150 grand orrery would correspond to no less than £15,000 today. 5

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the motion of the Moon always offering the same face to the Earth, or to illustrate the phenomenon of eclipses. Ferguson devised a mechanism that ensures the constant orientation of the Earth’s axis, which he called the mechanical paradox (Fig.  10.5). He observed that an arrangement of three gears in succession, the first and last with the same number of teeth and the first gear fixed while the other two can rotate on a plate, produces motion in which the third gear does not rotate with respect to a fixed reference. If, for example, this arrangement drives the Earth’s revolution around the Sun, the Earth’s axis remains constantly pointing to the celestial north. Ferguson’s scheme requires the three wheels to have the same number of teeth, but it is sufficient if the first and third wheels are equal, since the intermediate wheel does not affect the gear ratios. The mathematician Maclaurin owned an Orrery that aroused Ferguson’s interest, who designed and built his own. Although self-taught and with a modest knowledge of mathematics, Ferguson engaged in lectures, publications and courses to spread the knowledge of the new astronomy. To this end, he built planetary machines to illustrate specific topics, such as the Eclipsarium of 1756, to show the mechanism for projecting the moon’s or sun’s shadow on the earth, the places that would be affected and the duration. Many of his ingenious projects were published in 1773  in the volume: Select mechanical exercises: shewing how to construct different clocks, orreries, and

Fig. 10.5  Mechanical Paradox, Ferguson, 1773

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sun-dials, on plain and easy principles with several miscellaneous articles and tables. 6 The English Orrery’s gear trains were not published by these authors; some were obtained during restoration work, so it is not possible to know the precision with which the planetary motions were reproduced. Antide Janvier was fully aware of these limitations, in fact he wrote: 7 The other planetaries described in the Collected Machines of the Academy, tom. I (note B), in the works of Horrebow, tom. II (note C), in Nollet's Lectures on Physics, tom. VI, in Fergusson (Astronomy Explained), and which in England they call Orreries, from the name of Lord Orrery who had several of them built, do not fully achieve the purpose for which they were built, which must be to explain the system of the world down to the smallest detail, without neglecting any part or making new mechanical devices to go from one phenomenon to another.8

The Cometarium The desire to popularize new astronomical theories also led to a full understanding of Kepler’s second law: the segment (vector) joining the center of the Sun with the center of the planet describes equal areas at equal times. The limited eccentricity of the planets’ orbits makes it difficult to grasp this effect with a mechanical simulation. On 8 March 1732, John Theophilus Desaguliers presented to the Royal Society an instrument showing the change in velocity of a planet or comet, a highly original machine designed specifically to illustrate Kepler’s second law. Desaguliers contributed to the study of electricity by introducing the concept of conductive and nonconductive materials, he studied optics, in particular stereoscopic vision, and built scientific instruments: a figure fully integrated into the community of Natural Philosophers. Martin Beech (2002) published a description of Desaguliers’ machine. The machine was associated with the planet Mercury, that has the greatest eccentricity. The time scale was divided into 88 intervals, a rough approximation of the period of this planet. The machine was not born as a simulator of comet motion, which was not yet clear at the time. In fact, it was not until 1758 that  https://openlibrary.org/works/OL1139033W/Select_mechanical_exercises?edition=selectmechanical00 ferg. Accessed November 2022. 7  Janvier (1812, p. 69). 8  Tr. by the Author 6

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the prediction of the appearance of Halley’s comet was confirmed, thus proving the validity of the hypothesis of an elliptical motion advanced by both Newton and Halley. The mechanical simulation of elliptical motion was obtained by Dondi with oval wheels. Other mechanicians adopted the solution with circular orbits with the Sun positioned off-center. Desaguliers’ machine, on the other hand, reproduced an elliptical motion with an eccentricity of 0.67 – about three times Mercury’s eccentricity of 0.2056—in order to obtain a more evident effect of the change of velocity between aphelion and perihelion. The heart of the machine was a pair of elliptical gears rotating around their own focus according to the diagram in Fig.  10.6 left, in which we see the decomposition of the velocity vector into its two radial and tangential components. The two wheels are identical; the driving wheel rotates with constant angular speed around FD, transmitted by a crank mechanism. The driven wheel rotates around FP and its angular speed changes. An arm centered in FP supports a small ball representing the planet, which will move along the dashed elliptical trajectory. A model of Desaguliers’ machine was built by W. & S.  Jones, makers of scientific instruments in the late eighteenth and early nineteenth century (Fig. 10.6 right).

Italian Astronomical Machines When we think at the contribution of Italian scientist to astronomy the first protagonist is Galileo. He discovered the properties of the pendulum and introduced the use of the telescope to observe the sky. He rarely used other astronomical instruments. The second name that comes to mind is Gian Domenico Cassini, who went to Paris. The first astronomical observatory in Italy was built in Bologna in 1725, the Brera observatory in Milano was

Fig. 10.6  Left: Diagram of Desaguliers’ mechanism. From Beech cit. Right: Cometarium, by W. and S. Jones

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established by Ruggero Boscovitch (1711–1787) in 1764 and Padova had its observatory in 1767. Most of the instruments of these observatories were produced in England. Nevertheless, the cultural background was strong, cultivated in the Universities of Padua and Bologna in northern Italy. Strong as well was the wealth of noble families like Medici that ruled Florence, Gonzaga that ruled Sabbioneta, the Farnese in Piacenza and Parma, the kingdom of Sicily, the Republics of Venezia, Genova and many other duchies. A number of mechanics gifted with ingenuity and mathematical skills and knowledge of astronomy were leading figures of this period and participated in the developments of astronomy, suffice it to mention the Campani brothers.

Francesco Generini and Bartolomeo Ferrari Terrestrial and celestial globes of great quality were being built in Italy, partly due to Vincenzo Coronelli, the Benedictine cartographer working in Venice. Mechanized globes, such as Bürgi’s in 1594, had never been built. An exception is the work of Francesco Generini (1583–1663), who built a mechanized celestial globe in Florence in 1645, called Globo Andante [going Globe]. This project came at a time when the study of planetary motions was reaching its peak: just a few more years and Newton would define celestial mechanics. At the same time there was a desire to represent the sky with more complex machines. Generini was just working along the same way as Bürgi and Emmoser, who built their celestial Globes in the late 1500s, as Rømer (whose jovilabium dates back dates back to 1678) and Huygens (his planetarium was built in 1682). Generini was a sculptor and engineer at the court of Grand Duke Ferdinand II of Tuscany (1610–1670), a great patron of scientists and a collector of scientific instruments, which today are a large part of the collection of the Museo Galileo in Florence. Among the Grand Duke’s letters there are documents that demonstrates Francesco Generini’s collaboration with Vincenzo Galilei, the son of Galileo, and with Vincenzo Viviani to build a clock modeled after Galileo’s scheme. Generini published a booklet in 1645 to describe the functions of his Globo Andante (Fig.  10.7). Generini was convinced that no one had previously attempted to show the passing of time with a spherically shaped machine. He

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Fig. 10.7  Globo Andante by Francesco Generini

claimed that his globe was the first to show time in relation to the motions of the heavenly bodies. He writes: 9 The Diurnal movement in this our Globe is seen by the Revolving which in it is made from East to West, moving on the Poles of the World, with uniform  Drawing of Globo Andante di Francesco Generini Scultore, Firenze 1645. https://bibdig.museogalileo.it/tecanew/opera?bid=367959&seq=12. Accessed October 2022.

9

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Fig. 10.8  Left: Atlante Farnese. Right: Spherologium by Bartolomeo Ferrari

Motion, similar, & equal to the Revolving which is considered in the celestial equinoctial.

Fixed stars up to third magnitude are engraved on the globe, and the sphere can be opened to observe the inner mechanism. In addition, 24 lines indicate the hours, which Generini calls “artificial” hours; a circle at the latitude of Florence indicates the horizon, and two rings represent the tropics. In addition to the artificial hours, the machine shows the natural hours. The Sun and Moon move along two semicircles. Hercules, wearing the skin of a lion, 10 supports the globe and covers the weights that provide power to the machine. The machine can also be powered by water that flows along the legs of Hercules, or with a spring inside the Globe. Bartolomeo Ferrari was a mechanic and clockmaker from Bologna whose work has been studied by Silvio A.  Bedini (1966). He designed and built around 1680 a mechanized globe (Fig. 10.8 right) for the Duke of Sabbioneta Gianfrancesco Gonzaga (1646–1703). The machine was made of bronze and  The symbolic representation of Hercules is with a lion’s skin, called in Italian leonté, that recalls Hercules’ First Labor, the slay of the Neman lion.

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the celestial sphere, on which were marked the stars up to third magnitude, was supported by a figure of Atlas, 11 in analogy with the Farnese Atlas (Fig. 10.8 left). Ferrari usually built instruments for the Bolognese mathematician and astronomer Geminiano Montanari. The Duke Francesco Gonzaga commissioned Montanari a machine worthy of his rank, and Ferrari designed the instrument as a sculpture, a copper globe with 48 celestial images painted on it and with stars up to the third magnitude. The globe was enclosed by an armillary sphere with the horizon, meridian, equator and ecliptic. The whole work was supported by the mythological figure of Atlas sculpted in silver. On the knee of the Giant there is a dial to show the hours, minutes and seconds. Atlas has a pendulum suspended between the thumb and forefinger of the hand resting on the shield. Ferrari noted that the instrument would correctly indicate time until 1750; the full annual motion along the ecliptic was accomplished in 365 days and 6 h and required no corrections for leap years. It was designed for a latitude of 45°, thus suitable for the major cities of northern Italy, including Venice. Ferrari acknowledged his own limitations as an astronomer and was grateful for Montanari’s help; he also recognized the greatness of the Campani brothers, whose ingenuity and skill surpassed his own.

Bernardo Facini: The Planisferologio Farnese The Planisferologio Farnese, the work of Bernardo Facini testifies to the skills of Italian craftsmen in the construction of scientific instruments. This masterpiece of mechanical art and astronomical and mathematic knowledge is in the Musei Vaticani (Fig. 10.9); historians of horology and technology have long debated the author and date of construction. On this work, little known to the general public, there is extensive documentation. Original documents are preserved at the Biblioteca Apostolica Vaticana, while the first modern studies were made by G.H. Baillie (1941, 1942) in a series of articles that appeared from 1941 to 1942 in Horological Journal. Silvio Bedini’s study (1985) of 1985 delves into the origins and history of this instrument. Its structure and operation are also described in great detail by Ludwig Oechslin (1982), who in another publication considers this kind of astronomic clocks as models of the Cosmos (Oechslin, 1985).

11

 Bedini (1966, pp. 27–28).

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Fig. 10.9  Farnese Planisferologio. Right: the dial

This work had been attributed by the erudite Scipione Maffei (1675–1755) to the Bolognese astronomer Geminiano Montanari (1633–1687). Maffei confused it with another instrument called the spherologio mentioned in a small booklet of 1683. The correct attribution to Bernardo Facini was clarified when the difference between the Planisferologio and the spherologio was understood: two distinct instruments. Facini’s biography, and particularly his activities in Parma, was little known until notarized records of Facini’s family and life appeared following a reorganization of Piacenza’s notarial archives. Bernardo Facini (1665–1731) was born in Venice in 1665 to a family of small Bolognese nobility. Bernardo was probably educated according to the canons of the time at church schools, in which he learned about mathematics and astronomy. He also learned the arts of engraving, a skill that brought him into contact with Vincenzo Coronelli. With Coronelli he published in 1697 a compendium of navigation that contained methods for determining latitude, meridian, longitude, and the time of night or day. For these purposes Facini built various instruments such as nautical quadrants and compasses that he sold to Venetian shipowners, including the Dandolo family. A particularly complex instrument for solving various navigational problems, built with Coronelli, is now preserved in the Hermitage in St. Petersburg. It was

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sold to Tsar Peter the Great, who at that time was building a fleet of imperial ships (Fig. 10.10 left). He also built numerous mathematical instruments, like an instrument for calculating logarithms, built in 1714, engraved on both sides for finding the logarithm of a number and for the inverse operation (Fig. 10.10 right). Facini had achieved considerable fame for his skills as a mechanic and for his mathematical culture and knowledge of astronomy, to the point that in 1719 he was presented with Bartolomeo Ferrari’s globe that had come into the possession of one Battista Santirota of Venice, 12 probably purchased by Gonzaga while fleeing to the imperial army. The instrument was in very poor condition, but Facini valued it at three thousand small ducats. Santirota owed a debt to another Venetian, Pietro Rombenchi, and as collateral, while waiting to find the sum, he left the Atlas in Rombenchi’s hands. After numerous reminders to pay, Santirota obtained a final postponement and managed to borrow six hundred luigi, but when he brought Rombenchi the balance of the debt, he discovered to his disappointment that the Atlas had been sold to a merchant, Machera, for precisely the sum estimated by Facini, which corresponded to about 526 luigi. A quarrel arose between the two that still involved Facini, which caused him quite a bit of annoyance later on, but in any case, had brought him into contact with the Court of Farnese in Parma.

Fig. 10.10  Facini’s scientific instruments. Left: nautical astronavigation instrument. Right: logarithmic spiral 12

 Bedini (1985, p.73 ff.).

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Ferrari’s atlas had meanwhile been sold for far more than Facini’s estimated price. In fact, another Venetian citizen, Giuseppe Anzeloni, contacted Facini on behalf of the Duke of Parma offering him the opportunity of a job, the nature of which he did not immediately reveal. Facini was persuaded by Count Abate Jacopo Vezzi, a member of the Court of Parma, who again in great secrecy urged him to accept the job by paying him a large advance with the sole condition that under no circumstances should he ever reveal the sum received. Facini had a suspicion that the work was about the Atlas he had evaluated for Santirota. The opportunity to work for a Court tickled his curiosity and he accepted the assignment, expecting to be of short duration. He then moved to Piacenza, leaving his family in Venice, except for his eldest son Giambattista, who apparently had to be followed closely having “evil inclinations.” At that time, the Duchy of Parma and Piacenza was ruled by Francesco Farnese I (1678–1727) and Duchess Dorothea Sophie of Neuburg (1670–1748), who had married him on her second marriage. The Dukes mostly stayed at Colorno, and court life was among the most luxurious and expensive in Europe. Previous dukes had indebted themselves to European bankers to pay for their lavish parties, and nothing remained to improve living conditions in the duchy, not even for building bridges or roads. Duchess Sophie was considered a foreigner, her Germanic birth and upbringing alienating her sympathies. After all, it had been a marriage of political convenience, to tie the destinies of the Farnese with those of the Habsburgs. Francesco and Sophia initially undertook to reduce the costs of the Ducal court, while at the same time avoiding involvement in the wars of succession that were convulsing Europe. Bernardo Facini struggled to obtain Ducal protection directly; his work was the object of envy, not least because he had unfortunately confided the price that had been paid, unwittingly revealing a fraud perpetrated against the Duke. Envy at Court hampered his work and he had to seek the Duke’s protection in order to complete the restoration, safeguarding his reputation and satisfying the Duke. Facini managed to complete the restoration work in 1719. Nothing is known about the fate of Ferrari’s restored spherologio, which was probably sent to Spain, but there is no mention of it in the inventories of the properties of the Duchy of Parma and Piacenza, while a description of a silver Atlas holding a globe appears among the list of properties of the Spanish royal palaces in 1780. Facini would have liked to return to Venice but was commissioned by the Duchess to make an astronomical clock: the Planisferologio, which he completed in 1725. Facini built other instruments for the Duchess, and he also

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received an appointment as professor of mathematics at the University of Piacenza. He died in Piacenza in 1731 and had a funeral at the Duchess’s expense, to whom Facini’s daughter expressed her gratitude. In the years that followed, the Planisferologio was greatly admired; the Padovan naturalist and physician Antonio Vallisneri (1661–1730) described its merits and reported that its fame had reached even England, where no comparable machine existed. Vallisneri also reported that Facini was preparing a booklet with drawings to fully describe the machine, its construction and operation. Facini’s machine was bequeathed by the Duchess to her nephew Don Carlo di Borbone (1716–1788), who became king of Napoli and Sicily and later until his death king of Spain under the title Charles III, with a pledge to keep it forever in Piacenza. Recall that wars of succession raged in Europe. The War of Polish Succession after the death of Augustus II awakened the interest of the various kingdoms to settle estates. The Duchy of Parma was one of the places where the Habsburgs and Bourbons came into conflict, to avoid which Duke Francesco worked until his death. The war ended in 1734 with the recognition of Habsburg rule over the Duchy of Parma and Piacenza and Bourbon rule over Napoli and Sicily. The Farnese heirs then had to leave their properties and as early as 1732 tried to take the most valuable objects to Spain or Napoli. In 1734 the first transfers began, but the coffers were sent in part to Genoa (Ascione & Bertini, 2015), and many did not even leave. Many properties were sold to secure the Duchess’s income. In 1734 King Charles entered Napoli in triumph and immediately requested that the Planisferologio be sent to him, despite the constraint imposed by the Duchess. The instrument was sent first to Genoa and then by sea to Napoli, where it is mentioned among the royal properties in a 1759 inventory. Once in Napoli the machine caught the attention of the Royal Engineer Nicola Anito, who carefully examined it, probably dismantled it completely. Anito compiled a volume in honor of the King with 24 watercolor drawings (Fig. 10.11) of the Planisferologio di Parma (Anito, 1796), perhaps to support a petition for his own son. Silvio Bedini (1985) raises doubts about the attribution of this work to Anito, considering Anito’s modest skills in astronomy, mechanics, and mathematics. However, analysis of the document shows it to be the same hand signing Nicola Anito, who may have reproduced drawings or descriptions by Facini himself. Other mechanics and scholars examined Facini’s machine. Tommaso Felicetti, court watchmaker to King Francis I, in 1796 wrote a report (Felicetti,

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Fig. 10.11  A plate by Nicola Anito related to the thermally compensated pendulum.

1796) in which he described the astronomical indications of the machine and its state, emphasizing the inventiveness of its author and the quality of the machine, but makes no mention of the chime. In 1861 the reign of Francesco II Bourbon (1836–1894) fell, and the king took refuge in Rome, residing in the Palazzo Farnese, where he had furniture and decorations transferred from Naples, including the Planisferologio, which thus reached Rome for the first time. It remained in the Palazzo Farnese, abandoned for a long time until Count Alfonso Maria Borbone of Caserta, one of the last heirs of the Farnese Borbone family, who resided on the French Riviera, asked his friend the Duke of San Marino di Montalbano to have the machine appraised. He took it to the Dominican father Giambattista Embriaco, an accomplished watchmaker and director of a watchmaking school in Roma. Embriaco wrote a report (Embriaco, 1894) in which he notes that the chime no longer exists, perhaps missing since it was repaired by Felicetti; the small key mentioned by Nicola Anito is missing; the escapement organs are worn or broken and badly restored, with tin soldering; the ruby depicting the Sun is missing and the hands indicating astronomical minutes and hours and Italian civil hours are also missing. Finally, he notes that the semiprecious stones are not real but imitated.

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On the occasion of Pope Leone XIII‘s golden jubilee, the Count of Caserta decided to donate the Planisferologio, recalling the pope’s interest in science, who had in fact erected the Vatican Astronomical Observatory in 1891. And after an examination by the director of the Observatory (Rodriguer, 1903) was handed over in 1903 to the Hausmann firm for restoration. In a final report Ernesto Hausmann and associate Ermanno Frielingsdorf (Hausmann & Frielingsdorf, 1903) illustrate all the restorations including the remaking of the chime, an operation specifically requested by the Pope who wished to restore it to its original splendor. The Pope died just before the conclusion of the restauration the Planisferologio and it was then presented to Pope Pio X. This instrument is now preserved in the Musei Vaticani, Galleria Urbano VIII, after being kept in the Biblioteca Apostolica Vaticana until the 2000s. The most recent restoration work was carried out in 1982 by Ludwig Oechslin, who was thus able to make a detailed technical analysis of the mechanism (Oechslin, 1982). The instrument has an octagonal shape, like Rømer’s Eclipsarium and Huygens‘planetarium. It contains some important technical innovations that in the same years were also being experimented in England, such as the mechanism for maintaining power during winding (the remontoire invented by Jost Bürgi) and the original temperature-compensating pendulum, two solutions that highlight Facini’s inventiveness. The compensation pendulum of the Planisferologio consists of a parallelogram that changes shape as temperature changes, lengthening or shortening the effective length of the oscillating body. The instrument is richly decorated with engravings on the dial and back plate (Fig. 10.12). The engravings are the work of Antonio Friz (1696–1751),

Fig. 10.12  Left: Functions on the dial. Right: detail of the stones in the back plate

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from Piacenza, who describes work on the dial in a letter probably to the Duchess’s secretary in 1731. A later letter of 1733 informs that he began engraving the back of the instrument by order of Duke Carlo of Bourbon and that he would put his hand to the case of the machine (Bertini, 2015). On the front dial is engraved: «For the inexplicable glory of God under the most venerable heavenly auspices of Most Serene Dorothea Sophia Palatine Farnese first built by his own hand Bernardo Facini Veneziano, Piacenza, Anno del Signore 1725». On the back plate is engraved: So that in this work the first mobile moves the sun, the moon and the firmament with true motions, according to the accurate ephemerides, and to preserve the vigor of the double pendulum so that it does not suffer any alteration by cold or heat, let alone be subject to the verticality of the Machine, Bernardo Facini Veneto, professor of Mathematics and builder, invented many novelties by GOD’S GRACE. In Piacenza

In Fig.  10.12 right we can see some of the stones mentioned by Father Embriaco. The astronomical indications are all depicted on the dial, which is composed of a series of concentric circles. The machine is organized into two parts, a time train and a train for calendar and astronomical information. Let’s consider Fig. 10.12 left: the outermost ring shows the astronomical time, the inner one indicates Italian time, the center dial shows Spanish hours. The sun is represented by a small red stone; the moon is a small sphere showing the current phase by exposing a white or black part. The alignment of the sun indicator identifies the date of a possible eclipse. The two arms (setting and rising time) rotate independently and provides the indication of the length of the day over the course of the year, and the time of the beginning or end of twilight. The equation of time is read with the indication of the sun. As Oechslin points out, this astronomical machine is much more similar to the spheres of Finé or Baldewein than to the planetariums of Rømer or Huyghens (Oechslin, 1985). We are faced with a machine that still represents the Ptolemaic view of the cosmos, but in the form of an accurate instrument for measuring time. Facini is still bound to the Ptolemaic model, probably because it provides an easy solution to read the time and identify the position of the Sun and the Moon. In other words: on the dial we observe the sun rotating around the ideal center (the earth) as we observe it in nature; this holds also for the moon,

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that rotates around the ideal center. The purpose of this machine was certainly to provide the observer with accurate information, including eclipses, node passages, etc., which in the planetary machines depicting the cosmos in three-­ dimensional form is reproduced with Tellurium. Mechanically, the motion of the sun and moon is accomplished with an epicycle, in which the deferent is integral with the dial disc, and the epicycle rotates by dragging the sun or moon indicator. The data collected by Oechslin make it possible to compare the periods of the moon motions calculated and realized by Facini with those currently recognized (Table A.14). Oechslin’s painstaking study also pointed out that parts executed but nonfunctioning could have generated a reverse rotation of the starry sky with respect to the ecliptic, leading to a complete revolution after 25,404 years, corresponding to a precession cycle of the equinoxes, which is currently estimated at 25,786 years.

Francesco Borghesi and Bartolomeo Antonio Bertolla Finally, we must consider, among the most significant Italian achievements, the astronomical clock designed by Father Francesco Borghesi (1723, 1802) and built by Bartolomeo Antonio Bertolla (1702, 1789) (Bedini, 1966). Francesco Borghesi was born in Mechel in 1732, about 40  km north of Trento, then part of the Habsburg Tyrol. He was ordered to the priesthood in Salzburg and appointed parish priest first in Rumo, about 20 km north of Mechel, and later in his own home village. In Rumo, Borghesi met Antonio Bertolla, who was born there in 1702. At the age of 17, Bertolla went to Neulegenbach, on the western edge of the Wien Woods, as an apprentice watchmaker to Johann Georg Butzjäger, under the supervision of Christina Winz and Peter Wisshofer, master key-makers. Over the course of three years, as was customary, Bertolla lived in the master’s house and when he completed his apprenticeship, he received a license from the guild of lock smiths and master clockmakers J.G. Butzjäger, written in the neighboring county seat of St. Pölten, on 27 December 1722. This recognition enabled him to set up his own business in Rumo. Bertolla built an important longcase pendulum in 1765 (Lenner, 1999) for Prince-Bishop Clemente Sizzo de Noris (1706–1776), who was appointed to this position in 1763 by Pope Clemente XIII. The pendulum has an ebonized pear wood case 288 cm high, which keeps the winding for one year (Fig.  10.13 left). The clock shows the hours and phases of the moon, has a small alarm clock dial and is equipped with an

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Fig. 10.13  Left: Long case clock by Bertolla. Right: The dial of Borghesi and Bertolla first clock

on-demand repeater for hours, quarters and minutes. It was restored in 1926 and on that occasion some parts were removed, which were reconstructed during a new restoration in 1996. The repeater mechanism is quite unusual in a long case clock, evidently inserted to make it possible to tell the time at night without having to light a lamp, but simply by pulling a cord from the right side of the long case. This work clearly demonstrates the quality of Bertolla’s work, who was involved by Father Borghesi in an adventure that lasted several years. Borghesi was attracted by the ingenuity of mechanisms for measuring time and set himself the goal of building an astronomical machine, and devoted himself to the study of astronomy. He published a short paper Novissinma ac perpetua Astronomica Ephemeris Authomatica Theorico-Pratica [The most recent theory and practice for automatically and permanently determining astronomical

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efemerides] in 1763, which formed the basis for the construction of a first astronomical clock. The undertaking proved more complex than expected and Bertolla soon became discouraged, but Borghesi’s insistence and enthusiasm finally convinced Bertolla to complete the work after three years of work. It turned out to be an exceptional work, a true masterpiece that illustrated the motion of the moon and sun and other phenomena that occurred over time, even over centuries. Borghesi also undertook to write a detailed description of the machine and its functions as displayed in the dial (Fig.  10.13 right). Borghesi was also convinced that his experience with the first clock would enable him to build a second one that would display the motion of the two systems, Ptolemaic and Copernican. Borghesi described his astronomical model in another short treatise Novissimum Theorico-Praticum Astronomicum Authoma Juxta Pariter Novissimum MUNDI SYSTEMA [The most recent theorical-practical Astronomic automa, following the new SYSTEM OF THE WORLD] that bears the publication date of 5713 (i.e., 1764 according to a calculus from the origins of the world). This is a somewhat confused description of a Tychonic model that reveals Father Borghesi’s inability to conceive the motion of celestial bodies in abstract form. These publications by Father Borghesi were harshly criticized by an anonymous mathematician. This is referred in a letter cited by another scholar but of which a complete version has not been found. These criticisms point to errors in the design of the first clock built with Bertolla. These criticisms strongly affected Father Borghesi, who withdrew from his activities as an astronomy scholar and designer of celestial machines, returning to the duties of caring for souls. The second clock performs several functions, and the dial has a similar layout as the first one. Above the circular scales are a series of small windows in which appear the ruling planet represented with its symbol and astrological house; the dominical letter and the epact for the calculation of Easter; the Roman indiction; the Julian solar cycle (a 28-year cycle lasting 365.25 days, at the end of which the days of the week reappear on the same dates); the Julian golden number; the current era, or year; the current era or month; the sun in its epicycle; the 12 signs of the solar anomaly. The first ring represents the equatorial globe of the week and rotates clockwise. The second ring includes the measure with synodic period of the tides, the days of the mean age and the synodic period of the moon, the signs and their degrees for the mean distance of the moon from the sun.

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Other rings show the epicycle of the moon with the signs of the anomaly, the head and tail of the dragon to detect eclipses of the moon, 13 the degrees of the moon’s latitude and of some fixed stars, the firmament of the fixed stars, the signs of the zodiac and the degrees of the signs, the months of the year and the days of the months rotating clockwise during an average astronomical year. Lastly, there are two openings: for disconnecting the dial from the movement in order to carry out experiments and calculus and for adjusting the movement’s gear. The dial contains numerous engraved inscriptions, among which particularly interesting are the chronograms, phrases or inscriptions that contain letters expressing dates or epochs, e.g., in the inscription: FranCIsCVs I sIt pLan. DoMInator aeternVs the letters forming the chronogram are, in Roman numerals: C I C V I L D M I V whose numerical transcription is: 100 1100 5 1 505,001,000 1 5 whose sum is: 1764. Father Borghesi’s astronomical knowledge is modest in relation to his time, still closely linked to the geocentric view, albeit modified by Tycho Brahe. Even the scanning of time constantly referred to the Julian calendar has more the characteristics of mathematical amusement than utility. Both clocks were acquired by Empress Maria Theresa of Austria. Traces of the first clock have been lost, while the second is preserved at the Smithsonian Museum of History and Technology in Washington. Bartolomeo Antonio Bertolla’s workshop was given by the heirs to the Museo della Scienza e della Tecnica in Milano in the 1950s, where it was reconstructed with the original instruments, parts of clocks still under construction and drawings and diagrams of various projects. The creation of astronomical machines in Italy came a few years later than in northern European countries. On the one hand, the ties between European scientists in the sixteenth and seventeenth centuries were solid, exploited the network of academies, and enriched by frequent journeys, in which Italy was increasingly an important destination; but on the other hand, it seems that the scientific instruments necessary for astronomical studies were not produced in Italy, and in addition to this, while in northern Europe toward the end of the seventeenth-century the need emerged to divulge the new astronomy to lay people, in Italy this did not happen. This could be a consequence to the presence and influence of the Holy See, to whom the discussion on the nature of the heavens caused concern, and who put more effort into fighting the Lutheran heresy than into educating young people to science.  In arab astronomy the moon ascending node were located to the head of a dragon and its tail to the descending node. 13

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Borghesi and Bertolla’s clocks, as well as Facini’s, are conceived as automata, as machines for experimenting astronomical phenomena, not as luxury object or simple didactic aids like the Orrery we encountered in the English context, whose function was almost exclusively informative and aimed at giving a mental image of the arrangement of the stars. Borghesi and Bertolla’s automata are conversely instruments based on a mathematical abstraction of the workings of the cosmos, albeit anchored to the geocentric vision of the old astronomy.

11 France and Switzerland

In France, the most significant advances in astronomical machine-building techniques occurred after the mid-eighteenth century. French watchmakers placed great emphasis on the artistic quality of their work, employing cabinetmakers, goldsmiths, and jewelers to create fine objects worthy of the Royal Court. Major production was concentrated on the creation of clocks and pocket watches, also with numerous complications, and to a lesser extent toward astronomical clocks of very high quality but not suitable for scientific popularization as we have seen in England.

Claude Simeon Passemant Claude Simeon Passemant (1702–1769) was a merchant whose workshop was in the Louvre, and who designed and built scientific and optical instruments like microscopes, barometers, telescopes.1 The Passemant achievement that interests us most concerns the design and construction of a clock surmounted by a mobile sphere according to the Copernican system. This is a spectacular work (Fig. 11.1), whose 228 cm high gilded bronze case was built by the sculptor Jean-Jacques Caffieri (1725–1792). The clock and mechanics were built by Louis Dauthiau (1730–1809), who took about 12 years. In 1750 it was presented to King Louis XV, who bought it. It is now kept at the Palais de Versailles.  An interesting work by Passemant is his treatise on the construction of telescopes, which includes methods for producing large-diameter lenses (Passemant, 1741). 1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_11

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Fig. 11.1  Passemant. Astronomical clock. Right: Detail of the dial

We quote Dauthiau’s description in a small booklet published by Jombert in 1756, and transcribed by Janvier.2 This pendulum, surmounted by a mobile sphere according to the Copernican system, was presented on 23 August 1749 to the Académie des Sciences by Mr Passemant, the author of the calculation of the Sphere, for which it took him about twenty years. The members of the Académie, according to the report of Mr. Camus and Mr. Deparcieux, commissioners appointed to examine this Pendulum, certified that the revolutions of the planets are accurate; that not a single degree of difference from the astronomical tables will be found in three thousand years. Dauthiau, a watchmaker, executed and built it, taking twelve years. It was presented to the King at Choisy, on 7th September 1750. His Majesty, protector of the sciences and the arts, declared his satisfaction, ordered a new case according to the design of his choice, which was composed and executed by Monsieurs Caffiéry father and son, and which was again presented to the King at Choisy, on 20 August 1753, and was then transported to Versailles.  Berthoud (1802b, pp. 197–204).

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The sphere3 represents the different daily movements of the planets around the sun …. Their positions in the zodiac, configurations, stations and apparent retrogrades in relation to the Earth. On each circle bearing the orbit of a planet is engraved the time it takes to complete its revolution around the Sun. The Earth, during its annual revolution, also maintains its parallelism and sees the Sun travel through the signs of the zodiac and their degrees; the months and their dates, indicating the seasons, equinoxes and solstices. It also completes its rotation around itself in twenty-four hours, divided into 24 meridians; it has its own map where the main places on the globe are marked; there one can observe the rising and setting of the Sun, its transit through the meridian, the different elevations, the length of days and nights for each main place. The Moon revolves around the Earth and completes its revolution in twenty-nine days, 12 hours, forty-four minutes and three seconds; it shows its age and its different phases. It traverses the signs of the zodiac, indicates its eclipses and those of the Sun with precision, their locations, magnitude and duration. It distinguishes its different altitudes, as well as its rising, setting and meridian transit. The pendulum beats the second … This is an equation [of time] pendulum and marks true time and mean time. The pendulum chimes the hour and quarters of true time or solar time, repeating it at each quarter-hour or at will. The movement of the chime is spring-loaded with a fusee and chain; that of the pendulum is double-weighted, dropping only eight inches in six weeks; the weight is twenty pounds, and when reloaded the movement does not stop. The rod of the pendulum is bimetallic, of steel and copper … The movement it makes serves to move a point that indicates on a graduated circle fixed at the top of the rod, the different degrees of temperature, which forms a natural thermometer by the action of metals alone … On the front of the pendulum, below the dial, is a planisphere, an indication of the moon marking its age and phase. It also shows the day of the week, the date of the month, the name of the month, the date of the years … [and] leap years. The mechanics of the calendar of years is made in such a way that it could mark them for ten thousand years, should the pendulum exist for that length of time … There are three gears between the pendulum and the sphere, i.e., three pieces arranged in such a way as to be able to engage or disconnect the transmission of the pendulum’s movement when necessary. The first wheel is used to disconnect the one that drives the escapement … The second wheel serves to disconnect the  “One sees (says Mr. Passemant in a small printed work in which he speaks of this same Sphere) the rising and setting of the Sun for all the countries of the world; the days rising and setting regularly, the seasons following one another; the Moon rising and setting; the eclipses arriving when they reach the sky; one sees the stations and retrogrades of the planets and their direct movements, so that this machine shows the state of the Heavens at each instant. Historians have often mentioned the arrival of eclipses on the days of battles or great events; with such a machine one can determine the number of years elapsed and rectify the chronology” (Note by Berthoud). 3

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sphere from the movement it receives from the pendulum so that it can be moved by a crank and the movement of the Earth, Moon, etc. can be observed more clearly … The third wheel also serves to disconnect the sphere from the movement so that the movement of the Earth on itself takes place in twenty-­ four hours: in which case by means of the crank one can accelerate the different movements of the sphere to such a speed that one can observe … what would take several years … The mechanics of all these pieces are arranged in such a way that each movement can be separated, if necessary, although the whole is tied together. The number of wheels that make up the sphere is so simple that there are not even sixty wheels and pinions, few of which are inside it, which makes it easier to see and the whole thing more solid. The sphere has a diameter of one foot and is enclosed in a glass bowl. The box of the clock is entirely of mercury-gilt bronze; it has four faces of glass decorated with pleasing and well-finished figures, so that one can easily see all the mechanics of the work; its height including the crowning sphere is seven feet’. By Douthiau, clockmaker, &c 1756.4

Antide Janvier measured the gears for planetary motions by observing that the entire mechanism was driven by a wheel of 48 teeth (full rotation in 2 days) that sets the Moon in motion in the sidereal cycle. The next planet, Mercury is moved by the Moon’s gears and in turn moves the gears of the Earth’s annual revolution. All subsequent planets are again moved by the Earth’s gears. Finally, in order to obtain the lunar revolution in relation to the nodes (the draconic month), it is necessary to assume the lunar synodic cycle as the mover, whose relative gears Janvier was unable to determine. Table A.15 shows of the periods computed by Denis Roegel (2022); the difference to contemporary values is very small. The method used by Passemant to calculate the planetary periods is not known, but the precision obtained is nevertheless remarkable, as noted by the members of the Academie who had analyzed the project. The quality of this result was also recognized by Pierre Le Roy. Passemant’s planetary machine could be considered as belonging to the category of mechanical astronomical calculators, as the author Dauthiau pointed out; the precision of the instrument allows to estimate and check the dates of astronomical events such as eclipses. On the other hand, the extreme luxury of the work suggest that the quality of astronomical computation was just an attribute of a luxury machine for the King.

 All citations in this chapter are translated by the author from French.

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Antide Janvier We have already mentioned Antide Janvier several times and it is now the moment to learn more about this extraordinarily important figure in the history of both watchmaking and planetary machines. Unfortunately, outside the works on the history of watchmaking and technology in the French language, Antide Janvier is little mentioned; among the great watchmakers of the eighteenth and nineteenth centuries, he is the one who disappears behind Breguet, Le Roy or Raingo. In this chapter I try to place Antide Janvier’s personality and creations in their proper perspective. The Life  Antide Janvier was born in Lavans-lès-Saint-Claude, a village in the Jura mountains, on the first of July 1751. His father Claude Etienne, born in 1730, did not have a formal education and learned the watchmaking trade independently, becoming master watchmaker in 1763: it was certainly he who taught his son the craft during the cold winter days in the mountains of a region with a long tradition of watchmaking. Antide’s mother, Claude Francoise Tournier, born in 1723, belonged to the same family as Abbot Tournier, a well-known astronomer and mechanic, to whom his father entrusted Antide’s education, and who taught him Latin, mathematics, optics, the art of drawing, mechanics including mechanical turning. The knowledge of Latin was of great help to Antide, who was thus able to read and study works such as the writings of Huygens. Antide was also a reader of classics like Virgil and Horace, whose verses often appeared on the dials of his watches or as exergue in his writings. The mathematical teachings of Abbot Tournier, who died in 1766, were invaluable for the education of the young Antide, who years later, examining Passemant’s machine in his treatise Manuel Chronometrique (Janvier 1821), wrote that it could have taken him a few days instead of 20 years if he had used the method of continued fractions, which Janvier himself had learnt from studying Huygens’ writings. On the first of April 1764, Antide observed an eclipse of the Sun, which gave him a real passion for astronomy. Still impressed by the event, in 1765, at the age of 15, he began the construction of a mechanical planetarium, which he completed in the course of about 15 months. Proud of his work, he dared to compare himself to Pascal! This early work (Fig. 11.2) was described

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Fig. 11.2  Wooden planetarium Opus n.8, 1771. Remake of the first planetarium made at 15 years

by him in 1828 in a collection of machines he designed and built,5 where he writes: In composing this machine, I proposed to make evident the effects of the annual movement of the sun, combined with its diurnal movement, and the revolution of the earth on its axis; to mark at the same time the sidereal time, the average time, the true time, the duration of the day, the rising and the setting of the sun for any horizon; finally to represent the average movement of the moon in longitude and latitude; that of its nodes, its phases, its transits to the meridian, its rising, its setting, and its ecliptic conjunctions. … The clock which gives the movement to the whole machine is with seconds, regulated by a compensating pendulum: the escapement is with rest and with pins, a weight is the engine of it, and it is wound only every month.

In March 1768, at the age of 17, he had the audacity to present this machine to the Académie des Sciences, Belles-Lettres et Arts in Besançon. The learned members of this academy praised Antide Janvier: Mr Antide Janvier, of Saint-Cloud, having presented to the Academy a sphere he had constructed for the movements of an astronomy system, this company believed that, as it could not praise and encourage too much a young man of 17

 Janvier (1828, pp. 5–6).

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years of age to whom industry will do the honour of a consummate mechanic, it considered it as an act of justice to grant him the present certificate.

The city’s magistrates also honored Janvier for this work, and on May 17th, 1770, they named him ‘Citizen of Besançon’ (Trincano, 1943). During the next 3 years he perfected his training and participated, with his father, in the restoration and construction of clocks for the city of Besançon. In 1771, he built another large planetarium (3 feet in diameter) for public education, with boxwood gears, which demonstrated the motion of the planets with their irregularities and retrogrades, eccentricities, satellite revolutions and more. The construction with wooden gears was a tradition in the region, which took advantage of the availability of fine woods thus avoiding the cost of expensive materials such as brass. He then built a smaller, improved version of this machine, which could be dismantled and transported, and which was presented to King Louis XV at Fontainebleau in 1773 after a daring adventure, recounted by his great friend, Gabriel de Chénier. In short: in 1773 an important event occurred, the sighting of a comet, about which the famous astronomer Jerome de Lalande (1732–1807) wrote a short booklet in which he raised the doubt as to whether comets might one day strike the Earth. The writing caused bewilderment and fear among the people of Paris, who interpreted it as the announcement of the coming end of the world. This clamor fascinated Antide even more and he asked his father for permission to travel to Paris, taking his new transportable machine with him. He set off on foot and arrived in Auxonne, on his way to Dijon, and neglected to pay the customs duties imposed when leaving the region. On the way he arrived in Dijon and was asked at the customs office what he had with him. Proud and naive, he showed them his planetary machine, which was immediately confiscated for not having paid the duties previously. Desperate, he decided to reach Paris more quickly by taking advantage of the passage of goods wagons. Arriving at an inn, he told of his misadventure and a traveler advised him to contact Monsieur de Sartines, lieutenant-general of police. De Sartines, impressed by the young man’s desperation, worked to recover the seized object within a few days. Monsieur de Sartines suggested that Janvier present it to the King who was in the Castle of Fontainebleau. On reaching the Castle he managed to obtain an audience for the morning of November 3rd, 1773, at the King’s petit lever, in the presence also of the Marechal Duc de Richelieu (1696–1788) First Gentleman of the Chamber. Antide illustrated the machine to the King who, interested, asked for explanations. Richelieu hastily interrupted him, suggesting to the King that it was one of the usual scroungers. Antide, outraged, denied he was asking anything

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and accused Richelieu of lying, but of course this provoked the immediate arrest of the shameless young man who was sent by Monsieur de Sartines to lock him up in the Bastille. But de Sartines took the young Antide to heart and advised him to keep quiet about the episode. Janvier settled in Verdun where, after some years, he married his first wife, the daughter of his bookseller, Anne-Catherine Guillot, in 1783. During his stay in Verdun, Janvier taught mechanics and watchmaking and acquired a reputation for his great skill. In that year, the Comte de Provence, brother of King Louis XV and future King Louis XVIII (1755–1824), on his way to Metz, stopped in Verdun for a few days as a guest of the local bishop, who introduced him to young Janvier, who wrote «I had the honor of being presented to the Prince and of entertaining him every day by describing to him my astronomical machines housed in the Episcopal apartment.» In 1784, Janvier returned to Paris bringing with him two planetary machines—two movable spheres 4 inches in diameter—to have them gilded. De Lalande was using the same gilder and thus he learned about them. Janvier and de Lalande were members of Freemasonry and this certainly facilitated their meeting. De Lalande immediately took steps to introduce Janvier to the King, now Louis XVI, through Messieur de la Ferté, who was passionate about clocks and mechanisms. On April 24, 1784, the two machines were finally presented to the King, who immediately bought them, and shortly afterwards had him called to his service in Paris. From the end of 1784, Janvier therefore settled in Versailles at the Hotel des Menus-Plaisirs du Roi, the place where the artists worked for the King’s pleasures. In this place Janvier was able to work not only with serenity but above all benefiting from his knowledge and cultural exchanges with the most eminent figures in science and art of the time. The collaboration with de Lalande in particular was very fruitful for Janvier, who learnt even more about astronomy and created important mechanisms. He learned the use of logarithms to simplify calculus and helped de Lalande to identify an error in the computation of the lunar revolution in 1789, which he was able to demonstrate by constructing a demo mechanism. During this stay in Paris, Janvier, who suffered from deafness, was treated for a disease of the temporal bone, on which a hot iron was applied every day for 18 months by the Director of the Paris School of Surgery, M. Luis, to no avail. Only the intervention of the chief surgeon at the Hôtel des Invalides succeeded in easing his suffering. Janvier, despite his successes and qualities felt himself the object of envy from courtiers and other scholars, perhaps unjustly. In fact, he recounts in his autobiography that one day he found the second hand of a watch beating a

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quarter of a second had been removed. Janvier set back the hand, but he observed the same thing for several days, until one day the King noticed his irritation and asked him why, and he told him that it was he himself who had removed the hand because it reminded him that time was passing too quickly for him. Later Janvier discovered that during prayer the king took the hand out of a cupboard to observe it and remind him of the passing of time. 1789 was the year of the Revolution. In his autobiography, Janvier recounts how he learned of the plan to take the King and the Royal family to Metz. Having arrived at Versailles on October 4th to work on maintenance of the royal clocks, he noticed a turmoil among the guards and gentlemen. He is blocked from entering the Royal Cabinet to wind the clocks. He sneaks in through a back door with the help of a young valet. Here he discovers that during the previous turmoil they had changed the room, his mobile sphere had been moved and several tables had been placed in the center. On this table he found maps, drawings and escape plans for the King. The next day La Fayette arrived with the National Guard at Versailles, and on the 6th the King was taken to Paris, as it was no longer safe at Versailles. Jeanvier was fully royalist and an admirer of King Louis XVI, whose passion, Janvier specifies, was not mechanical locks but geography, clocks and astronomical machines. To satisfy this passion, Janvier designed and built the geographic clock (Opus n.180) (Fig. 11.3 right). Janvier himself, in 1832, recounts the episode that caused a serious disagreement with the Queen that still embittered him after years. Janvier writes that while waiting for the King, the Queen asked Janvier to explain the features of the clock and he allegedly referred to the ease with which he could tell the time in Metz. This mention by Janvier greatly irritated the Queen who glared at him, suspecting a malicious observation. She was still shaken at not having been able to escape from France in time. Later, while Janvier was again explaining the work to the King, who hastily decided to buy it, the Queen insistently sent a knight to summon his Majesty for Holy Mass, but the King continued to converse with Janvier to agree on a price. Eventually he asked Janvier to place it on a particular shelf. Sometime later, the King announced that he was renouncing the purchase, and Janvier believed the reason was due to the Queen’s displeasure at the episode. When the King was imprisoned on August 10th, 1792, Janvier was still reeling from the remorse of having offended the Queen and perhaps made the King believe that he was his enemy. He endeavored to obtain a pass from a valet who was in charge of the King’s wine cellar and tobacco at the Castle, who wished to bring him some comfort. He succeeded, and was held responsible to the extent that he had to accompany the valet, and on that last

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Fig. 11.3  Dial of the geographical pendulum. Opus n.180, 1791

occasion he was able to see the King from a distance, who may have seen him, but did not have the courage to approach him. Janvier’s attitude toward the Revolution was twofold, on the one hand he supported its values that tended to improve the living conditions of citizens, on the other hand he also strongly opposed the excesses that in the midst of

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Jacobinism resulted in vandalism against monuments and works of art. His friend Gabriel de Chénier wrote in this regard: While terror leaves most without the courage to fight, Janvier fights fervently against the savage barbarism that would like to wipe out the best achievements of the human mind.

And Antide Janvier wrote: We must form a human barrier to preserve knowledge and save the latest discoveries of science, thus fulfilling an important duty toward future generations.

When the National Assembly decreed the suppression of religious orders in 1790, Janvier proposed to set up a watchmakers’ town at Notre-dame du Mont-Dieu in the Ardennes, with the aim of creating jobs for mechanics, carpenters, decorative artists and to compete with the Swiss watch industry, thus reinvigorating a poor country. On August 1st, 1790, he sent a very detailed draft to the assembly. He writes: The enthusiasm for Freedom, the spirit of the public good animates all French people; and I come to propose to the National Assembly the project that I have dared to conceive, to populate a desert at my own expense, to enliven a poor country, to increase the product of industry, to found a city at no cost to the State.

This incipit reveals passion and determination, the intent to bridge a deep gap between French industrial capacity and that of Géneve, le Locle and La Chaux-de-Fonds in Switzerland, places very close to his homeland. In the project, he estimated the cost of land purchase and construction at 100,000 francs, which he pledged to underwrite immediately. The convent would be able to accommodate the artists’ families in their cells, and would have space for the already existing town hall, church and parish. Janvier also requested the right to exploit a forest to feed the furnaces of the enamelers, foundrymen and gilders. The project had nothing utopian about it, Janvier was well acquainted with the reality of the Vallée de Joux in Switzerland where a peasant society had been able to create and develop a successful industry, a refined and cultured business that had even given birth to Jean-Jacques Rousseau, a descendant of a family of Genevan watchmakers. Unfortunately, the proposal was not accepted, perhaps due to the too modest sum offered for the

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purchase, and in a short time the place became a military barracks and was later turned into a prison. In 1793, Antide Janvier was sent to the Jura to oversee the development of armaments and the installation of an aerial telegraphy system. And all this while he was constantly busy creating clocks with practical functions, such as tide predictions and measuring solar time alongside civil mean time. Of course, with the fall of the monarchy, the work and hospitality at the Hotel Menu-Plaisir also came to an end, and Janvier had to move. He was soon housed at the Louvre, where he continued to produce his works while incurring ever-increasing costs. The economic crisis had reduced the luxurious consumption of the nobility. His creations, on the contrary, were maintained at the highest level of quality and beauty. Janvier had to accept bankruptcy between 1810 and 1812. However, he immediately resumed his activity by completing more watches, which he had most probably kept hidden to avoid seizure and auction sale. Particular interest was aroused by a short treatise he published in 1811, Essai sur les horloges publiques, pour les communes de la campagne [Essay on the public clocks, for the communes of the countryside], which outlined the requirements and plans for a country clock, with the aim of enabling poor villages to equip themselves with a time-measuring instrument that was both inexpensive to build, easy to maintain and adjust, and sufficiently accurate for the needs of a working population. This treatise was also the result of discussions he had with Ferdinand Berthoud, who shared with him the task of studying the new decimal system of time division. In 1812 he published Des Révolutions des Corps Célestes par le Méchanisme des Rouages [Of the Revolutions of the Celestial Bodies by the Mechanism of the Gears]. This treatise, dedicated to his friend L.M. Waille, consists of three parts. The first is a translation from Latin of Christiaan Huygens’ description of his mechanical planetarium. For this work, Janvier had initially planned to reconstruct the machine, but limited himself to drawing new plates, which were later engraved by Jacques-Louis Leblanc (1774–1846), whom the author thanks in the work. In the second part, Janvier describes the planetary machine designed by Abbot Tournier. Finally, he publishes a project for a planetary machine «more complete than those built up to then». In 1818, Janvier married for a second time, apparently unhappily, followed by a separation that nevertheless financially strained him to the point that he had to sell his furniture, machines and even his clocks to Abraham-Luis Breguet (1742–1823), who affixed his name to them. Also, the Belgian watchmaker Zacharie Raingo (1775–1839) used movements made by Janvier and signed by himself or even his drawings and designs to build his astronomical

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clocks. With the Restoration in 1825, Janvier tried to return to Royal service, but the King Luis XVIII had no remembrance of him, who had been his watchmaker in Verdun in 1774. In 1828, he published a collection of machines he had designed and built, dedicating the book to a young watchmaker friend B.H. Wagner. In this work Janvier still described himself as the King’s clockmaker and mechanic even though he had not been reconfirmed (Janvier, 1828). Right from the dedication, one can read the bitterness of a man who, at the end of his life, sees the value of his works denied, and in thanking the young Wagner for having followed and assisted him since they met, he reminds him that he too must unfortunately suffer envy and betrayal. He struggled for a long time before obtaining a modest pension, and still continued to make watches, albeit less luxurious and complicated ones. He ultimately died in 1835, in near-poverty but supported and comforted by his few remaining faithful friends. Major Works  Janvier produced a considerable number of clocks, astronomical clocks and planetary machines, many of which have survived and are preserved in museums or private collections. He produced few pocket watches; the majority of his works are pendulum oscillator movements, very accurate and very elegant. A catalog of Janvier works has been published by Michel Hayard, from whose work I adopt the item numbering (Hayard, 2011). Before considering the astronomical machines, we must remember that most of Janvier’s clocks include elements related to astronomy, in particular the mechanism of the equation of time for which he devised innovative solutions, the indication of the lunar cycle or the seasons, and a particular clock called the Geographical Clock, which was the cause of the offence to the Queen. The majority of Janvier’s planetary machines are enclosed within an armillary sphere; the reason for this can be found in considering these works as educational tools, for which it could be useful to describe the planetary motions by placing them in the celestial reference system that is summarized in the armillary spheres. The presence of the armillary sphere also justifies the name by which they were called: sphère mouvant or mobile spheres. However, the purely educational character of Janvier’s works is contradicted by the accuracy with which the celestial motions were simulated: many of the machines he built are true mechanical simulators, analog calculators. After the first one, built in 1766, in 1771 Antide Janvier built a new astronomical instrument still made of boxwood with a diameter of about 70 cm. It is operated by a crank and illustrates the motion of the earth around the sun

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and the motion of the moon around the earth, which in turn rotates around its own tilted axis. The instrument thus shows revolution times and no other relevant astronomical phenomena. A metal copy from 1773 is smaller, about 22 cm. It is the one that was presented to the king in that same year and that risked imprisonment for Antide. Far more interesting and intriguing is the geographic pendulum built in 1791 (Opus n. 180 Fig. 11.3 right), which marks true solar time using the mechanism of the equation of time. In this mechanism, the equation of time is generated with a kidney-shaped cam, whose annual rotation modifies the height of the pendulum by slowing down or accelerating the clock. The hour is indicated in a small window at the top center and the minutes are shown above it. The dial is a fine enamel work by Joseph Coteau (1740–1812), representing a map of France with the administrative subdivisions decided by the National Assembly in 1790, and the arcs of longitude. The minutes are represented on the edge of a rotating circle. The time of each town can be read immediately by adding or subtracting from the minutes indicated by the clock the quantity marked on the ring. This ring has two subdivisions, on the outside in minutes, on the inside in degrees, with the longitude of Paris as the 0 reference. The projection of the regions is of a sinusoidal type, and in fact at the eastern and western edges the regions appear more compressed; the calculation for the projection was made by Janvier himself who prepared the diagram for the enameling miniaturist. In 1793 Janvier was asked to sell the clock to the National Museum, but negotiations stalled after a demand of 3200 francs. In 1806, Janvier offered the clock again, asking 4500 francs. After Napoleon’s approval, the clock was finally purchased and is now on display at the Château de Fontainebleau. Another watch that highlights Janvier’s great ability to translate astronomical knowledge into utility machines is the tide regulator of 1807 (Opus n. 307 Fig. 11.4 left). By the end of the eighteenth century, the cause of tides was well identified as the effect of gravitation due to the Moon and its positions relative to the Sun (see section “The Tides”). Newton’s theory had explained tides as a consequence of the gravitational forces exerted by the Sun and Moon. Pierre-Simon Laplace (1749–1827) around 1776 had in turn formulated a dynamic theory of tides that took into account friction, natural periods of variation in ocean basins. It was not yet the complete solution to the problem, which Lord Kelvin arrived at in 1860, considering all the factors of a periodic phenomenon. In Fig. 11.4 right we see a tide prediction machine built according to Lord Kelvin theory.

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Fig. 11.4  Left: Tide Regulator, Opus n.307, 1800. Right: Tide calculating machine, according to Lord Kelvin’s theory

Janvier was probably unaware of the Laplace studies, which required a much greater knowledge of mathematics than Janvier. The clock he produced is therefore an approximation to Newton’s theory. The problem is divided into several parts: first it is solved the determination of the synodic lunar cycle, to identify the new and full moon times when the joint effect of the Sun and Moon is at its maximum and thus indicate the phase of the moon. The gear 78 87 137 . . = 29, 5306 a perfect result. train for the moon synodic motion is: 18 33 53 The second part solves the daily motion of the Moon, that has a direct effect on the tides. To this result it is added the third part, that transmit a variable movement again with a cam. This cam has an irregular shape, different from the one for the equation of time. Its irregular shape derives from tidal historical data in a particular port, chosen as representative. A spring slides on the cam thus advancing or retarding an additional indicator. This clock is clearly an analog computer, simpler but substantially similar to the analog machines that trace the tides by decomposing the various phenomena into periodic functions, as Laplace and Kelvin discovered. An astonishing result, especially for someone who probably did not even know the sea, having always lived in the mountains of the Jura or in Paris.

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Between 1789 and 1801, Janvier created his masterpiece, Opus n. 322 (Fig.  11.5), to which he devoted special attention as demonstrated by the participation of artists already working for the royal family. It is a clock mounted on a piece of furniture built by Johann Ferdinand Schwerdfeger (1734–1818), a furniture maker for Queen Marie-Antoinette, the dials were painted and enameled by Joseph Coteau and Henri-François Dubuisson (c. 1749–1822), the bronze parts made by Etienne Martincourt (b. 1735), the creator of numerous clock cases. The work consists of a column on which is mounted a part with 4 dials, above which is the clock mechanism controlling the dials and on top the planetarium composed according to the armillary sphere model.6 The glass cage containing the gears is subdivided into three parts (Fig. 11.6 right): the upper part with the planetary movement gears, the central part with the chronograph mechanism (Fig. 11.6 left), and the lower part for the dial movements. The complete description of the machine and assembly diagram are published by Ferdinand Berthoud7. The work has four different dials on its four sides. The first dial shows the mean synodic revolutions of Mercury and Venus combined with the diurnal revolution of the sun. The second dial shows the mean synodic revolutions of Jupiter’s satellites, their successive configurations for all instants of the day, Jupiter’s parallax and the time of its meridian transit. This is a jovilabium, whose layout is depicted in a table from Antide Janvier’s work Des Révolutions des Corps Célestes.8 The third dial shows the sidereal and mean time of the Sun and Moon, with nodes, its apogee and eclipses (see Fig. 11.7 left and Fig. 11.11). The fourth dial shows the tides and the motion of the Moon. Janvier writes that all these data have «all the precision that can be expected from these means». These motions are reproduced in the planetary machine on top that includes the planet Uranus, discovered by William Herschel (1738–1822) in 1781 and named after him for a time. Janvier further writes: I set myself the task of linking the general phenomena of the tides to all these functions, and of indicating their time for sixty ports in the main places on Earth. And finally, to express all these revolutions in units of the same species,9 to bind them together with a single principle of uniform movement by means  This clock is currently not on French territory, part of a private collection closed to the public. Michel Hayard pointed out the short-sightedness of the French Government in preventing the export (Personal communication). 7  Berthoud (1802b, pp. 207–241 and Planche XX, XXI e XXIII). 8  Janvier (1812, p. 94 ff.). 9  Janvier does not mix synodic and sidereal cycles, but always uses synodic cycles. 6

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Fig. 11.5  Clock n.322, the Masterpiece of Janvier

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Fig. 11.6  The motor of the clock Opus n. 322. Right: Drawing of the Opus n.322, Planche XX

of the direct transmission of a wheel whose revolution is accomplished in 24 hours of mean time. To make the movement of this primitive wheel dependent on a particular gear whose driving force is regularized by a fusee, to leave this wheel with no other task than to carry out simultaneously all the revolutions I have indicated, and to subordinate the speed of this motor and its gear to the running of a clock.

The Jupiter dial (Fig. 11.7 right) is described in Planche VI by Janvier (Fig.  11.8). There are the four satellites revolving around the planet in the center, the pointers A and B run along the outermost scale, the ecliptic, divided into 360°, A shows the heliocentric longitude of the planet, B shows the geocentric longitude (as seen from Earth), the difference between the two values shown is the parallax (described with the diagram in fig. 6 of the Planche VI). C is the day pointer. D indicates the 24 h and E the day of the week. F indicates the time of Jupiter’s meridian transit, so this hand rotates at the same speed as the planet’s revolution and causes B to move, on which two thin wires are stretched to indicate the parallax. B oscillates with a gear K in fig. 2 whose pin acts on the alidade LL. This gear has a radius equal to the sine of 11° which is Jupiter’s maximum parallax. G indicates the current year and H represents the shadow cast by the planet opposite the sun. Fig. 2 shows the gears of the jovilabium, and fig. 3 shows the typical double cone assembly.

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Fig. 11.7  Opus n. 322. Left Sun Moon and eclipses. Right: Jupiter and satellites

For the motion of the Sun and the Moon (Figs. 11.7 left and 11.11) Janvier invents an original solution: inclined wheels. For the description of this solution see Fig. 11.9.10 To simulate the eccentric motion Janvier makes use of wheel AB (Planche XXI fig. 2) inclined 23° 28′ to the plane that represents the dial or the equator CD. This inclination is aligned to the line of the solstices. Moreover, to simulate the anomaly the radius of rotation of this wheel changes since a pin M slide into the slit NN (Planche XXI fig. 6). This assembly rotates the internal dial 5 (Planche XXI fig. 2) with the symbol of the Sun S (Planche XXI fig. 9). At the same time, the hand M (Planche XXI fig. 9) is driven at uniform speed by wheel X. In this way Janvier obtains the equation of time, a technique he also adopted for the lunar anomaly. About this invention Janvier writes: This machine, of an absolutely new type and more precise than anything that has been done so far, is the only one in existence; it is a model that has been proposed to the present generation for twenty years, without the author needing a patent to secure the honor of an invention that is truly his. Nowadays, in any

10

 Berthoud Tome II (1802b, p. 223 ff, and Planche XX).

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Fig. 11.8  Jupiter’s dial and satellites. Table.VI

kind of industry, enlightened men, those who see what they are looking at, can distinguish very well what essentially bears the stamp of quackery.11

The part of the machine dedicated to the clock, which can be seen transparently below the planetarium, is a real marine chronometer moved by a spring with a fusee and chain to maintain constant torque, détente escapement and thermal compensation of the length of the spiral. In addition, the balance is suspended on rollers. The design is similar to Berthoud’s chronograph published in 1773 in Traité des horloges marines. Another section of great interest concerns the anomaly of the motion of the Moon with the passages at apogee and perigee (Fig. 11.10). The solution includes the inclination of the Moon’s orbital plane by 5° and the inclination of the Earth’s axis on the ecliptic. The variation due to the orbital plane is realized with another ingenious choice of wheels with variable inclination. I will observe, in conclusion, that this part does not work uniformly; its movement is interrupted after 24 hours and the movement of an average day is accomplished in a few minutes. It seemed to me more useful and more curious to present the phenomena in this way and to preserve the picture of them for twenty-four hours under the eye of the observer. I do not believe, however, that  Janvier (1821, pp. 240–241).

11

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Fig. 11.9  Equation of time, clock 322. Table. XXI

these phenomena can be presented in any other way, since once the hour of the port has been established, the variations dependent on the respective coordinated situations of the Sun and Moon all act together; the change in twenty-­ four hours belongs to all of them, and one would not have a less verisimilar indication of the phenomenon with a continuous movement, than with the precise indication of the day of a jumping calendar.12

With these systems of gears, the elliptical motion of the Moon is combined with the synodic period, which is necessary for the calculation of the tides (fourth dial). The tides are regulated on the synodic revolution of the Moon and altered with the cam mechanism already described. The third dial presents the motion of the Sun, Moon and nodes. The nodes are projected on the equator, the transit of the Sun and Moon to the meridian are also shown. The smaller lower dial shows the calendar with month and day and the time with the equation of time. The phase and eclipse display are achieved with an ingenious construction of superimposed hands. The fourth dial presents the age of the Moon, its distance from the Sun in time and degrees, and the time of high tide in 60 ports. The display of the synodic motion of Venus and Mercury, indicates the distance in tense or leagues of Venus from the Earth.

12

 Berthoud Tome II (1802b, pp. 235–236).

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Fig. 11.10  Left: The inclination of the Moon orbit. The wheel varies its inclination by ±2.5°. Right: Top gears the Sun; bottom gears the Moon Both sides

Janvier’s Opus n. 322 (Fig. 11.11) is completed by a planetarium with a highly original design (Fig. 11.12 left). On the surface, it looks like an armillary sphere, but on closer inspection, one sees that the rings are simply place holders of the outer planets, on which are engraved the periods of revolution and average distances from the Earth in leagues, on the one hand, and the average speed of rotation, on the other. At the center of the four rings, which also include the planet Uranus, at the time still called Herschel, is the Sun, with the two inner planets Mercury and Venus, and the Earth with its axis of rotation correctly inclined by 23°, and around it the Moon, whose motion follows the inclination of its own orbit by 5°. A ring above the Moon has a marker with the positions of the nodes. Another ring below the Earth has an indicator for the hours of the day (Fig. 11.12 right). This examination of the gears of Opus n. 322 is completed with the periods and gears adopted by Janvier, starting with the revolution times of the celestial bodies that were consolidated by Jêrome Lalande in 1795 (Lalande, 1795). Michel Hayard13 has transcribed and revised all Janvier computation from a document by Janvier for Ferdinad Berthoud, in 1800. Berthoud, Coulomb  Personal communication.

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Fig. 11.11  Opus 322. Left: the dial of the eclipse. Right: detail of the indicator of the moon

and Delambre were appointed, by the National Institute, Commisioners to examine a sphère mouvant by Janvier. Their conclusion was: Our conclusions are that the class owes praises and encouragements to the citizen Janvier for the skill, the intelligence and the ingenious combinations which one notices in his moving sphere, and for the new way in which he represented the difference of the true time & the average time. Finally, we think that this machine must make us wish that its author completes promptly the works he is now occupied with. At the National Palace of Sciences & Arts, on the 11th pluviose, year 8 (January 31st, 1800).14

The wheel work and the periods of the Opus n. 322 are summarized in Tables A.17, A.18 and A.19.

14

 Tr. by the author.

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Fig. 11.12  Opus 322. Left: the planetarium. Right: Inner planets and Tellurium

We cannot overlook, in this concise presentation of Antide Janvier’s work, a model of an astronomical clock that has been reproduced several times by Zacharia Raingo (Fig. 11.13). Colonel Quill (1960) describes the procedure for setting the time and the correct planetary configuration, depending on location and date. It is a clock signed Zacharia Raingo (Fig. 11.14), but clearly copied from the work of Janvier, who, as already mentioned, had to sell his books, plans and documents when he went bankrupt. The work described by Quill is in Windsor Castle. Raingo also published a short treatise (Raingo, 1823) in which he deals with the subject of the construction of sphéres mouvent; reading it reveals a superficial knowledge of the principles of astronomy. In fact, he does not expound any calculus regarding gear ratios, while he dwells on the size of the planets and their distances from the Sun for machines of small construction. When compared with the sophistication of Janvier’s mechanisms, Raingo proves to be a disillusion. At the end of this overview, we have a much more accurate view of Antide Janvier, he emerges as an outstanding personality in the late eighteenth and early nineteenth-century. He cannot simply be classified as a watchmaker. With the watchmakers of his era, he shares the extraordinary ability to construct perfect movements both from a purely technical and aesthetic point of view. But these are not simple watches: they are planetary machines, into which he incorporates his profound knowledge of astronomy, cultivated through his friendship with Lalande. At a time when watchmaking had

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Fig. 11.13  Left: Clock with Tellurium, Janvier. Right: Clock signed Raingo. Musée International d’Horlogerie, La Chaux de Fonds

Fig. 11.14  Clock with Tellurium. Windsor Castle, Quill (cit)

already turned into an industry in Switzerland and England, and was also turning into an industry in France, Janvier remained a craftsman, almost a goldsmith, and in fact described himself as the ‘clockmaker to the King’.

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I have not been able to confirm from the documents examined the existence of an atelier in which mechanics and skilled workers collaborated, Janvier employed the work of top-class engravers, goldsmiths, furniture makers, miniaturist painters and enamelers, but his business was not organized as a factory. The French watchmakers of his time were certainly envious of the quality of his work and his success, Breguet, who, unlike Janvier, had an atelier of the highest quality, had contributed to the development of watchmaking science with innovative inventions and solutions, while Janvier emerged more as a keen observer of watchmaking studies with the ability to adopt the most innovative and effective solutions for his work. Perhaps this is why his contribution is not adequately appreciated in works on the history of horology.

François Ducommun Françoise Ducommun (1763–1839) was born in La Chaux-de-Fonds on 6 June 1763 to Abram, a master watchmaker, and Marie Anne Robert, herself the granddaughter of Josué Robert, watchmaker to the King of Prussia.15 Naturally, François continued the family tradition and soon became a mechanical watchmaker, appreciated for his ability to manufacture precision instruments for watchmaking. He took an early interest in astronomy and read Antide Janvier’s work Des révolutions des corps célestes, developing a passion for building planetary machines. In 1816, he embarked on the enterprise of building a large planetarium complete with all known planets, asteroids and satellites, so his work did not include Neptune, which was discovered in 1846, and Pluto, discovered in 1930. Ducommun was neither a mathematician nor an astronomer, so the work of designing the planetarium was not easy. He had to find the data for the periods of the celestial bodies he intended to represent, and above all the calculation of the gear trains. Probably, having read Janvier’s treatise, he resorted to the method of continued fractions, but there is no information about this. The work, which in addition to the planetarium includes a perpetual calendar, is moved by a crank. We are dealing with a model of the known solar system, not with an astronomical clock. The faithfulness to astronomical data can be seen in the functions realized, among which, in addition to the  Anne Jeanneret-de Rougemont: “Ducommun, François”, in: Swiss Historical Dictionary (DSS), version 14.02.2006 (translation from French). Online: https://hls-dhs-dss.ch/it/articles/041307/2006-02-14/ Accessed December 2022. 15

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Fig. 11.15  François Ducommun. Planetarium Globe. Left: Panorama. Right: the whole machine. Musée International d’Horlogerie, La Chaux de Fonds

accurately rendered periods of revolution, the precession of the aspides can be observed. The work was completed in 1817, as can be seen in the engraving François Ducommun 1817 Chaux de Fonds. The work immediately gained fame and many came to admire it at its creator’s home in rue Fritz-Courvoisier in La Chaux-de-Fonds. Ducommun demanded an obolus of one Swiss franc from visitors, which he then donated to the city’s poor. The planetarium has the aspect of a globe, the upper half part can be lifted to observe the planets and their positions with respect of the zodiac (Fig. 11.15). The panoramic view in the photograph16 highlights the size of the machine: the diameter of the Planetarium is 120 cm and its height is 140 cm when closed. On the edge of the cylinder that holds the planetarium are drawn the signs of the zodiac and celestial longitudes; the machine is protected by a cylindrical glass, above which is the hemisphere, painted with the main constellations that, when closed, will make the work appear as a globe. The material of the globe is cardboard painted in oil by Charles-Samuel Girardet (1780–1863), a painter, printer and engraver. The lower hemisphere holds the perpetual calendar mechanism; the year, month and day appear in two windows on the zodiac ring. The dates engraved on the ring of the calendar movement range from 1801 to 1899. The system consists of two modules: a calendar and the planetarium (Oechslin, 2018). The calendar mechanism moves three rings: the one at the left shows the time of day, on the middle the month and number of the day, its wheel has 366 teeth for the leap year; the third ring, right, bears the  The deformation depends on the reconstruction of the panorama from a moving sequence hand-made by the author. 16

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Fig. 11.16  François Ducommun. Planetarium Globe: the inner planets and planetoids. Musée International d’Horlogerie, La Chaux de Fonds

engraving of the number of the year. The calendar is operated with a crank that completes 24 h in one revolution, along its axis there are two jumping mechanisms, the first for jumping the leap day, the second for changing the day-­count to 365 or 366. The jump leap occurs every 4 years and every 100 years except the 400th. The crank acts also on the planetarium, the transmission of motion takes place by means of concentric hollow axes: from the outer to the inner, i.e., from the farthest planets to the nearest ones. The machine includes also the planetoids Ceres, discovered in 1801, Pallas discovered in 1802, Juno discovered in 1804 and Vesta discovered in 1807. These four planetoids or asteroids are located in an intermediate belt of the solar system between Mars and Jupiter (Fig. 11.16). The structure of the Ducummun’s planetarium derives from Antide Janvier’s designs. The moving bodies follow the synodic periods. To simulate the anomaly of elliptical orbits Ducommun puts the center of the orbits slightly eccentric. The eccentricity of planets is summarized in Table A.20, while the tropical periods are in Table A.21. The very small error with respect to Lalande’s planetary data suggests that the values adopted by Ducommun for the construction of his machine are precisely those of Lalande, while for the periods of the planetoids the source is Laplace 1808.

12 Blossoming in Germany and Austria: The Priestermechaniker

In the early eighteenth century, a series of conflicts took place: the War of the Spanish Succession, which ended in 1714. In 1721, the Northern European Wars ended with the Peace of Nystad. The War of the Polish Succession from 1733 to 1738 put Augustus III on the throne. From 1740 to 1748, the date of the Peace of Aachen, more conflicts broke out between Prussia, Spain, and France over the succession of Maria Theresa to the Austrian throne. Emperor Charles VI in 1713 issued an edict, Pragmatic Sanction, that changed the universally accepted Salic Law that precluded female inheritance, extending this right to female members of the Habsburg family. All these conflicts marginally involved Franconia, Bavaria, and Württemberg, which led to more stable local governments, in particular that of Karl Eugen II in Württemberg (1728–1793), who reigned from 1737 to 1793. Throughout Europe, a season of reform began, the Industrial Revolution slowly took shape and on a philosophical and cultural level, the Enlightenment. In Hessen, Baden-Württemberg, Bavaria, Bohemia, and Austria, an important production of astronomical machines and clocks flourished during the eighteenth century. The authors of these works were living in the southern regions of Germany, which favored the development of clock production in Württemberg and the Black Forest in particular and the development of a horological industry later on. These makers were monks or priests, both Supplementary Information The online version contains supplementary material available at https:// doi.org/10.1007/978-­3-­031-­30944-­1_12. The videos can be accessed individually by clicking the DOI link in the accompanying figure caption or by scanning this link with the SN More Media App.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_12

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Catholic and Protestant, who, with the design and construction of these machines, aspired to celebrate the Divine creation of a complex and harmonious Universe, now fully described and understood by astronomical science. Another reason for the flourishing of activity in these places is linked to the important work of reconstruction and restoration of schools, convents, and churches damaged during the Thirty Years’ War. We shall see the works of Johann Georg Neßtfells, Johannes Klein, the Augustinian friar David Ruetschmann David at St Cajetano, Michael Fras who took the name Frater Aurelius at St Daniel, the Augustinian Alexius Johann, Engelbert Wenzel Seige, a Cistercian friar and Philip Mathäus Hahn of the Lutheran faith. These religious also came into contact with the Scottish Benedictine, Bernard Stuart, who visited Regensburg and Salzburg on his travels. The works of these artists and scientists have been examined in detail by Ludwig Oechslin (1996), who called them Priestermechaniker, Mechanical Priests.

Bernard Stuart Bernard Stuart (1706–1755) was born in Scotland as Alexander, where he was educated until 1718. He continued his education in Regensburg, Shottenkloster St. Jacob, under the direction of his uncle Maurus Stuart. In 1725, after completing his theological studies, he entered the Benedictine order and took the name Bernard. In 1730, he became chaplain at the Nonnberg monastery in Salzburg. There he trained as a mechanic, watchmaker, architect, and mathematician. Stuart traveled in Europe from 1730 to 1739; as a Physicist, Mathematician and Architect he went to St Petersburg, Wien, and Augsburg. From 1733 to 1741, he taught mathematics at the University of Salzburg. In 1736, he was appointed Court Architect and collaborated on the construction of Leopoldkron Castle in Salzburg. In 1742, he went to Russia where he taught mathematics at the University of St Petersburg. He returned to Regensburg where he became Abbot in 1743. During a trip to Rome, he died in Ferrara in 1755. In 1731, he designed the mechanism of a luxurious clock for Archbishop Leopold Anton von Firmian (1679–1744), with a Boulle-style case. The builder was Jacob Bentele (1701–1773) and the case design by Johannes Kleber. Under the three dials, we read the inscription: «InVenIt et fIerI P.  BernarDVs StVart FeCIt In LaboratorIo MathesIs Professor sVo SaLIsbVrgensI», and in the central dial: «Ja. Bentele Mechan. Fecit. Jo. Kleber.

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Del: Salzburg. M. Schedl sculp». Under the dials, there is the coat of arms of the Archbishop Firmian (Husty, 1995). Firmian was highly interested into astronomy; the motion of the stars and knowledge of the calculation of time were so great that a constellation of the northern celestial circle was renamed in his honor as “Corona Firmiana”. As reported by franz Michael Vierthaler in 1799, the Corona Borealis was renamed Corona Firmiana. The clock (Fig. 12.1) is surmounted by the celestial globe, supported by the antlers of a couple of deer, and has three dials. The top dial shows the time in 24 h and the time in most important towns of the northern hemisphere. On

Fig. 12.1  Stuart. Astronomical clock. 1731

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the right dial, the hours are shown as Astronomic or Greek hour, Italian hour, Nürnberg hour and Deutsche hour. Moreover, it shows the length of days and nights and rising and setting of the Sun. It also shows a calendar with date and month. The left dial is dedicated to the Moon and shows eclipse, phases of the Moon, and the position of the Sun along the zodiac. The epact and golden number are computed to determine the date of Easter. An accurate analysis of this clock has been published by Peter Frieß (1995), with details on the gear trains. Some data on the periods as computed by L. Oechslin (cit.) are summarised in Table A.22.

Johan Georg Neßtfells Johann Georg Neßtfells1 (1694–1762) was born in Alsfeld, Hessen. Nothing is known about his childhood, his studies and his training years. He learned carpentry from his maternal grandfather Johann Sharff and became a master carpenter in the early 1720s. He soon moved to Wiesentheid to contribute as a carpenter artist to the construction and renovation of churches and convents in the region with the new local Baroque style. Important initiatives flourished in those years, and new monasteries and churches were founded, rebuilt, or restored. In Franconia, in particular, the abbey of Banz, near Bamberg, the Cistercian monastery of Langheim, and the Cistercian monastery of Vierzenheiligen assumed great importance. In these centers, Neßtfells carried out, as we shall see, many important works. Neßtfells decided to reside in Wiesentheid where he married. Beginning in 1731, Gregorius Stumm, the new abbot of the Abbey of Banz, decided to enlarge the building, repairing war damage and reconstituting the library by expanding the collection of scientific works. He then invited Neßtfells, known as a skilled cabinetmaker, to help with the work of beautifying the abbey. Neßtfells arrived at Banz in 1743. He was commissioned to furnish the library hall and the choir decorations. He also provided mechanical models for the scientific cabinet, made gnomonic and ballistic devices, invented a new wheel-cutting machine and a water-powered workshop machine and became also a master miller and the monastery’s headmaster builder. In his own words, his first stimuli came from observing the terrestrial and celestial globes and other precious objects in the library room, including Coronelli’s maps and globes. Hess writes:  The name Neßtfells is frequently changed in Nestfell, Nessfell or Neßfeld, possibly due to variations in spelling conventions between different regions or changes over time. 1

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His spirit had long been stretched toward something higher, and since, according to the unanimous judgment of his contemporaries, the strongest driving force behind all his endeavors was a well-founded ambition to be more than just a cabinetmaker, even early in his stay at Banz he had thrown himself with full zeal into scientific technology, particularly astronomy.2

He had the support of Brother Bonifaz Fleischmann, who taught him Latin and provided him with books on astronomy. Under his guidance, he built a small model of the celestial motions, but as soon as he felt confident enough, he disassembled it, taking only a gear train with him to Wiesentheid as a reminder. He had gained great self-confidence as evidenced by the anecdote that in order to follow the shortest path between Banz and Wiesentheid he crossed fields and woods, following only the directions of a compass. Neßtfells remained at the monastery in Banz for 9 years, sometimes returning to Wiesentheid as he worked out in his mind the creation of a wooden Copernican instrument. He completed the core work quickly by 1745, building the gear train for the planetarium driven by a pendulum clock whose winding lasted 8 days. He obtained the support of Graf Rudolf Franz Erein von Schönborn, lord of Wiesentheid, who summoned him to Wien to present Emperor Franz I with a simple instrument for geometry and astronomy, for which he received a fee of 200 ducats. The emperor commissioned Neßtfells to build a metal machine, based on the wooden model. Once completed, the machine was put on public display in Würzburg, in the house of Balthazar Neumann (1687–1753), a well-known architect of the Germanic Baroque. The work was extensively decorated with an elegant wooden stand and a glass cover through which the periodic motion and orbits of the planets could be observed. This machine (Fig. 12.2) was presented in 1753 to Emperor Franz I. The uniqueness and quality of construction impressed him so much that he awarded Neßtfells a gold medal and appointed him Königlich und Kaiserlicher Hofmechanikus [Royal and Imperial Court Mechanic]. The following year the planetary machine was displayed in the Imperial Court Library, next to the planetarium built by Rowley in 1723 for Prince Eugene of Savoy, purchased by Charles VI in 1737. Neßtfells also fixed this machine, which no one in Wien had been able to repair. In 1756, his wife Anna Christina died, whose three children with Georg had already died as soon as they were born; Neßtfells was thus left alone.  Hess (1908, pp. 19–20).

2

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Fig. 12.2  Neßtfells planetarium. 1753

Shortly thereafter, Count Adam Friedrich (1708–1779), the new Prince-­ Bishop of Seinsheim, asked Neßtfells to build a machine similar to the one in Vienna for the University of Würzburg, which he completed in 1760 together with sculptor Johann Peter Wagner (1730–1809). For this machine, he received a fee of 4000 guilders. In 1759, he converted to the Catholic faith and shortly afterwards fell ill. During his stay in Würzburg, he obtained the title of Astronomische Kunstmeister, under the name Joanni Georgy Neßfeld. It was during this period that, with the help of a faithful pupil, the layman Flosculus, he wrote a description of the Vienna machine, which was published in Bamberg in 1761.3 He died in 1762. The Vienna Planetary Machine  It is among the most comprehensive in representing the solar system. We have recalled that it was initially kept in the Imperial Court Library; later in 1768, it was moved to the K.K. Mechanisch-­ Physikalischen Kunstkabinette [Royal and Imperial Artistic Cabinet of Mechanics and Physics], established by Emperor Franz I by collecting instruments and objects from various collections, and also included the astronomi Kurzgefaßte, doch gründliche Beschreibung der von mir Johann Georg Neßtfell erfunden unde verfertigen accuraten Copernicanischen Planetenmachine, Nebst einer Erklärung Des vielfältigen Gebrauches, und Nutzens Derselben In der Astronomie, Geographie, Und Chronologie [Concise but thorough description of the accurate Copernican planetary machine invented and manufactured by me, Johann Georg Neßtfell, along with an explanation of its various uses and benefits in astronomy, geography, and chronology], Bamberg, Georg Andreas Gertner (1761). Digital version: h t t p s : / / p l a y. g o o g l e . c o m / s t o r e / b o o k s / d e t a i l s ? i d = T J - B t z e y j 4 A C & r d i d = b o o k - T J -­ Btzeyj4AC&rdot=1&pli=1. Accessed October 2022. 3

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cal clock of Bertolla and Borghesi. The Neßtfells machine was then moved by Maria Theresa to the Augustinergang of the Imperial Palace, and in later years it underwent various moves among the imperial palaces to arrive in 1886 in the Naturhistorisches Museum, where it is still kept today, in a room dedicated to meteoriteS. It is an object about 2 m high, the lower part of wood veneer is 1 m high and is in the Baroque style, with griffon decoration. The mechanical part consists of the astronomical machine contained in an octagonal glass case and a front dial showing the time and calendar. In the book Kurzgefaßte … (see note 3), the author among other things illustrates the transit of Venus in front of the Sun, a rare event that occurs with various periodicities: 105.5  years, 8 years, and 121.5 years, and was observed on 6 June 1761. The planets rotate according to sidereal and synodic periods; an epicyclic assembly simulates the change in velocity. In addition to the 5 planets, the machine includes Jupiter’s 4 Medicean satellites, also with sidereal and synodic rotation. The precession of the equinoxes with a period of 25,413 years is also simulated. The lunar, synodic, and sidereal periods include the precession of the nodes with a period of about 18.62 years, and the apsidal precession with a period of about 8.85 years. The lunar motion is represented on the dial, while the planetary motions appear in the planetarium, which can be seen in transparency from the protective glass. The Würzburg Planetarium  It has a different airiness and elegance of construction and is little taller and wider: 2.20 m and 1.15 m. Technically and astronomically, it is on the same level as the Viennese machine (Fig. 12.3). In contrast to the Wiener machine, the 1.08-m-high substructure is very austere, consisting of an octagonal glass theca, which merges into a roof, ending centrally with a horizontal octagonal lid also made of glass. The Rococo decorations found along the roof and those running along the edge of the planetarium case are not well integrated with the machine, probably a capricious whim of the machine’s purchaser, Prince-Bishop Adam Friedrich von Seinsheim. Between the six pairs of Corinthian columns are six small statues, allegories of the planets and the science of Urania (the muse of Astronomy). The whole imitates a fully Baroque Garden architecture. The periods of the Wiener Planetarium are shown in Table A.23.

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Fig. 12.3  Neßtfells Würzburg planetarium

Johannes Klein Johannes Klein (1684–1762) was born in Česká Kamenice in Bohemia. From 1742 until his death in 1762 in Praha, he was Rector at the Jesuit Clementine College in Praha where he taught mathematics and astronomy. He devoted himself to the construction of astronomical instruments and clocks and numerous planetary clocks. He also built a clock for the Sternwart observatory tower at the Clementinum, where he was honored with a large portrait, in which we see a geographical clock in the background. Among his most significant works are a geographical astronomical clock preserved in Dresden at the Mathematisch-Physikalischen Salon (Fig.  12.4 right) and three astronomical clocks preserved at the Clementinum in Prague: one is based on the Tychonic system (built in 1751 or 1756) (Fig. 12.5 left),

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Fig. 12.4  Left: Clementinum library, Praha. Right: Klein, Geographical clock. 1751–1756

Fig. 12.5  Left: Klein Tychonic clock. Right: Klein, Copernican clock. 1751–1763

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the second on the Copernican system (built in 1752) (Fig. 12.5 right), and the third is a geographical clock, built in 1732 similar to the one in Prague. They are all visually striking works in the Baroque style, made of ebony wood and with gilded bronze friezes. The astronomical clocks have an enameled globe; in general, the dials are silver. The Tychonic clock (Fig. 12.5 left) has a gear train that approximates the 365-day period of the year and the synodic lunar period to 29.306569 days. The Moon and the Sun rotate around the Earth at the center, and all the other planets rotate around the Sun. Some constellations are marked between the rotating stars and the 24-h (12 plus 11) scale. The dial of the Tychonic clock was probably designed by Ignaz Friz in 1751. In the Copernican quadrant (Fig. 12.5 right), the Sun is in the center, and some constellations are also engraved. The German and Italian hours are indicated on the back dial. The geographical clock (Fig. 12.4 right) depicts the calendar with a geographical globe and the hours are shown in two groups of twelve for day and night. The periods are approximated: the length of the day is 24 h and the length of the year is 365 days. The geographical clocks have a 24-h time display divided into two groups of 12. On the other side, there is a traditional dial with a 12-h division. The quality of these clocks is particularly high, yet they were not used for astronomical observations. Sybille Gluch (2021) in a study on the accuracy of astronomical clocks recalls that the observatory of the Clementine College tried to equip itself with regulators built in Vienna according to Graham’s method in 1758. The periods of planets of the Clementinum clock are from Oechslin, cit. (Table A.24).

David Rutschmann, Frater David a S. Cajetano David Rutschmann—(Frater David a S. Cajetano—1726–1796) was born in Lembach, near Freiburg am Brisgau, on the edge of the Black Forest, trained by his father as a carpenter. After his studies in mathematics, completed in 1746, his years of pilgrimage (Wanderjahren) soon took him to Wien, where he worked as a carpenter. In 1753, he took his vows and became member of the convent of the Augustinians of the Discalced Feet, Mariabrunn, and took the name Frater David a Sancto Cajetano. In 1760, he moved to the Augustinekirche in Wien. For about 5 years, he attended the teachings of

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Mathematics, Philosophy, and Mechanics given by the Jesuit Jospeh Walcher at the University of Wien. He died in 1796. In 1760, he began the construction of an astronomical clock, which he completed in 1769. It is 250 cm height, 77 cm width, and 49 cm depth. The wooden polished case has glazing on all four sides (Fig.  12.6). The clock movement has a Graham anchor with a second pendulum; the duration is about 32 days. This clock is signed “Fr. David a S.  Cajetano Augustini Discalc.Invenit et Fecit Viennae” and date “1769” on the front dial. On top of it, there is a central dial, 12 small dials arranged in a circle around a large central dial (Fig. 12.6 right). The small dials indicate starting from the bottom one in the center and proceeding clockwise: 1. Hours and minutes of the day

Fig. 12.6  Frater David a S. Cajetano astronomical clock. 1769. Uhrenmuseum, Wien

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2. Sidereal position of the planet Mercury, with a period of 87 g, 23 h, 14′, 4″ (87.2981) 3. Day of the week 4. Sidereal motion of the Moon with the time it takes to reach a node 5. Sidereal position of Jupiter, with a period of 11a, 314 g, 22 h (4332.8026) 6. Epact and golden number 7. Sidereal position of Saturn, with a period of 29a, 167 g, 22 h (10,760,5726). On this quadrant, there is also a double hand indicating the precession of the equinoxes, calculated to be 20,904 years 8. Dominical letter and solar cycle 9. Sidereal position of Mars, with a period of 1a, 321  g, 23  h, 31′, 53″ (687.2368) 10. Synodic period of the Moon, equal to 29 g, 12 h, 44′ (29.53055) 11. Anomalistic month, with a period of 27 g, 13 h, 16′, 35″ (27.5532) 12. Sidereal position of Venus, with a period of 224 g, 16 h, 48′ (224.7) The accuracy of these periods is summarised in Table A.25. The central dial has three rings. In the two outermost rings, it indicates Bohemian time and Italian time with a hand in the shape of the Sun. The third ring stands for the ecliptic and contains the zodiac symbols. Finally, the disk on top has a dragon-shaped hand that shows the position of the Moon relative to the nodes and possible eclipses. Two small windows on the left and right contain, respectively, the size of the eclipse of the Moon and the phase of the Moon. The large central dial has a hand with the Sun indicating day, month, sign of the zodiac, two hands for lunar nodes, two hands for full Moon and new Moon. A two-pointed hand indicates lunar apogee and perigee. Below the dials 83 towns are engraved; their time can be derived from the time pointed by the central hand. Four small circular windows indicate the year, which can add up to 9999. At the time of the photograph in the left, the year 2014 is indicated. On the back dial, solar time is indicated by the golden hand, the blue hand indicates mean time, and a third hand indicates minutes. The upper dial shows the day and in a small window the month. The values of the planetary periods adopted by David are very accurate and respect the data of the time. In Fig. 12.7, we can see a detail of the front dial and the intricate gears that move all indicators. The work is preserved in the Uhrenmuseum der Stadt Wien. In 1786, shortly before his death in Wien, he completed a second astronomical clock, less informative than the previous one.

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Fig. 12.7  Left: Detail of the front dial. Right: the intricate gears under the dial

Michael Fras – Frater Aurelius a S. Daniele Michael Fras, also known as Frater Aurelius a Sancto Daniele (1728–1782), was born in Thuringia in 1728. He joined the Augustinian Order of the Discalced Friars and took on his religious name. Although little is known about his life, he is known to have built pendulum clocks for the Imperial Court in Vienna. Fras passed away in Vienna in 1782. One of Fras’s most notable works is an astronomical clock built in 1769, which has been studied by Ludwig Oechslin (1996). The clock features a rich baroque case crafted by Johann Georg Dirr. In the writing Gründliche Erklärung eines astronomisch und systematischen Uhrwerks, welches P. Aurelius a S. Daniele, Augustiner Baarfüßer in dem K.K. Hofkloster zu Wien, dermaliger Lehrer der mathematischen Wissenschaften erfunden, und eigenhändig verfertiget hat4 published 1770 in Wien, Michael Fras provided a clear definition of the type of astronomic machine he aimed to construct, citing the work of P.M. Hahn and noting the failed attempts of others who had rushed or been inaccurate in their work. Fras then proceeded to describe his own astronomical clock in great detail, which can still be viewed today at the Bayerische Nationalmuseum.  Thorough explanation of an astronomical and systematic clockwork, which P. Aurelius a S. Daniele, Augustiner Baarfüßer in dem K.K. Court Monastery in Vienna, former teacher of mathematical sciences, invented and made with his own hands. https://play.google.com/books/reader?id=kEixU0qj_ JkC&pg=GBS.PP2&hl=it. Accessed November 2022. 4

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The large central dial of the Bayerische clock shows the signs of the zodiac and the indicator of the Sun, which rotates with a period of 365.243056 days. Also on this dial are 360° divided into 12 groups of 30, the typical subdivision for identifying astrological mansions. The other dials show: the lunar node cycle with a period of 6976.54191  days for the calculation of eclipses; the Draconic cycle, to which Fras gave the appearance of a dragon. Other information are the epact, the day of indiction, the day of the month, and of the week, the Italic and Babylonian time, and the period of the apsidal cycle of the Moon of 3230.783626  days. The globe at the top rotates with the daily period. In Fig. 12.8, we see the dial of another interesting astronomical clock, kept at the Akademie der Wissenshaften, Inssbruck. Paul Czermak (1898) published in 1899 an analysis of this machine, which displays the tropic month, the Moon’s apsidal and node cycles. Located on the top of the clock is a smaller dial that displays the civil hour. Directly below it, there is a fictitious sundial featuring a flat hand that imitates the movement of a gnomon’s shadow. The hand moves from the 6 o’clock position in the morning to the 6 o’clock position in the evening, before returning to its original position for the following day. To the sides of the top dial, there are two small dials, both with double hands. The dial on the left shows the declination and rising of the Sun, while the dial on the right shows the length of the day and night. To the left of the fictitious sundial, a longer hand indicates the day of the month, while a shorter hand indicates the month. The months are grouped into four sections for three normal years and one leap year. On the right side of the sundial, a longer hand shows the equation of time. The large central dial features the zodiac circle with its 12 figures, and five hands move along it. The dial is internally divided into 360° and externally displays the 365 days of the year, with the months labeled at the top and the days numbered every five. There are four divisions of days, one on top of the other, accounting for leap years and advancing by 1/4 of a day. The internal degree division has its zero point opposite to the external division, precisely on March 21, marking the spring equinox and indicating the longitudes of celestial bodies. The longest among the 5 hands features a gilded disc that symbolizes the Sun and completes one full rotation around the zodiac in a year. The shorter hand, attached to an eccentric disc, carries the Moon disc and accurately indicates the true length of the Moon through its revolutions. Meanwhile, a gilded hand connected to it by a star-shaped screw also moves along the zodiac. The fourth hand, which extends on both sides and bears gilded letters A (apogee) and P (perigee), indicates the Moon’s farthest and closest positions to

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Fig. 12.8  Astronomical clock by Frater Aurelius, at the Akademie der Wissenshaften, Inssbruck. 1775

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Earth. The fifth hand, represented by a wide gilded dragon, carries the dragon’s head (ascending node) on one side and the dragon’s tail (descending node) on the other, indicating the direction of the lunar orbit’s nodal line against the ecliptic. The apsidal hand requires 8 1/2 years and the dragon’s head 18 1/2 years to complete one revolution. By analyzing the position of these five hands in relation to one another, all the constellations of the Sun and Moon can be determined. Around the main clock face, four smaller ones are placed in the corners, each of lesser significance. In the upper left corner, there is an indication of 7 different musical pieces that a carillon plays one minute before the hour’s striking mechanism commences. At the centre of the large clock face mentioned above, below the concentric axes of the five hands, lies the following inscription: “Pater Aurelius a Sancto Daniele August. Discalc. p.t. captivorum curatus invenit et propriis manibus elaboravit. Viennae 1775”. Furthermore, an engraving is placed on the edge of the zodiac circle which reads: “Pater Aurelius Aug. excudit.”5 The planetary periods of Frater Aurelius clock are summarised in Table A.26.

Engelbert Wenzel Seige Engelbert Wenzel Seige (1737–1811) was a Cistercian monk, born in Bohemia and died in Ossegg, near Praha. His most interesting work, built in 1791, is a Weltmaschine, a machine of the cosmos. The machine (Figs. 12.9 and 12.10) can be observed from all sides. In the middle, a clock shows the annual calendar on the front dial, the time of day on the back dial. To the left of the clock is the planetarium representing the cycle of the day and, to the right, the cycle of the year. Above in the corners are four planetaries representing the satellites of Jupiter and the satellites of Saturn, again the daily cycle and the annual cycle. The fifth planetarium in the top center represents the entire system of planets. The arrangement of the rings recalls the armillary sphere but does not respect its orientation. The machine is preserved in the Technique National Museum, Prague. The periods of this machine are summarized in Table A.27.

 «Father Aurelius of Saint Daniel Augustinian, chaplain of captives, conceived and handcrafted the clock. Wien 1775». « Father Aurelius Aug. made». 5

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Fig. 12.9  Engelbert Seige. Weltmaschine, 1791. Technique National Museum, Prague

Alexius Johann Alexius Johann (1753–1826) was born at Steinach an der Saale, in the former Würzburg province, in the year 1753, November 11th. For a synthetic biography see Franz Christoph Arenz (1829).6 In August 1773, he entered the Augustinian Order; after completing his novitiate; he was sent to Freiburg in Breisgau to complete his higher studies at the university and after completing he was ordained a priest in 1777. Already a distinguished organist, he devoted himself to music, focusing primarily on the art of composition and reaching the point where he was able to compose several pieces of church music: Mass, Vespers, Completories, also some operas; promoted by the Superior Authority of Freiburg, he composed a Requiem, which was performed at the exequia for the Empress Maria Theresa in Munster under his direction. In 1781, he was called to Mainz. Not long after, he was employed as an official teacher at the Gymnasium. He maintained this position for 20 years  Digital version: https://www.dilibri.de/stbmz/content/pageview/2057212. Accessed November 2022.

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Fig. 12.10  Engelbert Seige Weltmaschine, 1791. Technique National Museum, Prague

Fig. 12.11  Alexius Johan. Weltmaschine. Left: Drawing of the machine. Right: The Clock in Mainz. 1802

with the complete satisfaction of his superiors until the dissolution of the Augustinian monastery. During this time, he drew up the plan for his first astronomical clock (Fig. 12.11). He calculated it mostly at night, since during the day was teaching. This clock was finished in 1802. Because of this work of art, the local prefecture sent the drawings to Napoleon and he received a pension of the domestic clergy, although he was not born on the left bank of the Rhine.

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After the termination of his monastery, he lived there for 7 years as a private priest. During this time, for a lover of art, he made the great astronomical clock, which is now in the Museum of the Diocese, and another great Copernican astronomical work. In 1809, still wishing to have a parish, he was entrusted with one in Heidenheim. Here he lived and worked as a pastor to the general satisfaction of his parishioners, until the end of the year 1821, from which he withdrew due to a noticeable decrease in strength, when he felt that in the future, he would no longer be able to carry out his professional duties properly; he died as a vicar in Dom on July 28, 1826. A description of the astronomical machine that includes also some drawings has been written by Arenz in 1829. The machine consists of many parts: a planetary clock with dials distributed on the sides of the base, a placed on the upper plane of the base, a celestial globe rotating with a daily period and around which rotate indicators of the angular position of the planets in relation to the zodiac. Planetary motions are represented both on the dials and around the celestial globe.

The celestial globe on top of the clock (Fig. 12.12) is inside an armillary sphere. On top of the clock and next to the globe there is a tellurium, on which are represented: an arc (that is the circle of daylight) that divides into two parts the surface of the sphere that represents the earth. Moreover, there is a pin which denotes the diameter of the earth. To indicate the phases of the Moon, the sphere of the Moon has the half toward the Earth gilded and the opposite half is black. A pin indicates the lunar eclipses. The Moon globe follows the inclined surface of its orbit by means of a shaft; thus, it moves upward over the ecliptic and downward when crossing the nodes. In Fig. 12.13, on the left, we see the layout of the calendar. In the center, there is the date, below left the day of the week and to the right the month. Below, the four small dials show the year as thousands, hundreds, tens, and units, on top left the Moon phase and on top right the distance of the Moon from the earth along its orbit. The dial on the right in Fig. 12.13 shows the Copernican system, where planets move along their orbits in the same way as in the Huygens planetarium. A detailed study by L.  Oechslin has demonstrated that all the planetary motions are implemented with an epicyclic gear system; their accuracy is reported in Table A.28.

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Fig. 12.12  Drawing and photograph of Johann’s globe. 1802

Philip Mathäus Hahn Hahn, who has been described as a pastor, an astronomer, an engineer and an entrepreneur, is a very special personality in the history we are examining. He always lived in Baden-Württemberg, carrying out his pastoral, scientific and technical activities in various small towns around Stuttgart. The Life  The second son of pastor Georg Gottfried Hahn (1705–1766) and Juliane Kunigunde Kaufmann (1711–1752), Philip Mathäus was born in Scharnhausen auf den Fildern on 24 November 1739. The Hahns were a wealthy family whose origins are documented as far back as the sixteenth century; his paternal grandfather, Matthäus Hahn (1670–1759), was a merchant and his maternal grandfather, Johann Philipp Kaufmann (1661–1748), was a pastor from Stuttgart.

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Fig. 12.13  Alexius Johan, Weltmaschine. Dials

Little is known about Philip’s childhood. He was initially educated by his maternal grandfather, who taught him Latin, Greek, Hebrew, and religion. At the age of eight, Philip developed a strong interest in astronomy. As he writes in an autobiography (Klagholz, 1989; Väterlein, 1989a): I began to observe the Sun, the path of the shadows cast by objects in the house, marked the positions from which it rose and set. And I noted the irregularities of the hours in which the shadows were repeated. Finally, I came into possession of a cylindrical sundial, the workings of which I did not really understand and which not even my father was able to explain to me, but I conducted experiments. I found in my father's library a description of the celestial vault with which I amused myself by learning, at the age of 10, to recognize the celestial constellations and to understand the movement of the Sun along the 13 constellations. At the age of 13, I was able to observe the different forms of sundials in a small book on sundials, owned by a Constable from Eßlingen, I copied some of them, … I gradually learned to make sundials.7

From 1749 to 1754, Philip attended the Latinschule Eßlingen in Nürtingen, where he studied mathematics and continued to cultivate his interest in astronomy, making observations and building sundials. During this time, he began a self-taught study of the Christian doctrine of Johann Arndt (1555–1621), which formed the basis of Hahn’s future Theology and Anthropology. Arndt is considered a forerunner of the Pietist movement.  Tr. by the author.

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In 1756, the Hahn family had to move to Ontsmettingen due to a disciplinary measure of the Consistory against their father Georg, who was accused of drunkenness. It was there that Philip met Gottfried Schaudt (1739–1809), with whom he soon became close friends and with whom he developed a lifelong collaboration. Schaudt had learned the art of making clocks and Philip and Gottfried’s technical skill enabled them to put their ideas into practice. In October 1756, Hahn enrolled at the Protestant theological seminary Tübinger Stift, a prestigious school where Wilhelm Schickard had also studied and taught 150  years earlier, and later educated great intellectuals such as Hegel, Schiller, and Hölderlin. During his theological studies, Hahn deepened his scientific knowledge by studying, still self-taught, the mathematical treatise of Christian Wolf (1679–1754), a pupil of Leibniz and exponent of the German Enlightenment. In 1758, Hahn was granted a 2-year scholarship by the Widerholt family foundation and was also paid thirty guilders to build a sundial for a church, thereby utilizing his mechanical and manual skills. Together, Hahn and Schaudt built several sundials and telescope lenses. In 1760, Hahn finally obtained his diploma as Master of Philosophy in Tübingen where he stayed for about a year to teach privately. In 1761, he was appointed Vicar in Lorch, near Schwäbisch Gmünd in the eastern part of Baden-Württemberg. He then moved to Herrrenberg and in 1764 obtained a Vicariate in Onstmettingen, thus succeeding his father. In Onstmettingen, Hahn set up a workshop in collaboration with Schaudt to build balances, astronomical clocks and machines for ‘the Glory of God’. In 1764, Hahn married Anna Maria Rapp (1749–1775), daughter of Schorndorf mayor Ulrich Rapp. The couple gave birth to six children, two of whom died as infants. All the sons, Christoph Matthäus (1767–1833), Christian Gottfried (1769–1831), Gottlieb Friederich (1771–1802), and Immanuel (1773–1833), developed mechanical and technical interests. Christoph Matthäus and Christian Gottfried, in particular, became skilled watchmakers, and with them Hahn started a production of watches that was commercially successful. In 1767, he built an astronomical clock in bronze and iron, which he presented to Carl Eugen, Duke of Württemberg. The Duke was so admired by the work that he ordered an even larger one for the Library of Ludwigsburg. During his stay in Ontsmettingen, Hahn built many sundials and pocket watches, and in 1769, he gave up the construction of balances to start building a large astronomical clock, called the Weltmaschine, and undertook the construction of a calculating machine, much admired by Duke Carl II of

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Württemberg (1728–1793), who became Hahn’s protector, calling him a ‘divine clockmaker’. In 1770, the Duke offered Hahn the position of professor of mathematics in Tübingen, but he refused, accepting instead to become pastor of the parish of Kornwestheim, where he lived in the rectory and set up a large laboratory. His brothers, Georg David Polykarp (1747–1814) and Ägidius Stephanus Gottfried (1749–1827), collaborated in the new business. In 1775, his young wife died and the following year Hahn married Beate Regine Flattich (1757–1824), who gave birth to eight more children. In 1781, the Duke offered a new, very lucrative position in the parish of Echterdingen, near Stuttgart. Here, Hahn mainly produced pocket watches and wrote a treatise on the construction of clocks. In 1779, he became a member of the Erfurt Academy of Sciences. He died of pneumonia on 2 May 1790. The variety of Philip Matthäus Hahn’s inventions and creations, the care he put into his work, the ability to teach not only his children, make him more than an innovator; he is also the one who contributed to the establishment of a precision mechanics industry in Württemberg. Writings and Works  Hahn wrote at least fifteen theological works, in which he expounds pietistic thinking, based on inner devotion, in contrast to Lutheran rigor and dogmatism. There are numerous technical writings on the construction of clocks and mechanical constructions in general, including: Beschreibung mechanischer Kunstwerke [Description of a mechanical artwork] published in 1774, in which he describes the balances and some astronomical machines; Von Verbesserung der Taschenuhren [For improving pocket watches] published in 1784. Very important is the work on the construction of balances based on Hahn’s invention, which made it possible to replace fulcrum balance (which measured mass by comparison with reference masses) with a balance that measures directly a weight on a graduated scale (Fig. 12.15 right). Various time-consuming and error-prone calculus are required when studying astronomy or building complex mechanisms such as clocks or planetaries. These are mostly arithmetic operations, possibly with logarithms. Blaise Pascal had already invented a mechanical calculator in 1642; Gottfried Leibniz also invented a machine capable of performing all arithmetical operations in 1772. Philip M. Hahn devoted himself to this problem, inspired by the machine

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Fig. 12.14  Left: Leibniz calculator, 1672. Right: Hahn’s calculator

invented by Lebniz.8 Hahn in 1779 writes on a magazine that the major difficulty was the construction of the carry-over mechanism. Unlike Lebiniz, whose surviving machine can be seen in Fig. 12.14 left, Hahn’s machine was so accurate that around five or six examples were produced by him (Fig. 12.14 right). After his death, his apprentices, his eldest son Christoph Matthäus, his brother-in-law Johann Christoph Schuster (1759–1823) and his friend Philip Gottfried Schaudt built several copies of the calculating machine of his invention. Another of Hahn’s collaborators, Jacob Auch (1765–1842) invented other solutions for the calculating machine, all of which are now part of the history of automatic calculation. Clocks and Astronomical Machines  In this field, Hahn developed great creativity and precision, adopting modern construction techniques that would facilitate the development of a watchmaking industry in Baden-Württember in the years following his death. Attention to the esthetic quality of pocket watches is evident, which we will find in astronomical clocks and planetary machines. From a technical point of view, the watches made by Hahn and his sons adopt an assembly with a full platine. Hahn imported the cylinder escapement from England, which enabled the construction of thinner and more accurate watches; this escapement was later adopted by other European manufacturers. Hahn tried to improve the precision of the cylinder mechanism; he also started studies for the construction of chronometers, but no achievements exist. He invented an original mechanism for repeating the chime, which was particularly useful during the night.  https://history-computer.com/philipp-matthaus-hahn/. Accessed May 2022.

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Fig. 12.15  Left: Pocket watches by P.H.  Hahn. Right: a balance invented by Hahn. Onstmettingen Museum

Fig. 12.16  Ludwigsburg Weltmaschine. 1769. Left: Right: Scaled-down copy of the original baroque case

Many pocket watches (Fig.  12.15 left) include complications that make them almost astronomical watches, including a calendar with the months and signs of the zodiac on the outermost scale, small dials indicating the time, the day of the week, the phase and age of the Moon, the minutes and the day of the month. Ludwigsburg Weltmaschine, 1769  Built in 1769 with the collaboration of Schaudt, this spectacular machine shows, in the central column, a series of dials indicating the time, the calendar and the year counter. On the left is a planetarium and, on the right, a celestial globe. The present case, made of walnut wood, was built after World War II. The original case was in the Baroque style, a copy of which is on display in the Ontsmettingen Museum (Fig.  12.16 right); in 1820, it was replaced by a mahogany case (Fig. 12.16 left).

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It took Hahn 18 months to build this machine while in Ludwigsburg. A description of this machine by Hahn has been published in 1774 in Stuttgart with the title Beschreibung mechanische Uhrwerkewelche unter der Direktion und Anweisung M. Philipp Matth. Hahns, Pfarrer in Kornwertsheim, durch sein Arbeiter seit Sechs Jahren verfertiget worden sind [Description of mechanical clockworks, which have been manufactured under the direction and instruction of M. Philipp Matth. Hahn, pastor in Kornwertsheim, by his workers for 6 years.]. It was moved several times over the decades until it was placed in the watch museum section in Württemberg Landesmuseum. The central dial has a calendar indicating the day, month, day of the week and the position of the Sun in the zodiac. The dial at the lowest position contains Bengel’s universal chronology, the hand indicating the current year. The planetarium on the left has all the known planets and the satellites of Jupiter and Saturn. The planetarium is surrounded by the ecliptic ring and the meridian ring; the ecliptic disc is engraved with the signs of the zodiac and the 360-degree division so that the position of the planets can be observed. Around the celestial globe, on the right, suspended from a curved string, revolve the planets and the Sun; the globe is surrounded by the ecliptic ring and the meridian ring; on the ecliptic disc are engraved the zodiac signs and the division into 360°. The motion of the planets adopts an epicyclic mechanism to simulate the anomaly. The periods of this machine, as computed by L. Oechslin (cit.), are summarized in Table A.29. Nürnberg Weltmaschine, 1770–1790  Philip Mathäus Hahn worked continuously on this machine for more than 20 years with the collaboration of his brothers, but it was not completed during his lifetime. Careful analysis revealed the later addition of Uranus’ satellite system to the 4 planetary systems. The baroque case for this machine no longer exists, and today it is mounted in a modern Plexiglas and metal system so that the mechanism is fully visible (Fig.  12.17). It is located in the Nürnberg Germanisches Nationalmuseum. The dials show: at the top the hours, minutes, and seconds; in the center the month, day of the month, day of the week and 24 h; at the bottom the year according to Bengel’s chronology. The clock mechanism has a period of one second and is driven by weights and motion is transmitted by gear wheels; the perpetual calendar is driven by a double-epicycle system.

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Fig. 12.17  Nürnberg Weltmaschine, 1770–1790

On the left is the system of the individual planets, operated by a crank, with their satellites, the Earth with the Moon, Jupiter with the four satellites, Saturn with the five satellites and Uranus with two satellites, which was added after 1790. On the right is the Copernican planetary system with the Sun in the center, the inner planets Mercury and Venus, the Earth with the Moon, and the outer planets Mars, Jupiter, and Saturn. Separately, in the front, the Geocentric system revolves around the celestial globe, with the Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn, and Uranus. The lunar and planetary nodes are depicted; the globe is oriented according to the celestial equator. The case should have been in the Rococo style; various drawings and watercolors exist. The work underwent many transformations, for example, the perpetual calendar originally used a five-pointed dial indicator, as we will see later in a tellurium; it was later modified by adopting an English-style movement. The system on the left of the individual planets is the same as in the Ludwigsburg machine. It can be seen, both in this work and in the Ludwigsburg Weltmaschine, that the periods of the planets differ between the planetarium and the geocentric globe, in which the error is particularly high, except for Venus (see Table A.30).

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Gotha Weltmaschine, 1780  This work was built by Philip Mathäus Hahn and his son, Georg David Hahn, during his time as parish priest in Kornwerstheim for the Physikalischen Kabinett Herzogs Ernst II. von Sachsen-Gotha-Altenburg (1745–1804). In the centre, the dial shows the 24 h, divided into daylight and night hours. The small dial at the top shows the calendar with the date, month, and day. At the bottom is the hour dial in Arabic characters and four small dials with Arabic numerals for the chapter ring. On the left a Copernican Tellurium, the Earth rotates around the Sun in 365  days 5  h and 49  min; it rotates on its own axis, tilted 23.5°, in 23  h 56 min and 4.5 s. Around the Earth the Moon rotates, indicating the phase. The rings represent the shadow line and the time of sunrise and sunset. On the right the Copernican system, Mercury rotates along an elliptical path, Jupiter and Saturn have 4 and 5 satellites, respectively. The periods of revolution of the Moon and planets are exact within 1000 years. A complete view of this spectacular machine is in Fig. 12.18. In the center, above, we find the celestial globe representing the motion of the celestial courses as observed from Earth, thus a geocentric system of apparent motion. The axis of the Globe, tilted like the Earth’s axis, makes a rotation of 1° every 72  years, simulating the precession of the equinoxes. The Sun’s

Fig. 12.18  Gotha Kopernikanische Weltmaschine, 1780

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circle has a diameter such that it casts a small shadow that extends for the duration of twilight at sunrise and sunset. The arrangement of the gears of the celestial globe is very compact and typical of the concentric axis solution. We do not report the planetary periods, which are almost identical to those of the other works; again, the Copernican planetary periods differ from those of the celestial globe. The periods have minimal differences from the Nürenberg Weltmaschine, so it is not worth summarizing them. Furtwangen Planetarium, 1774  It is a complete planetarium (Fig. 12.19) with a clock, illustrating the motion of the planets and the satellites of Jupiter and Saturn. It is enclosed in a glass case and has a pendulum clock with calendar. The planetarium is operated by a crank. Hahn built it during the Kornwestheim period in 1787. It was purchased by the Margrave of

Fig. 12.19  Furtwangen Planetarium, 1774

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Württemberg as a teaching instrument for the Collegium Illustre, today the University of Karlsruhe. In Fig. 12.20, we can see the outer planets and details of Saturn’s and satellites. It is exhibited in the Uhrenmuseum in Furtwangen. It has a diameter of 140 cm and a height of 122 cm. The planetary and satellite periods are summarized in Table A.31. Globusuhren  Hahn built several clocks surmounted by a celestial globe and with different dials. Some of these clocks have the tellurium mounted under the globe, such as the one from 1785 preserved in Darmstadt, Hessisches Museum. An example made in 1770 is in the Landesmuseum in Stuttgart. One dial shows the time, on a second dial is the calendar with a five-pointed indicator (Fig. 12.21 left), on a third dial there is a small tellurium (Fig. 12.21 right) and on the fourth dial there is Bengel’s chronology. The interest to these astronomical clocks is not on the implementation of the wheel work, rather on the layout of the different dials. Winterthur Tellurium  The machine is moved by a crank that acts on a calendar on the small central dial, on which one notices the use of a five-pointed indicator to correctly mark the day of the month. This method was adopted by Hahn in various astronomical clocks, like the above-mentioned Globusuhr of 1770. According to Oechslin’s studies, this machine is perhaps one of the earliest built, as is evident from his use of the 5-point date indicator, which was only used for a few other astronomical clocks and later abandoned. The

Fig. 12.20  Furtwangen Planetarium, 1774. Left: Outer planets. Right: Saturn and satellites

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Fig. 12.21  Globusuhr, 1770. Left: calendar dial with the five pointers. Right: Tellurium rotating with the hand. Landesmuseum Württemberg, Stuttgart

calendar is also indicated along the large disc on which the entire moving block of the Earth and Moon leans, representing the ecliptic. Due to mechanical backlash, this indication is not very precise, perhaps for this reason Hahn decided to duplicate it on the small dial (Fig.  12.22). One rotation of the crank corresponds to 24 h. This is definitely a machine created for educational purposes; Hahn’s handwritten description, preserved with the machine (Fig.  12.23) at the Uhrenmuseum in Winterthur, provides a very detailed overview of astronomical knowledge relating to Lunar and Earth motions, explaining the succession of seasons and the manifestations of eclipses.9 Oechslin notes that in this machine there are errors in the gear ratios and in the reversal of the direction of rotation, which affect the number of sidereal or solar days in the course of a year.10 We note that the planetary periods of these machines differ, even if only slightly, from each other; we could say that there was no progressive refinement, as if each machine was born by itself (see Table A.32).  The translation by the author of Hahn’s description is available in the web site of this book (See “Hahn. pdf ” in supplement). 10  Oechslin (1996, p. 198). 9

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Fig. 12.22  Hahn, Tellurium. Winterthur. Left: A short video of the motion of the machine. Right: Note the five pointers of the calendar for determine the current month (▶ https://doi.org/10.1007/000-­a97)

The Universal Chronology The origin and age are a relevant aspect of the vision of the Cosmos. Many scholars addressed this theme, and during the eighteenth century, it was part of the effort to reconcile the new astronomy with the Holy Writings. In some of the clocks built by P.M. Hahn there is a special dial (Fig. 12.24) that reports a chronology from the origin of the world. Hahn was using the chronology published by Johan Albrecht Bengel in 1745. The title of this work (Bengel, 1745) is: CYCLUS, sive de Anno Magno, Solis Lunae Stellarum, Consideratio ad incrementum doctrinae profeticae atque astronomicae accomodata [CYCLUS, or of the Great Year of the Sun, Moon

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Fig. 12.23  Hahn, Tellurium. Winterthur. 1745

and Stars. Consideration to increase prophetic and astronomical doctrine adjusted].11 Before Bengel, Kepler recognized the Hand of the Creator in the celestial harmony, expressed by the symmetry and elegance of the platonic solids and in the musical ratios hidden in the planetary orbits. Kepler also ventured into the study of chronology and published Chronologia a Mundo condito ad finem Politiae Iudaicae deducta [Chronology from the Foundation of the World Deducted to the End of the Jewish State] (Kepler, 2009a) written between 1610 and 1627. The chronology that Kepler constructs is based on the age of the Patriarchs as described in the Genesis and on other Books of the Bible. He determines the date of creation in the year −3993. To confirm this dating, and to go beyond his predecessors, Kepler applies his astronomical knowledge, considering the sky as a calendar. He assumes that at the time of the creation the planets were all set in an orderly geocentric arrangement. The ascending and descending nodes of all planets had a longitude of 0°. With a complex astronomical calculus, he found that the possible date of this planetary configuration was July 24th −3933. Isaac Newton too wrote Chronology of Ancient Kingdoms Amended,12 a work published in 1728 after his death. He exposes the chronicle of ancient  The translation from Latin of Bengel’s book is available in the web site of this book (See “Bengel.pdf ” in supplement). 12  https://archive.org/details/thechronologyofa15784gut. Accessed October 2022. 11

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Fig. 12.24  Dial of Hahn’s Globusuhr 1760 with Bengel chronology

kingdoms and makes use of astronomical calculus to demonstrate the correct dating of historical events. Bengel, a theologian of the Pietist tradition, was born in June 1687  in Winnenden, Württemberg, the son of a pastor and of the daughter of a high priest and member of the Consistory of Württemberg (Ehmer, 2008). He died in 1752  in Stuttgart, having occupied a prominent position in the Württemberg Parliament since 1748. Bengel was the initiator of biblical philology with the editing, in 1734, of an early critical text of the Novum Testamentum Grecum, the so-called Septuagint Version. He classified the variants of the text, introducing a scientific method of exegesis of the Holy Scriptures that was certainly original and inaugurated a modern strand of study. In Cyclus Bengel gives an interpretation of the Apocalypse of St. John. He adds the number 666 to the date 1143. Pope Innocenzo II died in 1143, and the new elected Pope was Celestino II: it was an epoch of turmoil for the Church and of anti-papal revolts. The year 1809 would thus be the end of a cycle and the beginning of a new one. Bengel goes further in his study and makes an effort to express the planetary periods as integer numbers. The total

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age of the world should be 7777 1/9 years, and he considers this number as seven millennia, each made of 1111 1/9  years. He calls this time interval chrono. From the chrono, he then derives all the planetary periods, comparing his results to those discovered at the time. This attempt to express cycles and periods with whole numbers was a complete failure, strongly criticised by his contemporary. On the other hand, the chronology was widely accepted. The Hahn’s chronology dial is divided into centuries interwoven with the chroni. Each century in the world calendar is associated with some historical event. The birth of Christ, for instance, falls in the middle of the world’s duration. The year −3343 since the creation of the world is the first year of Christ’s birth. Two hands indicate the dates dial. The longer one points to the two outermost scales, a century divided into 1-year intervals. The shorter hand points to the inner sector made of 4 circles, and it completes its theoretical turn in less than 8000 years. The innermost circle brings the names of the seven chroni. The next circle contains the major events of the seventy centuries. The next circles count the universal time in two circles corresponding to before and after Christ.

13 Chinese Philosophical and Mathematical Thought

Summarizing the development of mathematics and astronomy in China in a few pages is challenging, Anyway, a brief historical background is necessary, all the more so because of our scarce knowledge of the Chinese culture. I will necessarily limit myself to trace only an extremely superficial periodization of a history of more than 4000 years.

Kingdoms and Dynasties It is customary to date the origin of Chinese historical periods to around −2000, when the first social organizations emerged in kingdoms with a central administration. Before that time, archaeological findings show the presence of human groups distributed throughout the Chinese regions and dating back as far as −20,000.1 The Peking Man (Synanthropus Pekinensis) was identified in 1927. He lived in the early and middle Pleistocene and predates the Neanderthal. Windblown silt from the Gobi Desert area during the Quaternary period formed extensive löss deposits.2 The Löss Plateau favored the settlement of new peoples who practiced agriculture and animal husbandry, evidenced by signs of hundreds,  J. Needham’s work, in its first volume, provides a comprehensive overview of the history of China, as described in Needham and Tsuen-Hsuin (1958). For a more accessible read, Wood (2021) offers a less scholarly account. 2  Löss is a sediment produced by windblown sand or dry silt from desert regions or produced by soil erosion. In northern China, the Löss Plateau dates back at least two million years and consists of alternating layers of löss and soil. 1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_13

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perhaps thousands, of villages. Caves were excavated along the high walls of the Löss to house and shelter ancient peoples, such as those in the western Gansu region. The cultivation of cereals soon shifted to rice. In China, the presence of three great rivers—Yellow River (Huang He) in the north, Azure River (Yangtze) in the center, and Pearl River (Zhujiang) in the south—divided the country in two main areas, both with fertile land, fostering intense agricultural activity and a flourishing civilization. The northern region is roughly bounded by the Yellow River basin. The southern region lies in the basin of the Yangtze and Zhujiang (Fig. 13.1). Chinese Neolithic civilizations differed in different regions in pottery-making techniques and the breeding of different animal species. Xia Kingdoms (−2000 c −1600 c)  This is a legendary period. The mythical King Yu is said to have repaired the damage of the Yellow River flood, receiving for his bravery the Mandate of Heaven to rule. Recently, archeologists found traces of a dramatic flood in that epoch, thus confirming some stories.

Fig. 13.1  China’s main rivers. The Löss Plateau is roughly north of Xi’an and south of Yellow River

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Shang Dynasty (−1675−1046 c)  The Chinese Bronze Age develops from the −1600s, at the time of the Shang Dynasty and reaches very high levels of esthetic quality, with the construction not only of war weapons and chariots but also of vases, bells, and luxury items. To this same period dates the cultivation of grain, probably imported from the Middle East. The Shang period also documents the appearance of writing, practiced on bamboo tablets, animal bones, or tortoise shells. Several ethnic groups have been identified among the ancient Chinese populations: Turkic 3 from Central Asia in the north-­western regions, Tungusi in the north from the Siberian regions, Tibetans in the western regions, and populations collectively grouped as Thai, of southern and coastal regions. The contribution of these ethnic groups has enriched Chinese culture with a heritage of innumerable objects, traditions and rituals: bows and arrows, tattoos, Dragon mythology, spring festivals, and the use of drums are in fact common to all these peoples, and even the use of ropes with knots for counting is similar to the quipu of South America. This is evidence that the ancient Chinese world was not isolated, it had relevant communication with the Euro-Asian, South Asian, and island regions and even the American world. It is a history that, after about −1600, is fully and thoroughly documented in the annals of each kingdom. A summary of the dynastic s­ uccession in the historical period is summarized in Table 13.1. Zhou Dynasty (−1045 c. −256)  This period is followed by about 800 years of the Zhou dynasty. The peoples of the Zhou period are of western origin and spread through the Yellow River valley and the northern plains. The social structure was consolidated and reached a level of organization similar to that of feudal Europe. The nobility of the previous dynasty is incorporated into the new system by assuming powers in the various fiefdoms into which the empire is divided, allowing traditions to be preserved in a process of cross-­assimilation. The economy remains based on agriculture and the collection of tribute moves up the hierarchical chain of nobility to the imperial level. Acts of devotion are owed to the emperor by the nobility, who must periodically visit the capital Xi’An of the empire to pledge allegiance. The Zhou dynasty is divided into two periods: the Western Zhou (−1045 −771) and the Eastern Zhou (−770 −256), which in turn is subdivided into the

 Turkic peoples are ethnic groups from Eurasia and Central and North Asia including Siberia, who speak a family of languages that originated in East Asian regions from Mongolia to north-western China and spread during the first millennium B.C.E. to Central Asia. 3

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Table 13.1  Kingdoms and dynasties Ancient period Xia Kingdom Shang Kingdom Zhou Dynasty (feudal age)

−2100 c. −1600 c. −1600 c. −1046 c. −1045 c. −256

Imperial period Qin Dynasty Han Dynasty

−221 −206 −206 +220

Three Kingdoms

220 –265

Jin Dynasty

265 –479

North and South Dynasties Sui Tang Five Dynasties Period

420 –589

Song

960 –1279

Yuan Ming Qing Modern period Republic of China Popular Republic of Chian

1271 –1368 1368 –1644 1644 –1911

581 –618 618 –907 907 –960

1912 –1949 1949 today

Western Zhou Eastern Zhou Period of Springs and Autumns Warring States

−1045 −771 −770 −256 −722 −480

Western Han Xin Eastern Han Wei Shui Wu Sixteen Kingdoms Western Jin Eastern Jin Song

−202 +9 +9 +23 +25 +220 220–265 221–264 222–280 304 439 265 –317 317 –420 420 –479

Liao West Liao Northern Song Southern Song Xi Xia

937 1125 1125 –1211 960 –1126 1127 –1279 990 –1227

Republic of Chian (Taiwan)

1949 today

−480 −256

Period of Springs and Autumns until −481 and then the Period of Warring (or Fighting) States until the end of the dynasty in −256. It was not a time of peace; numerous conflicts arose both against invasion attempts by ‘barbarian’ forces and among the nobles themselves. In −771 Zhou Emperor Yu Wang was assassinated by the armigers of a minor feudal lord allied with the barbarians. His successor had to abandon the capital and the various feudal lords set up small kingdoms, making the Zhou empire de facto very small: Eastern Zhou.

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In the Zhou period, the feudal structure was divided into five tiers, with titles usually translated as duke, marquis, count, viscount, and baron. The population was also divided into groups of petty nobility, knights, scholars, peasants and farmers, craftsmen, and merchants. The community of scholars became increasingly influential, running for positions of service to power. It was in this era that the writings became associated with the author’s name. Confucius (−551 −479) lived in this period. He is considered the originator and advocate of the principles of good governance and established the guiding principles for the creation of the government bureaucracy, trained in the study of Confucian principles and selected on the basis of merit. In −318, the first academy, Jixia (Chi Gate Academy) was founded by King Xuan in the capital of the state of Qi (see Fig. 13.2), which attracted scholars from all regions. Teachers from different philosophical schools taught there. The period when the greatest development of Chinese philosophical thought began is coeval with the founding of Plato’s school, although there is no direct relationship between the two. Studies and the development of new techniques also flourished during this period, so much so that, by analogy with the periodization of Greek civilization, this era is considered the “classical” period of China. Between −480 and −221 (Warring States period, Fig.  13.2), wars led to a fragmentation of the feudal state. The feudal fragmentation was accompanied by the development of new weapons and continuous conflicts for the domination of new territories. These conflicts were incompatible with primary needs for the entire community, mainly the care of water regimentation and irrigation systems that required large labor forces. The Yellow River was subject to frequent and devastating floods, requiring the construction and maintenance of canals and dams. This was necessary to strengthen the central state against feudal powers that had been gradually weakening. The grandiose work on the construction of canal systems, serving as transport and irrigation routes for agriculture, was a key factor in strengthening the central power, pushing toward unification and the subsequent development of Chinese civilization. Qin Dynasty (−221 −206)  The Warring States period ended in −221 with the First Unification under the leadership of the Qin dynasty, and Shi Huangdi became Emperor of China. Under this dynasty, the system of state management was consolidated, the backbone of which was the bureaucracy formed according to Confucian principles. Fiefdoms were expropriated, and the nobility was forced to reside in cities. The Great Wall, a construction begun under the previous reign, was expanded, strengthening defense capabilities and reducing the risks of mixing

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Fig. 13.2  The Warring States

with neighboring nomadic peoples. The capital of the empire was Xianyang near present-day Xi’an; a rich tomb was built near the capital for the emperor, where the renowned Terracotta Army was found. After the death of the emperor, who was called the Great Unifier, his son was unable to continue his father’s policies and revolts broke out that ended after 8 years when the imperial army officer Liu Pang founded the first Han dynasty. Han Dynasty (−206 +220)  The Han period was marked by a partial return of feudal power, which nevertheless had to coexist with a central state that was impossible to eliminate and a bureaucracy that easily adapted to the return of the feudal system. The government was authoritarian, and the officials and feudal nobility acted cruelly following the principles of the philosophical school of legalist (see section “Philosophical Schools”).

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These excesses caused a resurgence of interest in Confucian principles, which guided methods of governance that respected basic ethical principles. Bureaucratic orthodoxy, known as Han Confucianism, was re-established and would continue for hundreds of years. During the Han Dynasty, in fact, the emperors realized that in order to be independent and protected from the conspiracies of the feudal nobility, the central government needed capable, competent, and learned officials, requirements that were independent of noble birth. A public career open to talent and totally meritocratic was born. Another expedient to avoid the return of the feudal system based on inheritance of property and titles was to entrust the emperor’s protection to figures unable to reproduce: eunuchs. They soon became a caste surrounding the emperor and his family, acquiring great power and wealth, to the point that many families offered their emasculated son to the Imperial Court to gain recognition and wealth for the entire family. During the reign of Emperor Wu Tong (−140 −87), considered one of China’s greatest emperors, the empire faced serious economic crises due to rising prices imposed by merchants, themselves subject to oppressive taxation and regulations, and to the costs of the wars against the population of the northern steppe and to conquest Korea. The officer Zhang Qian (−164 −113) was sent on a diplomatic mission to Bactria in −138 and after a 10-year captivity, he was able to explore as far west as the Persian Gulf. Returned to China, he brought great varieties of plants and other natural products. This explorer’s itinerary traced the first route of the ancient Silk Road. The last period of the Han dynasty is of particular interest as it is the era in which the Chinese sciences developed, in particular astronomy—which led to the definition of the structure of the Chinese Calendar—zoology, botany, and earth sciences. The invention of paper, improvements in pottery techniques, and the first porcelain creations, as well as improvements in textile techniques that were achieved more than a thousand years later in Europe, date back to this era. Toward the end of the dynasty, the central state began to lose strength. Palace and peasant revolts strengthened the generals who sought to reintroduce fiefdoms. Three Kingdoms (+220 –265)  The First Unification lasted about 400 years and ended with the Three Kingdoms, Wei in the north, Shu in the south-west, and Wu in south-east. The division of these kingdoms reflects the division of China’s main agricultural areas. Each of the kingdoms protects its own area of influence that provides the financial and food resources essential for maintaining power and running the state. The formation of these kingdoms also reflects

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the future course of economic development: initially, the northern regions along the Yellow River valley developed agriculture. Later agriculture ­developed also in the central regions along the Yangtze River. Finally, agricultural development in the south reached a level that allowed independence from the central state (Fig. 13.3). From then on, periods of fragmentation of the Chinese state alternated with periods of unification, partly overlapping, when different kingdoms united or split. Jin Dynasty (265–479) The First Partition was short-lived: it lasted from +265 to +280. The Second Unification brought together under the Jin dynasty the western regions from 265 and the eastern regions from 317. In the northern regions frequent local conflicts lead to alliances with tribes of Mongols, Huns, Turkic who gain power in some cities forcing the imperial court to move to Nanking in the central-eastern region. In the north between 304 and 535, Mongol, Hun, and Turkic dynasties alternated with three dynasties of purely Chinese origin. However, these invaders were assimilated into

Fig. 13.3  The Three Kingdoms

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Chinese society and culture much more than the Chinese assimilated the culture of the “barbarians.” Continuing conflicts led some of the population to find comfort in the Buddhist religion, which began to spread throughout the country. At the same time, Taoist culture, oriented toward speculation on nature and science, grew stronger and gave rise to scientific studies. Sui Dynasty and Third Unification (581–618) The Third Unification occurred under the two Sui dynasties. Yang Jien defeated the Northern kingdoms in 587 and established the new dynasty. To overcome the divisions between regions that had led to centuries of conflicts, Emperor Wen Di devoted great resources to building a system of canals to facilitate communications. A transportation route for goods and people was built to avoid long overland journeys and risky coastal shipping. The effect of this work was also to unify the agricultural capacity, removing the economic basis for the formation of opposing fiefdoms and kingdoms. The Sui kingdom was too short lived to have any influence on cultural development, but Confucianism was nonetheless strengthened in the Empire. Tang Dynasty (618–907) An unfortunate expedition against Korea provoked a revolution led by Li Yuan, from a family linked to the Sui and Turkic tribes. Li Yuan ascended the throne in 617: the first Tang Emperor. During the Tang dynasty, in 750, China achieved its greatest expansion, with the conquest of Manchuria, Korea and the extension of Chinese sovereignty into Xinjiang. After that a slow decline began, marked by the final defeat of the Chinese army by Arab forces in 751, which resulted in the loss of dominion over Xinjiang; a region that had formed a Buddhist religious barrier against the expansion of Islamism. Mongolia and the Uighur region also became independent, as did the ­southern Thai regions, whose principalities took over present-day Yunnan. In the North-East, the Tartars, who later founded the Liao dynasty (907–1125), conquered Manchuria and Korea from the Chinese Empire. Relations with the Tibetan peoples deteriorated to such an extent that in 787 an Arab-Chinese alliance was concluded, with an envoy from Harun alRashid, against the threat that also hung over the Caliphate and Muslim Central Asia. Despite military instability, China became a center open to foreign presence, including Arab, Syrian, and Persian scholars. All this favored the expansion of Buddhism, but already in 845, the Confucian bureaucracy, which saw the formation of a state within the state, destroyed 4600 Buddhist temples, forced the secularization of 260,000 monks, abolished 40,000 shrines and enslaved 150,000 Buddhist followers.

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During this dynastic period, the school system to train the bureaucratic class was strengthened and the system of laws was codified. Song Dynasty (960–1279)  From the point of view of the development of science, however, the next period is more important, that of the Song dynasty, which started the reunification following the fragmentation in 906 the period of the Five Dynasties. Despite its military weakness, the Song dynasty is considered the high point of Chinese civilization, the Golden Age. The economy, underpinned by great technological innovations, had reached a level of sophistication never reached before. The population reached more than 100 million people, whose prosperity was sustained by widespread rice cultivation and by coal used for heating homes and for making bronze and clay artefacts. A dense network of canals, the result of highly advanced hydraulic engineering, connected the main rivers. Ships of colossal size for the time, especially compared to those built in the West, allowed sea transport, enhancing trade. The capital was initially Kaifeng in Henan and later Hangzhou in Zhejiang, where the Grand Canal that connects major eastern cities with Beijing ends. Both cities were very populous, with a ruling class composed of the state bureaucracy and a middle class that we might call a commercial and artisan bourgeoisie. During this dynasty, there were great advances in science and technology. Among the leading scientists and intellectuals of the time were Su Song (1020–1101) (to whom we will return later) and Shen Kuo (1031–1095). Gunpowder, invented during the Tang Dynasty, was used systematically in military conflicts during the Song Dynasty. Siege war machines and new types of firearms were invented and also mounted on warships that were widely used in the great battles on the Yangtze River. This grandiose development ended in defeat by the Mongols led by Genghis Khan. For about 30 years, the military forces of the Song Empire managed to counter the Mongol advance, but in 1276 the southern capital Hangzhou fell and the naval fleet was completely destroyed 3 years later at the Battle of Yamen in 1279. Yuan Dynasty (1271–1368)  Kublai Khan (1215–1294), grandson of Genghis Khan (1162–1227), proclaimed himself emperor of China in 1271, opening the period of the Yuan Dynasty. Marco Polo’s travels and his stay at the imperial court lasted from 1271 to 1295.

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Ming Dynasty (1368–1644)  During the Ming Dynasty, Chinese science slowly declined, many books were lost, and the schools’ ability to transmit knowledge gradually waned. The effects of this decline, due to internal reasons, bear similarities to the collapse of Greek civilization at the time of the Roman conquest. On the other hand, it was also an era that initially engaged in exploration and travel. Zheng He (1371-1473) a eunuch of Islamic faith who was promoted to the position of admiral, created a massive fleet of giant junks with which he explored the Indian Ocean, traveling as far as the coast of Africa and bringing back unknown animals and products. The fleet was disbanded in 1433 after his last voyage because of excessive costs, just few years from the beginning of world exploration by the Western countries. China opened up to Western scientific knowledge with the arrival of Jesuit missionaries. Between 1587 and 1610, the Jesuit Matteo Ricci (Fontana, 1996),4 called Li Madou by the Chinese, stayed in the Celestial Empire (Fig.  13.4 left). To accomplish his mission of converting the Chinese to Christianity, he adopted local customs and rituals. He wrote moral booklets and translated Euclid‘s Elements, thus becoming an esteemed intellectual. His

Fig. 13.4  Left: Matteo Ricci, portrait by Manuel Pereira in 1610. Right: Ferdinand Verbiest  Through her biographical work on Matteo Ricci, Fontana captures the complexity and richness of Chinese culture during the turn of the sixteenth and seventeenth centuries, providing readers with a captivating window into this fascinating period of history. 4

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Fig. 13.5  Matteo Ricci’s map. Anonymous edition, probably 1604

fame led him to be welcomed in the capital and to dialogue with the ministers of the Court. Matteo Ricci is also credited with founding the Beijing Astronomical Observatory. The cultural success of the Chinese missionaries went so far that the Jesuit Ferdinand Verbiest (1623–1688) (Fig. 13.4 right), called Nan Huairen in Chinese, was appointed by the emperor as director of the Office of Astronomy in 1668. Ricci’s fame grew further when, out of respect for Chinese tradition, he drew a world map (Fig. 13.5) with China at its center, thus presenting the vastness of the world to the emperor and his ministers. Qing Dynasty (1644–1912) The next dynasty, Qing, ruled China from 1644 until modern times. At the beginning of the sixteenth century, European countries moved to colonize China. First the Portuguese who obtained the rule of Macao, which was maintained until 1999. Later, England and United States tried to open trade legations. The most dramatic years were Opium War between 1839 and 1860, when England undertook to sell opium in the Chinese market, provoking the opposition of Emperor Daoguang (1782–1850). In 1842, a first treaty granted to England the dominion over Hong Kong. The Second Opium War ended in 1860 with a treaty that obliged China to leave the opium trade free and to grant customs exemptions for other goods. Finally, the Republic of China was founded in 1912, which eventually became the People’s Republic of China on October 1 1949.

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Philosophical Schools The philosophical schools of ancient China developed from −700 to −200 during the period of the Warring States: Confucianism, Daoism, Mohists and Logicians, and Legalists.5 I will briefly characterize these schools by considering their contribution to the development of science and technology and their contribution to the formation of social and political thought. Confucianism was dominant in ancient China but made no contribution to scientific knowledge. Daoism, considered an enemy by the Confucians, is more interested in speculation about Nature. It corresponds in some aspects to pre-Aristotelian thought, and forms the basis of all Chinese science. Confucians accepted feudal society, while Daoist strongly opposed it. The Legalists devoted themselves to the codification of laws and are largely responsible for the transition from pure feudalism to the typically Chinese form of the feudal-bureaucratic state. The ideology of the bureaucratic state was in fact a synthesis of Confucian and Legalist principles. The Mohist school was interested in technical issues, while the thought of the Logicians had a speculative philosophical character, similar to that of the Greek Sophists. Confucius  Confucius (−552 –479) was born in −552  in the state of Lu, present-­day Shantung, into the Kong family descended from the Shang imperial house of the state of Song. He was called Kong Fuzi, i.e., Master Kong, Latinized as Confucius. He devoted his life to the development and dissemination of a philosophy that advocated the importance of harmonious social relations. He has long sought official positions that would allow him to put his ideas into practice. In −496 he was exiled and traveled with his disciples, meeting with feudal princes to whom he expounded the principles of his school. He devoted the last 3 years of his life to writing and educating his disciples. Confucian thought was a product of an era of violent conflict, of chaos, in which smaller states became battlegrounds between more powerful ones and the population fell victim to the violence of war. The name Period of Springs and Autumns was given to this era in a book of chronicles attributed to Confucius himself, which recounts events from −770 to −481. There were no laws and social order derived from the force and arbitrariness of the nobles. They, when not at war, devoted themselves to hunting and pleasures, burdening the shoulders of the people entirely. A time, therefore, when  Needham (1956, p. 1).

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human life had no value. Confucius’ preaching was thus revolutionary for his time. Confucius’ greatness lies mainly in having established schools when the only education was military. They were schools open to anyone and not just the nobles or the rich, so they were based only on merit. The quality of his teaching is witnessed by one of his best students, Tseng Shen,6 who writes that, to follow the Dao, three thing are most important: to remove any trace of violence in your attitude, to express serenity on your face, to remove all vulgarity from your speech. In another passage, the Master defines perfect love:7 to behave with others as if they were your guests, to recognize the sacrifice of people working for you, to give no cause for resentments, do not do to others what you would not want done to yourself. This last principle is common to all religions. In revealed religions it is God himself who imposes it; in Confucian thought this principle descends from the coherence of the whole ethical system. Other principles also descend from this, leading to a political function of Confucianism. The government had the responsibility to look after the welfare and happiness of its citizens, thus following an ethical behavior. In these principles, the system of power management, which began at the end of the feudal era, was rooted. A bureaucracy, educated in schools, selected on merit through official competitions, assists the king and performs the functions of public government, respecting natural law. In Western philosophy, ethics and politics are two distinct domains; in Confucius, there is no distinction, but it lacks the notion of positive law that we will find in the School of Legalists. Confucius’ teaching also contains the principle that the power of the prince, and later the emperor, derives from the will of the people expressing the will or mandate of Heaven. Confucianism did not establish itself as a religion; only in later centuries did a cult toward the Master arise and many temples were built in his name, where rituals were held in honor of the great thinker. Dao  It is impossible to describe Daoism in just a few pages. A concise introduction to Daoist thought can be found in the aforementioned work by Needham. The Daoist school, whose origin is attributed to Laozi (−570 −?), was responsible of the studies of chemistry, mineralogy, botany, zoology, and pharmaceutics in East Asia. Taoist scientific thought bears similarities to that of  Ibidem p. 6.  Ibidem p. 7.

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the Greek Presocratic and Epicureans. Daoist observation of nature was guided by an experimental and empirical method, not a systematic one. Totally missing was Aristotle’s classificatory intent and systematization of a body of scientific terms. Daoist thought is deeply aware of the universality of change and transformation, perhaps one of its greatest insights. The oneness of nature is also encapsulated in the concept of Yin and Yang, which is usually described in a simplified way as the relationship between feminine and masculine. The Daoist themselves were not immune to transformations, to the point of converting the agnostic naturalism of their origins into a religious mysticism and eventually into a Trinitarian theism consisting of: (1) the transformation of proto-Daoist experimentalism into a magical practice aimed at predicting fortune; (2) primitive communitarianism into a path to personal salvation; and (3) anti-feudalism into egalitarian secret societies with anti-foreigner and anti-­ dynastic tendencies. According to J. Needham8 it was not the Confucianism that blocked the development of science, rather the economic and social system of feudalism imbued with a pervasive bureaucracy. Thus, Daoism left no room for the growth of scientific elements. Between the second and thirteenth centuries, the empirical nature of technological achievements was emphasized by Daoists. The spread of the Buddhist faith inhibited the further development of Daoist philosophical thought. Mohist The Mohists represented the pacifist chivalrous component of Chinese feudalism, interested in the scientific methods and experiments that emerged from war technology. The moral principle that guided them was respect for the weaker, which they extended from interstate politics to the behavior of individuals. Their studies for military defense led them to develop knowledge of mechanics, architecture and hydraulic engineering. These practices prompted the Mohists to take an interest in basic scientific methods and studies of mechanics and optics, which are considered the earliest records of science in ancient China. The founder of this school of thought, Mozi, lived between −470 c. and −380 c., and was thus a contemporary of Democritus, Hippocrates and Herodotus. The school he founded was divided into various strands, and many turned their speculative interest to ethical problems. This is evident in the expression Great Togetherness which is used to characterize the period in history when the  Ibidem p. 162.

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government was led by wise men, and everyone exercised respect for others, practiced virtue, valuable things were not hoarded, and there was no theft or treachery. It was a kind of golden age, after which the Dao eclipsed. The disuse of the Dao principle meant that empire became a matter of family inheritance, men’s love was reserved only for their parents and children, and things of value were accumulated for private benefit: the period of Lesser Tranquility. Needham notes9 that the expression Great Togetherness is the title of a famous book on socialism by Khang You-Wei (1858–1927) and was also used by Mao Zedong (1893–1976), thus becoming a watchword of Chinese communism. Logicians  The Logicians were part of the Mohist school, but they were interested in different subjects, similar to those of the Greek Sophists; their schools are also located in the fourth to third century, after which the school of Logicians died out. The main exponents were Hui Shih and Gongsun Long. Gongsun Long’s book is in the form of a dialogue; among the few surviving fragments one recognizes the theme of universals as opposed to concrete objects, which leads one to reflect, for example, on the relationship between form and color, with a paradoxical argumentative procedure. A classical paradox was that of the “horseness”: a white horse is not a Horse, even if horses of color white can exist, so horse and white color can be associated, but a white horse cannot be a Horse in itself. The logicians considered also the problem of infinity in the same epoch that was also investigated by Greek philosopher. An example is the sentence that if a stick one foot long is cut in half every day, it will still have something left after ten thousand generation. Legalist  The first elaborations of criminal codes date back to the sixth century, the legalist school, in turn, was organized around the fourth century, originally in the northeastern Qi state and later in the three Han, Wei, and Zhao states into which it was divided during the third century, at the climax of the Warring States period. The fundamental idea of the legalists was that the complex of customs, ceremonies, and compromises administered paternalistically according to Confucian principles were inadequate for a strong, authoritarian government. The cornerstone for the legalists was thus the positive law fixed a priori, to which everyone in the state, from the ruler himself down to the humblest slave, had to submit, on pain of severe and cruel punishment. Han Fei, who  Ibidem p. 168.

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died in −233, organized the principles of legalist into a philosophical order, and his writings have come down to us. He developed the thought of the master Shang Yang, who advocated a strict regulation of private production and trade, and the fundamental role of agriculture. Law is the foundation of the authority and is the basis of government; it is what forms the people. If the law is strong the country is strong, writes Han Fei, if it is weak the country will be weak. This entails severely punishing every smallest infraction, without exception. In the army, the responsibility is not individual. Armies were organized in squads of five soldiers, and if one of them was killed, the other four were also beheaded for allowing it. A reporting system was created, and failure to report was punished severely, even by torture. One story illustrates this: Prince Chao of the Han State got drunk and fell asleep exposed to the cold, so a crown guard covered him with a cloak. When the Prince woke up, he asked who had covered him, and since no one answered, he put the crown guard to death, following the principle that transgression from duty was worse than mere negligence. The legalist principles were adopted during the Qin Dynasty, becoming the fundamentals of the centralized power during the first unification. Jurists were aware of the conflict between a theoretically well-constructed positive law, on the one hand, and ethics and fairness, and even common sense, on the other. The conflict between the law of legalists and the ethics of Confucians is well illustrated in the ethical dilemma of whether a son should conceal his father’s crime or expose it by providing evidence against him. Confucius had decided that filial piety should prevail over the state law. Han Fei, on the other hand, as a legalist, argues very insistently for the opposite view. However, the Confucian position prevailed over the legalist position and was transmitted to posterity in a classic treatise: Filial Piety (Xiaojing) published during the Warring State period. During the Qin Dynasty, the activities of merchants were subject to high taxes and despised, in contrast to the high esteem in which warfare and agriculture were held. The effort to regulate every aspect of social life by law went so far as to standardize units of measurement and even the size of chariot wheels. Everything was thus expressed in numbers both to control agricultural production and to determine punishment. But the legalists’ philosophical principle was exhausted in mechanistic materialism, which simplified reality with mathematics and metrology. The legalists blatantly failed to take complexity into account, unlike the Confucians, who instinctively left room for the organic character of man and society.

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The dominance of legalist thought was gradually defeated over the course of two centuries. Law was again incorporated into Confucian ethics and later emperors justified their mandate on a principle of natural law, i.e., norms of behavior considered universally moral. A side effect was that norms concerning measurements and standards were abandoned leading to chaos that lasted until contemporary times. The most important aspect that characterizes all these schools is a philosophical thought in which metaphysics is completely absent. For example, the notion of time and space in Chinese thought has none of the abstraction that connotes the same concepts in Western thought. Marcel Granet (1971) clarifies how these two concepts have nothing abstract, but are mental representations of material places and historical periods. Even the notion of Ying and Yang has nothing metaphysical about it, Granet further observes that: Yin and Yang cannot be defined pure logical entities nor as simple cosmogonic principles. They are neither substances, nor forces nor genera. ... Chinese thought, completely dominated by the idea of efficiency, moves in a world of symbols made up of correspondences and oppositions that one only has to set in motion, when one wants to act or understand. ... The category of sex proves its effectiveness in linking human groups. It imposes itself as the principle of an overall classification. Then the totality of the contrasting aspects that make up the society formed by men and things is arranged in two opposing bands of male or female peculiarities. Symbols of sexual oppositions and commonalities. Yin and Yang seem to conduct the concerted race in which these aspects call and respond to each other like so many emblems and signals. ... The Chinese do not like to classify by genus or species at all. They avoid thinking by means of concepts that, placed in an abstract Time and Space, define the idea without evoking the real. To defined concepts, they prefer symbols rich in affinity; instead of distinguishing in Time and Space two independent entities, they place the games of their emblems in a concrete sphere constituted by their interaction: they do not detach Yin and Yang from the social realities of which these symbols evoke the rhythmic order.10

The philosophical thought is grounded in society itself and is strongly pragmatic in nature, as indeed emerges from Confucian thought, which codifies a social ethic based not on religious or metaphysical principles, but on the tangible principle of the welfare of the social community through permanently good oriented behavior.

 Granet (1971, p. 109).

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Natural philosophy, understood in the sense of the scientific study of nature that took shape in Europe with the Renaissance, is also absent from Chinese thought. Platonic idealism, which permeated Western thought, was absent in China, as Matteo Ricci observed when explaining the foundational arguments of the Christian religion. Granet reminds us that Chinese science does not go as far as formulating the concept of genus/species that we encounter in Aristotle, the foundation of a classificatory approach to scientific knowledge. Botany or zoology writings are not based on a scientific approach as we are used to considering, but on a practical approach. For example, plants and herbs are described for their use, food or medicinal, not on the basis of morphological, let alone reproductive characteristics. On the other hand, in Confucian thought, we encounter the concept of the Ladder of Souls, a classification that constitutes almost a cosmological vision. Aristotle recognizes in plants a vegetative spirit, in animals a vegetative and sensory spirit, and in humans, in addition to these two, he recognizes a rational spirit, thus configuring a gradation of living beings. The -fourth-century scholar Hsüng Chhin (Gongxi Chi), one of Confucius’ pupils, sets out a ranking in which at the first level is water and fire that have a subtle spirit (something similar to the Greek pneuma), plants have life but no perception, animals also have perception but not the sense of justice that is proper to humans.

Mathematics In our imagination, Chinese mathematics is immediately associated with the vision of a merchant counting quickly with an abacus, and this exhausts our idea of Chinese mathematics. But counting and computational technique is only a minimal aspect of Chinese mathematics.11 In the study of the history of Chinese mathematics, some questions of particular importance emerge. Was the nature of this mathematics an art of logical reasoning or a technique of computation? Was it simply arithmetic or was it also concerned with number theory? Was geometry a discipline in its own right or was it only in the service of topographic surveying? We must remember that there are numerous ancient texts on mathematics, some of which can be considered as theoretical and research texts, others as manuals and still others as formularies. If we neglect these different types, we  For an extensive discussion of Chinese mathematics see Needham and Tsuen-Hsuin (1956), Martzloff (1997). 11

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might be inclined to regard Chinese mathematics as a mere technique and mathematical treatises as a mere mnemonic aid. Many of these manuals do not contain demonstrations. Some theoretical texts contain demonstration as only sketches and are not set out in the axiomatic-deductive form known to us. This does not imply that the authors did not know the demonstrations or did not know how to set them out in a complete form. In fact, oral tradition has always played a very important role in Chinese thought. The earliest documents of Chinese mathematics are much later than the appearance of mathematics in Egypt or Babylonia. The first writings are in the period of the Warring States, ending around −200, when Greek mathematics, in particular geometry, had already acquired its most important results. The presence of elements of Western mathematics in Chinese texts, however, is a confirmation of cultural and commercial relations between East and West. Mathematical Writings and Dynasties  Most works on mathematics appear with the Han dynasty (Western Han −208 −9). There are indications that the 16-volume works The Rules of Calculus (Suanshu) and the 26-volume Rules of Calculus of Xu Shang (Xu Shang suanshu) were published early in this period, but they have not survived and are quoted in later works. No Chinese ­mathematical documents dating between the eighth and eleventh centuries have come down to us, apart from those found in Central Asia, particularly those hidden and sealed-up in Dunhuang caves around the year 1000. The oldest surviving works, also from the Han period, but by an unknown author, are The Arithmetical Classic of the Gnomon and the Circular Path of Heavens (Zhoubi suanjing) and Rules of Calculation in 9 Chapters (Jiuzhang suanshu), which we will refer to hereafter as The Nine Chapters. The first of these books, Zhoubi suanjing, is in fact a treatise on cosmology that gives to myths a mathematical treatment. The authors make use of decimal notation and set rules for addition, subtraction, multiplication, division, and for extracting the square root of any number. They know how to find Pythagorean triples like (3,4,5) and (6,8,10). The ratio of the circumference to the diameter of the circle (the well-known π) is approximated to 3. They also knew how to manipulate properties of similar triangles. The second book, The Nine Chapters, is a handbook, dealing, in addition to the arithmetic outlined above, with elementary geometry and methods to compute the surfaces and volumes of regular and irregular polygons and polyhedra. It also deals with algebra with solution of linear systems including the use of negative numbers. Some scholars have advanced the conjecture that it was a work addressed to land surveyors and tax officials of Han administration. We are thus faced with a mathematics developed for administrative

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purposes, for agriculture, and for tax calculation. These methods are also needed in astronomy to compute the calendar. From the third century of the current era and up to the sixth, theoretical mathematics was improved introducing logical reasoning. As we shall see, computational approximations were no longer an empirical process of trial and error. A period of great technological innovation began: geographical maps were drawn on the basis of a rectangular grid of equidistant straight lines (Pei Xiu, Minister of Public Works, third century); Tao Hongjin (456–536) wrote a treatise on pharmacopoeia. The astronomer Yu Xi (307 c.–338) discovered the precession of the equinoxes, and in the fifth century, Zu Chongzhi (429–500), mathematician, astronomer, and engineer invented various machines and mechanisms. In the period of the Third Unification, with the Sui (581–618) and Tang (618–907) dynasties, mathematics was officially taught in the schools for the preparation to the official exams. The material was collected in several books, collectively called The Ten Computational Canons, which actually collects 12 works. We can find, for example, tables for multiplying numbers from 1 to 9. This tables omit the calculus b × a when a × b has already been computed. In general, these are elementary mathematical problems, although they include solutions to problems that are part of number theory, such as the hundred-­ fowl problem and the problem of the remainder (see later). Little is known of the period from the tenth to the twelfth century, the works only contain allusions to elementary mathematical problems, although there are references to texts (disappeared) dealing with more complex problems. Between 1247 and 1303, new works were published. It was a period of mathematical revival, a kind of Renaissance. In this period, China came into contact with Arab culture, a consequence of the Mongol invasions and the establishment of the Yuan dynasty (1260–1368). Guo Shuojing (1231–1316) an astronomer and calendarist expounds the fundamentals of spherical trigonometry. After this flourishing phase, Chinese mathematics entered in a period of decline, particularly during the Ming dynasty (1368–1640). The tradition of oral teaching was lost, leaving the transmission of knowledge to only written text. Soon this knowledge became incomprehensible, lacking any formalization and terminological convention. This problem is still an obstacle to the contemporary study of ancient works. A revival of studies occurred with the arrival of the Jesuits in China, who introduced Western mathematics. It was Matteo Ricci who edited the first

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Chinese translation of Euclid’s work, from the Latin version by Christophorus Clavius. Later developments are beyond the scope of this book. Instead, it is useful to return to examine some central topics in ancient Chinese mathematics. Main Authors After the First Millennium  A mention of the few authors whose biography and works are known and briefly described by Jean_Claude Martzloff (1997) helps to understand the context in which mathematical studies developed and to grasp their differences from European scientists.12 Li Zhi (1192–1279)13 was the son of a secretary to an official named Hu Shahu, was born in present-day Beijing, obtained a doctorate in Luoyang in 1192, and was employed as Assistant Magistrate in the district of Gaoling14 in 1230. Due to the war, could not take office and became Governor of the prefecture of Jun.15 When the Mongols conquered Kaifeng in 1233, he escaped the massacre thanks to the intervention of Yelü Chucai (1190–1244), a highranking official who had passed into Mongol service. In 1234, he retired to hermitage in Shanxi and came into contact with various intellectuals. During this period, he began to write his main work Ceyuan Haijing (Sea mirror of circle measurements), a 12-volume treatise containing a single geometrical figure that is used to describe and solve 170 problems. In 1237, the future Mongol emperor Kubilai sent emissaries to ask his advice on how to govern, how to organize exams to recruit officials, and how to interpret earthquakes. In 1264, Emperor Kubilai admitted him to the newly established Henlin academy so that he could edit the official annals of the dynasty. He soon resigned from the post for health reasons and died in 1279. The analysis of Li Zhi’s unique geometric figure (Fig. 13.6) highlights the conceptual framework of Chinese geometry. The sides of the triangles are not denoted by letters as in Western geometry, but by names that evoke the idea of a city map. The triangles edges are integer, and they form a Pythagorean triplet, for example, the triplet (17, 8,15) where 17 is the length of the hypotenuse. From this triplet, one obtains infinite right-angled triangles by multiplying the triplet by a constant. The entire book is a long list of 692 formulae concerning the area of triangles or the length of segments. The formulae are  See also Needham and Tsuen-Hsuin (1959) and Granet (1971).  Martzloff (1997, p.143 ff). 14  Present-day Shenxi Province. 15  Present-day in Henan. 12 13

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Fig. 13.6  Illustration from Li Zhi’s book, called “figure of the round city”

stated in a straightforward manner without any logical explanation of any kind. These formulae did not have a geometric interest to Li Zhi, they are of interest only for the role they play in his computational technique. The 170 problems he tackles in the later part of the book are of the type: two men walk along certain streets around a circular city—such as the one depicted in the figure—in which a circle is enclosed in a right-angled triangle. Two men cannot see each other since they are hidden by the circular walls. The question is always to determine the diameter of the city, given the distance they have walked to see each other, a distance that is invariably 120 bu (units of length in steps).

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When dealing with other similar problems, the role of the edges of the triangle can be swapped, configuring a problem that mimics the previous one. This approach takes the form of a literary artifice, so this text has a structure that is called parallel prose (pian wen), a literary genre that constraints metrical, and syntactic structure. It is difficult for us to grasp these aspects without an in-depth knowledge of the Chinese language; however, a mathematical knowledge exposed in a literary style emerges from the study of the experts. This rhetorical style most likely had a mnemotechnical function. Qin Jiushao (ca. 1202–1261) was born in Anyue16, his father had been admitted as an official in 1193 and held various positions in the local administration. Qin Jiushao writes In my youth I lived in the capital17 so I was able to study in the school of Astronomy; later I was instructed in mathematics as a resident student.

It is known that he held the position of Defence Commander from 1233. In 1234, the Mongols invaded Sichuan, and Qin Jiushao writes At the time of the clashes with the barbarians, I spent many years at the remote frontier, without worrying about my safety among the arrows and stone bolides, I endured danger and misery for ten years.

He became gentleman of Court at the prefecture of Jiankang,18 but he soon left that place in mourning for the death of his mother. It is believed that he has since devoted himself to writing the work Shushu Jiuzhang (a rewriting of The Nine Chapters, Jiuzhang suanshu) in which he deals mainly with problems of astronomy and studies calendar cycles.19 It also deals with celestial chronology and celestial calamities, to make what Greeks called parapegma. Another problem that Qin Jiuschau tackles is the point field (jian tian): how to determine the area of a quadrilateral made of two triangles with common base. The solution that Qin Jiuschau devises avoids a calculus that exploits the geometric properties of the two triangles. It is based instead on the solution of a fourth-degree equation whose coefficients depend on the  In present-day Sichuan.  In present-day Hangzou. 18  Present-day Nanjing. 19  Martzloff (cit.) observes that Gauss extended the application of his study on the problem of remainders to the field of chronology. Specifically, he used his findings to determine the day of the year that corresponds to the golden number and Easter-related indications. 16 17

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dimensions of the figure. Qin Jiuschau also invented a method for solving polynomial equations, similar to Horner’s 1819 method. Zhu Shijie  (1240–1314). Almost nothing is known about the life of this mathematician. He lived toward the end of the thirteenth century and wrote the work Siyuan yujian (Jade Mirror of the Four Elements) in 1303 (Fig. 13.7), contributing to the advancement of algebra. He tackled problems with up to 4 unknowns, and dealt with problems of architecture, finance, military logistics using elementary algebraic techniques, which he mastered to the point of drawing Pascal’s (or Tartaglia’s) triangle (Fig. 13.7 right page) representing the progression of the coefficients of the binomial

(a + b)

n

with n ≤ 8. The book presents numerous geometric problems for the subdivision of plane figures or solids. For example, how to subdivide a trapezoid into equal parts with straight lines parallel to the base, or the subdivision of a disc into equal parts with a bundle of parallel chords. These may seem like mathematical games, but their solution requires knowledge of algebra and also a method that corresponds to Newton’s finite difference interpolation technique.

Fig. 13.7  A page from Jade Mirror of the Four Elements

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Cheng Dawuei  (1533–1606) is considered the greatest scholar of Chinese arithmetic. One of his descendants, in the preface to a reprint of his main work, Suanfa tongzong (General Source of Calculus Methods), wrote: In his youth, my ancestor Cheng Dawei was very academically gifted, but although he was well versed in study, he continued to practice his profession as an honest local agent without becoming a scholar. He never left behind the classics and the ancient tadpole script20 but was particularly gifted in arithmetic. In the early years of his professional life, he visited the fairs in Wu and Chu. When he found books that mentioned ‘square fields’, ‘hulled grains’, he never considered the price before buying. He questioned respectable old men who had experience in the practice of arithmetic and gradually and tirelessly formed his own collection of difficult problems.21

There is nothing original about his writing; it is a reworking of earlier works. It is mainly a treatise on the use of the abacus, but considers questions on the mystical meaning of numbers (such as magic squares, trigrams, and musical pipes). The computational rules are written in verse to make them easier to memorize, and perhaps this and the high literary style made them successful at a time when the Jesuits were introducing Western mathematics to China. Some Themes in Chinese Mathematics  Chinese mathematics is not structured into independent fields of study as in the West starting from the Renaissance. Above all, mathematics was not a formal discipline. I will recall her some classical themes. Numbers and Numbering  In 1899, thousands of turtle bones and shells with inscriptions were discovered in the northwestern region of Xiaotun, where the city of Anyang in Henan is located today, near the capital of the Shang Dynasty (−fourteenth, −eleventh century). The inscriptions on these findings (many thousands more were found in the twentieth century during various excavation works) have been interpreted as divinatory texts. Some contained numerical information, such as numbers of animals, people, shells,  Tadpole writing first appeared when the Confucius school was demolished in the second century. The name derived from the tadpole shape with large-headed characters and tails (kedou). It was distinguished from insect writing (sawgoek). However, “tadpole writing” may simply be a popular way of referring to the ancient writing of the Chou period. https://en.wikipedia.org/wiki/Tadpole_script Accessed November 2022. 21  Martzloff (1997, p. 159). 20

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Fig. 13.8  Left: Numerals on bones. Shang Dynasty. Right: rods numerals, Warring States period

days, months, sacrifices, and war expeditions. From examination of the signs drawn, two numbering systems were identified, the first of which is a mixed decimal and duodecimal system leading to a sexagesimal system (like the Babylonian numbering system). Some scholars believe that quipu knots were used before this numbering method (Jacobsen, 1983). The second type of numbering is a decimal system (Fig. 13.8 left). Two versions can be identified, the more complex one seems to indicate particular numbers greater than ten with a multiplicative notation: the multiplier can be drawn above or below the number representing the multiplicand. These notations also appear in bronze artefacts. In the period of the Warring States (−480–221), numbers were engraved on coins with sticks as in the decimal numbering (Fig. 13.8 right). It should be emphasized that this numbering looks positional, but it is not so, because the value given is closely related to the context. In common parlance, we find a similarity, when we say that a commodity costs “two thousand and five” meaning from the context 2500, while “two thousand and five” in a statistical survey indicates precisely 2005. Inserting spaces between numbers resolves context ambiguities. As for zero, there is no definite evidence to indicate that this entity was known in China, even considering the dual use of the symbol 0 as an indicator of the order of magnitude when placed at the bottom of the number (positional numbering) or as a number in itself. Calculation Tools This numbering system lends itself to a calculation method based on counting sticks arranged on a table. The sticks should be

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specially shaped, such as triangular-section bars or small prisms, to prevent them from slipping or mixing up. Besides sticks, the abacus was the most widely used instrument. With the abacus one can perform basic arithmetic operations and calculate quadratic and cubic roots. Merchants have always made systematic use of this tool, performing long calculations with great speed and agility. Matteo Ricci observed the skillful use of this tool and tried to teach the Western method of calculation based on writing numbers, which he called brush calculus,22 adopting symbols—completely absent in ancient Chinese mathematics—to represent operations. We have seen in section “The XVIII Century” in Chap. 2 that the adoption of specific symbols by Leibniz and Newton to denote advanced mathematical operations made it easier to develop infinitesimal calculus. Numerical and Algebraic Problems We limit here to examining a few mathematical problems. Prime Numbers  The notion of a prime number had no theoretical explanation. It was taken into consideration when trying to arrange a number of beans in a rectangle. Non-prime bean numbers can be arranged in rows and columns with two factors, while prime numbers can only be arranged in a single row. For instance, 15 can be arranged as an array (3 × 5) while the prime 17 cannot. Root Extraction  Historians of Chinese mathematics note that the topic was widely presented in the various works, with each author proposing his own solution. In Chinese mathematical language, this operation is described as a particular type of division. Liu Hui (third-century current era), in a commentary on the Nine Chapters, considers the operation as an iterative subdivision (kaifang in Chinese) of the square. Suppose we want to find the side of the square in the lower right-hand corner of the figure (Fig. 13.9) whose area is A. To do this, it is necessary to subdivide A in successive steps into regions that Liu Hui calls yellow, red, and blue. In the first step, the yellow square is defined so that it has the maximum area x2 corresponding to (for example) the square of the number of hundreds (or thousands) of the radicand. The side of this first square is the first digit of the square root. This leaves an area (A − x2) in the shape of an L (denoted gnomon in the figure).  Fontana (1996, p. 121).

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Fig. 13.9  The Liu Hui method for finding the square root

In this subdivision, the squares at the corners will always be yellow, and the rectangles that make up the L will be red on the inside and azure on the outside. In the next step, the yellow square of side y and the two red rectangles of area xy and yx are subtracted from the area of L and a new L is determined, in A − x2 figure (a). The inequality y ≤ is used to determine the value of y, that 2x is found by trial and error. At each step, the yellow is the color of perfect squares. The method is also suitable for the extraction of the cube root, imagining that a cube is divided using the same procedure. Martzloff notes that the geometric analogy presented by Liu Hui is very similar to that described by Theon of Alexandria (ca. +350) in a commentary on Ptolemy’s Almagest.23 The Hundred Fowls Problem  This is an example of a typical arithmetic problem that is solved empirically but can be formulated using systems of equations. It involves finding how many fowls of three types can be bought by spending a total of 100 qian knowing that the cockerel costs 5 qian, the hen

23

 Martzloff (1997, pp. 227–228).

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costs 3 qian, and 3 chickens cost a total of 1 qian. The solution to this p ­ roblem was found empirically by trial and error. With Western notation, this problem can be written with the system of equations: 1 5 x + 3 y + z = 100 3 x + y + z = 100

Solved by taking, for example, z from the second equation and substituting it into the first to arrive at the relation: 7x + 4y = 100 from which y = 25–7/4 x. Since y must be an integer, then it will be a multiple of 4, so we can write x = 4t, y = 25 − 4t, z = 75 + 3t, with t = 0, 1, 2, 3. Chinese mathematicians did not provide a theoretical justification for this calculus, they only provided solutions obtained empirically, while the solution concerning the problem of remainder in the integer division was only demonstrated by a Chinese mathematician in 1861.24 Martzloff (cit.) points out that this problem is found in the mathematical literature of the same period, Indian (Šrīdharacārya ca. 850–900), European (Alcuin ca. 735–504), and Arabic (Abū Kāmil ca. 900).25 The Problem of Remainder  The determination of the calendar needs to solve a problem of congruence (Caire & Cerruti, 2015). The Chinese calendar during the Zhou Dynasty starts when the midnight, the winter solstice and the new moon coincide. This event has a 60  years  cycle. The dates of equinoxes and solstices for celebration and feasts are calculated from this zero. The zero day was called shangyuan, and if the winter solstice would happen r days after the shangyuan and s day from the new moon, then that year was N years after the shangyuan. To solve this problem, a system of congruences must be solved: aN ≡ r mod 60   aN ≡ s mod b where a is the number of days in a tropical year (365), and b is the number of days (28) of a lunar month. The mathematician Sun Zi (third century) describes the problem:  Ibidemp. 308.  Ibidem p. 309.

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if a number is repeatedly divided by 3, the remainder is 2 id it is divided by 5, the remainder is 3, if it is divided by 7, the remainder is 2. Which is this number? There is also an anecdotical version, known as the emperor secretly counts his soldiers: if he counts them in groups of 3, then 2 remain out; if he counts them in groups of 5, then 3 remain out, if he counts them in groups of 7, then 2 remain out. The equations in this case are  x ≡ 2 mod 3   x ≡ 3 mod 5  x ≡ 2 mod 7 



Sun Zi provides one solution: x = 140 + 63 + 30 = 233

The general solution is26 23 ( mod 105 )



The generalization of this problem is known as the Chinese Theorem of Remainders,27 and the solution has been found by the general Qin Jiushao, in 1247. The importance of this theorem goes well beyond the simple practical problems above; it is part of the problem of cryptography in modern computer system. Interpolation  Chinese calendarist had to interpolate astronomical data to determine the dates of astronomical events. They invented a linear interpolation method, as always described as an example on specific case. This method was already known to the mathematicians of the Southern Song and Yuan eras, while in the West the problem was not solved until after the fourteenth century and was completely solved with eighteenth-century mathematics and the development of the concept of function. Since much of the mathematical knowledge had been forgotten after the fourteenth century, when the Jesuits arrived, Chinese mathematicians learned these methods again, but often failed to recognize what their predecessors had already discovered. 26 27

 Ibidem.  Ibidem, p. 312.

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Problems of Geometry and Trigonometry  For Chinese mathematicians, geometric entities are concrete objects, so a line is the edge of a field and a plane figure is the shape of a city or even a cultivated field, while a point is the end of a stick. However, Needham28 identified a Mohist canon, Mo Chin, that contains definitions of geometric entities. For example, the definition of a point is as follows:29 The line is divided into two parts and the part into which nothing remains [can no longer be divided] and forms the extreme end [of the line] is a point. A point can be at the end [of a line] or at its beginning such as [Regarding its invisibility] nothing is similar to it.

These definitions resemble the first and third definitions in Euclid’s Elements Book I I A point is that which has position but not dimension III The intersection of lines and their extremes are points

The above canon also contains definitions of lines of equal length, parallels, rectangle, circle and circumference, and the definition of volume. It is possible that the study included an axiomatization of geometry like that of the Greeks, but the Mo Chin text is very damaged and incomplete. Apart from this partial evidence, Chinese mathematics approaches geometry, as mentioned, from a quantitative and empirical point of view, like the works of Heron of Alexandria in the first century. Chinese geometers were familiar with plane figures such as squares, rectangles, isosceles and right triangles, trapezoids, rhomboids, circles, and circle segments rings. The solids known and studied were cube, parallelepiped with square and non-square surfaces, pyramid, pyramidal frustum, prism and wedge, cylinders, cones and conic sections. Liu Hui was also one of the leading scholars of solid geometry and commentator of the ancient work Gnomon and the Circular Path of Heavens (Zhoubi suanjing). Pythagorean Theorem  The Pythagorean theorem was well known, particular names were used to denote the base, height, and hypotenuse of a right-­ angled triangle. There is a demonstration in Zhoubi suanjing. The diagram in Fig. 13.10 belongs to a later commentary and probably dates back to the third  Needham and Tsuen-Hsuin (1956, p.91 ff.).  Ibidem.

28 29

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Fig. 13.10  Demonstration of the Pythagorean Theorem. The diagram on the right is known as the hypotenuse diagram and demonstrates the Pythagorean rule for numbers 3,4,5. The diagram on the left shows how a square of size 3 fits into a square of side 5

century. Liu Hui called this figure “the diagram that gives the relations between the hypotenuse and the sum and difference of the other two sides, where the unknown can be found from the known.” Evaluation of π  In Egyptian and Babylonian culture, it was common to take the number 3 as an approximation of the number π. Other approximations were 3.1604 and 3.125. Approximations with more decimal were also known in Han times. Around the year 130, the approximation 3.1622 was calculated. In the third century, during the Three Kingdoms period, astronomer and mathematician Wang Fan obtained the ratio 142/45 = 3.15(5). In the same period, Liu Hui tackled the problem by inscribing a regular polygon in the circle and calculating its perimeter using the properties of right-angled triangles. With a hexagon, he arrives at the value 3.14. But he also calculates two more accurate values, the minor one being 3.141024 and the major one

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3.142704, slightly more accurate than the value 22/7 = 3.1428 determined by Archimedes around −250 using a polygon of 96 sides. Toward the end of the fourth-century, Zu Chongzhi obtained the value 355/113 = 3.14159292035398, which was equaled in the West in the sixteenth century.30 Geometric Coordinates  In the Alexandrian period, points on Earth and in the sky were identified with latitude and longitude coordinates. After the collapse of the Greek civilization, the use of coordinates was forgotten until the Renaissance, while in China the use arose independently and developed steadily. In China, data were collected and represented in the tabular form, which is also a coordinate structure: the columns can be considered abscissae and the rows are the ordinates. After all, the chessboard is also a structure of orthogonal coordinates. The basic principle of Chinese mathematics is to transform a geometric problem into an algebraic calculus problem. A geometric coordinate system relates geometry and algebra. We can therefore say that the use of coordinate systems in China predates by several centuries their appearance due to Descartes in the seventeenth century. Trigonometry  We have seen how trigonometry originated with the ideas of Hipparchus, and subsequent developments, leading to the first concepts of spherical trigonometry, were due to Menelaus and Ptolemy. The full development of trigonometry was due to Arab scholars. Chinese mathematicians are not familiar with trigonometry. In the eleventh century, they make use of the concept of the chord of an arc, but do not develop specific terminology to denote the main trigonometric functions and do not associate the meaning of sine and cosine with the names of the main sides of the right triangle. In the thirteenth century, to improve calendar calculation capabilities, the need to understand spherical trigonometry emerged. Guo Shuojing wrote a specific treatise on the subject, when he probably came into contact with the Arab world. Fig. 13.11 D indicates the point of the summer solstice, OCD is the ecliptic plane, AOB is the equatorial plane. The trigonometric problem is to find the formula to convert from equatorial to ecliptic coordinates. CMNK

 Needham and Tsuen-Hsuin (1956, pp. 100–101).

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Fig. 13.11  Conversion of spherical coordinates by Guo Shou-Ching

is a rectangle by which Guo Shuojing calculated the coordinates of D knowing the arcs AB and AC.31 After this work, no further works can be traced until the arrival of the Jesuits, when Matteo Ricci introduced western trigonometry. Chinese mathematics after Jesuit teaching gradually approaches that of the Western world, to which, however, the innovations of the eighteenth and nineteenth centuries are attributed. Concluding this brief overview of Chinese mathematics, we would like to recall an observation by Needham.32 They evaluate Chinese mathematics as fully commensurable with the pre-Renaissance achievements of the other medieval peoples of the Western world. Greek mathematics was undoubtedly at a higher level for the more abstract and systematic character, as seen in Euclid. On the other hand, in algebra Greek achievements were weaker than in China and India. Other scholars, including Alfred N. Whitehead, observe that Greek mathematics was very advanced on the more abstract concepts, which in the West were deepened by Weierstrass, Cantor, and Cauchy in the nineteenth century, those related to the notion of infinity and infinitesimal, which was a theme strongly present in Greek philosophical thought. Greek mathematics, on the other hand, appears weaker on the aspects of elementary mathematics developed in Chinese mathematics. Interest in 31 32

 For an explanation of the method see Martzloff (1997, p. 331).  Needham (1956, p. 150 ff.).

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elementary mathematics is central to Chinese thought, but Chinese mathematicians did not participate in the transition from application and practice to the realm of pure theoretical and abstract thought. However, we have seen that the problems to which mathematics seeks to provide solutions are common to all human cultures, and the methods that have contributed most to the advancement of mathematics have been tackled everywhere and approximately in the same historical epoch; the mutual influence between Chinese and Western mathematical thought in the earliest times remains to be demonstrated with concrete evidence, while the influence is now fully demonstrated from the Middle Ages onward.

14 Chinese Astronomy and Astronomical Machines

In early Chinese culture, the universe was conceived as an ethical unity. This cosmology was coherent to the organicist conception of the Shang era. Mankind was considered an integral part of the Universe, and its role was to maintain the harmony of the universe by following the rules of the nature. One of the basic rules was the succession of seasons, which was described by the calendar. Agriculture was the main activity, and the passing of the seasons governed its timings. The calendar was the tool in the hands of the emperor who received from the sky a mandate to rule the people. The acceptance of the calendar by all who submitted to emperor’s power continued unabated until the twentieth century. Astronomy and calendar science have always been orthodox disciplines, cultivated within a Confucian framework, unlike other sciences that emerged from Daoist or Mohist thought. The activity of the astronomer and the calendarist is at the service of the Emperor, it is part of government’s activity and conducted within the imperial palaces. It was not a personal activity dictated by the desire for knowledge. The cosmological view of ancient China is strictly related to the nature of the system of power of the kingdoms and the empire: the Emperor is the Son of the Heaven, he rules the four regions of the Earth that correspond to four region of the sky, the four seasons. This division into four parts is expressed in the figure of the square, which we find in the Imperial Palace. A substantial difference in the methods of Chinese astronomy is evident when one examines the encounter with the Jesuit missionaries who brought Western astronomy to China. The latter bases its observations on the motions of the stellar regions around the pole, it is a circumpolar astronomy. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_14

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observation of the circumpolar stars does not depend on the heliacal rising, since during the night they are always visible. Greek and medieval European astronomy, on the other hand, recorded motions in the regions of the ecliptic, observing the heliacal rising of the stars or the rising or setting of the constellations of the zodiac. Initially, this difference led Ricci and his brethren to fail to understand the methods of Chinese astronomy. Given this methodological difference, we can now identify the topics of ancient Chinese astronomy: the cosmological theories, the mapping of the sky and its coordinates, the understanding of the maximum circles of the celestial sphere, the use of circumpolar stars to detect the meridian transit, the study of eclipses, the development of astronomical instruments, and the comprehensive recording of important astronomical events.

The Image of the Cosmos The first image of the cosmos (Fig. 14.1) dates back to the time of the Warring States (−480–221) and is described in −180. It is known as Gai Tian [canopy heaven] cosmology: the sky is a hemisphere covering the earth, which in turn is imagined as an inverted bowl, made of two concentric domes at a distance of 80,000 li. The ecumene of the earth is at its center, Ursa Major is at the

Fig. 14.1  Gai Tian Cosmology

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center of the sky, the rain falling on the earth flows toward its edges where it forms a circle of ocean. The vault of heaven rotates from east to west and drags with it the sun and moon, which also rotate but very slowly in the opposite direction. The rising and setting of celestial bodies are only an illusion and they never pass below the base of the earth. The sun illuminates the different regions of the earth like a beacon of light that shifts its direction. The sun is a circumpolar star that continually illuminates one part of the earth or the other; its distance from the pole varies throughout the seasons, following parallel declination circles. This cosmological vision gives rise to an imaginary subdivision of the sky into 5 regions (Fig.  14.2), called palaces. The polar region represents the Imperial Power, and the four regions are related to the seasons. We can imagine that the regions are subdivided in the sky by circles that in western astronomy are called colures. This subdivision of the sky reveals a perfect symmetry, typical of the Daoist thought (de Saussure, 1919–1920). Besides to this structural description of Chinese cosmology we must consider also the relationship between the sky and myth and religion, as for the constellations in the western cosmology. An example of this link is the painting on silk of Fu Xi and Nü Wa , discovered at the Astana Graves in Xinjiang, a burial site from third to eight centuries (Fig.  14.3 left). The silk paint is dated around the second half of eight century (Tang Dynasty) and represent Fu Xi and his sister Nü Wa joined on a body of a serpent. They represent the

Fig. 14.2  The regions of the sky. The division of the equator is projected on the horizon plane

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Fig. 14.3  Left: Fuxi and Nüwa. Painting on silk. Right: The map of 28 xiu and grouping of mansions. To the left and to the right the ideograms of spring and autumn. The central sector denote (top and clockwise) day, evening, night, morning. The red arcs delimit the four quarters (the four seasons). The hours are indicated in the inner circle in two groups of twelve (day and night)

foundation myth. The couple, the only survivors of a great flood, were commissioned to repopulate the world by forming a large number of clay figurines, brought to life with divine assistance. Fu Xi is holding a compass and Nü Wa a square ruler, which are the symbols of the Universe. The Heaven is round (the compass) and the Earth is square (the square ruler). Round and square, male and female are symbols related to the Ying-Yang principle. In the asterisms painted on the sides, we can recognize the Big Dipper and the Pleiades. Fu Xi is considered as the first legendary Emperor, honored as the first to establish law and order, inventing ropes and nets for fishing. The transition from the idea of the hemisphere to the idea of a complete celestial sphere, comparable to the ideas of Eudoxus of Cnidus, probably emerges around −400, but a full description of this model is due to an important astronomer: Zhang Heng (78–139), who in a commentary on the armillary spheres describes the cosmos as an egg that has the earth inside it as the yolk, much smaller. The circle of the sky is divided into 365.25°. Half of the sky is above the earth and half is below. He goes on to identify the two poles, the regions of the sky whose stars are visible and those whose stars are invisible because they are at the antipodes. Zhang Heng also realizes that the sky is

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infinite. This theory, known as Hun Tian [entire heaven], recognized the primary role of celestial spherical coordinates. The model of infinite empty space became established during the late Han period (+25 + 220) and was named after the Xuan Ye school. The basic concepts are the sun, the moon, and the stars float freely in empty space, where everything is condensed vapor. This model identifies the irregular and retrograde motion of the planets. The Pole star is considered as being fixed; the Great Bear never disappears below the horizon in the west like the other stars. The seven moving stars travels eastwards, the sun turns 1° every day and the moon 13°. This theory does not try to explain the moving star motion, as in the Aristotelian theory of homocentric spheres. It lacks any justification for the causes of the motions and of its configuration. Chinese astronomers do not investigate these matters, they concentrate on observations. The main observation concerns the sun, which unfortunately with its brightness does not make it possible to tell in which portion of the sky it is at any given moment. Egyptian astronomers first, and later the Greeks, tried to overcome this problem by attempting to fix the time of heliacal rising of certain stars. In Egypt, the heliacal rising of Sirius heralded the arrival of the floods of the Nile. This approach to the problem led spontaneously to refer the positions of the Sun relative to the constellations located in the ecliptic region, i.e., in the zodiac, an approach that Needham calls ‘by conjunction’.1 Chinese astronomy has adopted a different method based on ‘opposition’ of the Sun to the stars visible at night. This method is based on the systematic observation of circumpolar stars that never set. By being able to identify the meridian transit of a circumpolar star, it is possible to know the night time (remember that the nocturnal time is based on this very principle). In this way, the star that is in opposition to the Sun every day will transit the meridian exactly at midnight. Chinese astronomers identified the solstices long before the Alexandrians. The beginning of the year is set at the winter solstice, while Alexandrian astronomers based the chronology of the year on the spring equinox. Maps of the Sky and Constellations  As early as the -XIV century, the equatorial circle was subdivided by hourly circles passing through the pole (maximum circles2), thus segmenting the sky into regions, delimited by two  See Appendix “Positional Astronomy”  A maximum circle is the intersected of the sphere with a plane passing through its center. In this context, the circles correspond to meridians. 1 2

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maximum circles passing through the pole and any star for which the culmination or transit time can be determined. The number of regions is 28, called xiu, and are related to the lunar cycle. The number 28 is a good compromise between the synodic and sidereal periods. The xiu are the mansions of the Moon and Sun. They are organized into 4 groups of 7, placed in four equatorial zones, called palace, corresponding to the 4 seasons. Each xiu has an associated asterism. We can see (Fig. 14.3 right) that each group of 7 mansion is organized with three mansions in the center and two mansions on either side (external circle). With this map we can illustrate the opposition method used to determine the Sun location. In fact, the ideogram near East indicates the spring equinox, which is in the eastern mansion at hour 6 in the morning (hours are indicated in the internal circle). In the evening, at hour 6, the Sun will be in the opposite mansion, that of the autumn equinox, indicated by the symbol near West (de Saussure, 1919–1920). The names of the four palaces (the seasons) have mythological names: in the East the Blue Dragon, in the South the Vermilion Bird, in the West the White Tiger, and in the North the Black Turtle. Unlike Western cosmology, these names do not correspond to asterisms interpreted in figurative form. The years were named after 12 animals, according to a legend attributed to Buddha.3 Scholars have investigated the origin of the organization of the celestial vault into the 28 xiu, a subdivision shared with India and Arabia. Excluding Arabs, whose observation date after seventh century, it is still an open question if India was first. Nine of the 28 asterisms are common with the Indian tradition, and eleven others share the same constellation but with different stars. On the other hand, the connection of the xiu system with the observation of transit of the circumpolar stars leads one to believe that the Chinese system is the oldest. In fact, the opposition method is absent from the Indian system. Furthermore, an Indian hymn from the -XIV century (contemporary with the xiu oracles) mentions the word nakshatra that denotes the stars as a whole, while references to individual mansions can be found in works from four or five centuries later. Scholars have also investigated the relationship between the Chinese and Babylonian systems. Cuneiform tablets from Ashurbanipal times show that

 Buddha had summoned all the animals but only twelve showed up. He therefore decided to name the years with these names according to the order in which they presented themselves: mouse, ox, tiger, rabbit, dragon, snake, horse, goat, monkey, rooster, dog, pig. 3

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Babylonian astronomers observed the heliacal rising of the stars.4 Therefore, the two methods developed independently. Chapters on astronomy are found in official dynastic records from the -II century onwards with the work Shiji (Historical Record) by Sima Qian (-145 -86). The astronomical chapters contain star catalogues that are copies of other lost. One catalogue date back to the era of the Warring States (−476–221) and is attributed to three schools led by three great astronomers who lived between −370 and −270: Shi Shen of the Qi state and Gan De of the Wei state. The third, whose real name is not known, is called Wu Xian, probably a minister of the Shang dynasty. These catalogues were compiled about two centuries before that of Hipparchus. The astronomer Shi Shen wrote the work Thien Wen (Astronomy) and the astronomer Gan De the treatise Thien Wen Xing Chan (Astronomical Stellar Predictions), while Wu Xian’s work bears just his name. The royal astronomer Chen Zhuo of the state of Wu (Three Kingdoms period 220–265) constructed a map (unfortunately lost) of the stars and constellations according to the studies of Shi Shen, Gan De and Wu Xian. It contained 254 constellations, 1283 stars, 28 xiu, and 182 other stars for a total of 283 constellations and 1565 stars. Between 424 and 453, another astronomer Qian Lezhi constructed a planisphere by marking the stars identified by the three great astronomers in different colors: those of Shi Shen were colored red, those of Gan De in black, and those of Wu Xian in white or yellow.5 Thus, we are in the presence of prolonged sky observation, with periodic updates of older maps. These are veritable catalogues in which each constellation has associated: (a) the name of the asterism, (b) the number of stars it contains, (c) its position with respect to the neighboring asterism, and (d) the measurements in degrees (referred to 360.25°) of the main stars. These measurements included the hour angle (right ascension) of the main star measured from the first point of the xiu to which it belongs, the distance from the north pole.6 Since the celestial latitude of the main star is also given in some texts, we have a clue to the contamination with Greek and medieval astronomy that adopted ecliptic coordinates. Comparing the polar star coordinates

 Needham (1956, pp. 252–259).  Needham (1959, p. 263). 6  Recall that Chinese astronomy adopted an equatorial reference system, in which the positions of celestial bodies are identified with right ascension (AR) and declination, with the only difference that they used the complement of declination, i.e., the distance to the north. 4 5

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to present day, and considering precession allowed to determine that these ancient catalogues date back to −2400 c.7 The Dunhuang star chart shows these constellations. The city of Dunhuang is located in Gansu in northwest China, at the crossroads of two important Silk Road routes. It is in an oasis in the Taklamakan Desert, and the city was the first trading center encountered by merchants from the west. It was also an ancient Buddhist pilgrimage site and a military garrison along the oldest route of the Great Wall, built during the Han dynasty. The star chart was found in 1907 by the archaeologist Aurel Stein, among 40,000 other manuscripts and the first printed books, preserved in one of Magao‘s caves, the caves known as the ‘thousand Buddhas’. It is a very thin roll of paper, 3940 mm long and 244 mm high, written on one side only. It is divided into two parts. The first part contains a text of about 80 columns of astrological and meteorological predictions with 26 drawings of clouds of different shapes (Fig. 14.4 left). The notes below the clouds describe the possible events announced by their shape. They could be meteorological events (like in the parapegma), or the announce of risk of illness or the arrival of enemies. The second part is 2100 mm long and contains a star atlas organized into 12 vertical maps (Fig. 14.5 top), each with an accompanying text, followed by a map of the circumpolar region, ended with a column of text and the

Fig. 14.4  Dunhuang Map. Left meteorological prediction with shape of clouds. Right: The drawing of an archer  Ibidem, p. 249 ff.

7

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Fig. 14.5  Dunhuang Map. Top: the complete map. Bottom. On the left, the lunar month 5, which includes the Canis Minor, Cancer, and Hydra. On the right, the lunar map 4 with the constellation Orion. Note the colors of the different authors

drawing of an archer representing the god of lightning, dressed in the style of an imperial official (Fig. 14.4). The 12 maps cover the 28 xiu, each asterism is indicated by its name. The first xiu correspond to the month of February. Every strip of the chart shows the constellations visible in the xiu. The black stars appear to be taken from Chen Zhuo‘s catalogue, the red ones are probably by Shi Shen, and the stars are all marked with the same size, so there is no brightness indication. In Fig. 14.5 bottom two xiu, on the right the Orion constellation can be easily recognized. The circle of stars on the right was named The soldiers’ market and the constellation with a U shape (under the reddish left leg of Orion) have the name the soldiers’ latrine. There are between 1339 and 1359 stars grouped into 257 asterisms. A thorough analysis shows that all stars are visible from the latitude of 34°N of the

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historical capitals. The stars reported are up to magnitude 6.5, proof of the very keen eyesight of Chinese astronomers. The celestial positions are given in degrees, which reveals that astronomical observations were the basis of astrological use. The map of the circumpolar region has further cosmological value in that it symbolizes the Purple Palace with the Celestial Emperor in the Pole, circled by his family, servants, and military officers. The constellations in this region are named as element of the Imperial Court. The Big Dipper, for instance, is the Chariot of the Emperor. From a scientific point of view, this map presents a cylindrical projection of the asterisms of the mansions and an azimuthal projection of the circumpolar region, two projection techniques in use in modern cartography. In the in-­ depth study of this document by Bonnet-Bidaud et  al. (2009), there is an evaluation of the accuracy of the projections. The authors compared the co-­ ordinates of the stars of magnitude >3 detected on the map with the actual right ascension and declination, corrected for precession to the year 700, and obtained a correlation of more than 0.9, which is very high considering the measurement inaccuracy on a digital map scan. Bonnet-Bidaud et al. believe that the document is older than the 940 estimated by Needham. In favor of this different hypothesis, there are several evidences. The paper is made of mulberry fibers, very thin and expensive, similar to that in use until the end of the Tang dynasty (618–907). It is a paper certainly suitable for works dedicated to the Imperial Court. In the accompanying text there is a sentence that translated literally is ‘your servant Chunfeng says …’. Unfortunately, the date and author’s name are missing. This brings to the conclusion that this map was compiled by the astronomer Li Chunfeng (602–670). Other elements, such as the style of writing and the position of the stars and pole in the map, lead these authors to date the map to between 649 and 684, during the Tang dynasty. Chinese astronomers were able to detect stellar positions with a precision comparable if not superior to that of the astronomers of the Alexandrian period. The association of the name of the asterism with the positions of the stars is an original conception. In the oldest documents of Western astronomy, the Dendera Zodiac dated around −50 and the Farnese Atlas dated around −100, the constellations are represented only with symbolic figures and without precise positioning of the stars. In the star catalogue by Hipparchus, the number of stars identified was about 850, in the Almagest there is a list of 1028 stars. In this Chinese chart, there are 1464 stars accurately position. The Chinese observations were probably even more accurate, for Zhang Heng wrote:

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North and south of the equator there are 124 groups always shining. 320 stars have individual names. There are 2500 in total, not counting those observed by sailors. Of the very small ones there are 11,520. And all of them have an influence on fate.8

Astronomical Events  Besides the catalogue of stars, Chinese astronomers recorded a large number of astronomical events. Eclipses  The earliest eclipses, dating to the Shang period, that have been found on bone inscriptions are: −1361, −1342, −1328. -1311, −1304, −1217. The only reported solar eclipse is −1217. But when theory of eclipses did appear in China? That moonlight is a reflection of sunlight was already understood by Parmenides in the -V century, and Anaxagoras explained the cause of lunar eclipses also in the -V century. In China, this was clarified in later times. Shi Shen believed that the eclipse was due to a prevalence of the Yin over the solar Yang. In turn, Gan De speaks of eclipses as emerging from the sun itself, misinterpreting sunspots. The theory of the influence of Yin still survived in the first century, in particular the skeptical philosopher Wang Chong (25–100) continued to support the theory of the influence due to an intrinsic rhythm of the luminous bodies, against the hypothesis that the eclipse was due to the masking of one celestial body towards the other.9 In the Song period, the philosopher Chu Hsi wrote in 1180 a clear description of the cause of the eclipse. At the end of the lunar month there is an eclipse of the Sun when there is a conjunction of the Moon and the Sun at the same east-west degree (AR) and on the same north-south line (declination). In this configuration the moon covers the sun and this causes the eclipse. Similarly, when there is a full moon, and when it is in opposition with the sun at the same degree and along the same line, the Moon is protected by the Earth and there is an eclipse of the moon.10

In the treatise on the seasons Tso Chuan in −542 there is a list of 37 eclipses from −720 onwards, almost according to Ptolemy, who in the Almagest gives  Needham (1959, p. 265).  Ibidem p. 413. The author observes a similarity between the subject matter they are discussing and the philosophical ideas of Titus Lucretius Carus. 10  Ibidem p. 416. 8 9

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the date of −721 as the first eclipse recorded. Since the Han dynasty, the dates of eclipses have been recorded regularly. A total of 925 solar and 574 lunar eclipses were counted up to 1785. The partial match between occidental and Chinese observations are probably due to several factors. First of all, solar eclipses are not visible everywhere, unlike lunar eclipses. The sky is often overcast and the lunar eclipse is not easily visible. In addition, eclipses are considered announcements of negative events, and in order not to offend the emperor, the recording of an eclipse may be canceled. For example, after the death of the emperor Gao Zu (−256 −195), known also as Liu Bang, the first emperor of the Han Dynasty, the Empress ruled cruelly for some time, to the point that an eclipse that never happened was announced in −186. Chinese lists of eclipse are therefore partly unreliable, given the political value of the announcement. Nevertheless, the accuracy of the records grew over time. As far as the prediction of eclipses is concerned, we can note that the recognition of the inclination of the moon’s orbit occurred in 206, when the calendar indicated its inclination to the ecliptic as 6°. In later years, an official terminology was established to indicate the start, end, and maximum time of the eclipse and to distinguish between complete, partial, or annular eclipse. Novae and Supernovae  The appearance of a nova is written in one of the oracular bones of the -1300s records an astronomical event. Chinese astronomers later called it a guest star, a kind of unwelcome participant in the life of the Empire. The inscription reads ‘on the 7th day of the month, a great new star appeared in the company of Antares’. The year is not given; however, the event is estimated to be between −1339 and −1281.11 The nova recorded only by Chinese astronomers in 1054 gave rise to the Crab Nebula. In the Essential Elements of Song History (Song-hiu-yao), we read on 27 August 1504 «I humbly observe a host star that has appeared during these nights; above it is a faint yellow twinkle». By 1506, the host star had disappeared. Chinese astronomers recorded in total nine novae. Historical records of novae and supernovae have been used by contemporary astrophysicists to correlate them with radio emission sources. As for the nova near Antares, the hypothesis cited by Needham that it might be the source 2C 1406 has not been confirmed; the other nine historically documented novae have been observed by Tycho in1572 and Kepler in 1604.

 Ibidem p. 423 ff.

11

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Comets, Meteors, and Meteorites  The passage of comets is one of the fields in which Chinese astronomy has collected a great deal of data. Between −613 and +1621 no less than 3722 comets have been recorded. The first comet registered on Babylonian tablets dates back to −1140. Chinese astronomers recorded not only the sighting but also the different positions of their path. For example, the comet of 1472 was detected in Europe by Regiomontanus, who reported a single position of it on January 20 near Arcturus. Paolo dal Pozzo Toscanelli (1397–1482) tracked its trajectory from January 8 to January 26, providing a simple description but no quantitative data. In contrast, more than thirty positions are indicated in Chinese texts for the entire visibility period of just over a month. These data, described with great precision in reference to the nearest stars, made it possible to accurately determine the trajectory of this comet. The comets were clearly distinguished from the seven moving stars, with more terms denoting their characteristics, e.g., sweeping star, or sailing star, or long star, or even blazing star like a candle. The first observation in China of Halley’s Comet dates back to −240 with certainty and is recorded again in −87 and −11. Its appearance of 1066, the date of the Battle of Hastings, is also present in Kepler’s accounts.12 Systematic observations revealed the phenomenon that comet’s tail is always oriented in the opposite direction from the sun. Meteor showers, moreover, are associated to comet passages. Chinese astronomers have recorded, since −687, more than 200 pages of data of meteor showers. Sunspots  The last type of astronomical event, which demonstrates the observational capabilities of Chinese astronomers, is sunspots. First observed in Europe by Galileo through a telescope, they were initially thought to be caused by the passage of small planets. Galileo thought they were a kind of clouds of the Sun since he observed their rotation with the celestial body in about a month. The astronomer and alchemist Liu Xiang (−79–8) observed and described sunspots from −28. Since then, 112 major sunspots have been recorded. Their shapes and sizes have been compared to an egg, a coin, a feather, or a peach. It is possible that sunspots were also observed in earlier times, in fact the term wu meaning crow or black is often found associated

 The Halley’s Comet was essential to confirm Kepler’s theory of elliptical orbits and to consider the comets as part of the Solar System. 12

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with the sun as in this sentence: «scholars know that there is a three-legged crow in the sun».13 The Motion of the Planets In the Khai-Yuan Chan Ching (Treatise on Astrology of the Khai-Yuan Kingdom) written between 718 and 729, we see the ancient names of the planets dating back to the -fourth ­century. Each planet is associated to a cardinal point and one of the five elements14 (Table 14.1). Chinese astronomers had identified the direct motion, called shun, and the retrograde motion called ni of the planets. They fixed a terminology to describe the rising chu, the advancing chin, the change of direction fan, the retreating thui and finally the setting ju. Through careful observations they had identified planetary periods, distinguishing between sidereal and synodic periods. Table 14.2 shows the periods noted at different times by Chinese astronomers. These values demonstrate that astronomers achieved more and more accurate measures, comparable to those obtained by Greek astronomers in the same era. However, the study of the celestial path, which fascinated the Alexandrian astronomers, was completely absent. This does not mean that they did not have a mental image of these paths. In fact, the motion was described, as said, by many terms. Needham note that in a conversation transcribed in 1190 two philosophers discuss the large and small circular paths of the sun and moon and the much larger paths of the planets and fixed stars. One of them believed that retrograde motion was only apparent and depended on the different velocities of the various moving bodies. He suggested that calendarist should realize that retrograde motions called ni and thui motions were also progressive like shun and chin.15

Table 14.1  Name of planets. From Needham, cit. p. 398 Jupiter

Sui xing (the Year-star)

East

Wood

Mars

Ying guo (Fitful Glitterer

South

Fire

Saturn

Chen xing (the Exorcist)

Centre

Earth

Venus

Thai pai (the Great White One)

West

Metal

Mercury

Chen xing (the Hour-star)

North

Water

 Ibidem. p. 435.  Ibidem. p. 398. 15  Ibidem p. 400. 13 14

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Table 14.2  Sidereal planetary periods. From Needham, cit. p. 401 Mercury 88.97 (d)

Venus 224.7 Mars 686.98 Jupiter (d) (d) 11.86 (y)

Gan De -IV cent.

68

585

Eudoxus -II cent.

110

570

Suma Chien -I cent.

Saturn 29.46 (y)

400 260

626

28 12

28

Liu Hsin -I cent.

116.91

584.13

780.52

11.92

29.79

Li Fan +85

115.881

584.024

779.532

11.87

29.51

The Arrival of the Jesuits and Western Astronomy We can now better understand how the Jesuit missionaries contributed to Chinese mathematics and astronomy. When Matteo Ricci met Chinese scholars, he quickly realized that they computed eclipses by a much more inaccurate method than the one he studied. Jesuit Sabbatino de Ursis (1575–1620), an astronomer in the Jesuit community in China, had exactly calculated the solar eclipse of December 15, 1610. This fact prompted the Emperor to initiate a reform of the official calendar. The Jesuits sought to remedy the absence of a geometric explanation of planetary motions. Stereographic projection methods, also useful for cartography, were unknown. As already said, Ricci edited the first Chinese translation of Euclid’s Elements. The Jesuits also introduced misconceptions into Chinese astronomy, such as the theory of homocentric spheres, but they popularized the telescope. They introduced more advanced methods of algebraic computation and techniques for instrument construction, including division into linear and angular scales and the use of micrometer screws, which were fundamental to precision instruments. The Aristotelian and Ptolemaic doctrine of celestial spheres was opposed to that of the Xuan Yeh school of bodies floating in empty space. Jesuit science also hindered the spread of the heliocentric conception and propagated an erroneous theory of the precession of the equinoxes, which Chinese astronomers regarded with caution. The Jesuits did not understand the importance of the equatorial and polar coordinate system and did not understand the 28 xiu division of the lunar mansions, which is independent of the duodecimal division of the zodiac.

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They probably did not know that Tycho Brahe had also introduced the equatorial coordinate system that allowed him to improve the accuracy of his measurements. As a result, the Jesuits imposed the construction of ecliptic armilla in place of the equatorial armilla. The Jesuit missionary was keen to establish a collaboration between the Chinese and the astronomers of seventeenth-century Europe. The Jesuit Johann Schreck, known as Johannes Terrentius (1576–1630), who had been a member of the academy of the Lincei before coming to China, wrote a letter (Terrentius, 1993) to Kepler in 1623, where he reports on the task of improving the calendar and asks for recent astronomical works on eclipse calculus and the theory of the moon, in particular the writings of Galileo and Kepler.16 Kepler’s answer (Kepler, 1993), in which he responds to Terrentius sentence by sentence, was published as “Commentatiuncula” in Regensburg in December 1627, and extended by an appendix written on 15 January 1630.17 However, the Commentatiuncula never reached Father Terrentius, the actual addressee of Kepler’s letter. This mixture of modernity and conservatism was assimilated by Chinese astronomers, just as they had done in the past, mixing with contributions from multiple invasions and ethnic contaminations. The Jesuit Schall von Bell (1591–1666), in 1645 wrote the Treatise on Astronomy and the Science of Calendars according to the New Western Method. This work was warmly welcomed and became the reference work for Chinese astronomers. Note the expression New Western Method, the emphasis on New Method reveals that, for Chinese philosophy, the overcoming of ancient conceptions was not based on cultural domination but on the acceptance of the scientific methodology that had been established in the West, particularly with Galileo. As Needham write, the Jesuits could insist that the natural sciences born with the Renaissance were primarily ‘Western’, but the Chinese clearly understood that they were primarily ‘new’.18 In 1669, Verbiest oversaw the renovation of the observatory in Peking. The old instruments from the Yuan or Ming era were removed from the astronomical platform on the eastern wall of the Imperial Palace, and new instruments were installed: (1) a simple ecliptic armillary sphere supported by four dragon heads, (2) a simple equatorial armillary sphere mounted on an arched dragon back, (3) a large celestial globe whose horizon is supported by four  pp. 257–314.  This was the first text to be printed in the new press in Sagan and at the same time presented to Wallenstein as a New Year’s gift. 18  Needham (1959, p. 449). 16 17

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pedestals, (4) a horizon circle for azimuth measurements, (5) a quadrant supported by a vertical staff, (6) a sextant on a single pedestal, (7) an alt-azimuth quadrant, (8) an elaborate equatorial sphere, and (9) a small celestial globe. In a report sent to the journal Nature in 1889 we read19: Mr. Thomas Child, who has just returned from Pekin, has sent us very beautiful photographs of the two interesting old astronomical instruments at the Pekin Observatory. These instruments are the most ancient of the kind in the world, having been made by order of the Emperor Kublai Khan in the year 1279. They are exquisite pieces of bronze work, and are in splendid condition, although they have been exposed to the weather for more than 600 years. They were formerly up on the terrace, but were removed down to their present position to make way for the eight instruments that were made by the Jesuit Father Verbiest in 1670, during the reign of the Emperor K’ang Hsi, of the present dynasty. (Fig. 14.6)

Fig. 14.6  Beijng Observatory, view from the terrace

  Nature (1889–1890) Vol.41, Nov. 21, 1889. p.  6. https://archive.org/stream/in.ernet. dli.2015.228314/2015.228314.Nature-1889-1890_djvu.txt. Accessed July 2022. 19

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The Tapestry of Astronomers (Fig. 14.7), produced by the French Bauvais manufactory in the early eighteenth century, depicts Jesuit and Chinese astronomers engaged in the study of the heavens at the Imperial Office of Astronomy. The figure with the long white beard is most probably Johann Adam Schall von Bell (1591–166) (Standen, 1976).

Mechanics Astronomers and mathematicians worked in the exclusive service of the emperor and were a component of the imperial bureaucracy. The secrecy of their work was total, guaranteed even by being housed in the imperial palaces.

Fig. 14.7  Tapestry of the Astronomers, detail. Beauvais Manufactory. Residenz München

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Many skilled craftsmen were also housed in the palaces. They worked in the imperial workshops and arsenals. The organization of workers is described in Kaogongji (Register of Artisans), that is a chapter of the Register of Institutions and Rites of the Zhou dynasty, supposedly compiled during the Han dynasty (-IV century). In the Register, we find six classes of workers and among them hundreds of craftsmen. Craftsmen were classified in the treatise as stone and jade workers including quarrymen and carvers; pottery workers, including vase and tile makers; wood workers, including bow and arrow makers, furniture makers, construction supervisors, carpenters, and agricultural tool makers. In addition, there were canal builders and hydraulic engineers, and finally metal workers, vile alloy smelters, precious alloy smelters, bell smelters, sword makers, plough blade makers, and makers of measuring instruments. The list is very long and shows, on the one hand, the precision with which the imperial bureaucracy censored productive activities and, on the other, a careful division of labor.20 There was also a hierarchical organization that started from planning and decision-making through leadership, management, and execution. Lords and princes made decisions about a work, following the principles of the Dao. Ministers and officials were responsible for execution. Transportation was the job of merchants and travelers. Women worked on textiles, and peasants were in charge of land and agriculture. The invention of machines was the task of men of ingenuity and their tradition was maintained by skilled men, and those who continued for generations were called artisans. It is not merely a register of activities or an arbitrary classification, but principles of organicity that regulate the roles of different members of social classes. The organization into guilds or corporations was absent in ancient China. This form of aggregation, which made artisans autonomous from central power, first emerged in the Italian Renaissance and then spread throughout Europe. Guilds, in Europe, were intermediate social bodies between ordinary citizens and the government. In China, it made its appearance when systematic trade relations developed with European countries. Merchants had to register with the imperial bureaucracy in order to trade. It is, however, an entirely different form of aggregation from that of the West, also for the reason that merchants were considered the lowest level of human activity and strongly despised.

20

 Needham (1965, p. 16).

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To conclude, we have seen that mathematical knowledge, in particular geometry, was far from being based on the methodological principles founded by Euclid. Hence, the connection between geometry and mechanics, as characterized by Newton,21 did not exist in China. Even the graphical representations of geometric problems are rare. In the West, technical drawing originated in the Renaissance with Leon Battista Alberti (1404–1472) who wrote the treatise De Pictura previously the use of depiction was limited to simple machines or the description of geometric problems. It was after the mid eighteenth century that technical drawing spread in the West, giving rise to a true engineering discipline. Drawing to depict machines or buildings in ancient China was approximate, certainly not a guide for construction.

Astronomical Instruments and Machines Recall that in China, the observation of the sky was mainly used for preparing annual calendars. While in Europe the aim of astronomical studies was the understanding and prediction of planetary motions, in China this need was secondary, so the astronomer’s instruments were in a certain sense simpler, but not necessarily less precise. Shadow Measurement, Gnomons, and Masonry Observatories In 2003/2004 in Xiangfen County, Shanxi province, an archeological observatory was discovered at Taosi. Taosi is considered the capital of the mythical King Yu, of the Xia period. Taosi ancient observatory forms part of the Taosi archaeological site, one of the most famous of the about eighty sites of Longshan Culture (c. -3000-2000). Archaeologists have found a round foundation- pit in the center and three concentric rammed-earth circles on which pillars were erected. As viewed from this center, the series of 11 pillars with 10 slots and the 2 pillars with 1 slot appeared to form a single line of 13 pillars and 12 slots. Calculations and experimental observations indicate that 4000 years ago the sun would have risen at the June solstice in the northernmost slot. December solstice would have risen in the slot next to the southernmost pillar. Slot no. 7 (counting from south to north) could possibly have been used to determine the spring and autumn equinoxes: nowadays the sun can be seen as rising in this slot on March 18 and September 25 (see Fig. 14.8), close to vernal equinox.  See Sect. “The Cause of Celestial Motion: Isaac Newton”.

21

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Fig. 14.8  Left: Taosi observatory, the pillars (reconstructed), and the center of observation (glass covered). Right: A drawing of the layout of the pillars

The simplest form of the gnomon was a stick driven into the ground to observe the movement of the shadow cast by the sun and detect the passage of time. There are writings from the -600s that describe astronomers observing the sun’s shadow and noting the length of the shadow to identify solstices and equinoxes. The estimation of the ecliptic inclination, as detected by the shadow angles between the two solstices, probably dates back to the fourth century, but the only certain document dates back to +89: the astronomer Chia Khuei writes that in winter the shadow angle measures 115° and in summer 67°, their difference halved gives an angle of 23.66°. Ptolemy arrived at the value of 23.5° in about 143 and Liu Hung in 173.22 Two astronomers Nankung Yüeh and the Buddhist astronomer I-Hsing (683–727) between 721 and 725 conducted a measurement campaign23 at nine locations between latitude 17° 4′ in the far south (in present-day Vietnam) and 40° near present-day Lin-Xian in Shansu. The result of their measure is 7973 li, about 3500 km.24 The quest for precision also led Chinese astronomers to devise new instruments. In 1276, during the Yuan dynasty, a giant masonry gnomon was built in the city of Gaocheng. It was a construction for measuring the length of the Sun’s shadow as it passed. The current structure was restored in the Ming era. The shadow of the gnomon is projected onto a graduated scale, called a shigui, 12.62 m long, which has three water channels to check its horizontality. An even older astronomical tower survives in Korea in Gyeongju, called the Cheomseongdae (star-gazing tower) built around 632–647. The tower is 5.7 meters wide at the base and 9.4 meters high and is topped by a square structure.  Needham (1959, p. 287).  Ibidem, pp. 292–293. The longitude was probably 107.15° east and the identified locations are different although relatively close. 24  Let’s recall that de Maupertuis’s expedition to measure the Meridian dates back to 1739. 22 23

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Measuring Time: Sundials  A peculiarity of the oldest Chinese sundials is the orientation of the plane onto which the shadow of the gnomon is cast. As we have seen, Chinese astronomy is based on an equatorial/polar orientation. This led Chinese astronomers to construct sundials whose projection plane is parallel to the plane of the celestial equator and the gnomon is perpendicular to it and points towards the celestial north. These instruments are mentioned in writings dating back to around −100. There are many variants of portable sundials, the simplest of which is shaped like a small box. When the lid is opened, a thin wire casts a shadow on a graduated scale. The box often contains a compass for correct orientation. More complex portable compasses show the course of the sun for different latitudes. It is very likely that these types of sundials are the result of contacts with Arab civilization. Measuring Time: Water Clepsydrae  Measuring time with sundials provides exact solar time only at the place of observation. Clepsydrae, on the other hand, provide a position-independent measure of time. Chinese clepsydrae were water powered. They appeared as early as the seventh century, and many variants and refinements of them were built. One problem with clepsydrae is maintaining a constant water flow. In fact, during emptying, the fluid pressure decreases with a consequent reduction in the outflow rate (Fig. 14.9 left). To overcome this limitation, clepsydrae with cascading containers were introduced as early as the first century of the current era, and more complex solutions were introduced between the third and sixth centuries. Some clepsydrae had a mechanism with a weight to stop the water flow, a mechanism of great interest for astronomical use. The Chinese clepsydrae were precise enough to detect the irregularity of the length of the day during the year. Their accuracy, however, was still insufficient to detect the equation of time, which was not measured accurately in Europe until the seventeenth century, after the invention of clocks, with an error of a few seconds. A clepsydra built in Canton in 1316 remained in operation until the early 1900s. So-called combustion clocks were used to measure short time intervals or as alarm systems. They can be simple cones or sticks of incense (Fig. 14.9 right). More complex combustion clocks are bronze or silver tablets in which a labyrinth-like path (called shou or longevity) is deeply engraved. Burned incense travels through the labyrinth until it runs out. Jesuit missionaries also described these instruments in their reports, observing various uses as alarm clocks. The burning time was carefully calculated. A metal object could be

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Fig. 14.9  Left: This drawing depicts a roukoku, a type of water clock. The history of water clocks is written in Chinese. It states that they were invented by the Yellow Emperor, a mythical emperor of around −2600. Water clocks existed in China before the current era. Right: combustion clock. When the calibrated incense sticks finish burning, a small bell chimes to alert

hung from the burning material, and when combustion exhausted the support, the object would fall into a metal bowl that resonated like an acoustic alarm clock. Armillary Spheres  There is a substantial difference between the Western and Chinese armillary spheres. We have already seen that the celestial reference system in use in Hellenistic times was the ecliptic or zodiacal system, while the one used by the Chinese was the equatorial system, which was introduced to Europe by Tycho Brahe. The greater simplicity of the equatorial system allowed the construction of large machines. (Fig. 14.10 right). The equatorial armilla is much better suited to measuring celestial coordinates and is therefore primarily an observational instrument. Probably the simplest primitive system consisted of a rotating circle representing the meridian or celestial equator, over which a fiducial element, the alidade, was slid to identify the hour or declination angle. Complete armillae are described from the mid -I century, near the time of Hipparchus. The ecliptic ring was added in 84 by the astronomers Fu An and Chia Khuei. The armilla comprising the horizon and meridian rings was built by astronomer Zhang Heng in 125. He was the first to provide the instrument with a waterpowered motion mechanism. Since then, the construction of armillae has been perfected, they are partly made of iron and bronze, the supporting axes are made of steel. Also included are a ring oriented like the ecliptic and an observation tube instead of an alidade.

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Fig. 14.10  Left: Zijinshan Observatory. Equatorial torquetum. The first part (top right) has the equator ring with axis parallel to the earth axis and pointing to the north pole and the declination ring with an alidade (missing). The second part (bottom left) has horizon and altitude rings. It combines an equatorial and alt-azimuth armillae. Right: Equatorial armilla, probably designed by Ferdinand Verbiest

Around 720, Emperor Xuanzong (685–762) of the Tang Dynasty ordered an armillary instrument to be built in order to correctly calculate eclipses. I-Hsing took on the task and to measure the inclination of the moon’s orbit with the necessary precision he invented a mechanism to track celestial motion. This mechanism was driven by a water clepsydra with an escapement, very similar to that created by Su Song.25 The use of this instrument is described in the Chin Shu (History of the Chin Dynasty) where we read that the observations were done by two astronomers working together. One was inside the building and measured the position of a star from the armilla clock. The second astronomer, outside, checked the position of the star, to synchronize the clock with the starry sky. This description confirms that the armilla was used for observations and to calibrate the measurement of time, two essential functions for calendar calculation. The apogee of armillae construction was during the Northern Song dynasty, but after the Tartar conquest in 1260, the technology declined and over time understanding of the use of the instruments was lost until the arrival of the Jesuit missionaries. They also constructed Ecliptic and Altazimuth armillas and large torquetum (Fig. 14.10 left).

 Needham and de SollaPrice (1986, p. 77).

25

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The movement system of Chinese armilla anticipates the adoption of the equatorial telescope mount (see Sect. “Instruments to Guide the Observation”), proposed by Robert Hooke in 1670 and actually built from the following century until Fraunhofer’s refinements in 1824 (King, 1955). Armillas and other astronomical instruments are preserved not only in Beijing but also at the Zijinshan Observatory in the Purple Mountains National Park near Nanjing. Time and Hours  During the Han (−206 +220) and the Southern (Liang) (502–557) dynasties the division of time changed. The hours were subdivided in three ways. A duodecimal subdivision, a centesimal subdivision, and a night subdivision suitable for guard shifts. A cycle of 12 pairs of hours was established at the time of the Han Dynasty. Day and night were divided into 12 double hours (called shi), the day begins at 11  a.m. and continues with 1  a.m., 3  a.m., 5  a.m. and so on until the 9–11 p.m. interval. We are thus faced with a cycle identical to our 24 hours, aggregated in pairs. The day, from midnight to midnight, was divided int 100 parts (called kè) that correspond to an interval of 14 minutes 24 sec. This subdivision underwent many reforms. Emperor Gao Zu of the Han dynasty ordered a reform and the division was increased to 120 to find a compromise with the 12-hour subdivision. After his death it reverted to 100, and the reform of Emperor Wu (502–549) reduced the number to 96. Around the year 560 there was a return to the division into 100 parts. The subdivision of night watch (called Gēng-diǎn) was different in each region and depended on the local day-night cycle. It is a variant of the centesimal subdivision. The cycle begins at sunset and terminates at dawn and is divided into five parts. The duration of each interval was not uniform, given the different length of the night between summer and winter. Historians believe that the duodecimal subdivision and the nocturnal subdivision of the guards are of Babylonian origin and in fact originated at the same time. Conversely, the centesimal subdivision has a typically Chinese nature corresponding to decimal numbering. Su Song Machine  It is commonly believed that clocks are an invention of the Western world and that the Chinese did not know about them. Certainly, one of the reasons for this belief is the accounts of the Jesuits. Matteo Ricci

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brought mechanical clocks with chimes that aroused great interest among the senior Chinese officials he frequented. However, we have already seen that many armillae built in previous centuries were equipped with mechanisms to track the movement of the celestial sphere. Some of them could be considered as clocks. Before describing the level of sophistication of the time-measuring instruments that had already been built by the year 1000, we must introduce a very special personality: Su Tzu-Jung called Su Song. Su Song was born in 1020 in a locality in Fukien province. In the course of his career, he held numerous public offices, gradually progressing into a bureaucratic career. Early in his career, he was in charge of preparing imperial edicts and advising the emperor and key ministers. During the empire of Yingzong (1063–1067) of the Northern Song dynasty, Su Song became Supervisor of the staff of the Ministry of Finance and received an honor reserved for the most incorruptible officials. During this period, Su Song was associated with a conservative Confucian party. He often received special assignments, including trips abroad, particularly to the northern countries in the Liao Kingdom of the Mongol peoples. These trips gave him the opportunity to learn about different calendar systems and to develop an interest in astronomy. During his visit to the Liao Court, Su Song had to pay formal homage on the solstice day, but his calendar scheduled it one day earlier than the local calendar. This episode prompted him to study with interest local methods of calendar calculation. He argued heatedly with Mongolian astronomers in an attempt to make up for the bad impression. He managed to pull through by acknowledging the better accuracy of the Mongolian method. He observed that the difference would only become apparent if the delay fell at midnight, emphasizing the conventional nature of the start of the day. On his return home, he reported the episode, highlighting the errors of the imperial calendarist, which were punished.26 Approximately twelve years after his diplomatic assignment in the Northern countries, he was promoted to the position of Vice-Minister to the Right of the Minister of Personnel and at the same time Senior Director of the Imperial Chancellery. His scientific work also took place during this period. In 1086, the emperor issued an order for the examination of existing astronomical instruments and for the construction of an astronomical clock that could equal or surpass those  Needham (1965, p. 447).

26

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built at the beginning of the dynasty or earlier in Tang Dynasty times. Su Song was asked to suggest the name of a qualified person to do the job. He wrote a document for the Emperor explaining why he was proposing mathematician and engineer Han Kung-Lien, who was not a member of the Astronomical Bureau but an official of the Ministry of Personnel to which Su Song himself belonged. Su Song had by then reached the age of 75, and had been awarded numerous titles: Grand Protector of the Army, Marquis K’ai-Kuo of Wu-kung. He was also one of the tutors deputed to the imperial heir. When he died in 1101, he left a collection of literary works. Su Song had not only knowledge of astronomy and calendar science. In 1070, he wrote the Pen Ts'ao T'u Ching (Illustrated Pharmacopoeia), a treatise on pharmaceutical botany, zoology, and mineralogy, which contains valuable information about iron and steel metallurgy in the eleventh century and the therapeutic use of drugs such as ephedrine. Su Song’s Clock  In 1088, a fully functioning wooden pilot model was completed for Imperial approval. Two years later, the armillary sphere and celestial globe created by bronze casting were ready. In 1092 he began to write the monograph Xin Yi Xiang Fa Yao (which can be translated as The New Instrument) explaining the operation of the machine, which was completed in 1094 and officially presented. The astronomical clock built by Su Song and his collaborators in 1090, in the capital of Henan, Kāifēng, had a mechanism driven by a water wheel that rotated an armillary sphere at the top of the tower and a celestial globe on the upper floor inside; in Fig. 14.11, we see the original drawing. The time was indicated by the movement of five superimposed wheels bearing the hour figures. The height was between 30 and 40  feet (9–12  m). A wooden model was built in 1088 while in 1090 the celestial globe and armillary sphere were completed in bronze. The tower was disassembled in 1275 during the invasion of the Jurchen, and brought to Beijing, but nobody could assembly it again. Su Song’s description was reprinted several times until the mid-nineteenth century. Needham, Ling and de Solla Price translated this work and in particular the third chapter that contains the detailed description of the mechanism (Needham et  al., 1986). The escapement system is explained in Fig. 14.12. The motion of the whole machine comes from the great wheel, and it is transferred with axis and gears to the hour display wheels, to the celestial globe and finally to the armilla

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Fig. 14.11  Original drawing of Su Song clock tower. On top the armillary celestial globe. On the right the water reservoir. In the background, the large wheel that works as an escapement. In the foreground the wheels the display the time

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Fig. 14.12  Scheme of the escapement wheel. The wheel is made of scoops that can rotate individually on a fulcrum with a counterweight. In this way, each scoop, once filled with water, rotates and pours out the water. Once emptied, the scoop returns to its normal position. The scoops are mounted on the spokes of the wheel on which the other levers act to stop and release its rotation. Each scoop is filled in about 24 seconds. The block (02) stops the wheel. Water pours from the constant level tank (08) into the paddle (12). When 12 is filled, it moves the lever (13) rising, the counterweight (15) and lowering the side (13) of the lever (14). This action pulls the chain (06) thus raising the stop block (02) through the lever 04. The wheel is now free to rotate while the scoop is emptied leaving free the levers (14) and (04) to reset. Level (04) is attached with a spring to the stop block (02) that return to the stop position. The lever (01) acts to avoid a recoil during the wheel rotation. This whole “tic” process takes place in an instant

on top of the tower. The gearing for the motion of the armillary sphere and the celestial globe is based on wheels of 600 teeth and pinions of 6 teeth, so that 100 rotations per day of the driving axle produce one rotation per day of the armilla and globe. One step of the rotation cycle corresponds to a time division of 2′ 15″. A full reconstruction of the machine is in Fig. 14.13.

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Fig. 14.13  Reconstruction of Su Song tower. Top right: the hour display wheel. Bottom left: the celestial globe in the interior of the tower. Bottom right the armillary sphere on top of the tower

 hinese and Western Mathematics C and Astronomy The element that most distinguishes the two worlds is the choice of celestial coordinate system used also to build the armillary sphere. We have already noted that scholars today believe that also Hipparchus made his first star catalogue in equatorial rather than ecliptic coordinates, but in the Hellenistic era the ecliptic coordinate system was universally adopted.

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Chinese armillae were equipped with a sighting tube to align the star correctly. Western ones, when used for observation, were equipped with an alidade only. The equatorial mount allowed the tracking of a star with a single rotation around the polar axis, as in modern telescopes. These technical features were perfected from the first century onwards, coeval with Hipparchus, but completely independently, despite the already existing trade between China and the Hellenistic West. A second very important aspect of Chinese astronomy concerns the absence of mathematical analysis of celestial motions. Mathematics was used for practical problems; mathematicians were not interested to theoretical reflection and the knowledge of geometry was limited. In addition, the purpose of astronomical observations was the construction of the calendar, so astronomers were mainly interested to solar and lunar motion. The use of the equatorial coordinate system has a cognitive effect. In the West, reference to the ecliptic led to pay special attention to the motion of heavenly bodies in relation to the zodiac. Equatorial reference leads to turning attention to the position of stars in an absolute reference system in the celestial vault. Astrology in the west was based on a mathematical model of the motion of planets, Sun and Moon. The position of the moving bodies was considered as the cause of a person’s character. In Chinese culture, Daoist principles led to consider human character influenced by the totality of the surrounding nature, hence the lack of interest in imagining a causal correlation related only to the positions of the planets. Chinese astronomy was, as well, interested in the constellations of the ecliptic, but it was confined to the association of the positions of the Sun and Moon in an observable and recognizable scenario, not expressed by numbers. In China, the astrolabe is an unknown instrument, partly because the stereographic projection on which it is based had not been developed in Chinese trigonometry. While in the West the astrolabe proved to be of great use in navigation at sea, to the extent that its evolution led to the invention of the sextant, in China the use of the magnetic compass largely met the needs of orientation at sea and when crossing desert regions. The Chinese navy during the Yuan Dynasty essentially explored the eastern coasts and southward and westward to the Indian Ocean and to the Arabian Gulf, so the determination of the longitude was not a need, the compass was a sufficient tool. The Renaissance astrolabe and astronomical clocks had a popularizing value in that they offered the general public a representation of the cosmos. The study of the sky, moreover, has continuously changed the vision of the Cosmos. The idea of a Cosmos as an integral part of humanity and nature and

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the very structure of the organization of power: centralized and hierarchical did not modify Chinese cosmology. The figure of the astronomer was at the sole service of the emperor, the Son of the Heaven, to whom the annual calendar was used to determine the rites and functions of government. There was no need for the merchant, craftsman, or peasant to know the date of the equinox or solstice or the celebration of a festival. The direct observation of the sun or moon was enough to mark the day or month. Regarding the measurement of time, in China, the water clock was perfected into an effective timepiece. Despite the use of water clocks was widespread throughout the world, it was only in China, around the year 1000, that an escapement mechanism for counting time and its fractions was invented. And this was at least 300 years before the inventions of the first mechanical clocks in Europe. In water clocks, the measurement of time is still related to the measurement of a quantity of matter, in contrast to the counting of oscillations, which engaged European watchmakers for a long time in experimenting with the best solutions for the escapement. The oscillation watch in fact poses the problem of decoupling the escapement from the transmission of motion, a problem completely absent in Su Song’s escapement. From a theoretical point of view, the Su Song escapement is in fact composed of a mechanism that gradually accumulates energy to cause a release, without, during this process, interfering in any way with the transmission of motion thus satisfying one of the major requirements of clock escapements.

15 Design of a Simple Planetary Machine

Background Before starting the construction of my tellurium, I conducted thorough research on eighteenth-century machines, as described by Ludwig Oechslin in his book (Oechslin, 1996), which included technical drawings and gear train data. Oechslin also explained the innovative technical solutions adopted by the Priestermechaniker to replicate the anomalies caused by elliptical orbits. Additionally, I consulted the book by King and Millburn (1978), which also provided valuable information on gears. Furthermore, I referred to the book by Dubois (1849), which contained a description of the Huygens planetarium, along with the method of continued fractions for computing gear ratios. While I could have easily replicated one of these projects, I decided against it and instead reviewed current-day astronomical data on planetary periods. Based on this research, I decided to design my tellurium from scratch. Planetary periods are now recorded with remarkable precision and are frequently re-evaluated. With such advancements, achieving high accuracy, sufficient for the prediction of astronomical events, is possible. Nonetheless, it is my belief that the accuracy of mechanics is limited compared to what computational methods can achieve today. Through computational models, Supplementary Information The online version contains supplementary material available at https:// doi.org/10.1007/978-­3-­031-­30944-­1_15. The videos can be accessed individually by clicking the DOI link in the accompanying figure caption or by scanning this link with the SN More Media App.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. L. R. Marini, Imago Cosmi, Astronomers’ Universe, https://doi.org/10.1007/978-3-031-30944-1_15

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problems in planetary kinematics and dynamics can be solved with a degree of accuracy that has been demonstrated in the achievements of space exploration. In principle, the periods obtained by the masters of astronomical clocks and machines were very accurate compared to present-day data. However, no studies have been conducted on the efficacy of the end result. Is it true that a Weltmaschine by Philipp Mathäus Hahn can run for 72 years and demonstrate a precession of the equinox of 1 degree? Is it also true that a machine by Antide Janvier can continue operating for years and produce a cumulative error that corresponds to the theoretical result? The accuracy of a handmade mechanical instrument is limited. My opinion is that these machines cannot run continuously for more than a few years before they need to be cleaned and maintained, thus interrupting the celestial motion they simulate. This is also true for crank-operated machines: to make them move for the equivalence of hundred years they must be operated with a relatively high speed and the risk of failure increases. For my machine I had to consider this aspect and limit the accuracy to an error of a few minutes per year, considering that, giving my experience, accuracy of construction is considerably limited unless computer-aided manufacturing instruments (CAM) are used. Moreover, the mechanical simulation of anomalies today is just a curiosity. To observe on a planetary machine, the anomaly is very difficult, it can only be identified when the planet’s movement is displayed on dials and indicators, where numbers can be directly read. I designed my tellurium with a CAD (computer-aided design) system, drawing the machine assembly in three dimensions and simulating the kinematics with gear calculation programs. In the course of the design, I had to modify the gears in order to introduce transmission trains to make them compatible with the dimensions of the machine and the direction of rotation. I will also give some hints about construction problems that may reveal unexpected difficulties. Finally, I will describe a particular functionality that was not initially planned, but which I wanted to realize in the course of the work: to control the motion of the machine by means of a computer program that activates a stepper motor. In order to do this, I wrote a program to calculate the date of astronomical events and an interface between a tablet and the motor controller. This makes it possible to drive the machine to position on the desired date, showing the planetary configuration that the sun–earth–moon system will have at that time. With this solution, I have connected mechanical and computational simulation into a hybrid model.

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About Gears  Leaving aside historical aspects, let us recall that as far as the construction of a planetary machine is concerned, the fundamental concept of gear theory concerns reduction ratios. In fact, it is a question of reducing the rotational speed of a driving wheel so that at the end of the chain of gears the rotational speed is significantly lower and equal to the desired period. As we have seen, the ratio between the speed of the driving wheel and the final speed is expressed by a rational number, which in turn can be broken down into products of whole numbers. In a gear train, a distinction is made between a driving wheel and a driven wheel. In a speed-reduction train, the driving wheel is called a pinion and has a small number of teeth, generally between 6 and 16. The driven wheel has a higher number of teeth than the pinion, and the reduction ratio is the result of the ratio between the two numbers of teeth. Given Z1 the number of teeth of the driving pinion and Z2 the number of teeth of the driven wheel, the reduction ratio is Z2/Z1, e.g., with Z1 = 8 and Z2 = 32 the reduction ratio is ¼. In other words, to obtain one revolution of the driven wheel, 4 revolutions are needed by the driving wheel. To ensure that the transmission of motion is frictionless, so that no energy is wasted, wear of gears shall be avoided. In systems that perform heavy service, lubrication is used, as in the case of an automobile’s transmission system. In clockwork mechanisms, lubrication of wheel teeth must be avoided and limited to axle pivots rotating in small holes. To this end, the profile of wheel teeth must comply with specific shapes, set by industry standards. The fundamental concept is that gear teeth must not slide. The driving and driven teeth are in contact at one point, which must rotate together with the driving wheel without the two surfaces slipping. Two types of profile for gear teeth have been identified: the epicyclic profile—generated by the rotation of a circle around a larger circle—typical of watch and clock mechanisms, and the involute profile for heavy service machines. Planetary machines have characteristics similar to watches, the energy to generate the motion is very small. Ancient machines, on the other hand, had teeth with a triangular profile, as we have seen in the Antikythera mechanism and Dondi‘s Astrarium. The geometry of the modern profile is in Fig. 15.1, where pitch diameter/ circle, internal diameter/circle, outside diameter/circle, addendum, and dedendum are represented. The number of teeth Z and the pitch diameter P are in a relationship that determines the module M of the cutter required for grinding. In the English standard, the relationship is P = M × Z, so adding the addendum to P we get the outside diameter, which is the diameter of the wheel blank before griding.

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Fig. 15.1  Geometry of cycloidal gears

Another peculiarity of the English standard is that if the role of the driving wheel is exchanged with that of the driven wheel, the profile of the gear changes. In my own design, I have sometimes had to resort to this, but I have not made the change, which is certainly necessary for a gear that must be driven by a weak force. In my design, I adopted the English standard BHS (particularly used for pendulum clocks) for epicyclic gear profiles, as I have commercial gear cutters made to this standard.1 In the English standard, the pinions have a profile that depends on the number of teeth, so different milling cutters are needed for 6, 8, 10, 12, 16 teeth. A gear train can also be configured to produce an epicyclic motion, which approximates planetary motion. The main schemes for realizing an epicyclic gear train are described by Ludwig Oechslin.2 We have also seen the pin&slot mechanism in the Antikythera machine,which generates the speed anomaly, and variants of this method in Dondi’s, Baldewein’s, and other renaissance machines.

Design Constraints A project is never free of constraints, and I would say luckily, as then the variety of possible solutions would be too vast. I identified major constraints from the beginning and gradually discovered new ones that forced me to fine-tune my design. The first constraint is the precision with which I can make a machine that simulates planetary motion. The first is temporal precision, i.e., the period of  P.P. Thornton. https://ppthornton.com. Accessed June 2020.  Oechslin (1996, pp. 151–157).

1 2

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rotation or revolution corresponds to astronomical values, but small approximation can be acceptable since it will not be a perfect simulator of planetary motion. The space precision constraint is perhaps the first to be defined. Size limits make impossible to reproduce the distance ratios between the Sun, the Earth, and the Moon and the dimensional ratios between the different diameters of the celestial bodies, for example, if we set Earth—Sun to100 mm, the Earth–Moon distance would have to be 0.0025 mm. Another set of constraints relates to the construction tools available and the associated costs. Time and Space Precision  The study of the many planetaries and Orrery constructed over time has revealed a continuous search for precision in order to mechanically simulate the periods of celestial bodies. On the one hand, precision in astronomy is a requirement of planetary mechanics and astrophysics studies. In these two fields, precision has completely different requirements. Whereas in astrophysics time is measured in millennia and distance in light years, in planetary mechanics distances are measured in Astronomical Units or its fractions. Planet’s periods have accuracies of the order of a second. Although we often find the period of planets or satellites with values to the hundredth of a second, the irregularities of planetary motions, due to the complex interactions between mutual gravitational fields, give them non sense for a mechanical simulation. One field in which precision must reach microseconds is GPS satellite positioning, where the relativistic effects of moving satellites must be compensated to avoid position detection errors on the order of a few kilometers. From the point of view of mechanical simulation with planetary machines, precision may have different requirements. The machine may have as its purpose the schematic illustration of planetary motion in the solar system, in other machines there may be a desire to make predictions, with particular interest in eclipses and tidal cycles. If the purpose is purely illustrative, a period accuracy limited to a few days is more than sufficient. If the purpose is forecasting, then the accuracy must approach the second. Let us consider the duration of the synodic lunar month, which is approximated as 29.5306 days, and consider an error of +2 sec per month. During an entire cycle of precession of the Moon nodes, which has the duration of 223 synodic months, the accumulated error will be 446  sec, approximately 7.4 minutes. This approximation can still be considered acceptable if the prediction of the eclipse is limited to the detection of the day. If the delay were to increase to 6.5 minutes/synodic month, the period would be about 29.5358, and after 223 months the accumulated delay would be 24.15 hours, just over

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a day. It is well known that the approximation to 29.5  days of the lunar period, adopted on many wristwatches and pendulum clocks, leads to an error of one full day in 4 years, which involves the calibration of the mechanism. It is also worth remembering that astronomical observations from the 1700s onwards were assisted by pendulum clocks that were gradually more accurate. The reading of time was determined by the escapement of the clock, which in the regulators was generally one alternation every second. It is only in recent decades that measurements are marked by a time that has the accuracy of 1 nanosecond or 10−12 sec. in atomic clocks. Another critical aspect in precision machining is the backlash between gears (Fig.  15.2 left). Transmitting the motion with gears has an intrinsic backlash due to the very shape of the teeth, which in theory should rotate without friction and teeth would be in contact to a single point. To avoid the onset of friction and wear the distance between the teeth is not null but kept to a minimum. In mechanics, this distance gives origin to the backlash. If the direction of rotation is reversed, there is a time interval during which the two gears settle, and we can observe a hysteresis cycle due to the backlash (Fig. 15.2 right). In order to have negligible backlash, it is necessary to arrange the gears so that the primitive circles are perfectly tangent to each other, as we have recalled from the fundamentals of gear theory, but this is almost impossible for handmade wheels. The presence of the backlash has consequences on the accuracy of the machine’s positioning. If one wanted to use the planetarium to simulate a celestial phenomenon, one would have to ‘zero’ it, i.e., arrange it in the exact position at the current date before activating the movement to reach the target date. During zeroing, the backlash must be eliminated, i.e., the gears must be fully engaged as soon as the motion is started. As for spatial accuracy, it is impossible to reproduce the size and distance of celestial bodies to scale. Therefore, size and distance will only have a symbolic function.

Fig. 15.2  Left: backlash. Right: hysteresis of backlash

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As for the axial orientation of celestial bodies, it can be approximated to one angular degree, since greater precision has no particular effect on the machine’s ability to simulate motion. Let us now consider the elliptical shape of the orbits. The eccentricity e of an ellipse with axes a (major semi-axis) and b (minor semi-axis) is defined as the ratio between the semi-distance d between the foci and the major semi-axis a: e=

d ;e = a

(a

2

= b 2 ).

The eccentricities of the planetary and satellite orbits are summarized in Appendix “Positional Astronomy”, where distances are in astronomical units. In the tellurium, I fixed to 30 cm the diameter of the ecliptic. If we take this value as the major axis, the minor axis would be 29.99  cm. It then becomes of little use to set up an epicyclic motion to simulate an elliptical orbit. It is sufficient to mark the positions of perihelion and aphelion on the ecliptic itself. Finally, as far as temporal precision is concerned, the use of gears with epicycles would have allowed the solar and lunar anomaly to be simulated correctly. Again, I preferred to sacrifice this precision, believing that the main purpose of my project was to illustrate the mean motions. Tools and Cost  The working tools in turn impose other constraints. For example, the maximum diameter of a wheel depends on the swing of workpiece over the lathe bed of the machine used for gear cutting, while the number of teeth is limited by the subdivisions of the dividing wheel used to advance the gear cutter at each step. Another constraint is the length of a piece to be machine turned, which depends on the distance between the centers. We will look again at these constraints when discussing the instruments and tools.

Gear Computation A critical step in the design of a mechanical planetarium concerns the calculation of the number of teeth that make up the gear train for each movement of the celestial bodies. In this project, I wanted to keep the gear trains for each body separate to make the overall structure of the system easier to understand.

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The astronomical framework, which we have already described, has highlighted the values of the periods of rotation and revolution, which I recall in Table 15.1. The rotations and revolutions of the Sun, Earth, and Moon are multiples of the duration of a day, and it is therefore a matter of calculating five gear trains that optimally approximate these values. In the project, I do not consider the Tropic, Anomalistic, and Draconic month; furthermore, I preferred to use the Saros period instead of the precession of the lunar orbital plane to illustrate the possibilities of eclipses. Finally, as we shall see better, I have not implemented the precession of the equinoxes. Assuming that one rotation of the first drive wheel corresponds to a 24-hour day, we will have to find out how to break down the periods. Computation of the Reduction Ratio: Historical Background  The problem of gear train calculation is central to the field of watchmaking. While the calculation for a 12-hour and 60-minute gear train is simple and solvable with elementary methods, when dealing with complications such as moon phases or even more so for astronomical clocks, the problem becomes more complex. To understand this, let us start with the calculation of the synodic lunar period, equal to 29.53059  days. We can approximate the average synodic lunar period to 29.5306 days, so we need to calculate a gear train with the ratio 29.5306/1, in other words we want every 29.5306 days the moon to make one complete revolution, driven by a crank in which 1 revolution corresponds to one day. If we could build a wheel with 29.5306 teeth that moves a ‘finger’ that triggers the moon, we would be fine. But in general, gears have an integer number of teeth, not a decimal, and the ‘finger’ trick is also no good, because the moon would move with a click, instead of continuously. In tower and wristwatch clocks, the usual solution is to consider the approximate period of Table 15.1  The periods considered in the project Period (days) Sun rotation Earth rotation Earth revolution Moon revolution

Saros cycle Moon orbital precession (nodes) Equinox precession

Sidereal month Synodic month Tropic month Anomalistic month Draconic month

25.38 1 (24 h) 365.2564 27.3217 29.5306 27.3216 27.5545 27.2122 6585.360 (18.03 years) 6738.383 (18.45 years) 25785.9133

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29.5 days, thus driving the moon wheel with a wheel of 59 teeth (29.5 × 2) which in two revolutions describes two lunar cycles. However, this solution contains an approximation error of 0.06 days in two months or 0.03 days per month which corresponds to 0.7344  hours/month. This error accumulates over a year to 134.1198 hours which corresponds to 5.588325 days. Thus, in four years, we will have a delay of 22.3533 days, almost an entire lunar cycle and this in clocks with a moon phase will force us to adjust the position of the moon dial. To solve the problem, we must find two integers whose ratio gives precisely the decimal number 29.5306. Suppose, for example, that the final gear has 5 teeth, we should then find the ratio: 29.5306∗ 5 / 5 = 147.563 / 5

We still have a decimal number in the numerator. The only solution is to look for an integer that is as close as possible to the ratio we are trying to solve. Numerous methods have been devised, and in the history of watchmaking a particularly interesting one is the Stern–Brocot method. Achille Brocot (1817–1875) is a well-known name among watchmaking experts. He was the inventor of a mechanism for adjusting the length of the pendulum with high precision and of a recoil-free escapement with stones, both of which were widely used in clocks in the second half of the nineteenth century. He also devoted himself to theoretical studies in connection with the calculation of gears. A detailed exposition of the method is presented by Brian Hayes (2000). This method is basically a procedure for calculating the approximation with continued fractions of a rational number representing the reduction ratio of a gear pair, so let us come to the continued fractions. The Method of Continued Fractions Continued fractions were used in ancient times by the Hindus to solve linear equations from the fifth century onwards. The Hindu civilization had the opportunity to collect many results from Greek mathematics of the Hellenistic period also by virtue of Alexander’s conquest campaigns. Continued fractions reappeared in mathematics in 1572 when Bombelli, in his Algebra, used them to solve square roots. In England, John Wallis and William Brouncker (1620, 1684) used continued fractions to represent real numbers, in particular Brouncker, who was the first president of the Royal Society, gave this approximation to the number 4/π (Eq. 15.1).

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4 = 1+ π 2+

1 9 25 2+ 49 2+ 2 +…

(15.1)

The result of Eq. (15.1) is 1.197718631178707, while a more accurate approximation with continued fraction would be 1.27323954473516. This representation in Eq. (15.1) immediately brings out the meaning of the name of this method: we see that the approximation of the number is the result of the sum of an integer part and a fractional part composed of “cascading” fractions, which could go on forever. One can decide to truncate this sequence of fractions at some point and find the rational number one is looking for by solving the sequence of fractions from the truncation backward. The formalization of continued fractions is due to Euler, who published De fractionibus continuis dissertatio in 1744, which not only allowed him to use them to solve differential equations but also to express a rational approximation to the constant that takes his name: Euler number e. The method of continued fractions makes it possible to find a rational number (expressed as the quotient of integers) that is the best approximation to a real number. Since the development in continued fractions of a real number is infinite, this method is accurate to the desired tolerance and lends itself very well to implementation with an algorithm or on spreadsheets. In the field of watchmaking, the method of continued fractions was proposed by J.A. Lepaute in his treatise (Lepaute, 1767). Before him, Christian Huygens used precisely the method of continued fractions to calculate the gears of his planetarium, as reported by A. Janvier (1812), P. Dubois (1849), H. Amin (2008), D. van den Bosch (2018). The basic principle for applying this calculation method to determine the number of teeth of wheels and pinions is to consider the numerator and denominator as a wheel/pinion pair. It must of course be checked whether their value is suitable for the wheel or sprocket construction. If the numbers are too large and are prime numbers, then the solution found is not satisfactory and a less accurate approximation must be sought. If, on the other hand, the numbers are large but not prime, they can be further broken down into prime factors generating a series of wheels/pinion pairs.

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Formally continued fractions are expressions of the form: x = a0 +

1

, 1 a1 + a3 + …

(15.2)

where a0 is an integer number and a1, a2, a3, … are integer numbers >0. This sequence of integers can be written as in Eq. (15.3):

[ a0 ;,a1;,a2 ;,a3 …],

(15.3)

where the symbol “;” separates the first element. Let us consider, for example, the number 415/93 = 4.462365591397849 which we can approximate as 4.4624. A rough first approximation can be obtained by truncating the number to the first decimal place, obtaining 4.5. This approximation is called first convergent. We can consider this result as the sum of the integer 4 with a decimal part, which, expressed as a fraction, is ½, so from Eq. (15.2) we get Eq. (15.4):



4.5 = 4 +

1 2

(15.4)

but this number is too large, we can reduce the quantity to be added to the integer part by reducing the divisor, e.g., by adding 1/6 to 2 and we get Eq. (15.5): 4.4615 = 4 +

1

(15.5)

1 2+ 6

this is the second convergent, but it is still less than the result due. We then try to increase the partial quotient by adding 1/7 to 6, obtaining Eq. (15.6): 1

4.4615 = 4 + 2+

6+

(15.6)

1 1 7

This result can be written in the form [4; 2, 6, 7]. These numbers can be used to construct a quotient of integers whose ratio is exactly the number

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considered. And this ratio in turn can be seen as the number of teeth of pairs of gears that realize the desired reduction. Let us look at the procedure for going from the continued fraction to the ratio of two integers. Let us call by {nn} the succession of numerators and by {dn} the succession of denominators. The succession {an} contains the coefficients of the decomposition into continued fractions [4, 2, 6, 7] = [a0; a1, a2, a3], with n = 3. We can write in recursive form the series of calculus, we have to do to generate the quotient that approximates the number (with n we denote the numerators, with d the denominators) as in Eq. (15.7):



n0 = a0 ; n1 = a0 .a1 + 1;…nn = nn −1.an + nn − 2 d 0 = 1; d1 = a1 ;…d n = d n −1.an + d n − 2

(15.7)

If we apply this procedure to the previous example, we obtain Eq. (15.8): n0 = 4; n1 = 4.2 + 1 = 9; nn = n1.a2 + no = 9 ∗ 6 + 4 = 58; n3 = n2 .a3 + n1 = 58 ∗ 7 + 9 = 415

d 0 = 1; d1 = 2; d 2 = d1.a2 + d 0 = 2 ∗ 6 + 1 = 13; d3 = d 2 .a3 + d1 = 13 ∗ 7 + 2 = 93

(15.8)

We then have the succession of quotients that gradually approximate the desired number better and better (Eq. 15.9):



= 04

4 9 58 415 = ; 4.5 = ; 4.615 = ; 4.4624 1 2 13 93

(15.9)

If we were now to make a gear train for this ratio, we would be faced with a new difficulty: the number 415 is too large for a gear wheel. We try to decompose these two numbers into prime factors, obtaining: 415 = 5 × 83 and 93 = 3 × 31. The numbers 5 and 83 are now two reasonable values for gears, while 3 is still not suitable, but we can multiply the numerator and denominator by two, obtaining Eq. (15.10):



415 10 83 = × 93 6 31

(15.10)

Now the values 6, 10, 31, and 83 are reasonable values for wheels and pinions.

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The Russian mathematician Alexander Khinchin (1894–1959) proved that continued fractions constitute the best approximation of a real number with a rational number. The approximation we obtain by decomposing a real number into continued fractions is obviously more accurate than that obtained by simply truncating the real number to the few decimal places. In the case where the rational number to be approximated is already a rational number, the sequence of continued fractions is finite. Fortunately, the ratios that we have to translate into gear trains for a planetarium are rational numbers (remember that the periods of celestial bodies are already approximated by rational numbers, which therefore have a finite number of decimal digits), so from a theoretical point of view there are no particular problems: we will always obtain a finite sequence of convergent. Gear Trains Computation of the Tellurium  I have written a simple program using Matlab©, listed in the appendix, that computes the continued fraction decomposition, finds the prime factors of numerator and denominator and print a table with the result. Lunar Synodic Motion, Comparison of Different Gears  The examination of the lunar period and the realization of the ideal gearing is a good example of the problems that must be considered in the design of a planetary machine. Contemporary measurements of the mean lunar synodic period give the value 29.530589, and we have already seen that 29.5306 is a fully acceptable approximation for a mechanical gear system with an error of −0.95  s/day. Some authors choose a period value with a larger number of decimals. In particular, I quote Popkonstantnović et al. (2014) and studies published in the Horological Journal in December 1985 and March 1986 by A.G. Bromley (1990). Both studies are based on the method of continued fractions. Bromley’s work certainly suffered from the limited capacity of electronic calculators of his time, which did not make it easy to calculate with high precision (at least to the 13th decimal place) and to determine prime factors. I have also considered Partridge’s work (1991), whose gearing is the same as that of Bromley, but using a differential mechanism. Table 15.2 contains the periods of the mentioned authors, and the optimal gearing. All these values were computed with continued fractions. When we examine this table, we note that the search for the most accurate period cannot be separated from its feasibility. Gears with 251 or 173 teeth can be difficult to cut; moreover, they generate an error of the order of 0.12%. Limiting the target to the approximate ratio of 29.5306 on the other hand,

NASA/JPL My approx. Janvier Popkonstantinoviç Bromley Partridge

29.530589 29.5306 29.530589 29.530582959 29.53058979 29.53058912

Period

51,649 49,700 51,649 20,760 502 502

Value 13 2 13 2 2 2

29 2 29 2 25 251

137 5 5 7 71 137 2 3 5 173

Prime factors

Numerator

1749 1683 1749 703 17 17

3 3 3 19 17 17

Convergent

Error vs. NASA/JPL

11 53 29.5305891 3 11 17 29.53060012 −0.0011 11 53 29.530588908 0.000000 37 28.5306 −0001109 29.52941176 0.117715 29.52941176 0.117715

Prime Value factors

Denominator

Table 15.2  Comparison of different factorizations for moon synodic period

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gives an error of the order of 0.011% and the gears will have a maximum of 71 teeth. Janvier’s gearing is impressive, with 137 teeth for the largest gear and a practically zero error he could approximate the average synodic period of NASA/JPL. To sum up, the period that approximates to the mean value 29.5306 can be obtained with a relatively simple train in which the gear with the largest number of teeth is 71. Bromley’s result requires a wheel with 109 teeth. All three of these decompositions have an error per lunar month