Teaching the Quadrivium: A Guide for Instructors 1949822346, 9781949822342

Table of contents :
Contents
Preface
Part I. Geometry
1. Instruments
2. Procedures
3. The Foundation of a Science
4. Proofs with Triangles
5. Parallels
6. The Composition of Quadrilaterals
7. Ratio
8. The Golden Ratio
Part II. Arithmetic
9. Counting
10. Numbers in Themselves
11. Demonstration with Natural Numbers
12. Primes and Relative Primality
13. Linear Diophantine Equations
14. Numbers in Themselves, Revisited
15. Relations Between Numbers
Part III. Music
16. Sound
17. The Monochord
18. The Tone
19. Approximation
20. The Diatonic Genus
21. Gregorian Modes
Part IV. Astronomy
22. Observation
23. Plane and Spherical Trigonometry
24. Principal Solar Events
25. A Refined Solar Model
26. Terms of Time
27. Elements of Lunar Astronomy
28. Stars, Fixed and Moving
Part V. Beyond the Quadrivium
29. Physics
Index

Citation preview

PETER ULRICKSON

TEACHING THE QUADRIVIUM

T H E C AT H O L I C E D U C AT I O N P R E S S

Copyright © 2023 the catholic education press All rights reserved ISBN 9781949822342

Exsultavit ut gigas ad currendam viam.

In memory of Ian.

Contents Preface

Part I

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1

Geometry

1 Instruments 1.1 Week 1 Plan 1.2 Liberal . . . 1.3 Care . . . . . 1.4 Patience . .

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3 3 4 5 6

2 Procedures 2.1 Week 2 Plan . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Cleverness . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 10 11

3 The Foundation of a Science 3.1 Week 3 Plan . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 15 17

4 Proofs with Triangles 4.1 Week 4 Plan . . . 4.2 Speech . . . . . . 4.3 Reading . . . . . . 4.4 Prudence . . . . .

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19 19 22 23 24

5 Parallels 5.1 Week 5 Plan . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Obvious . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 31

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5.3 5.4

Rivals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 The Composition of Quadrilaterals 6.1 Week 6 Plan . . . . . . . . . . . 6.2 Analogy . . . . . . . . . . . . . 6.3 Elements . . . . . . . . . . . . . 6.4 Cost . . . . . . . . . . . . . . . .

32 34

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37 37 39 41 43

7 Ratio 7.1 Week 7 Plan . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Forgetfulness . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Categories . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 48 49

8 The Golden Ratio 8.1 Week 8 Plan . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Beauty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Death . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 55 57

Part II

59

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Arithmetic

9 Counting 9.1 Week 9 Plan . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Alexandria . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 63 65

10 Numbers in Themselves 10.1 Week 10 Plan . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Mystery . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Redundancy . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 69 71

11 Demonstration with Natural Numbers 11.1 Week 11 Plan . . . . . . . . . . . . 11.2 First . . . . . . . . . . . . . . . . . . 11.3 Whole . . . . . . . . . . . . . . . . . 11.4 End . . . . . . . . . . . . . . . . . .

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75 75 77 79 81

12 Primes and Relative Primality 12.1 Week 12 Plan . . . . . . . . . . . . . . . . . . . . . . . .

85 85

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12.2 Unending . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . .

87 89 91

13 Linear Diophantine Equations 13.1 Week 13 Plan . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 97 99

14 Numbers in Themselves, Revisited 101 14.1 Week 14 Plan . . . . . . . . . . . . . . . . . . . . . . . . 101 14.2 Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . 103 14.3 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 15 Relations Between Numbers 15.1 Week 15 Plan . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Adrastus . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 111 113

Part III

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Music

16 Sound 119 16.1 Week 16 Plan . . . . . . . . . . . . . . . . . . . . . . . . 119 16.2 Rhythm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 17 The Monochord 123 17.1 Week 16 Plan, Continued . . . . . . . . . . . . . . . . . 123 17.2 Endurance . . . . . . . . . . . . . . . . . . . . . . . . . . 125 17.3 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 18 The Tone 129 18.1 Week 17 Plan . . . . . . . . . . . . . . . . . . . . . . . . 129 18.2 Canon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 19 Approximation 133 19.1 Week 17 Plan, Continued . . . . . . . . . . . . . . . . . 133 19.2 Rational . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 20 The Diatonic Genus

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20.1 Week 18 Plan . . . . . . . . . . . . . . . . . . . . . . . . 141 20.2 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 20.3 Universality . . . . . . . . . . . . . . . . . . . . . . . . . 145 21 Gregorian Modes 21.1 Week 19 Plan 21.2 Week 20 Plan 21.3 Life . . . . . . 21.4 Classes . . . .

Part IV

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Astronomy

149 149 150 151 153

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22 Observation 157 22.1 Week 21 Plan . . . . . . . . . . . . . . . . . . . . . . . . 157 22.2 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 23 Plane and Spherical Trigonometry 161 23.1 Week 21 Plan, Continued . . . . . . . . . . . . . . . . . 161 23.2 Week 22 Plan . . . . . . . . . . . . . . . . . . . . . . . . 162 23.3 Computation . . . . . . . . . . . . . . . . . . . . . . . . . 164 24 Principal Solar Events 165 24.1 Week 23 Plan . . . . . . . . . . . . . . . . . . . . . . . . 165 24.2 Repetition . . . . . . . . . . . . . . . . . . . . . . . . . . 166 25 A Refined Solar Model 25.1 Week 24 Plan . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Week 25 Plan . . . . . . . . . . . . . . . . . . . . . . . . 25.3 Certitude . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 169 170 171

26 Terms of Time 173 26.1 Week 26 Plan . . . . . . . . . . . . . . . . . . . . . . . . 173 26.2 Grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 27 Elements of Lunar Astronomy 177 27.1 Week 27 Plan . . . . . . . . . . . . . . . . . . . . . . . . 177 27.2 Week 28 Plan . . . . . . . . . . . . . . . . . . . . . . . . 179 27.3 Maculate . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

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28 Stars, Fixed and Moving 181 28.1 Week 28 Plan, Continued . . . . . . . . . . . . . . . . . 181 28.2 Quirks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Part V

Beyond the Quadrivium

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29 Looking Back and Looking Ahead 187 29.1 Week 29 Plan . . . . . . . . . . . . . . . . . . . . . . . . 187 29.2 Week 30 Plan . . . . . . . . . . . . . . . . . . . . . . . . 188 29.3 Exsultavit . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Index

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Preface The quadrivium is and is not a tradition. On the one hand, mathematical and philosophical writings over a period of centuries testify clearly to this fourfold division of fundamental scientific learning, whose legendary origin is Pythagoras himself. As the Roman Empire weakened and split, new authors and translators took up these ancient arts and passed them on to the Middle Ages, during which they remained influential. Yet the quadrivium is not a tradition. It has not been handed on. Explaining why would be difficult and speculative, but some of the forces are clear: algebra, utility, Copernicus and Kepler and Newton, polyphony, the age-old craving for novelty. The book A Brief Quadrivium offers the contemporary student an accessible, unified introduction to the mathematics of the quadrivium. In a small way, then, it is a part of that already-existing quadrivial tradition. This book, Teach the Quadrivium, exists so that the tradition can once again be traditional in the second sense—so that it can once again be handed down. Teaching, while rewarding, is often difficult. It is especially difficult to teach something unfamiliar. This situation faces most of those who wish to teach the quadrivium today. This book assists teachers in two ways. The first is mundane; there are plans for what students should read and do each day, and suggestions for how to assess them weekly. The second is more elevated. Various essays, coordinated with the simultaneous work of students, help teachers frequently reorient themselves to broader themes in the history and philosophy of mathematics, even as they teach technical specifics. The separation between the mathematics of A Brief Quadrivium

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and the essays of this book is deliberate. The practical reason for the division is that students who face too great a variety of material are confused by it. They find it difficult to discern what is essential and what is peripheral. The younger students for whom A Brief Quadrivium is written must first learn a manageable quantity of mathematics, without distraction. Beyond mere volume, though, there is a deeper principle. Due order must be observed. It is not appropriate to ask a student to learn something and simultaneously to stand in judgment over it, to offer a critique or comparison. Those activities are important, but they cannot supplant or even be contemporary with the acquisition of fundamentals. The readers of this book vary in their backgrounds and interests, and the essays are intended to vary correspondingly. Readers can be divided using a couple of relationships. One is the relationship of the teacher to the student. The other is the relationship of the teacher to the mathematics. There are three likely relations between the teacher and the student. The first is the ordinary one, of a teacher and a whole classroom of students in a school. The second relation is parental; some parents school their children at home. The third relation is that of identity. In the case of an adult studying independently, teacher and student are the same. There are three likely ways that the teacher will relate to the mathematical material. First, the teacher might view technical mathematics as something foreign and unfamiliar. In this case, the teacher probably has a more humanistic education. The task now is to integrate mathematical learning into that literary base. In a second case, the teacher might have substantial technical mathematical or scientific training, but this training has been too narrow. In that case, the teacher can use mathematics as a way to gain a broader view of the life of the mind. The third case is that of the pragmatist. Sometimes we wish simply to get things done. For this kind of teacher, what is especially important is that students learn the material and things run smoothly. I hope that the scheduling suggestions help. Given the two independent groups of three qualities, there are in principle nine different kinds of readers of this book, each with particular interests and needs. There should be something in the

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essays for institutional instructors, parents, and autodidacts; for humanists, engineers, and pragmatists. If an essay is not to your taste, feel free to move on to the next one. In mathematics, as in many endeavors, discipline and regularity contribute greatly to success. You and your students can succeed in the study of the quadrivium. Follow the plan here, and enjoy the way.

In festo Sancti Bonaventurae Anno Domini nostri Iesu Christi 2022

Part I

Geometry

Funes ceciderunt mihi in praeclaris.

1 Instruments 1.1

Week 1 Plan

Overview: The purpose of the first week is that students become skillful in using compass and straightedge. The work is straightforward, and everything the students need is in the text. They can work independently. Looking Ahead: Next week students will put these early, basic skills together in order to perform interesting constructions. Explore the Puzzles at the end of Chapter 1, and the Procedures of Chapter 2. Think about what other sorts of constructions you might ask students to explore if they have more time next week. It is fine if you do not know how to perform the construction, and the thing you choose (e.g., trisect an arbitrary angle) might even be impossible. Stronger students will enjoy the challenge of exploring beyond the textbook. Notes: 1. Be sure that stronger students do not rush. They must take the time to do things well, even if the tasks are simple. 2. Encourage students who struggle at the beginning. Five whole days are given to these elementary uses of the compass and straightedge. It is fine if students are not particularly successful in the first few days. 3. Strive to work on all of the tasks each day, even if you must

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reduce the number of repetitions somewhat. Assessment: If you have enough time, require each student to complete each task while under supervision. If your time is limited, select a couple of the simpler tasks and a couple of the more complicated ones. Correct students who do things sloppily or who rush. Be attentive to the slight displacement of the straightedge needed to produce a line through given points, and check to see whether the compass radius remains the same for the entirety of the circle drawn (it might tend to get larger).

1.2

Liberal

Why does someone memorize a poem? To get a good grade? For fun? To impress? The distinction between theoretical and practical knowledge is an old one, and it makes good sense. Some knowing is for its own sake. Other knowing is for bringing about, for making or preserving some thing, for ministering to our humble, humbling needs. Poetic speech has freedom, grace, needlessness. Questions of utility might seem base in contrast. Yet memorizing a poem can be useful; it can serve some purpose. The quadrivium makes up four of the seven classical liberal arts. The single term “liberal art” joins in two words the tension of freedom and service. These disciplines are “liberal” because they are for persons freed from the consideration of material need, and they are “liberal” because they have a proper finality within themselves. They are, however, also “arts.” They make things; or better, they let us make things. This opening chapter, on Instruments, prepares the student to practice a craft. The workman must know his tools and use them skillfully. Such words evoke labor, and rightly so. There is work to be done. Does this labor lead us away from the liberality of these arts? The “-vium” in quadrivium (and trivium) comes from “via,” way. There are two scales at which to consider ways, directedness, service, in this study of mathematics. On the one hand, the smaller

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scale, we can consider direction or order within mathematical study itself. On the other hand, the larger scale, we can reflect on the place of mathematical study in the direction or order of the whole life of the mind. First, consider directedness within mathematical activity. Our senses play an essential role in all our thought, including in our mathematical thought, abstract though it is. This makes it fitting to ask students to begin with an experience of the mathematical objects that they will treat in their study of geometry. They sense the geometric objects, and they make them. This making activity is not for itself; it is a preparation for the more purely contemplative mathematical activity still to come. Second, consider the direction of intellectual life as a whole, within which mathematical study plays only one part, important though it is. Many wise thinkers, over many centuries, have thought that the specific disciplines of the quadrivium are especially suited to prepare us to seek and acquire wisdom. Mastery of a single discipline does not suffice to make us wise; even multiple disciplines are insufficient, since intellectual achievement alone does not make a good life. Whatever the good life is, the life of wisdom, mathematical excellence will be only a part of it. Mathematics serves this perfection, and does not constitute it. One can serve a noble purpose without thereby becoming servile. The free character of mathematical study becomes, in a remarkable way, the source of its subordination to an even higher end. Why memorize a poem? Having done it first for itself, we find that it can also be for something else.

1.3

Care

Our study of the quadrivium begins with attention to small details. The first exercises in the use of instruments encourage care in simple things like the production of points, lines, and circles. Careful drawings make the ideas of geometry clearer. They are also beautiful. To practice care yields more than just a superficial or aesthetic benefit. Many errors arise in mathematics through carelessness. Eager to grasp something (or worse, to show off our knowledge) we

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gloss over some essential detail, one that turns out to be the heart of the matter. At times the quadrivium will not surprise or excite your students. The base angles in an isosceles triangle are equal. This does not ordinarily astonish us as something unexpected. We study this not to find something new, but instead to learn the way that the familiar fact rises from the foundation of geometry. To know this relation between the base angles requires that we carefully attend to what is said and to what is not said. We must not be sloppy with our language (is it part of the definition, or not?) or presumptuous about what has been established (what sorts of arguments about triangles can I make?). Careful work is beautiful, and it protects us from error; it also satisfies, and this is a final reason to practice care. You rightly desire to understand all the material in A Brief Quadrivium, all the rich trove of mathematical insight into the world’s structure, revealed and taught to us by history’s greatest minds. We won’t ever know it all, though. By practicing care in all our work, we discover the real satisfaction to be found even while we are still on the way.

1.4

Patience

To study A Brief Quadrivium in its entirety is a challenging task, one that will take some time. Things that take a long time require us to be patient. It is good to know in advance some of the points requiring patience. • Practicing the actions: Before formal, logical, mathematical study, we will begin with fairly mechanical, instrumental actions. These are simple to describe, but not always simple to perform. Sometimes, for example, when you try to make a circle, the compass will not work well (or you won’t use it well) and what you get won’t look like a circle. With time, and patient practice, this will occur less often. • Going from actions to explanations: After a couple of weeks of instrumental work, we will make a dramatic shift. Our goal will be not to make but to explain. Doing this is a dramatic advance in human thought. It means that we are concerned not only with

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what is useful, with what works. Instead, we strive to grasp what is and to give a comprehensible account of it in our own words. This is a different sort of mathematical activity than making and computing. Adopting the new perspective demands patience. • Understanding the statement given: The early propositions that we prove will be fairly straightforward and clear. Later assertions are more obscure. How do we shed light? Consider these questions when reading a statement. Do we understand all of the terms? Do we distinguish between the hypothesis and the conclusion? Do we have some grasp of a relation between the things given and the things sought? After patient work, reviewing, rereading, reconsidering, we eventually grasp the statement considered alone. The task of understanding is not then complete, as clarified by the next couple of items. • Understanding the proof of a statement: Understanding a mathematical statement is just the beginning. The heart of mathematics is in the proof, the explanation of why a statement follows from others that have been accepted. It takes time to understand a proof. How does this one thing follow from a previous one? Am I forgetting something important? How can I keep all the details in mind? Is this account really persuasive, compelling, complete? These questions will come to mind as you wrestle with mathematical proofs. • Understanding how the various statements form a single, structured body of thought: It is not enough to grasp a proposition in isolation. Instead, it is also necessary to understand it in relation to other statements, and ultimately in connection to the course of study as a whole. Why is this proposition being put here, and not elsewhere? What statements are used in the proof? Why are we bothering with it at all? Does this even really need to be proved? Didn’t we prove this already? The answers to these questions will not always be immediate. Be patient, continue at a modest pace, and step back from time to time to reconsider the ground that you have covered. • Understanding how the various disciplines—geometry, arithmetic, music, astronomy—themselves amount to one single thing,

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one way composed of four ways: At the culmination of your study, you and your students will begin to see that quadrivium is a single whole. Do not presume that this conclusion will come immediately, and do not feel bound to insist on it. Perhaps it will arise later in life, with time for reflection, and after encounters with other kinds of mathematics and natural science, or with philosophy and theology. There is no rush.

2 Procedures 2.1

Week 2 Plan

Overview: The purpose of the second week is that students use the compass and straightedge to complete some interesting geometrical constructions. Looking Ahead: Next week marks a dramatic shift, one that persists for the duration of the course. Instead of doing and making, the focus is on knowing and speaking, on giving proofs. This might be quite new for you. It is good to spend time now reading ahead so that you are not caught flat-footed. Notes: 1. Draw students’ attention to the way that some procedures are a part of other procedures. 2. The modular character of the procedures (see previous note) is a key to remembering them. Do not think about all the steps for each procedure. Instead, think about discernible sub-procedures that can be separately remembered. 3. The relationship of one procedure to another is like the relationship of one proposition to another. We break down our thinking into clear, separable chunks, and then assemble them to understand new things. 4. Observe that the statement of the task (in contrast to the corresponding procedure) never involves the names of points. The

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explanation of the procedure by which we carry out the task involves, for convenience, the use of names. These names are in no way essential. It is good to practice explaining the procedures with different choices of names, so that students distinguish clearly between the things (objects of geometry) and the names chosen for them. 5. Strive to work on all of the procedures each day, even if you must reduce the number of repetitions somewhat. 6. Pay attention to the data—the things given—and how the task is stated relative to the data. For instance, the procedure about tangents asks not for an arbitrary tangent, but rather one that passes through a given point. Assessment: If you have the time, require each student to complete each procedure while under supervision. If your time is limited, select a couple of the simpler tasks and a couple of the more complicated ones. Be sure that the instruments are used carefully, as in Week 1.

2.2

Utility

What is the use of all this? Students, parents, administrators, passers-by: any of them might raise the question, and not only about mathematics but also about poetry, geography, or biology. Knowledge of mathematics is often useful. Basic arithmetic helps us to keep accounts straight. More complex mathematics goes into the analysis of things like bond prices and bus schedules. There are even “useless” applications of mathematics, in arcane, elaborate models in physics, studied without any evident practical application. These are some of the most immediate uses of mathematics. Another use of mathematics is less direct, less explicit. We hear that mathematics teaches us to think clearly, to think critically, to solve problems, to make arguments. On this account, the specific details of the mathematics we do are not as relevant; what matters is the intellectual muscle we gain along the way. A story is told that one of Euclid’s students inquired about the use of geometry. Euclid said to his slave, “Give him threepence,

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since he must make gain of what he learns.” These questions are not new. Consider a more contemporary, more complex story of mathematical utility. Mathematicians spent centuries investigating the foundations of Euclidean geometry. This lead to the development, in the 19th century, of non-Euclidean geometry. This abstract, seemingly nonsensical geometry contributed to Albert Einstein’s development of general relativity in the early 20th century. By the late 20th century, general relativity had a practical use for ordinary people. The clocks used in the Global Positioning System (GPS) satellites require relativistic corrections to provide accurate locations. The utility in this case is a happy byproduct, centuries later, of more purely theoretical thought. I recommend avoiding much discussion of utility, whether of the specific kind (say, calculating things useful in engineering) or the general kind (teaching how to think). If we make too big a deal of either kind of utility, we run the risk of misleading students, frustrating and discouraging them. It would also be an error to emphasize too much the free or “useless” character of pure mathematics. Here is one way to respond to the question about usefulness. “Wise people have often thought it good for students to spend time learning what we are now learning. I trust those people, and I hope you trust me. Let’s spend some time taking their advice seriously and attempt in good faith to complete this course of study. Perhaps, at the end, we’ll have an idea of why it is good.”

2.3

Cleverness

Mathematics tests and develops our cleverness. It is no surprise, then, that cleverness shapes the sense many people have of the nature and use of mathematics. The clever inventions of recent centuries highlight the productive power of mathematics and natural sciences in technical endeavors. Computers, spaceflight, electron microscopy: these are all impressive, and in some way mathematical. I presume that you, who are reading this book and teaching from A Brief Quadrivium, have been shaped by a culture that emphasizes cleverness in mathematics. While I acknowledge that cleverness has a place, I encourage you to avoid giving much attention to clev-

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erness when teaching the mathematics of the quadrivium. Let me say a bit about why, and explain how my attitude is reflected in the structure of the book. The tasks that students complete throughout the book are labeled as “Exercises” and not “Problems.” This use of language is deliberate. When we encounter a problem, the point is to solve it, using whatever is at hand. Do I need to get over a wall? I might use a rope, a ladder, or a friend. An exercise, on the other hand, is more determinate. The point is not just to arrive at the end state, but to complete the whole motion. After one repetition of a back squat or bench press, the weight is right back where it started. No change occurs in the external state of things. The change occurs within the person who performs the exercise. Let’s consider a concrete example. One of the exercises in the next chapter says “Copy the five postulates.” This is about as far from cleverness as we can go—an activity that is purely scribal. The reason to include this exercise and others like it is that there are some things that students should not try to come up with on their own (at least in the beginning). Millennia of tradition lie behind the Euclidean postulates. Students need simply to learn them, to practice producing them. Only after long familiarity and deeper study do we become capable of offering any sort of criticism, analysis, or alternative. I recognize that cleverness has its place. Long ago, when Plato wrote his Republic, this was already the case. In Book VII, Socrates observes that people who are clever in mathematics are often clever in other areas of knowledge. In addition, he suggests that training in mathematics can make students sharper. I do not intend to deny these statements. Instead, I wish to show that the quadrivium offers exercises accessible to a wide range of students. Cleverness plays no role in completing them. They are a straightforward, unambiguous way to transmit a valuable inheritance.

3 The Foundation of a Science 3.1

Week 3 Plan

Overview: In the third chapter, the third week, we move from tangible, executable physical procedures to speech. This shift will persist for the whole book. Looking Ahead: This week we get some important terms and principles, but do not do any proofs. Proofs are the heart of mathematics. Take a look at Chapter 4 to see what they look like. Some of it might be familiar from a geometry class you took in high school. By remembering where we are going (proofs) you can remain oriented as we talk about preliminary, foundational elements this week. Notes: 1. There is relatively little material in Chapter 3. If time remains on a given day, use the remainder to revisit the procedures of Chapter 2, or for general exploration with the compass and straightedge. Students can take the time to play. 2. Do not worry if things seem too abstract, and if it seems like students (or you) do not really understand what is being said. This has to be done now, but it will make more sense after you have completed your whole study of geometry. Just follow the plan for the week and do the best that you can. 3. Pay careful attention to the definition of “right angle” (this is a focus of Day 3). Note that there is nothing numerical in it. It is

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purely geometrical, involving only equality of angles. Day 1 1. Read aloud the first section, Terms, of Chapter 3. 2. Discuss questions that arise. Acknowledge that this will be somewhat confusing. Pupils might be discouraged or frustrated. Do not try to rush them. Clarity will arise as we continue to move forward. 3. Complete Exercises 1 and 2. Day 2 1. Read aloud the second section, Postulates, of Chapter 3. 2. Spend time beginning to memorize the five postulates. This can be done by reading aloud repeatedly. 3. Reread the definition of “right angle” in section 3.1 Terms. 4. Complete Exercise 3 (copy the postulates). Be sure this is done neatly and exactly. It can be done repeatedly. Day 3 1. Read aloud the third section, Proof, of Chapter 3. 2. Complete Exercises 5 and 6. 3. Continue to work on memorizing the postulates, and attempt Exercise 4. Have the student check for any errors and correct them. 4. Have students draw a picture illustrating the definition of “right angle,” and have them use the picture to explain the definition. 5. If desired, complete Exercise 3 again. Day 4 1. Reread the whole chapter. This can be done together aloud, or by each pupil silently.

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2. Discuss new questions that have arisen, or have pupils explain what they now understand better. 3. Repeat Exercise 4, giving more time for study if it is needed. Day 5 1. Repeat Exercises 1, 5, and 6. Consult the chapter as necessary. 2. Have the student select procedures from Chapter 2 and practice them again. 3. Be sure that at this point each pupil can state the postulates exactly from memory. Spend time reviewing them. Assessment: • State some undefined terms. • Define “right angle.” • Define a term using undefined terms. • List the postulates exactly from memory.

3.2

Motion

The phrase in medias res means “in the midst of things.” This phrase originates in the Roman poet Horace. Horace wished to describe one way that good stories can be told. Listeners find themselves immediately, at the outset, in the middle of the tale. What does this have to do with mathematics? We too must throw ourselves into the middle of the story. Things are already underway. Here is what I mean by that. We could spend a long time explaining exactly what is meant by “definition.” We could spend more time talking about how people prove things, and determining what sorts of things even admit of being proved. Then we could talk about what it means for certain things to be principles of reasoning, exploring what axioms and postulates are. These things are all good. They are all necessary, even. I do not think, though, that they should come first. It is impossible for most people, and especially for younger students, to make any sense of such conversation if they

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have no examples in mind of what these sorts of things look like concretely, in a specific mathematical situation. What does this have to do with Chapter 3? Here is one example. A Brief Quadrivium does not give a substantial account of what a postulate is. There is a brief note, and then there are some postulates. Students will memorize them, and use them. There is no general “theory of postulation” in the book because I believe most people can only fruitfully discuss postulates in general after becoming familiar with them in some specific setting. We have to keep moving, from the very beginning. This is how language works. Somehow, when we speak, we are confident that our words signify, that they make our thoughts intelligible to others. There are two ways that we could err in our motion. One is an excess, haste. The other is a defect, slowness. As you study and teach the quadrivium you must continually examine: are we learning things deeply enough? And on the other hand, are we presuming to grasp things more deeply than our preparation warrants? Superficial learning is a sign of haste, but protracted discussion might also reveal a fault. In the first century AD, the Roman author Quintillian wrote a work called The Institutes of Oratory. It influenced the way that people thought about liberal education throughout the Middle Ages. Quintillian considers an objection to the study of grammar, one that might well be transferred to mathematical speculation: will it not lead students to a pedantic, narrow focus on minutiae, weakening the mind and distracting them from more important things? This is a serious objection, and Quintillian grants that the danger indeed exists. His response, though, is not to abandon all study of grammar. Instead, Quintillian says that this kind of study injures only those who linger in it, but not those who pass through it. We must consider the foundations of mathematical thought in a similar way. We cannot skip them, and we must pass through them, but we also must not tarry in them.

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3.3

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Existence

Do triangles exist? Some readers will be fascinated by the question and want to explore it in depth. Others will not see the point. This note offers some thoughts on “existence” when we speak about mathematical things. I hope that it will also clarify Euclid’s formulation of his postulates. When we study geometry, our minds really grasp things. Geometry is not an empty word game. It is difficult, though, to say exactly what and where and when those things are. When we talk about a triangle, the triangle is not the lines on the page. The marks indicating the lines are material things, and mathematical things are not material things. Although it is not material, the triangle is still real, in some way, somehow. Take a look at the first postulate: “to draw a line from any point to any point.” This wording is a bit awkward. Here are a couple of alternative formulations. First is one that emphasizes personal action in time and space: “Given two points, I (or you, or we) can draw a line that passes through them both.” Second is one that emphasizes independence from human action: “For each pair of points there exists a unique line passing through those two points.” Why would we choose Euclid’s statement and not one of the other two options just given? One reason is tradition; it has worked for a while, so it is reasonable to continue using it. Tradition is a good justification, but let’s see if we can uncover something on our own, too. What is a defect of the first formulation? By speaking of action in time, the process of drawing, it leads us too much towards physical things. There is nothing wrong, of course, with attention to physical procedures. The proper sphere of mathematics, however, involves separation from tangible, physical, material things. And it certainly does not depend on the particular people involved. What is a defect of the second formulation? The second alternative grants too much, perhaps, to mathematical things. Mathematical things are in some way both “out there” and “in us.” When we say they “exist” we do not mean it in the same way that we mean, say, “my friend exists.” Given that the “existence” of mathematical things seems to involve us in some way, it is reasonable that the

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postulates reflect this, and refer briefly to the action involved in production. The infinitive “to draw . . . ” strikes a balance between the too personal (“I can draw . . . ” in a particular place and time and manner) and the too reified (i.e., “the thing is there, unqualifiedly, without relation to people”). I believe that philosophical discussions of mathematical existence will be most fruitful when the interlocutors have a substantive, familiar grasp of a range of mathematics. The study of A Brief Quadrivium offers students good preparation for such conversation.

4 Proofs with Triangles 4.1

Week 4 Plan

Overview: This chapter offers the first proofs of the book, involving the simplest geometric shapes. Looking Ahead: We only use the first three postulates right now. The fourth and fifth postulates will be used in the next couple of chapters. Notes: 1. The purpose of a proof is to explain how one statement follows from other statements. It is like an essay. The purpose of a proof is not to find or to make a number or shape. 2. It is fine to let this material spill over beyond one whole week. There is a bit of extra room in the next week to allow for this. Take the time to allow students to master these early proofs. 3. Repetition is important. Students should reread proofs, copy them, and attempt to give them from memory, repeatedly. Day 1 1. Read Sections 4.1 and 4.2. 2. Talk with the students about the difference between proof and procedural instructions. Compare the proof of Euclid’s Proposition I.1 to the instructions for Exercise 1 of Chapter 2. Ask the

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students to explain how the two activities, drawing and explaining, are different. 3. Complete Exercises 1, 2, and 4. Day 2 1. Reread Section 4.2. Answer questions that arise, and inquire whether it seems clearer when seeing it for a second time. 2. Read Section 4.3. 3. Have students illustrate Side-Angle-Side Congruence by drawing triangles. 4. Compare these three things—accepting an axiom, proving a proposition, accepting a proposition without giving a proof of it. 5. Complete Exercise 4 again, exchanging roles (asking and responding) if possible. 6. Attempt Exercise 3. If desired, repeat Exercise 2 before attempting 3. 7. Discuss the attempt at Exercise 3. 8. Complete Exercise 6. Day 3 1. Review the statements of Propositions 9 (I.1), 11 (I.3), and 12 (I.4). 2. Read Section 4.4, at least through the conclusion of the proof of Proposition 14 (I.5). 3. Answer questions about the proof of I.5. Observe that it is not possible to keep everything in the mind at once in such a long proof. Instead, we must proceed step by step, marking our progress as we go. It can be helpful to add marks to a diagram to indicate things that have been shown to be equal. 4. Have the student make a copy of the proof of I.5, including a labeled diagram.

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5. (If continuing on to Proposition 15 (I.6) today) After reading the proof of I.6, talk about reductio ad absurdum. Present examples from ordinary speech in which we take as given something which ultimately does not hold. (e.g., “Suppose mom were at the store. Then she would have taken the van. But the van is here.”) Non-mathematical examples will, of course, not be perfectly demonstrative. 6. Attempt Exercise 7. Day 4 1. Review the proof of Proposition 14 (I.5). 2. (If Proposition 15 (I.6) was not considered yesterday) After reading the proof of I.6, talk about reductio ad absurdum. Present examples from ordinary speech in which we take as given something which ultimately does not hold. (e.g., “Suppose mom were at the store. Then she would have taken the van. But the van is here.”) Non-mathematical examples will, of course, not be perfectly demonstrative. 3. Compare a statement and its converse. One example: if a shape is a square, then it is a rectangle. Another example: if a person is a resident of New Jersey, then he is a resident of the United States. Observe that I.5 and I.6 are converses (the term “converse” itself need not be emphasized) and emphasize that these are distinct assertions, each of which requires its own proof. 4. (optional) Have students form statements and converse statements about things in ordinary life. Ask that they produce some in which both the statement and the converse are true and some in which only one direction holds. 5. Read Section 4.5 through the proof of Proposition 18 (I.13). 6. Complete Exercises 8, 10, and 12. 7. (optional) Complete Exercises 9 and 11. Day 5 1. Read the rest of Section 4.5.

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2. Have students attempt to give the proofs of Proposition 19 (I.15) and Proposition 20 (I.16) after reading and discussing them. Give occasional assistance, while letting them strive to remember what they have just read. 3. Complete Exercise 17 and attempt any remaining exercises that are of interest to the student. Assessment: • Prove Proposition 9 (I.1). Justify statements in the proof with reference to definitions and postulates. • Prove Proposition 14 (I.5) or Proposition 15 (I.6) (select one).

4.2

Speech

You might find that Chapter 4, and subsequent chapters, have more words than you expect. Why so many words? Proofs are the heart of mathematics. Computations and equations, things that probably played a big role in your own mathematical education, are subordinate to proving. A proof is an explanation, a verbal account, and so it necessarily involves words. Mathematics is not simply verbal, it is oral. It involves spoken words. I say this for two reasons. One is theoretical: I want to clarify what mathematics is. The other is quite practical: I want students to have success as they study. Let’s consider the mathematical spoken word from a theoretical perspective. Words that are spoken (in contrast to those that are written) are clearly relational; they involve a speaker and a hearer. Whenever we speak with other people we take into account who they are, what they know, what they like. The objectivity, the impersonality, of mathematics do not entirely displace this more relational aspect of speech. A proof can be treated as a purely logical entity. Does the asserted relation between the hypothesis and the conclusion indeed hold? Does the account of this relation given in the proof succeed in exposing the nature of the relation? These are, in a sense, absolute questions, questions whose answer requires no context.

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A proof is also relational, though; it is a rhetorical act and not simply a logical one. Euclid’s Proposition I.1 has a number of proofs. Are any of them wrong? No. Are any of them the best? No. The fittingness of the proof cannot be judged in the abstract. We must consider it as something designed to communicate to some person. So much for theory. What about practical consequence? Here are two points to keep in mind. First, students should try to say aloud the things that they read. A Brief Quadrivium is written with the intention that students will read it aloud. This is a good test of understanding. Can you read the words aloud and say them as if they are really yours? Try this, slowly. When you begin to falter, you know where you need to study and think more. Second, students should realize that there is no one perfect proof. There is no shell game, no trick, such that a petty, hidden reason can upset an entire solution. Giving a proof simply means giving a good explanation. With a bit of time and practice, students will have a decent idea of what it means to give a good mathematical explanation. Those who study mathematics sometimes go about fearfully asking “Is it right?” They are afflicted by self-doubt and wonder whether the teacher has some trick up his sleeve. Perhaps you were such a student. With a few weeks studying the proof-based mathematics of the quadrivium, these fears should subside, replaced by greater confidence. Can I give a good explanation? It is a straightforward question. Sometimes we can say yes, unreservedly. Sometimes the answer is definitely no. And sometimes we can say, “Well, it is a bit fuzzy to me, but here is the best explanation I can give right now.”

4.3

Reading

The word “lecture” has a bad reputation in some circles today. The lecture is said to be oudated, ineffective, passive. No doubt there are some lectures that fail in these ways. The word “lecture,” though, is quite simple in its root. It simply means “reading.” A lecture is a reading together.

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A lecture is not merely declamation. The lecture is shaped and enriched by the questions students ask as they inquire, with their instructor, into the meaning of the timeless writings being read. The course of study that I propose for you in A Brief Quadrivium, and in this book, is that of a lecture. Why emphasize reading? One reason is to clarify that there is not something clever or secret hidden beyond the printed words. There are not any tricks that I am concealing, or merely gesturing towards, and no esoteric mysteries. My intention is that you and your students simply read the book carefully. If you come to a point of difficulty anywhere along the way, the first thing to do is to read again, perhaps aloud. Attempt to utter the words as if they were your own, and see where you get stuck. Another reason this is a lecture is that I am not coming up with the material out of thin air. A Brief Quadrivium is a commentary on and distillation of the writings of Euclid, Ptolemy, Boethius, and a few other authors. The things that they wrote are worth understanding, and I want to help you and your students understand those things. Both A Brief Quadrivium and this book are born of my own reading, as I strive to make those authors intelligible to you so that their profound thinking can bear fruit anew. Boethius, Ptolemy, and Euclid were themselves also compiling and commenting, in addition to creating. By taking up this study together, we join them and those on whom they relied, all contemplating together the order revealed to us in mathematical things.

4.4

Prudence

Prudence is the virtue by which we choose fitting means to attain a good end. The means that lead to a good end depend on many circumstances that vary with place and time, and so prudence cannot be reduced to theorizing about wholly abstract cases, divorced from all particulars. Prudence is inescapably personal. While there can be discussion about the prudence of a course of action—and indeed taking suitable counsel is often a part of prudence—this deliberation never eliminates a fundamental hiddenness and obscurity in the judgment of prudence.

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The logical character of mathematics does not displace prudence from mathematical teaching. The order in which the subjects are presented, the things omitted or emphasized, the exercises recommended, all of these are related to the logical, but also involve prudence, judgment, in a particular time and place. Let me clarify, briefly, some of the points in which I have had to make decisions, decisions that I hope are prudent. By discussing these points and sketching my reasons here, I hope to delineate the more personal, subjective, and hence provisional aspects of A Brief Quadrivium. The division of fundamental mathematical knowledge into the four disciplines of the quadrivium goes back many centuries, perhaps to the Pythagoreans. How should the four be taught, though? Should they be taught simultaneously, sequentially, mixed in some way? Answers do not arise in an obvious way from the tradition. Boethius writes about the primacy of arithmetic. This primacy is in an absolute, philosophical sense, though. To speak of that order, the order of being, is different than addressing the order of learning. In Book VII of Plato’s Republic, the disciplines of the quadrivium are presented in the order arithmetic-geometry-astronomy-music. Just as in the case of Boethius, this is not necessarily a chronological order for learners. Why will we study the disciplines one at a time, and not all at once, or in pairs? It is good for learners to be presented with a small, definite amount of material that they master before moving on. The real relations that exist among the disciplines, relations that deserve reflection and discussion, are best appreciated when students already grasp each of the disciplines substantively. We can speak of the unity of the quadrivium. What parts, though, are being brought together? These must be known, as intelligibly distinct parts. This leads us to take the sequential approach, seeing each part in turn. Why the specific order? It seems good to begin with the “pure” disciplines of arithmetic and geometry before the “applied” ones of music and astronomy, so that students first master the basic tools. For the purpose of introducing proofs, geometry is superior to arithmetic, since arithmetic is more closely related to computation, doing, and use, while geometry lends itself quite naturally

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to explanation and postulation. Quadrivial astronomy is the most technically demanding and so deserves to go last, when we are best prepared. It also works well in the final place because the requisite preparatory observations take time. Astronomy works well in the final place for a deeper reason, too; having begun with simple instruments in our hands, looking down, we end by raising our eyes to a timeless and unchanging font of wonder, the heavens above. First we took something up, and finally we are taken up. Now for the way we go about geometry. Students begin A Brief Quadrivium with a compass and straightedge and instructions. Euclid’s Elements does not begin this way. In fact, the Elements contains no reference to physical devices. Why have I put more tangible, procedural material first? Briefly, this approach makes it clear that the words of the book correspond to something real in the world. The words are not an empty, arbitrary game. We should also note that we do not know the way Euclid’s Elements was originally presented. What sort of work had students already done? It makes sense to suppose that classical geometry presumed a whole educational culture, with practices appropriate for various times and ages. Geometry did not consist solely of one book. Chapter 4 skips over the proofs of some early Euclidean propositions, and even omits mention of some. This is, again, a pedagogical choice. We might have chosen instead to make the study of geometry consist of the whole of Euclid’s Book I, omitting nothing. Or we might have chosen to go further, including in their entirety additional books of the Elements. In the second case, it would be necessary to limit the time spent on the other disciplines of the quadrivium. The decisions I have made about what to include fully, what merely to mention, and what to omit are based on my experience of what young students are ordinarily capable of doing, and my judgment of what is needed to produce a study of the quadrivium that has four clear, balanced parts as well as a coherent unity. A final point about prudence regards ratio, a concept that ties together all four parts of the quadrivium. We introduce ratio towards the end of our study of geometry, using Euclid’s Book VI, about ratio in geometry, and occasionally using the abstract ratio from his Book V. This is again a pedagogical choice, deviating from the textual order in the classical source. In more computational portions

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of astronomy, we follow Ptolemy by abandoning the strict language of ratio, though unlike him we use decimal and not sexagesimal numbers. These are only some of the decisions made in composing A Brief Quadrivium. Why consider these choices here? The quadrivium is, on the one hand, a clearly discernible historical thing. On the other hand, though, as something we wish to pass on today, it is also something incomplete, indeterminate. You should know both that A Brief Quadrivium represents the substance of ancient mathematical thought, and also that its specific contours as a contemporary course of study involve many decisions.

5 Parallels 5.1

Week 5 Plan

Overview: This short chapter introduces the use of the Fifth Postulate, which lets us conclude that two lines intersect based on their relationship to a third line. Looking ahead: Next week covers parallelograms, and culminates in the Pythagorean theorem (which talks about the equality of regions made from squares). It is important to observe that numbers (i.e., what we call “area”) are not used in the classical, Euclidean, purely geometric formulation of the Pythagorean theorem. What is done right now, with parallel lines, is the key to the next chapter’s non-numerical approach to “area.” You might want to glance ahead to get a taste of the purely geometrical treatment of area. Notes: 1. You might have extended your study of Chapter 4 beyond one week. That is fine. Some review of that material is built into this week. 2. It is important that students can distinguish between assuming that alternate interior angles are equal (I.27) and proving that alternate interior angles are equal (I.29). Day 1 1. Study Proposition 21 (I.27) and Proposition 22 (I.29).

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2. Discuss the difference between the two propositions. What is assumed as a condition, and what is concluded, in each case? 3. Carefully review the statement of the Fifth Postulate and be sure that students can give it from memory. 4. Copy the two proofs (I.27 and I.29). Day 2 1. Review Propositions 21 (I.27) and 22 (I.29). 2. Read Proposition 23 (I.30) and the preceding discussion about the term “transitive.” 3. Complete Exercises 6 and 7. 4. Attempt Exercises 4 and 5. Day 3 1. Study the final two proofs of the chapter. 2. Complete Exercises 1, 2, and 8. Day 4 1. Review the entire chapter. 2. Give as many of the proofs from memory as possible. 3. Ask students to say which propositions require the Fifth Postulate and which do not. Day 5 1. Review all of Chapters 4 and 5. 2. Review specifically the proof of Proposition 20 (I.16) in preparation for assessment. Assessment: • State the two propositions about parallel lines and interior angles determined by a transversal. (i.e., I.27 and I.29)

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• Distinguish between the two propositions: which one requires the Fifth Postulate? • Prove each of the propositions (I.27 and I.29). • Prove Proposition 20 (I.16).

5.2

Obvious

Math teachers occasionally encounter students who find their statements “obvious” and thus pointless. Will you? Whether you have this experience depends, among other things, on the temperament of your pupils, their prior education, and the culture of your school. In Chapter 5, and throughout the quadrivium, there are things that might come across as “obvious.” If this objection arises, how do we respond? Consider Proposition 23 (I.30), the transitivity of parallelism. Some students might not understand the point of justifying this statement. Is it not simply evident? Why bother with a proof? You will need to make a judgment about whether a student is genuinely inquiring or simply stubborn. The following remarks might help you with the inquiring student. The word “obvious” arises from the prefix “ob” (in front of, toward) and the familiar “via” (way). A thing is “obvious” when it is “in the way.” We do not need to go out of our way to see it. Let us grant, then, that a statement like Proposition I.30 is “obvious.” Instead of taking the word in a dismissive sense (i.e., negligible), take it closer to its root. To say that the statement is obvious, that it is “in the way” or “in the path” leads to a question. Having spoken of a way: what is the way, the path, the via? The answer is that the proof is the path. The axioms (another name for postulates) are the point at which we begin. The proof is the way that we walk from them, and the end of the journey is in the conclusion that was to be shown. We want not only to arrive but also to know how to get there. Once we establish axioms, everything else is determined. All the rest is, in a sense, obvious, to a perfect mind. A mathematical statement either follows from our axioms, or it doesn’t. We have no choice. Seen in this light, the “rhetorical” character of mathematics,

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as opposed to the “logical,” becomes clearer. The accounts we give must be artful. That is the only thing we can contribute. All of the freedom, all of the room for human ingenuity, comes in the way that the principles and the conclusion are related through clear speech. The logical connection itself is not up to us. Mathematical conclusions alone suffice at times, independent of their relation to clear first principles. Engineers constructing a bridge have no need of a proof. They simply need reliable mathematical statements that will help them make something. They need conclusions, not explanations. They do not even need flawless conclusions. Approximations suffice. I say this to grant that there are settings in which proofs are irrelevant. We need not imagine that the real good of pure mathematics is an all-encompassing one. Ordinary life acquaints us with the difference between the destination and the road taken. There are many ways to travel from Georgia to Maine. One is to fill up your gas tank and head north. Another is to buy a plane ticket. There is the Appalachian Trail though, too. The journey by foot is hardly straightforward. It is not “obvious.” Why do people—many, every year—bother taking the trail, and not the plane? To ponder this is to gain some appreciation for why we prove in mathematics.

5.3

Rivals

The same thing can be said in many ways. Some changes in logical form are small, while others are more drastic. In the latter case it might take some time and thought before you recognize that two formulations are equivalent. Euclid’s Fifth Postulate is the most complicated of his postulates. Mathematicians have, throughout the intervening centuries, examined how it might be restated. Here is one alternative, called Playfair’s axiom. “Given a line and a point not on that line, there is a unique line passing through the point and parallel to the given line.” The whole force here is in the word “unique,” meaning that there is only one. (It is possible to prove, from the first four postulates, that there must be at least one. Produce a perpendicular to a perpendicular, and then use Proposition I.27 to conclude that the lines are parallel.)

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Playfair’s axiom is not a straightforward rephrasing of Euclid’s Fifth Postulate. Instead, the claim appears different on first glance. It turns out, though, to be equivalent, and this equivalence is itself a mathematical proposition, something that must be proved. To prove the equivalence, two things are to be done. The first is to show that Playfair’s axiom is a theorem of ordinary Euclidean geomtetry. The second is to assume only Euclid’s first four postulates, along with Playfair’s axiom, and then to show that Euclid’s Fifth Postulate is a theorem of this logical system. This kind of demonstration is within reach of stronger students who study A Brief Quadrivium, though it is not treated in the book. When we consider the equivalence of distinct sets of axioms we are engaged in what can be called meta-mathematical activity. This kind of thinking clarifies the meaning of first principles by comparison. It is a mathematical activity that also borders on the philosophical. If there is no logical difference, why bother distinguishing? Why not adopt Playfair’s axiom and abandon Euclid’s Postulate? We need to consider the relative, personal, non-logical aspects of the two statements. Charles Dodgson is probably most familiar as the author, under the name Lewis Carroll, of Alice’s Adventures in Wonderland and Through the Looking Glass. Writing those stories was a hobby. Professionally, Dodgson taught mathematics at Oxford. As a mathematician, he argued for the Euclidean approach to geometry in a book called Euclid and His Modern Rivals. In this dramatized and occasionally humorous account of a rather dry subject, Dodgson considers, in extensive detail, alternative geometry texts proposed by his contemporaries. The use of Playfair’s axiom is one of the alternatives he analyzes. What objections does Dodgson raise to the use of Playfair’s axiom? He offers a few; we will limit our discussion to one. He notes that the ingredients of Euclid’s postulate—two lines, a transversal line, and two interior angles—are “all clear positive conceptions” (emphasis in the original). On the other hand, the use of Playfair’s axiom puts the beginning student in difficulty because: Playfair requires him to realize a Pair of Lines which never meet, though produced to infinity—a negative conception, which does not

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convey to the mind any clear notion of the relative position of the lines.1

We see Dodgson’s pedagogical concern, that the language used in instruction convey clear, graspable notions, definite mental entities, to ordinary students. This single objection hardly closes the case. We might still reasonably consider using Playfair’s axiom, or some other axiom, in the place of Euclid’s Fifth Postulate. Even Dodgson himself did so in later work. Euclid and His Modern Rivals contains more, though, than this kind of technical analysis of specific statements. It also offers more general remarks about education, remarks that help us think about when and why to deviate from received standards. Dodgson notes that a single convention, widely adopted, offers a political good. We come to share something with our fellow citizens. Adopting the Euclidean order and number means that people can speak clearly to each other about a single body of knowledge, and that the thought of many preceding centuries, filled with references to the Euclidean standard, remain intelligible to us. You might encounter, either in person or in writing, a mathematician or scientist who suggests that Euclidean geometry has been replaced, superceded, maybe that it is in some way naïve. Do not be discouraged. Such rivalries among thinkers are not new, and they go back far beyond the 19th century. Euclid’s geometry is good, and that is enough for us.

5.4

Diagram

It is impossible to say just what I mean.

Perhaps you know the line above, from T.S. Eliot’s The Love Song of J. Alfred Prufrock. Sometimes we search without success for the right words. Do mathematicians suffer this affliction, though? They would seem to be immune. What else but mathematics offers us so perfect an experience of exactitude? Surely mathematics, with its precision, stands apart from all obscurity, all fuzziness. 1 Charles

1885.

Dodgson, Euclid and His Modern Rivals, 2nd ed. London: Macmillan,

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This is largely so. In practice, however, even mathematics does not arrive at perfect explication, and can only hint at it. When the time comes to set down suitable words, they are hard to find, or too complicated to be useful. In place of words, or alongside them, we use pictures, sketches, diagrams. One example of incompleteness in mathematical explanation comes in this chapter, in the term “alternate interior angles.” That term was not defined. It would be silly to take it as an undefined term, though, since it is not so fundamental as something like “point” or “line.” Between defining and leaving wholly undefined lies a third way; we gesture towards something that could be defined, without ever making such a definition explicit. Diagrams aid us in making these gestures. A good way to say what “alternate interior angles” are is to point at the diagram. The very first proposition of Euclid’s Elements is famously reliant on a diagram. When the two circles have been produced with the given segment as radius, and endpoints as centers, we refer to a third point, one that arises as an intersection of the two circles. What justifies this reference? If you ponder carefully, you will find that the intersection does not follow in an explicit, verbal way from the postulates. You can see the intersection, though, if you look at the diagram. Some commentators, both recently and long ago, have claimed that this is a defect in Euclid. They say that he ought to state some definite further principle by which we could conclude that two circles so situated do indeed intersect. I do not see Euclid’s use of a diagram as a failure. There are some specific, more technical, mathematical and pedagogical reasons for my inclination to trust his approach. There is something bigger, though. When I look at the Elements as a whole, I see a striking, meticulous ordering of a complex body of material, an ordering that makes both logical and instructional sense. This global view leads me to think Euclid must have had a pretty good reason for proceeding as he did. He might have made a mistake, might have overlooked something. My guess, though, is that he could give a far better answer than you or I can for why the proof of Proposition I.1 is a good one. (A practical aside: avoid telling students there is any “controversy” about reasoning from diagrams on a first pass through Euclid. They will be able to appreciate such things when

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they have greater mathematical maturity, through completing the whole study of the quadrivium.) “It is impossible to say . . . .” Eliot’s Prufrock begins with an extended quotation of another poem, Dante’s Divine Comedy. Eliot’s difficulty is Dante’s too; throughout his journey from anxious, solitary darkness to harmonious, celestial radiance, the eloquent Florentine poet cannot stop speaking of his own inability to say what he has seen. The dramatic tension—how do we express in words what has been impressed upon our senses? —is a part of every life, and lingers even in mathematical terminology. In its Greek origin, the word theorem refers to a thing beheld; it is the object of a vision.

6 The Composition of Quadrilaterals 6.1

Week 6 Plan

Overview: The highlight of this week is the Pythagorean theorem. This significant theorem will be fundamental in later work. We will make frequent use of it when studying astronomy. The theorem is likely familiar to you; keep in mind the difference between knowing the statement and knowing the proof. Looking Ahead: Next week takes a turn away from pure geometry, to the more general notion of ratio. Our plan follows Euclid to a degree; the Pythagorean theorem concludes Book I of Euclid’s Elements, and ratio is Euclid’s theme in Books V and VI. Take time now to examine how ratios are defined and how they are compared. These are elementary but demanding topics, and the perspective of Chapter 7 is likely to be new to you. Notes: 1. Proposition 29 (I.35) undergirds this whole chapter. We use it to see that two quadrilaterals are the same. 2. You are familiar with the word “area,” meaning a number that says how much space a region takes up. We do not use area in this chapter, but instead directly decompose regions into geometric shapes (triangles). 3. If you wish to look up more details about this geometrical approach to area, the relevant term is “content.” We are showing

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that regions have, in modern parlance, “equal content.” 4. You might find other proofs of the Pythagorean theorem. Keep in mind that if they look shorter and simpler, they probably hide a lot under the surface. Day 1 1. Read Sections 6.1 and 6.2 through Proposition 29 (I.35). 2. Discuss the first paragraph of Section 6.1. Do the three things listed there make sense to your students? 3. Complete Exercises 1, 2, 3, and 4. Day 2 1. Reread yesterday’s material, and complete the rest of Section 6.2. 2. Discuss the notion of “equality” indicated in Proposition 29 (I.35). How is this different from what we did before? See the essay on analogy below. 3. Talk about how the statement of Proposition 20 (I.37) is tricky. What word needs to be clarified in order to be sure that the statement is correct? (It is the word “equal,” used in the broader sense introduced in I.35.) 4. Complete Exercises 5 and 6. Day 3 1. Review the previous discussion of equality of quadrilaterals, distinguishing it from our earlier notion for triangles. 2. Read Section 6.3. 3. Complete Exercise 8. 4. Attempt Exercise 7. Day 4 1. Reread Section 6.3 again, paying careful attention to all details in the proof of the Pythagorean theorem.

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2. Complete Exercise 8 (again). 3. Attempt Exercise 9. Day 5 1. Review how the notion of equality was generalized in Proposition 29 (I.35). 2. Review the proof of I.35. 3. Review the proof of the Pythagorean theorem. 4. (Stronger students/Extra Credit) Complete Exercise 10. Assessment: • Prove Proposition 29 (I.35) from memory. • Prove Proposition 34 (I.47) from memory. • Extra Credit: discuss the use of Postulate 4 (all right angles are equal) in I.47.

6.2

Analogy

To understand a word’s meaning, we must often look into its context. What does it mean to say “I’m blue?” This depends on whether you are playing a board game, or just got out of a cold lake, or are feeling glum. Understanding the context often requires us to ask and not just listen. This makes oral communication clearer than written words, at least when friends try to know each other’s minds. Lawyers and contracts are a different story. The word “same” admits multiple senses. Consider the question Are you from the same family? Siblings might answer one way in youth, and differently later in life when married. Their children, who are cousins, might answer either way, too, depending on who is asking, and when and where the question is being asked (at school? at a family reunion? another cousin? a reporter?). Families are certainly real, but “same family” requires some further specification. In mathematics, too, we can inquire about sameness. Are the two shapes the same? Consider that question in these cases.

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• Two circles of the same radius in different places on a page. • A big circle and a small circle. • A true circle and a roughly circular shape, the circumference of which is slightly wobbly. In each instance we can think of reasons to say that the shapes are the same, and reasons to say that they are not the same. If a word is used in entirely different ways, we say that it is used “equivocally,” and if used in the exact same way we say that it is used “univocally.” Between univocal and equivocal uses, there is an intermediate place. In that case, we say that words are used “analogically.” This means that the uses of a word are related, but still different. In the study of four-sided shapes, we encounter a notable instance of analogical speech in the words “same” and “equal.” Sameness for triangles, in Chapter 4, was a fairly strict, rigid notion. The sameness or equality we use for quadrilaterals is looser. This looser meaning also makes sense for triangles, and we use it in Proposition I.37, and not the stricter one. We can replace analogy at times with univocal speech by choosing different terms to distinguish the various notions. You might find in other books that triangles are “congruent” if they correspond via rigid motion (i.e., like in Ch. 4) and that they “have the same area” or “have equal content” if they are equal in the looser sense of I.37. A richer mathematical vocabulary helps us make clear distinctions and unambiguous statements. Euclid’s choice of a single term for multiple senses of equality must be seen as a pedagogical choice rather than an accident. Perhaps it forces us to think carefully about the whole proposition before presuming to understand it after a quick reading. I have chosen to follow him by including some verbal ambiguity, though I also use the word “congruent” at times, since it might be familiar from other geometry classes you or your students have had. The Greek word Euclid uses for sameness is present even in our language today, in words like isosceles and isobar (a line on a meteorological map indicating locations of the same atmospheric pressure) and isomer (a chemical term for compounds made from the same constituents).

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Pondering mathematical sameness hints at metaphysical questions. What are mathematical things? Where are they? How would we determine that they are the same? When we speak of “two things” being “the same,” do we really speak of “two things” at all, and not just one?

6.3

Elements

Euclid compares plane regions by breaking them down into triangular pieces. This is the heart of Proposition I.35, which is foundational for the end of Euclid’s Book I, the content of our Chapter 6. Working with triangles rather than using the numerical notion of “area” might appear to be a needless complication. This essay explores the geometrical approach to area. Euclid’s book is called Elements. Material can be called elementary in two ways, with respect to the order of learning, and with respect to the order of being. In the first sense, elementary things are appropriate for those who are learning, who are beginning. In the second sense, elementary things are those at the base, the foundation, of a whole field. The elementary particles of contemporary physics are foundational, but not easy to grasp. Euclidean geometry can be called elementary in both senses of the word. Two points determine a line. This is Euclid’s first postulate. Three points determine a triangle. If we think of the points as being in space, we see that they determine a whole plane, too. There is a bit of imprecision in that statement. It holds not for three points in general, but only for three points that are not in a single line. A triangle is the simplest properly planar shape. There is nothing intrinsically numerical in a segment, beyond its own unity, and that it has two endpoints. There is no preferred way of measuring it, considered in itself. That means that studying triangles and parallelograms by numerical “length” or “area” is foreign to geometry, strictly speaking. Area is not an elemental property of these things. The notion of comparing parallelograms by disassembling them into triangles, the idea of Proposition I.35, allows us to work within the world of geometry by passing to the most basic elements, rather than leaving the world of geometry through the use of number.

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Triangles play a role in Plato’s dialogue Timaeus, which explores the creation and order of the world and the place of mathematics in our descriptions of it. At Socrates’ prompting, Timaeus gives a lengthy speech whose topics range from the cosmos as a whole to the human body and its diseases. Along the way, he speaks of the Platonic solids, those solids (such as the cube) that can be constructed by joining, along their edges, copies of a single regular polygon. These solids are seen to correspond with the elements fire, air, water, and earth. Timaeus explains how two simple triangles can be used to construct the faces of all the Platonic solids except one, the dodecahedron, whose faces are regular pentagons. There is something awkward in Timaeus’s account of these elemental triangles, no doubt an intentional feature of Plato’s artful writing. We do not have the space to explore the complexity of this issue here. For now, we can note two complementary things. On the one hand, speaking purely mathematically, certain decompositions and assemblies are clear and exact. On the other, our grasp of the mathematical order that underlies, or at least relates to, physical things, remains ever imperfect. From Plato and ancient science we can skip ahead more than two millennia, to the early twentieth century, to another problem of decomposition and rearrangement. In 1900 the renowned mathematician David Hilbert presented a list of significant mathematical problems. His list would influence much subsequent research. One of Hilbert’s problems relates to cutting and reassembly, but in solid figures (i.e., three dimensional ones) rather than the plane figures of Euclid’s Book I. Consider a regular tetrahedron, a solid shape determined by four points in space all at equal distances to each other. Each face of this solid is an equilateral triangle. Hilbert’s third problem asks whether the tetrahedron can be cut up into some number of solid pieces, perhaps a huge number of them, which can then be reassembled to form a cube. This cube would, of course, have the same volume (volume being the numerical analogue in three dimensions to area in two dimensions) as the original tetrahedron. While some of Hilbert’s problems remain open today, this problem was solved almost immediately after Hilbert posed it, by his student Max Dehn. Dehn showed that the tetrahedron cannot be cut

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up and reassembled into a cube. In other words, while a cube and a tetrahedron might have the same volume, we cannot use division into a finite number of pieces to show this. Instead, it requires the sort of limiting procedures first developed by Archimedes, procedures that ultimately lead to calculus. Dehn had to flee Germany in 1939. Passing through Scandinavia, then crossing Russia by train, he eventually arrived in the United States, where he only found a permanent position after some time. The institution at which he landed, Black Mountain College, was not renowned for its mathematics; Dehn was the only mathematician ever to teach there. Some of his contemporaries found the juxtaposition of the outstanding mathematician and the humbler college jarring, but Dehn loved his time at the school, on whose grounds he was buried, and where he taught not just mathematics but Greek, too. Dehn was a thoughtful teacher, said to have used a Socratic style, and he delighted to talk with his students while walking in the woods, enjoying all of nature’s elements.

6.4

Cost

We often fail to appreciate things that come to us too easily. This is no reason, though, to create needless labor. What is the place of “cost” in the study of elementary geometry? The Pythagorean theorem, the classic, ubiquitous statement about how the sides of a right triangle relate to each other, marks the culmination of the first book of Euclid’s Elements. It comes as the penultimate proposition of the book, which concludes with the converse. Euclid’s arrangement means that we must pass through fortysix antecedent propositions before arriving at this great summit. While A Brief Quadrivium omits many of the intermediate Euclidean propositions, we keep the order. The result is that we need a few chapters of preparatory work before we arrive at Theorem 34 (I.47). Hundreds of proofs of the Pythagorean theorem exist, and you will easily find many of them. One original proof was even given by an eventual President of the United States. During his time as a congressman, James Garfield discovered and published a proof using trapezoids. By his own account, it was found “in mathematical amusements and discussions” with fellow members of Congress.

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In contrast to the lengthy development of preliminary matter in Euclid, many of the other proofs available are quite short, consisting of little more than a labelled diagram with a bit of accompanying text. These short proofs might lead students to wonder whether the Euclidean path is too costly, an extravagant mental gymnastic that obscures instead of clarifying. This is a reasonable concern. It is good to have many roads by which to travel from one place to another, and it is good to have many proofs of the same proposition. Euclid himself gives a second proof of the Pythagorean theorem. At the end of Book VI, using the technique of ratio and proportionality, Euclid proves the Pythagorean theorem anew. In fact he goes beyond this, and proves a generalization, which includes the Pythagorean theorem as a special case. The advantages Euclid’s Book I proof holds against other proofs is that it is entirely elementary and entirely explicit. If you examine many of the other proofs available, you will see that they involve statements that require substantial justification. They are not really shorter paths to the conclusion, but rather merely portions of paths whose full length would require a development just like Euclid’s. Many proofs involve some sort of cutting, often called dissection. Euclid’s proof is also a kind of dissection proof. One way to remember the structure of the proof is that the altitude dropped from the vertex of the right angle, and continued across the square on the hypoteneuse, divides the square on the hypoteneuse into regions that are separately equal to the squares on the two legs of the triangle. Having divided the large square in this way, we just need to remember how to show that its parts are equal to the smaller squares, for which we use Proposition I.35 and its consequences. Some breakfast cereals are advertised to be “part of a complete and balanced” meal. If we examine the associated photo closely, we see that the phrase “part of” does yeoman’s work. The complementary elements would seem themselves to constitute a whole meal. Short diagrammatic dissection proofs of the Pythagorean theorem are similar; they are “part of” a complete and balanced proof, and not the whole thing. One reason to praise Euclid, then, is that he has given us a proof where all is paid upfront. His approach appears expensive, but only because there are no hidden costs.

7 Ratio 7.1

Week 7 Plan

Overview: This chapter introduces a key concept, ratio. Ratio is the thing that ties together all four disciplines of the quadrivium. Our approach is to think about ratio first through geometry, but we will also use it in arithmetic. Looking Ahead: Next week, Chapter 8 on the golden ratio, concludes the study of geometry. Plan ahead now for a unit assessment, and give students reminders that it will be coming. Next week’s material will probably seem easy in comparison to the work done this week. For now, persevere, knowing that the effort will pay off. Notes: 1. You might find it helpful pass back and forth between the formal language of Chapter 7 and your familiarity with fractions. The ratio a : b can often be treated like the fraction ba , although we should be careful about what sorts of things a and b are. 2. At the same time as you rely on your intuition about fractions, be careful to consider what is new in, or at least more clearly brought out by, the notion of ratio. 3. The exercises are the key to grasping this topic. Students must think about copying things out repeatedly. The way they will think about this is by doing it.

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4. Avoid all temptations to measure segments using a ruler with units marked. 5. Euclid’s Elements first presents abstract ratio (Book V) and then ratio in plane geometry (Book VI). I have chosen to mix the two so that the abstract notion appears more tangibly. 6. The technical sense of ratio that Euclid records in Book V of his Elements seems to come originally from the mathematician named Eudoxus. When we wish to emphasize that we are working with ratio in the sense of Eudoxus, it is sometimes useful to say “Eudoxan ratio.” 7. We will use ratio fairly strictly in geometry, arithmetic, and music, but once we get to astronomy we will use decimal numbers and abandon our scrupulous adherence to the language of ratio. 8. Notation and Terminology: The way to read A : B is “the ratio of A to B.” Day 1: 1. Read Section 7.1. 2. Complete Exercises 1, 2, 3, and 4. 3. Discuss familiar examples in which ratios involving different kinds of things are comparable. For instance, the ratio of one animal’s weight to another’s, in comparison to one animal’s height to another’s. Day 2: 1. Reread Section 7.1, and read Section 7.2. 2. Complete Exercises 5, 6, 7, and 8. 3. Recall that there were two ways to talk about equality of triangles, a strict way and a loose way. The comparison of triangles in Proposition 42 (VI.1) involves the looser sense. In other words, if the bases are the same, the triangles are the same under some decomposition into smaller pieces. They might not have the same angles.

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4. Proposition 43 (VI.2) is phrased tersely. It is important to read the statement, look at the diagram, and skim over the proof, then go back to the diagram to nail down exactly what is being discussed. Do not simply wrestle with the statement of the proposition. Day 3: 1. Read Section 7.3 through the proof of Proposition 46 (VI.19). 2. Be sure to read things like AC : AB as “the ratio of AC to AB” when reading the proof of Proposition 46. Do not fall into passive absorption of the symbols, or sloppy use of language. Practice reading aloud. 3. Make a diagram illustrating Proposition 46. 4. Complete exercises 9, 10, and 11. Day 4: 1. Complete Section 7.3. 2. Note that the proof of Theorem 47 (VI.31) involves some work with ratios—work that is in Exercise 17. Take a look at this exercise to get the general idea, even if students are not ready to complete the exercise itself. 3. (optional) Attempt Exercises 13–17. Day 5: 1. Reread the whole chapter. 2. Illustrate Euclid’s generalized Pythagorean theorem as in the textbook, but use a pentagonal figure or an oddly shaped quadrilateral on each side. 3. Complete Exercises 19, 20, and 22. 4. (optional) Attempt Exercises 13–17 and 21. Assessment: • Have students complete Exercises 2–7. Optionally, create new versions of these exercises with different lengths—you can use a ruler to arrive quickly at the desired ratios. • State and prove Proposition 42 (VI.1).

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Forgetfulness

Can virtue be taught? Plato’s dialogue Meno explores this question, and uses a mathematical example along the way that can help us think about the study of Eudoxan ratio. Meno, from Thessaly, speaks with Socrates, in Athens. Meno’s slave is with him. The discussion of virtuous action leads to an inquiry into learning itself. Socrates wants to show how learning involves bringing to mind something that we somehow already possess, that it is a kind of recollection. In order to show this, Socrates asks questions of Meno’s slave, who is uneducated. The puzzle that Socrates and Meno’s slave investigate is a mathematical one. Suppose we have a square. How do we produce another square that is twice as large? Meno’s slave first proposes to double the side of the given square. Socrates, by asking questions, leads him to conclude that such a square is too big; it is four times the size of the original. The slave next tries a shorter side, one and a half times as long. This square is also too large, as the slave discovers with a bit of prodding. Socrates then leads the way by producing a helpful diagram. He draws a square arrangement of four squares, and produces another square within them by drawing suitable diagonals. Since each of the diagonals cuts each square in half, as the slave can see, and the additional square is half the whole, being the combination of halves of all the parts, Socrates guides the untutored attendant to the discovery that the square on the diagonal is twice the original. The exhibition seems to show that there was something in the servant already; Socrates extracted it only through questions. The friendly interrogation revivifies a dormant power the slave possessed at the outset. Socrates then invites Meno to join him in searching, in a way like the slave did, for what virtue is, and not merely how it is acquired. Setting aside this old philosophical text, let us attend to our more immediate task, the study of ratio. Here we need the complement to memory: a virtuous forgetfulness. The abstract ratios of Euclid’s Book V and the geometric ratios of Book VI are closely related to fractions and their familiar arithmetic. Fractions must be left behind, though, in order to grasp Eudoxan ratio.

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A fraction, a rational number, is a single entity. You are accustomed to treating it as such a thing. A ratio is also a single thing, but it is a single thing that involves two other things, related but distinct. This distinctness in the generating objects is important. The habit that you must “forget” for a time, while studying ratio, is the notion that there is a number associated to every geometric object. Instead, you must hold in mind the two distinct magnitudes, and then consider the ratio that joins them as a third thing, something additional to the two objects. The association of a number with a geometrical object only comes about through the choice of a unit. Choosing a unit is arbitrary. Once we have done so, we can become mired in complications to show that our purely geometrical conclusions are independent of the choice of unit. In his conversation about squares, Socrates directs us to pass from numbers and units and names to the underlying things—segments, sides, diagonals. The Meno dialogue reminds us that we might already have all that we need, for mathematics and for virtue, although we will need to do some searching within. There is another side to knowing well and doing well, though; we have all acquired habits worth losing. The Greek word for truth is aletheia. The initial letter functions just as it does in our words agnostic and amoral, indicating that aletheia is being without lethe, forgetfulness. The river Lethe, a river of oblivion, was a feature of ancient mythology. While its classical valence is ambiguous, Dante gave the stream of forgetfulness a decidedly positive place, setting it atop the mountain of Purgatory. That the newly purified might rise to the heavenly spheres, they must first bathe in its waters and forget their sins.

7.3

Categories

If even a pure mathematician refers to something as “abstract nonsense,” the situation must be dire. The middle part of the 20th century saw the development of an area of mathematics known as “category theory,” whose abstruse character won it this teasing denomination. At first these upper reaches of contemporary mathematics seem far from the humbler lowlands of ancient Greek ratio, but the newer developments usefully illuminate the older ones.

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When we take Eudoxan ratio seriously, we keep separate the objects that make the ratio, and consider the ratio as a third thing that joins them. A category (in the contemporary mathematical sense of the term) is a more complicated gadget than a ratio, but shares some features. Categories consist of “objects”—things—and “morphisms”—relations between things. Categories account for both aspects—things and their relations—but keep them separate, just as the ratio keeps its parts separate. The category theorist then goes farther, and thinks of categories as being, themselves, objects. What are the relations between them? They are called “functors.” Categories and functors between them are already at a fairly high level of abstraction, but we have not yet arrived at even the original purpose of categories, to say nothing of subsequent elaborations. Given two categories, and two functors (relations) between them, we can consider relations between the two functors. Such a relation is called a “natural transformation.” Category theory was first developed to talk about these natural transformations between functors. A reasonable number of readers will have had an introduction to something called “linear algebra” while studying mathematics, science, or engineering. The next few paragraphs are addressed especially to them. Recall that a “vector space” is made out of vectors, which can be added together and also rescaled by “scalars.” When first encountering “linear transformations,” maps between vector spaces, we usually see matrices. A matrix describes how a linear transformation works by saying what it does to a special collection of vectors, called a “basis” of the vector space. A basis is not itself an intrinsic feature of a vector space, but instead arises from a choice. Other choices are possible, just like we can choose various units for measuring length. Given a vector space, we can construct another vector space called its “dual space.” This is the space of linear transformations from the given vector space to the scalars. For finite dimensional vector spaces, there is no essential difference between the original vector space and the dual space. More formally, the two are “isomorphic.” This relation between the two, while it exists, is only constructed ad hoc. If we repeat the dual space construction, though,

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we get something more. The dual of the dual of the original space is isomorphic to the dual, and so isomorphic to the original. The additional feature present here is that there is a special morphism, a linear transformation, from the original space to the dual of the dual. This linear transformation does not require us to pick a basis and write out a matrix. It comes for free. It is “natural.” The dual space example just given comes from the paper General Theory of Natural Equivalences (Transactions of the American Mathematical Society, 1945) by Samuel Eilenberg and Saunders Mac Lane, which inaugurated the study and use of categories. The first few paragraphs are accessible without anything beyond linear algebra, and the paper is freely available online. Within the first pages you can encounter the “naturality squares” that are ubiquitous in categorical discussions. These squares are often found in media images of mathematicians, in which scholars meditate in front of chalkboards covered with complicated formulas and figures. All of this might seem abstract, and it is. There are a couple of points of common ground between this piece of modern mathematics and our study of an old scientific tradition. One is that mathematicians in all ages have found it useful to talk about relations between relations (yes, both) between things, maintaining careful distinctions of these levels. Another is that this sort of talk comes across as needlessly abtract. Fear not; the taste can be acquired.

8 The Golden Ratio 8.1

Week 8 Plan

Overview: While last week was mostly about using geometry to understand ratio, we now use ratio to understand geometry. Specifically, we look at the famous golden ratio from two perspectives.

Looking Ahead: You should have a unit test on geometry soon, likely this week. Continue to plan for that and to give students reminders. Next week the next discipline, arithmetic, begins. The first week is fairly light, so it might be a good time to make sure all students are up to speed, or to accommodate other curricular demands. This is also a good time to start planning ahead for astronomical observations—see Exercise 11 in this chapter. Notes: 1. The first perspective on the golden ratio comes from Euclid’s Book II. One way to think about what is happening in Book II is that Euclid is showing that rectangular regions behave in a way that is compatible with ordinary arithmetic (which uses numbers). 2. The second perspective on the golden ratio involves ratio directly, unlike the first. If we think about how to prove that proposition using modern language, it is that the area of a region is the product of the sides. That gives Proposition VI.14.

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3. I have chosen to avoid discussion of the Fibonacci sequence and other of the more recent perspectives on the golden ratio. If you choose to discuss these points, be sure not to displace the two Euclidean perspectives. Day 1: 1. Read Section 8.1 2. Review Section 6.2 and talk again about “area.” 3. Complete Exercise 1. Practice repeatedly. Day 2: 1. Reread Section 8.1. 2. Read Section 8.2. 3. Review Section 7.2, focusing especially on Proposition VI.1. 4. Review Exercise 1, and complete Exercise 2. Day 3: 1. Review Section 8.2. 2. Read Section 8.3. 3. Complete Exercises 3 and 4. 4. Review the Procedure from Chapter 2 on how to circumscribe a circle about a triangle. Day 4: 1. Return to Section 8.1 and study the proof of Proposition 52 (II.11) more closely. It should be clearer with the time that has elapsed. 2. Complete Exercises 5, 6, and 7. Day 5: 1. Reread all of Chapters 7 and 8. 2. Complete Exercises 8, 9, and 10. 3. Complete Exercise 11—start planning ahead for astronomy.

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Assessment: • State two ways of defining the golden ratio. One involves equal parallelograms. The other involves ratio directly. • Given Proposition II.6 (indicated in Figure 8.2), prove Proposition 52 (II.11). • Construct a regular pentagon using compass and straightedge.

8.2

Beauty

Euclid alone has looked on Beauty bare.

Edna St. Vincent Millay won fame as a poet at the age of 20, when she entered a contest sponsored by The Lyric Year. Her fourthplace finish in that competition was a great controversy. Ten years later, at which point she was an established name in the literary world, Millay published the poem whose first line is quoted above. Among her elegant words about the vision to which the ancient geometer invites us, Millay also offers an admonition. She says “let all who prate of beauty hold their peace.” A danger lurks; we can be hastily presumptuous when we speak of aesthetic delight, and by this presumption fail to rise above ourselves. The poem continues “and lay them prone upon the earth and cease/to ponder on themselves.” This posture, prone upon the earth, features also in the contest poem, Renascance, that first made her name. “So here upon my back I’ll lie/And look my fill into the sky.” In this older poem the contemplative pose generates a more ambivalent outcome. The narrator, with senses open to all that is in the world, comes to be oppressed and overwhelmed by the suffering within it. “Ah, awful weight! Infinity/Pressed down upon the finite Me!” After a death and rebirth, refreshed by rain, she experiences new joy, confident that she can find the divine presence in every thing. In both poems, Euclid alone and Renascence, Millay shows that an encounter with the transcendent is not merely pleasant, though it might please. Mathematical beauty is both “luminous” and “terrible” in her account. If we would speak of beauty in the subjects

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we teach and learn, we do well to keep these both in mind, that the beautiful is delightful but also holy. In the presence of something holy we rightly fall silent. Shakespeare writes of silence in a more mercantile vein. My love is strengthened, though more weak in seeming, I love not less, though less the show appear. That love is merchandized whose rich esteeming the owner’s tongue doth publish everywhere.

The Bard notes that, however beautiful our words, they cheapen through overuse. A love that remains strong and true need not shout the fact, year after year. As the couplet has it, I would not dull you with my song. This chapter, the final chapter of our study of geometry, involves a ratio whose aesthetic properties have often been touted. I neither propose nor presume any definite aesthetic responses, here or elsewhere in A Brief Quadrivium. There are two reasons: one is historical, the other is principled. The historical reason to avoid a lengthy discussion of this ratio’s purported pulchritude is simple; such discussion is not a feature of the classical sources of the quadrivium. In Euclid, in particular, this ratio—what he calls division of a segment in “extreme and mean ratio”—is treated technically, just like the rest of his material. This special ratio allows, among other things, the construction of a regular pentagon, the construction students learn here in Chapter 8. While reasonable claims can be made about the use of the golden ratio in ancient art and architecture, it seems that it was not until Luca Pacioli’s Divina proportione, illustrated by Leonardo da Vinci and printed in 1509, that the ratio’s relation to beauty was treated at length in any text. The second, more principled reason has to do with student ability. Phi, the golden ratio, does indeed possess many remarkable properties, and the ratio reveals itself in created things in striking, often surprising ways. These properties and appearances, though, are often not things that typical students of A Brief Quadrivium are prepared to treat in a properly mathematical way. To give one example, the convergence to phi of ratios of consecutive elements in the Fibonacci sequence requires, for a genuine mathematical treatment, some theory of infinite sequences and limits. Rather than a

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superficial account of these fascinating matters, then, I have chosen stick with what we can do thoroughly. We may speak about beauty: with our students, with our colleagues, with our friends. We may find it, too, as we apply ourselves to ancient disciplines. As we do this, let us beware lest our speech lose the sobriety born of a sacred encounter.

8.3

Death

Carl Friedrich Gauss died in Göttingen on February 23, 1855. Born in 1777, he had lived to the age of 77. It is impossible to summarize in a brief span Gauss’s incredible contributions to mathematics and the natural sciences. During his life, Gauss was usually ahead of the game. A legendary instance comes from his school days. A teacher had hoped to keep the students occupied, demanding that they sum the numbers from 1 to 100. The young Carl did this immediately. You and your students will learn the trick in the next part of the quadrivium, Arithmetic, in Chapter 14. Later, when he was a man, fellow mathematicians often found that he had beaten them to the punch. János Bolyai discovered non-Euclidean geometry, and his father, an acquaintance of Gauss, shared the news with the famed mathematician. To the sorrow of the Bolyais, Gauss said that these were things he had himself investigated, years before, but kept private since he did not think there was a suitable audience for the results. Gauss’s motto, Pauca sed matura, means “Few, but ripe.” It would be difficult to call his contributions to “the queen of the sciences” a paucity; Gauss’s collected works exceed a dozen dense volumes, and his achievements in one area were unsurpassed for 200 years. (Manjul Bhargava, the mathematician who finally developed Gauss’s ideas further, won a Fields Medal, the top prize in mathematics, in 2014.) Despite all this, the motto still reminds us of our limits; a well-pruned tree bears better fruit. Wise thinkers in many generations and many traditions have advised us to meditate on our coming deaths. This recollection helps us live well. Here in Chapter 8 we butt up against a limit; our study of geometry will go no further. Earlier chapters have already noted the boundaries of this course. We examined multiple

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proofs of Euclid’s first proposition, but none for his second. The construction of a regular pentagon, the satisfying pinnacle that we reach here, still bears about it the marks of our finitude. We only get the construction and not the proof. If you and your students have the opportunity and desire to study Euclidean geometry beyond the limits of this text, one good way to do so is through Books III and IV of Euclid’s Elements. These books treat circles and the relationships between circles and other geometric figures. A high point of Book IV is the proof that the construction of a regular pentagon truly does produces a regular figure, not simply an approximation good enough for the naked eye and our coarse physical instruments. A small number of students might go even further, from ancient mathematics to more modern developments. One avenue is the discovery that led Gauss to dedicate his life to mathematics, a remarkable result that is closely related to the classical work of this chapter. By the age of 21, Gauss had found that it is possible to construct a regular 17-sided polygon using only a ruler and compass. Moreover, he showed that even more extravagant polygons were possible. In general, he showed that the polygons constructible with compass and straightedge are exactly those that arise from a special family of prime numbers. His result contains two sides, possibility and impossibility. Whether you stop with A Brief Quadrivium, or turn from it to the old classics like Euclid, or even go beyond Euclid to the more recent genius of someone like Gauss, your mathematical learning will be bounded. It is to be measured not with respect to the whole of mathematics but with respect to the whole of a human life, laden with both potency and responsibility. The great German faced his own difficulties and sorrows while winning glory: onerous Napoleonic taxes, the early deaths of his first wife and their third child. He was constrained, too, when passing from this life. The renowned mathematician, who also made a living as a surveyor, had requested that a regular 17-gon be inscribed on his tombstone, an enduring tribute to his astonishing youthful achievement. The city councillors demurred; the stone masons claimed it was too difficult. When we keep his great mathematical tradition alive, we remedy in some small way this pusillanimous injustice.

Part II

Arithmetic

Eloquia Domini, eloquia casta.

9 Counting 9.1

Week 9 Plan

Overview: This week we pass from geometry to numbers. Like in Chapters 1 and 2, we begin with playful activity to accustom ourselves to the objects involved. Later, we will give proofs. Looking Ahead: This is a light week, so it is a good time to deal with administrative items. Music and Astronomy each involve prerequisites that are logistically complicated. You need a monochord for music, and students must make observations prior to beginning astronomy. Take the time this week to be sure that those things will be ready in time. Notes: 1. While this week is a break from formal proofs, it is not a break from explanatory activity. Students should be able to give reasons for the various estimates that they make. 2. Some students will tend to want excessive exactitude, and thereby bog down. Encourage them to move along briskly, writing down their estimates, with the promise that they can return to earlier steps to modify them after taking a first pass through the whole process. 3. Other students will estimate too hastily, producing numbers that simply “seem right.” Encourage them to focus on a small piece, in which they can produce something fairly exact. For instance, if

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estimating blades of grass, measure out a specific one square inch region outside and have them count exactly how many blades of grass are within it. Estimation will then play a role, as that little region is used to approximate the whole. Day 1: 1. Read Sections 9.1 and 9.2. 2. Complete Exercises 1, 2, and 3. Day 2: 1. Reread Sections 9.1 and 9.2. 2. Complete Exercises 4, 5, 6, and 7. Day 3: 1. Review the estimates that were made on the first two days. Offer critiques. Consider these questions. • Where do there seem to be overestimates, where are there underestimates? • What important qualifications were omitted or not given due weight? • What is the range of reasonable estimates? How narrow can it be made? 2. Complete Exercises 9 and 10. Day 4: 1. Complete Exercises 11 and 12. 2. Have students use the results of Exercises 11 and 12 to make plans for the future. Suppose that they are assigned an essay of 500 words, or 5000 words, on a given topic. How should they plan to complete the essay? What time will be needed for what parts (reading, thinking, writing, revising)? They can write down a specific schedule for completing this work. This could even be done with a real assignment, if they have a concurrent writing class.

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Day 5: 1. Go outside and make estimates involving things that are seen. How far away is that distant building? How tall is that tree? How many trees are in that park? How many gallons of water in that pond? How many toilets in that office building? How many offices? As much as possible, students should come up with the questions, and they should evaluate the estimates themselves. Assessment: There are many questions that can be asked along the lines of those given in this chapter. Feel free to create your own. Here are a couple that can be used. • How many books are in the local public library? • How many words does a person speak in a lifetime? • How many people’s names will you ever learn?

9.2

Games

Plato’s admirers and detractors alike concede that his writings span a wide ground; it has been said that the history of philosophy is a series of commentaries on Plato. Even Plato’s most ardent disciples, though, cannot claim that he went so far as to write about the atom bomb. The Academy’s founder got his mathematical ideas in Italy, from the followers of Pythagoras, while avoiding political turmoil at home. Two millennia later, turmoil in Italy and beyond led a different thinker, Enrico Fermi, to seek refuge even further abroad. The winner of the 1938 Nobel Prize in Physics ended up in the United States, and by the summer of 1945 he was in New Mexico for the trial detonation codenamed Trinity. At the time of the explosion, Fermi stood with other scientists at a close, but safe, distance from the blast. When the shock wave reached them, Fermi dropped some small bits of paper and watched what they did as they fell to the ground, driven by the wind that the detonation caused. He then briefly did a mental calculation and told those who were with him his estimate of how strong the explosion had been. Fermi’s result turned out to be quite close to the one derived from complex instruments stationed near the bomb.

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Enrico Fermi was able to make this rapid, accurate estimate because he had learned over many years how to see the key features of a given system of quantities, making good guesses by relating unknown things to those that were more familiar to him. He was a genius scientist, but his method can be taught, even if we will not ever have his physics prowess. The puzzles of this opening chapter are in some way like Fermi’s calculation. One difference is that they do not involve knowledge of physical parameters, like the acceleration due to gravity or typical air resistance. Those are things that students pick up when they study physics or chemistry. This book’s estimates involve things accessible without prior technical training. This essay opened with Plato, and the observation that he knew nothing about nuclear weapons. It turns out that he did, however, know something about what Fermi did at the test. In Plato’s late work called Laws, a figure called “the Athenian” speaks about mathematics and education. He advises that a society should use games to introduce children to arithmetic. He says that the Egyptians do this, with puzzles about how to distribute garlands or arrange wrestling tournaments. The youthful arithmetical games of Plato’s Laws are not deadly, but Fermi’s bomb is. War and death are, though, under the surface of Plato’s dialogue Timaeus, introduced earlier in an essay on geometry. Socrates explains at the beginning of Timaeus that he wants to understand what the best city looks like when it is in conflict with other cities. The dialogue of Timaeus takes place the day after a discussion of the best city in the abstract. Now Socrates wants to understand how this good polity would respond when faced with the strains brought on by the inevitable limitation and strife of life in the world. Socrates notes that a superficial sophist, someone who wanders from place to place teaching convenient tricks of argumentation for pay, will not be able to present how the virtuous statesman would act in such a situation. Being rooted in a particular place both limits and frees. The artful speech of the rhetorician is hemmed in by the humbling realities of a concrete place. At the same time, though, the speaker is elevated by this very limitation, since only the encounter with these natural limits will direct his theorizing away from the

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fanciful and towards the truly good. There are no right answers to these questions of this chapter. It is also true that mathematics is exact. These two statements are both true. The finiteness that encumbers us leads us to battle with others for the resources of the earth. In times of peace we strive for another kind of mastery, discovering new and clearer ways to speak about the order within created things. Even as our speech becomes more accurate, it ought to remain playful, as we keep in mind the memory of our natural limits.

9.3

Alexandria

Somewhere in Spain there once stood a statue of Alexander the Great. In the 60s BC, while the youthful Julius Caesar was serving the Roman Republic in Hispania, he encountered that statue. The man who would found the Roman Empire grew discouraged at the thought of the Greek conqueror. He pondered how little he had himself accomplished, while Alexander at his age had already conquered the world. Alexander founded as well as conquered. One of his foundations is the city of Alexandria, at the mouth of the Nile, where Africa’s longest river empties into the Mediterranean. Alexander had been tutored in his youth by Aristotle. Alexandria would become a center of learning, eventually housing a library renowned throughout the ancient world. Alexandria was home to Euclid and Ptolemy, two of our main sources for the quadrivium. We have little biographical information about Euclid. One thing we do know, though, is that he founded a school of mathematics in Alexandria and taught there, likely from about 300 BC to 270 BC. It seems that he had previously studied mathematics in Athens, with the successors of Plato at the Academy. One way that we get an idea of when Euclid lived is through Aristotle’s writings. Aristotle often uses mathematical examples and analogies, and in some places Aristotle’s perspective on geometry is clearly pre-Euclidean. This establishes a limit on how early Euclid might have flourished. At the other end of the range, one ancient source says that Apollonius, a great mathematician who investi-

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gated curves like parabolas and ellipses, studied in Alexandria with students of Euclid (but not Euclid himself) in the mid-200s BC. Ptolemy lived centuries later, when the Roman Republic had become the Roman Empire. He wrote his Almagest, the source we will follow for astronomy, around 150 AD. Like Euclid, Ptolemy worked in Alexandria and wrote in Greek. Both Euclid and Ptolemy composed textbooks that were so successful, so compelling, that they eliminated the works of their predecessors. Hipparchus, for example, invented trigonometry and created scientific astronomy by uniting Babylonian observational data with Greek demonstrative mathematics, yet we know his contributions through Ptolemy’s writings rather than through his own works, only one of which is extant. A great library came into being at Alexandria around the time of Euclid or shortly thereafter. The library contained the equivalent of about 100,000 books, though at that time the texts were stored as scrolls. One of the early chief librarians was a scholar named Eratosthenes. Eratosthenes will come up again both in our study of prime numbers and when we talk about the size of the earth. Julius Caesar accidentally burned part of Alexandria, possibly damaging the great library, during his civil war of 48 BC. Two centuries later, during Ptolemy’s life, the institution was headed towards decay. The last scholars recorded to be associated with the library lived about one hundred years later, in the middle of the third century AD. The exercises of this chapter ask students to consider Alexandria’s ancient library. The scholars who labored there collected and preserved an immense store of learning even when the cost of producing written texts was high. It is good to reflect on the striking accomplishments of those who have come before us, and to be inspired by them. The library’s ultimate demise also reminds us that no matter what impressive physical structures are built, ancient wisdom is kept alive only by the deliberate effort of those who wish to pass it on.

10 Numbers in Themselves 10.1

Week 10 Plan

Overview: Last week dealt with computations; this week deals with definitions. These definitions will be the basis for subsequent proofs. Looking Ahead: Next week we return to proofs. Take a look ahead now to get a sense of what they will look like. There is some additional complexity next week, as we will also look at what it means to say that certain mathematical statements are false, using counterexamples. Notes: 1. Observe that the definition of “even” does not involve time or human activity. We do not say “when you divide . . . .” Instead we say “there is . . . ,” we do not talk about changing things. 2. It is possible to prove that every natural number that is not even is in fact odd. One way to do this involves something called the well-ordering of the natural numbers. Briefly, if there were some natural number that were neither even or odd, there would be a least such number. Its predecessor is either even or odd, etc. 3. The sources of quadrivial arithmetic (Boethius, Nicomachus, and others) are intersted in the “figurate” numbers that we use here (triangular) and later.

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Day 1: 1. Read Section 10.1 and 10.2. 2. Begin Exercise 1. 3. Complete Exercise 2. 4. Students can explore triangular numbers, attempting to list as many as possible. Day 2: 1. Reread Sections 10.1 and 10.2. 2. Read Section 10.3. 3. Continue working on Exercise 1. 4. Complete Exercises 6 and 7. 5. In later work we will see a specific method (the Sieve of Eratosthenes) for generating primes as needed in Exercise 6. For now, have students simply find primes by examining each number individually and checking for divisors. Day 3: 1. Reread Section 10.3. 2. Read Section 10.4. 3. Complete Exercises 3, 4, and 8. Day 4: 1. Reread the whole chapter. 2. Redo Exercises 6 and 8. 3. Attempt Exercise 5. 4. Exercise 5 is a challenging activity. Showing that definitions are equivalent requires students to think at a more abstract level. At one time one definition must be taken as given, and at another time another must be taken as given.

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5. The reason Exercise 5 involves integers and not just natural numbers is that the number 1 is a tricky case (it is the successor of 0, but 0 is not a natural number). We will call 1 a number, and specifically an odd number, but as was mentioned earlier, some ancient sources would not call 1 a number. (Do not be too concerned about this distinction. It is not foolish, but it is also something that is essentially independent of what we will be doing.) Day 5: 1. Complete the weekly assessment, and take a look ahead to the next chapter in preparation. Assessment: • Define “even number” and “odd number.” • Write down three triangular numbers. • Give the first 10 powers of 2. • State a prime number between 30 and 40. • Define “perfect number.”

10.2 Mystery The monad holds seminally the principles which are within all numbers.

The quotation above is from Iamblichus, in a work called The theology of arithmetic. Iamblichus discusses the numbers from one through ten. Three, for example, is especially beautiful, beyond the beauty of other numbers. Five is particularly associated with justice. Iamblichus falls into the class of thinkers called Neoplatonists. The first of the Neoplatonists, Plotinus, wrote around 250 AD, and Iamblichus himself was born at that time. An important later Neoplatonist named Proclus wrote in the fifth century AD, and one of his works is a commentary on Book I of Euclid’s Elements. Iamblichus wrote a biography of Pythagoras. In it, he says that Pythagoras used the Pythagorean theorem to illustrate justice; it is something consistent (like justice) amid the infinite variety of triangles. Iamblichus also relates the legend that a Pythagorean

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named Hippasus died, cast into the sea by the gods, because he divulged the mathematical secrets of the Pythagorean sect. These tales of Iamblichus present the symbolic use of mathematics and the mysterious hiddenness of the symbols. Mathematical mysteries might be appealing to some readers, and off-putting to others. They are certainly far from the computations of an algebra classroom. To come to grips with the old line of thinking, it is useful to compare mystery in numbers to poetry. When poetry appeals to us, it grasps us in an immediate and inarticulable way. At the same time, though, poetry also presumes preparation that we can rationally describe. To enjoy poetry we need a grammatical foundation; we need to know both words and things. We must know the things to which words refer, lest the words be mere sounds. And we must know how to say the words, how to pronounce the syllables, which syllable to accent. We should know, too, how to spell the words. The appreciation of poetry begins, then, with a mundane slog to acquire vocabulary, to spell correctly, and to speak with good pronunciation. Only when these are in hand will be be able to enjoy certain special combinations of sounds, rhythms, or images. Learning the quadrivium does not require us to affirm our faith in any esoteric mysteries, and it does not even require us to discuss what other people have written or believed about such things. Instead, we simply acquire a basic mathematical vocabulary, learning both the words and the things to which they refer. Any attempt to discuss symbolic understandings of number that lacks a substantial mathematical foundation will be stillborn. In the 20th century, the novelist Walker Percy explored themes of alienation and mystery in both fictional works and essays. In one of his essays, the chapter “The Loss of the Creature” in The Message in the Bottle, Percy tells the story of a couple from the Midwest who go to Mexico. The travelers first visit the main sights. What they find there are other people like them. They want, instead, to find something more “authentic.” Somewhere between Guanajuato and Mexico City, they stumble accidentally on a small village. The villagers are practicing a religious ritual, dancing for rain, petitioning the corn god. The American travelers are delighted; they have found their authentic Mexico. They can barely contain

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themselves, and wish that their friend, an anthropologist, were with them. In Percy’s story, the anthropologist does return with them. Percy notes something remarkable. When the three are there, together, the couple do not look at the villagers. They look instead at the anthropologist. They want him to authenticate their experience. They are alienated—from the place and from themselves—and need someone else to affirm their encounter with something real. No matter what you and I do, we can never participate in the rain dance. We do not believe in the corn god, and we are not from the village. There is a difference, though, between Percy’s story and an encounter you might have with number symbolism. The matter is not one of religious ritual so much as one of poetry. I do not want to serve as the anthropologist, authenticating an encounter with “ancient number symbolism.” I want to give you, instead, a chance to speak like a native. I want to be a linguist, helping you learn a language. Later, when you are fluent, you can decide for yourself whether you find the allusive use of numbers appealing.

10.3

Redundancy

The carpenter is advised to measure twice and cut once. Redundancy can be a strength, then, and not simply a logical flaw. Let us look at redundancy on two scales, large and small. Within our study of arithmetic as a whole there is redundancy—the material of this chapter is repeated in a later one. And at the smaller scale, redundancy plays a role in one of the exercises in the current chapter. We will begin with the smaller scale. An exercise asks students to count rectangular configurations. This kind of counting amounts to asking about the number of ways to factor the number, which is itself a question about what is called the “prime factorization” of a number. Prime factorization means breaking the number down into pieces that are not further reducible, analogous to the chemical statement that water is H2 O, assembled from hydrogen and oxygen. The notion of “rectangular configuration” is ambiguous. These words alone do not indicate whether or not we should orient the

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rectangle in a particular way. In kindergarten, you might have heard of two ways to fold a paper: “hot dog” and “hamburger.” The reason we need this terminology is that a bare rectangle does not possess what the mathematician would call a “distinguished” side. We can, of course, speak of a “shorter” side, provided we are not dealing with a square. At that point the ambiguity is eliminated. That is how we can use words to distinguish between folding hot dog-style and hamburger-style. Now on to the particular exercise. We will consider a simpler version. How many rectangular configurations of 6 objects are there? A first answer to this question is “one.” That would be the configuration in which there are two rows of three objects, which can also be seen as three rows of two objects. A second answer to the question is “two.” This answer arises if we consider a 2 × 3 configuration to be different from a 3 × 2 configuration. In other words, we can consider “oriented” rectangles. This is not a pointless distinction. Consider two groups traveling. One is a group of three couples. The other is a group of two young families, each with a single child. In both cases there are six total people, but the “factorization” of six implicit in the arrangement of, say, train or plane seating will affect the comfort of these travelers and their neighbors. There is a different way to get the answer “two.” That is if we consider one row of six objects as a rectangular configuration. That might seem like cheating, but asking whether or not this belongs amounts to clarifying the definition of “rectangular arrangement.” Combining the two “two” answers, we see how the answer might be “four” too. If we consider ordered/oriented configurations, and we allow for configurations whose length or width is one, then we get 1 × 6, 2 × 3, 3 × 2, and 6 × 1. Now let’s turn to the larger scale. The material of this chapter is repeated in Ch. 14. What is the point? Our introduction to arithmetic is similar to the beginning we made in geometry. The genuinely Euclidean material only started in Chapters 3 and 4, after students got familiar with geometry through tangible constructions with physical instruments. We are doing something similar now. Chapter 9 refreshed basic skills of computation and estimation and also emphasized moving in a light, playful way. Chapter 10 is a bit

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more technically demanding, but it only deals with definitions, not proofs. All that we do at this point is look at various numbers and give names to special properties they have. In geometry, once we had gotten familiar with some proofs, it was possible to return to the basic procedures of Chapter 2 and examine them more deeply, proving that the procedures really did accomplish what we said they did. Things will go similarly with arithmetic. After a few chapters that introduce proofs with natural numbers, we will return to the things that are defined in Chapter 10 and prove statements about them, in Chapter 14. Finally, there is even some redundancy in the essays of this book. We saw earlier that “sameness” or “equality” could be used in different senses, when we explored the decomposition of parallelograms in Chapter 6. You might profit from returning to that discussion now, and comparing with the notion of “sameness” for rectangular configurations. Is a 2 × 3 rectangle the same as a 3 × 2 rectangle?

11 Demonstration with Natural Numbers 11.1

Week 11 Plan

Overview: We now return to proofs as we did in geometry, this time with natural numbers. We also get a sense of what it means to say that our statements are true by looking at when similar statements are false. Looking Ahead: Next week includes a proof that there are infinitely many prime numbers. This proof is more challenging. If you are not familiar with it, you should take time to examine it now in preparation. Notes: 1. Some of the statements in this chapter are “obvious,” such as that the sum of two even numbers is even. The purpose of doing these proofs is not to surprise students with a new result but to become accustomed to the form of basic arithmetic proofs. 2. Repetition is helpful. Students can simply write out the given proofs, verbatim, repeatedly. This is a good way to study. 3. I deliberately avoid using the symbol “=” where it might appear natural. Instead, we say “is the same as” and keep the various quantities clearly distinct. This is because many students have the habit of thinking of “=” as a command or operation, rather than

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as an assertion that might be true or false. You might choose to use “=” when writing things out, but do so only with care. 4. Keeping the “goal” in mind is helpful. This is explained in the marginal note in Section 11.1. For more complex proofs this method is less useful, but for the simpler proofs about divisibility it helps students see where to go. Day 1: 1. Read Section 11.1 2. Practice with the proofs, discuss them, copy them. 3. Complete Exercise 1. Day 2: 1. Reread Section 11.1. 2. Read Section 11.2. 3. Attempt to give the proofs from Section 11.1 from memory. 4. Complete Exercises 2 and 3. Day 3: 1. Read Section 11.3 2. Pay attention to the definition of divisibility. Recall the remark about evenness—it does not involve human activity, i.e., “When you divide.” 3. Given a pair of numbers, say whether the first divides the second, the first is divisible by the second, or neither. (Be sure students can distinguish between “divides” and “is divisible by.”) 4. Complete Exercise 4. Day 4: 1. Read Section 11.4. 2. Complete Exercise 5.

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3. Discuss how you would disprove statements like this: “Everyone in this family is right-handed.” Compare that to the counterexamples discussed in Section 11.4. Day 5: 1. Reread Section 11.4. 2. Read Section 11.5. 3. Complete Exercises 7, 8, 10, and 11. 4. (optional, more difficult) Complete Exercise 9. This proof will be less formal, and can assume familiar features of divisibility. Assessment: • Prove a selection of Propositions 58, 62, 64, 65, and 66. • Complete Exercise 6.

11.2 First The puzzle about the chicken and the egg reminds us that there can be various ways to speak of something being “first.” Do we wish to speak about time, about the perfection of a thing, about something else? Let’s start by considering the disciplines of the quadrivium. Ancient writers like Boethius affirm that numbers are “prior to” geometry. This statement is not about the passage of time. Instead, it refers to a kind of hierarchy in what things are. We can count objects that we don’t perceive as being extended in space (colors, pairs of friends, possible outcomes) but once we consider things that are extended in space we can conceive of counting them, either because they are actually multiplied or because, being material, they are conceivable as repeated. Aristotle goes so far as to say that arithmetic is “more exact” than geometry, because it has fewer first principles. He says this at the beginning of his Metaphysics. The absolute priority of arithmetic over geometry suggests that it might be reasonable to reverse things, when considering the chronological order, the order in which to teach and learn. It is sometimes said that “what is first in the order of intention is last in the order of

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execution.” Supposing that arithmetic really is more fundamental, it might be fitting to take an indirect route to get there. This is one reason for the order of our study. Things that are prior and clearer in themselves are often posterior and more obscure according to us. It seems good that students become acquainted with proofs in geometry, where the axioms have a clear role. Axioms are more obscure in natural numbers. This is borne out historically. Arithmetic axioms are present in a way in Euclid’s Elements (see also his Data) but were really only fully fleshed out by Giuseppe Peano around 1900. Let’s now narrow down our investigation of priority, thinking only about numbers. Our study exclusively treats natural numbers, those numbers that we use to count. The arguments we use will extend fairly easily to integers, which include zero and negative whole numbers. It takes some work to be able to say exactly what a negative integer is, though analogies with things like debt are familiar. Subsequent, larger classes of numbers include (in order) the rationals, the reals, and the complex numbers. Questions like, “Does x + 4 = 1 have a solution?” or “Does x2 + 1 = 0 have a solution” are only meaningfully answered once we choose which of these levels we want to work at. Sticking with natural numbers is historically fitting, but it also corresponds to the structure of mathematics even when it is given its most contemporary, abstract foundation. Integers, rationals, and real numbers are all assembled from natural numbers, which truly are more “natural” and more fundamental, even when using a set theoretic basis for arithmetic specifically and mathematics generally. We have seen the priority of arithmetic among the disciplines, and of the natural numbers among all numerical things. Let’s finally consider priority in the natural numbers themselves. We are accustomed to saying that the first natural number is 1. You might be surprised to find that accomplished mathematicians in the past said otherwise. Some said that 2 is the first natural number. Others would even say that 3 is the first natural number! How can we make sense of this? Let’s first think about why someone would say 2 is “first.” When we start to count, we begin of course with 1. Suppose, though, that there were no multiplicity at all, that there were only unity. We

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would never begin to count at all. So one way to make sense of the idea that 2 is the “first” natural number is that multiplicity or repetition is the grounds for ever even beginning to count. Seen in that light, it makes more sense to say that 2 is first, in the sense of being a kind of cause. How could we say that 3 is “first?” This is a bit more of a stretch, but we can still make sense of it. When we have only two things, we have mere difference, distinction. When we have three things, on the other hand, we finally have a fully realized multiplicity. Not only do we have distinctness, but we also have distinctness of distinctness (i.e., there are multiple differences, not just one difference). This might appear to go too far; observe, though, that it does not seem so crazy in ordinary language. We say “both of them went to the store” and thereby indicate exactly two people. We say “all of them bought tickets for the baseball game” and thereby indicate how many people? When we say “all of them” we mean a generic multitude. Not at least two—we mean at least three! So in a way even our contemporary English grants a kind of primacy to three.

11.3

Whole

A number is somehow both multiple and one. Mathematics involves actions that we take as well as things that simply are. The actions relate especially to the multiplicity, and the simple being especially to the unity. Emphasis on computation and procedure in early years can make it difficult to see that there is anything other than our actions, anything other than the “pieces” that we apparently “assemble.” To understand the proofs that will come in subsequent chapters, it is helpful to reflect on the “wholes” that lie beneath the surface of our activity. People sometimes use the equation 2+2 = 4 for rhetorical purposes, wishing to give an example of irrefutable mathematical truth while making an analogy to something else that the hearer is to find obvious and irrefutable. The equation is, of course, true; I do not intend to deny the sum. It is not, however, a good example of mathematical truth. The encounter that you and

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your students have had with proof in the first part of this book should have given you a deeper appreciation for the place that explanation plays in mathematics. A bare equation like 2 + 2 = 4, without of any complementary framing speech, is lacking in comparison. Aristotle often uses mathematical analogies in his philosophical writing. He does not, as far as I know, ever use 2 + 2 = 4 (in a suitably ancient formulation of that expression) to make any of the analogies, though. That is a further suggestion that equations like this are not at the heart of mathematical truth. Consider the equation 1421 = 42 + 1379 which also asserts simply that one number is the sum of two others. This is a true sum, but bad for rhetorical purposes. Why? There are a couple of things that make 1421 = 42 + 1379 much less accessible and compelling, to us. The first trouble is that it is written “the wrong way.” The “answer” is on the left, the first thing you read. The “problem” is on the right, the second thing you read. Calling these “answer” and “problem” refers to the customary exercises in arithmetic that fill our early mathematical years. There is another trouble with the equation, beyond the order of the terms. The second problem is the size of the terms. They are too big. We cannot really take them in in a single glance; our only access to them is indirect, by way of computational procedure. The expression 2 + 2 is the name of a thing. It is not really a command, though it serves as one in an elementary school classroom. The statement 2 + 2 = 1 + 3 is just as good, just as true, as 2 + 2 = 4. The same single thing, “fourness,” can be described in a variety of ways. Even as we describe the one thing in these various ways, nothing in it changes. I can call the sun both “the yellow orb” and “the body that heats the earth” and both of these accounts point to the same thing. In the study of arithmetic that we are beginning to undertake, students will need to work with expressions like m + n. They need to understand that this is not a command, it is not a task, it is not a problem. It is a single thing: “the sum of m and n.” Modern axioms for arithmetic might include the notion that “ad-

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dition is commutative,” (or this might be a theorem derived from simpler axioms). Symbolically, this is indicated as m + n = n + m. If we phrase this in words, though, it seems to become tautological. “A thing composed of two parts is composed of those two parts.” More precisely, “a thing composed of a first part and a second part is composed of the second part and the first part.” When we formulate it that way, in words, we see that we are not dealing with something that is specifically mathematical. It applies instead to all being. It is a more general principle. We see, then, that a good way to illustrate mathematical truth is by saying something like “The Pythagorean theorem is a consequence of Euclid’s five postulates” rather than by saying 2 + 2 = 4. The former is properly mathematical, the latter is not. The second one is simply an instance, in mathematics, of a more general principle about things being unchanged even as we call them different names or describe them by different properties. In our study of arithmetic, we assume familiarity with the natural numbers and their sums and products. Building up arithmetic from the axioms called the Peano axioms means that one does not take those things for granted. Instead, one assumes only a very little bit; essentially, the only thing assumed is the successor relation, that each number is followed by another number. It is then possible to define the sum of two numbers, using the successor relation. In that setting, 2 + 2 = 4 becomes a theorem. No one will find it persuasive, though, if you attempt to make an argument by saying “That is as obvious as the fact that 2 + 2 = 4 is a theorem of Peano arithmetic.”

11.4

End

It is good to begin with the end in mind. When we know where we are going, we can hope to get there. The conclusions that must be established in our basic proofs in this chapter are things like “the number is even” or “the number is divisible by 5” or even “the number is divisible by d.” Attention to the definition of the relevant term helps us to reformulate the conclusion in a more concrete manner, a manner that makes proving easier. Showing that a number is “even” means showing that it is twice

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some other (whole) number. A proof of evenness then requires me to hunt down this half. In hunting down the half, I might use various names for the whole, rearranging the names until one of them makes evident exactly what the half is. Students can get confused during these basic arithmetic proofs. They might be unsure about what constitutes a complete argument. One thing that will help students stay oriented is ensuring that they know all of the definitions exactly. The other thing is closely related; they must write down the goal that follows from the definition. The proof is complete when they arrive at the goal. Focusing on the goal is a common reminder in many situations. To sail a boat efficiently, one should pick a point on the horizon and continue to aim at it even as other things come into the field of view. When a motorcyclist faces an obstacle in the road, he should look not at the obstacle but at the path he wants to take, because you go where you are looking. Goals, attention, and mathematics come together in the writings of Simone Weil. Weil was a philosopher and author most active in the 1930s, and she died in 1943 at the young age of 34. Weil was occupied with Marxism and labor movements in her youth before a religious conversion that turned her to mysticism in the final eight years of her life. Her brother, André Weil, was one of the 20th century’s great mathematicians. One of Simone Weil’s short works is called Reflections on the Right Use of School Studies With a View to the Love of God. In this essay, she explains that all our intellectual labor can bear spiritual fruit, since it can teach us attention, and attention is a foundation for true prayer. Weil observes, at the same time, that attention is not physical exertion. Indeed, if we find that our bodies are physically tense during study, there is a good chance that we have not given our full attention. How do we give our attention if it cannot be forced? Weil’s striking recommendation is that we must cultivate something negative, the suspension of thoughts that are too particular, too narrow. The problem, she says, is that we get in our own way with an excessively active mind, when we should instead be waiting for our mind to be penetrated by things. How can we balance this advice with what I said earlier about

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keeping a goal in view? The two things—attention to the goal, and a passive receptivity to things—are two parts of the same intellectual act. The proofs themselves are often short, especially in this chapter. We hold in our minds the beginning and the end, and wait for the middle. As we work towards the goal, we must remain sufficiently sensitive, inwardly, so that we can be aware of the moment in which particular pieces of reasoning cohere together to become a single whole, uniting the hypothesis and the conclusion. This moment of insight might be accompanied by mild physical relaxation or a sigh, and it is not something that we acquire by effort. Instead, it arrives freely, as a gift. The themes raised by Weil are also present in the poems of the spiritual writer St. John of the Cross. John wrote in Spain in the 16th century. His teachings are for those who have advanced far along the way of prayer, but we can still get something from them by which to reason about mathematics and learning by analogy. In The Ascent of Mount Carmel, John writes about how a holy person can discern when to go from more active kinds of prayer to more passive. The answer he gives is simple. We cease to walk when we reach the end of our journey. We can only stop, though, if we have kept the end in mind.

12 Primes and Relative Primality 12.1

Week 12 Plan

Overview: Primes are the building blocks of the natural numbers. They are “first” (i.e., prime) according to multiplication and division. Looking Ahead: The next chapter includes some tangible puzzles to familiarize students with an abstract problem. You can offer physical demonstrations of these puzzles. If this is something you wish to do, plan now and prepare your materials in advance. Notes: 1. If you will be reusing the book, scan the table used for the Sieve of Eratosthenes and make copies. 2. The theorem on the infinitude of primes is demanding. It is important that students correctly distinguish the various levels of reasoning. Here is an outline of the form of reasoning: Given a collection of primes, we produce a number. That number itself is either prime or composite. In either case the original collection of primes must have omitted one prime. 3. The Sieve of Eratosthenes and the Euclidean algorithm for the greatest common divisor are good procedural complements to the abstraction of Theorem 69 (IX.20). Students who find the theorem difficult can enjoy the more practical, step-by-step work of the algorithms. Both activities are important in mathematics.

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Day 1: 1. Read Section 12.1 2. Complete Exercise 1: execute the Sieve of Eratosthenes using the included table. 3. Attempt Exercise 2. 4. Attempt Exercise 3. (Note that a number must have one factor that is not greater than its square root. Why?) Day 2: 1. Reread Section 12.1. 2. Read Section 12.2. 3. Complete Exercises 5 and 6. 4. Observe that in Exercise 5 we look at specific collections, but the proof involves an abstract collection, not a specific one. Day 3: 1. Reread Section 12.2 and discuss the proof of the infinitude of the primes. 2. Read Section 12.3. 3. Complete Exercises 7, 8, and 9. Day 4: 1. Read Sections 12.4 and 12.5. 2. Complete Exercises 11 and 12. Day 5: 1. Execute the Euclidean algorithm using large numbers (with three to five digits each) chosen at random. 2. Review the proof of the infinitude of the primes. Assessment: • Prove Proposition 69 (IX.20), that there are infinitely many primes. • Use the Euclidean algorithm to compute the greatest common divisor of 153 and 391. Show all steps.

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Unending

In 1862, Abraham Lincoln signed the Homestead Act. In 1988, Kenneth Deardorff of Alaska became the last American to acquire his land via homestead. Raised in Los Angeles, Deardorff went north seeking open space. Millions of acres were occupied and farmed in the years between Lincoln and Deardorff. There was much open land in the United States in 1862, and some remains today. However much land we have, though, it is and always was finite. The infinite is a tricky subject, one that has provoked thinkers in every age. Mathematics often drives the philosophical discussion. There are good reasons to be careful when we start talking about the infinite, since it is easy to do so sloppily. We need not shy away, though; it is a subject worthy of further investigation. Before talking about the infinite in general, let us examine the specific notion of the infinite in our proof of the infinitude of the primes. The exact statement given in A Brief Quadrivium is “The collection of all prime numbers is not finite.” Euclid’s formulation in the Elements can be translated fairly strictly as “Prime numbers are more than any assigned multitude of prime numbers.” Euclid’s phrasing appears a bit odd to us, but it gets closer to the heart of the proof. How could we possibly show that prime numbers are infinite? You should take some time to reflect on how challenging this is. Consider this thought experiment. Suppose we had a machine that would immediately tell us whether or not a number is prime. We can then feed in the numbers 2, 3, 4, . . . , 998, 999, . . . for as long as we want, in each case finding whether or not it is a prime. Suppose we do this for a year. What do we know after a year? We will have a long list of numbers that are prime, but we will know nothing about larger numbers. This will still be the case if we work for a decade and not just a single year. No matter how far we keep going in our search for prime numbers, the list that we have will not, of itself, tell us whether or not there are any more to come. The brilliance of Euclid’s proof is to turn things around. We do not try to grasp directly at something that is infinite. Instead, we look at finite things, and show that all of these finite things possess a certain property. More specifically, we show that each “assigned

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multitude of prime numbers” (you can also call it a “finite collection of prime numbers”) must omit some other prime number. The proof tells us how to track down what this “missing” prime number is. The proof is general. It does not show that only one specific collection of primes is incomplete. Instead, it shows that any finite collection of primes is incomplete. Therefore, the collection of all primes must be infinite. The negative assertion, a claim about unlimitedness, arises through positive assertions about finite collections of prime numbers. In Greek “the infinite” is “to apeiron.” The prefix “a” functions just like our “in” at the beginning of “infinite.” It means “not.” The way we get at the infinite is by saying what it is not. Aristotle’s treatment of the infinite reflects this linguistic hint. For Aristotle, the infinite exists potentially rather than actually. This is a point in which some writers find Aristotle at fault. If those critics’ ideas are pressed, though, it seems that Aristotle’s distinctions will stand. Galileo raised a paradoxical point. The square numbers 1, 4, 9, 16, 25, . . . are clearly a part of the natural numbers, but not the whole, since some numbers (3, for instance) are not squares. This would suggest that there are fewer square numbers than natural numbers. On the other hand, every natural number can be squared, so every natural number corresponds to a square number. This would suggest that the squares and the natural numbers have the same size. This paradoxical situation reveals that we need to be careful when we attempt to compare sizes of infinite mathematical things. Georg Cantor would develop a good notion of comparison about a decade after Lincoln signed the Homestead Act. When Kenneth Deardorff, the last homesteader, received his title from the Bureau of Land Management, it was for 80 acres. He had chosen his site so that he would have a lot of room—his nearest neighbors in the 1970s were 50 or 60 miles away, well over the horizon. In a sense, then, his homestead was “infinite.” Practically speaking, it did not have any evident boundary or limit. On the other hand, his homestead is decidedly finite; the government, and perhaps his neighbors, will be sure to remind him of this. While “the infinite” in the abstract has been historically the most contentious point for philosophical discussion, infinitude of some kind is inescapable when we do mathematics. The abstraction of

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mathematical thinking leads us to things that are not fully definite. One simple example comes from arithmetic, in this chapter. Students will soon be able to prove that “The square of an even number is divisible by 4.” What is “an even number” in that statement? It is certainly something definite; a number, not a rhinoceros, and an even number, not an odd one. Yet it is not fully definite. Students err if they think of it as referring to, say, 100, rather than to even numbers generally. Somehow, we are able to think about “an even number” without thinking about “that even number.” Plutarch’s Life of Romulus tells a number of stories about the founder of Rome. When Romulus established the boundary of the city, he yoked a bull and a cow and plowed a furrow, indicating where the wall should be erected. Those who were with him cast the clods of dirt inside the city’s new limit. At some points, Romulus and his neighbors lifted the plow out of the ground, leaving a break in the trench for the future placement of a gate. People will go in and out; even the boundary has limits.

12.3 Memory The proof of the infinitude of the primes is the first hard proof in our study of arithmetic. It can be memorized and understood, but those will take some work. This gives us an opportunity to reflect on the place of memory and how to use it. Ancient authors writing about memory technique (“mnemotechnics”) distinguished between memory for words and memory for things. One fruit of our study of mathematics is that students can experience this distinction in a clear, direct way. If they try simply to memorize immediately all the words of a proof or even a definition, the task is quite difficult. Taking the time to ponder the meaning eventually makes the parts clear. At that point the thing itself, the idea, can be remembered, rather than just the words that put us in touch with it. There is more work up front to get past the words to the thing, but the early effort pays well in the end. One old memory technique involves using a sequence of places within a building. The locations need to be traversed in a specific order—there is a first, a second, a third, and so forth—and they must be spaced apart, so that there is no confusing one for the other.

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Striking images are placed in each of these places that help the memorizer recall the relevant point. Your students might find this memory technique useful for something like giving a speech. Our own language reflects this kind of memory, when we say things like “in the first place.” It might be difficult to memorize a proof using the technique of a “memory palace,” though students are welcome to try. The architectural technique illustrates more general principles of memory that can be useful both for mathematics and other disciplines. We need to keep things in order. We need to keep them distinct. We need to play with the ideas creatively so they make an impression on us. Erasmus of Rotterdam, a great Renaissance humanist, discusses memory techniques briefly in his work On the method of study. He recognizes that technical methods like the “memory palace” can have a place, but he is not particularly enthusiastic about them. Erasmus gives a simple account; the three features essential for memory are understanding, system, and care. Understanding is straightforward as a principle, but in practice it can be easy to shortchange. We must take time to think things through thoroughly. Mathematics presents particular difficulties. When we seek to understand ordinary, non-mathematical writing, we get some sense of the time needed from the volume of text. Some writers are of course denser than others. With mathematics, though, appearances can be deceiving. Even a very short bit of mathematical explanation can take a long time to understand. Until the moment of enlightenment comes, all appears dark. The effort must be made no matter how laborious and protracted the struggle. By “system” Erasmus means that we can recover the whole from a fragment. Recovery like that requires understanding; we must know not only the parts but how they relate, too. We can support the order that understanding imparts by technical means. These are things like numbers, locations, letters of the alphabet. We can connect naturally occurring sequences to sequences in the things we remember, and thereby have a way of moving from one piece to the next. The final point Erasmus identifies to strengthen our memories is care. Let us consider an objection. There are some students who, in their words, “don’t care.” Their use of “care” regards sentiment:

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how they feel. This is not Erasmus’s care. Instead, he would simply have us read—repeatedly, slowly, and with attention—those things that we choose to remember. This puts care within the reach of us all, whether or not we take an immediate liking to our subject.

12.4

Algorithm

While many of our terms (theorem, lemma, arithmetic) are of Greek extraction, “algorithm” is different. It comes from Uzbekistan. Abu Ja’far Muhammad ibn Musa Al-Khwarizmi wrote his Compendium on Calculation by Completion and Balancing around the year 800 AD. The name “Al-Khwarizmi” suggests that he came from Khwarizm, the city in modern-day Uzbekistan now known as Khiva. When his name was transliterated in Latin it became algorismus, and mixing with the Greek arithmos led to English’s hybrid “algorithm.” The word “algebra” comes from the Arabic al-jabr (roughly, “completion”) in the title of Al-Khwarizmi’s book. The Euclidean algorithm certainly goes back a thousand years before Al-Khwarizmi, as far as Euclid, and perhaps earlier. Aristotle refers to a method of “reciprocal subtraction” that seems to indicate something like this technique. The way that the algorithm appears in Euclid’s Elements is slightly different than the presentation in A Brief Quadrivium. The differences deserve a brief exposition. First of all, Euclid never sets out the algorithm as instructions, or as a series of steps to follow. He refers merely to “the less continually being subtracted in turn from the greater” in the statement of a proposition. What he means by those words is revealed in the subsequent proof, where Euclid illustrates how this repeated subtraction is iterated. It is through understanding the proof that we extract a general procedure from Euclid’s text. A second difference is that Euclid divides his treatment of the algorithm into two cases. The first case supposes that two numbers are relatively prime, and concludes that after repeated subtraction (i.e., the execution of the algorithm) a unit will be left. The second case supposes that the two numbers are not relatively prime, and concludes that after repeated subtraction the greatest common divisor will be left. This division reflects the fact that Euclid treats the unit separately, so that the number 2 can be thought as the first

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properly numerical “number,” a notion discussed earlier. It would make little sense, from this perspective, to speak of the unit “dividing” a multitude, since it is only by measurement with the unit that it becomes possible to speak of the multitude. Euclid’s proof has an interesting mixture of definiteness and indefiniteness that seems calibrated to teach most clearly. On the one hand, he treats the algorithm in a general way that applies to an arbitrary pair of numbers. On the other hand, in order to prove that the algorithm really yields the greatest common divisor, he illustrates this by considering a somewhat specific computation. He supposes that the algorithm terminates after three steps, the least number of steps in which the full nature of the algorithm can be shown. Euclid demonstrates that the divisibility passes through each of these specific steps. There is definiteness in the number of iterations, and indefiniteness in the numbers chosen. I have chosen a definite example for A Brief Quadrivium because I believe it will be more intelligible to the typical reader, though I consider the Euclidean approach quite attractive as well. Regarding algorithms more generally: A Brief Quadrivium emphasizes proofs and explanations over computational methods. Part of this emphasis is due to the nature of the mathematical material itself, and part is due to the strengths and weaknesses of the age in which we live and teach. We certainly can and should praise those who have succeeded in developing remarkable methods of computation, even if those methods will not be objects of our current study. Many great mathematicians have been interested in computational techniques and practical applications. Al-Khwarizmi’s algebra was written to help with managing things like the division of inheritances and the digging of canals. Blaise Pascal was proving novel theorems of geometry at the age of 16 and later wrote his philosophical Pensées; he developed a mechanical calculator to assist his father with tax collection. In the 20th century, John von Neumann was a child prodigy who made jokes in Greek at the age of 6, published mathematical research at the age of 19, and at 30 joined Albert Einstein as one of the first professors at Princeton’s Institute for Advanced Study. Von Neumann’s capacity for abstraction—the mathematics that underlies quantum mechanics, for instance—did

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not prevent him from contributing to the very concrete engineering challenges involved in developing the first electronic computers. Euclid did well to unite his algorithm with a proof of its correctness. We should neither be lost in the clouds of abstraction nor so fixated on procedure that we lose sight of anything else. In December 1994, Intel recalled its Pentium processor because of a subtle computational error. The cost to the company was roughly $475 million. Thomas Nicely, a professor of mathematics, discovered the flaw. He was investigating prime numbers.

13 Linear Diophantine Equations 13.1

Week 13 Plan

Overview: This week explores how relative primality and greatest common divisors can be principles for other kinds of mathematical knowledge. Looking Ahead: The next chapter revisits the terms of Chapter 10 and uses them demonstratively. Take some time this week to look back at those definitions. Notes: 1. The puzzles in this chapter will be fun for some students. They are not “realistic.” The mathematics involved, though, is genuinely important in applications whose scope is beyond our current study. 2. If you look up linear Diophantine equations elsewhere, you will probably see a couple of differences. One is the use of variables. The other is the use of the = symbol. There is no mathematical difference in the results. The way things are written here is for the sake of clarity and explicitness, and to help young students develop good habits of mathematical speech. 3. The notion of “primality” mentioned in Section 13.3 is quite general. It can be used even with more complicated mathematical things like polynomials.

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Day 1: 1. Read Section 13.1. 2. Have students try the puzzles in the section independently. 3. Observe the marginal note in 13.1.1: the intention of these puzzles is not to come up with a strange way of physically manipulating the cups. The imagined instruments are to be used simply. The puzzle consists entirely in combining them. 4. Try to create new puzzles. Day 2: 1. Reread Section 13.1. 2. Read the beginning of Section 13.2, stopping before the section labeled “Sufficiency” (subsection 13.2.1). 3. Complete Exercise 2. Day 3: 1. Reread the first part of Section 13.2, and complete the section by reading about “Sufficiency.” 2. Complete Exercises 3, 4, and 5. Day 4: 1. Read Section 13.3. 2. Mimic Exercise 5 with other puzzles. 3. Complete Exercise 6. Day 5: 1. Review the entire chapter. 2. Practice using the Euclidean algorithm in reverse. 3. Complete Exercises 7 and 8. (Exercise 8 is a bit harder.) 4. Complete Exercise 1.

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Assessment: • Define “linear Diophantine equation.” • Determine whether a given linear Diophantine equation has a solution or not, and justify your answer. (Pick one used in the book, or create one yourself.) • Find a solution to a linear Diophantine equation, either by a direct search or by the Euclidean algorithm in reverse. • (Extra Credit) Use the Euclidean algorithm in reverse to find a solution to this problem: express 1 as an integral linear combination of 289 and 304.

13.2 Unknowns Algebra is not a part of the quadrivium. The linear Diophantine equations we study in this chapter bring us close to algebra, but we deliberately hold back from passing fully into that frame, which is a later development. This essay sheds some light on what your students deal with when they study algebra. Algebra also likely played a big role in shaping your own perception of mathematics. When we speak of the “unknown” or the “variable” in mathematics, we use the word in a different way than in other kinds of speech. Numbers such as two and three do not change. They cannot pass from one to the other. A piece of clay can be at one time spherical and then reshaped in the form of a cube. The change in that case happens in the clay, not in sphericity or cube-ness, each of which is separate from the clay. Similarly, there is no mathematical thing that is a potentially-but-not-yet-number that later becomes number. The clay, in contrast, is at one time formless, and then an artist makes it look like a giraffe. When we consider an expression like x2 , what is meant by x? It is hard to say. The best answer is that it is a placeholder for a number. When we are given some number of objects, we can consider a square array of objects, having the given number of objects on each side. That is a general possibility; it is not restricted to any definite natural number. As a result, we talk about the generic expression x2 and not simply specific squares like 22 and 32 .

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Now consider an equation such as x2 + 10 = 6x + 2. This might make you nervous if you were uncomfortable with algebra. You need not worry. We will proceed very simply. The equation happens to have two “solutions.” One of them is 2, and the other is 4. Check that these are in fact solutions. This means that when we understand x to refer to the number 2, the statement (that the left side is the same as the right side) is true, and similarly with the number 4. The signification of x in the equation alone is, however, indefinite. Nothing in the equation itself tells us that x would refer to either 2 or 4, though the equation does eliminate other possibilities. To call our equation an “equation” is somewhat complicated. The two sides are unequal if we suppose x to refer to the number 3. Compare that equation to a different situation. The mathematical expression ( x + 1)2 = x2 + 2x + 1 also uses the equal sign. This assertion holds, though, no matter what value x is understood to take. It imposes no constraint on the reference of x, and it tells us something about the way that addition and multiplication are related to one another. This use of the symbol = can be called an “identity.” The unknowns used in equations like the one above involve a puzzle, and the problem for the student is to find a solution to the puzzle. Unknowns, or variables, also come up in a different way in mathematics, when we are interested not in finding but instead in defining. More specifically, we use variables when we want to define functions. Functions will come up later in more depth; for now, a function is a rule assigning a unique output to each input. The inputs and outputs are ordinarily numbers. If I want to define the function that assigns to a given input the square of that input, it is nice to avoid using words. We give a name to the function, often f , and say that f ( x ) = x2 . Note that in f ( x ) = x2 there is no constraint being imposed on x. Instead, this “equation” is really giving a definition. We are saying: “when we say f we mean the squaring function.” By convention, writing the name of the function f and then a number (or symbolic expression) in parentheses right next to it means “the output of f corresponding to the input contained in the parentheses.” So f ( a + 3b) is a way of saying a2 + 6ab + 9b, which is just another

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version of ( a + 3b)2 . Algebra and functions came into use after the quadrivium was established. Euclid does have what we might call a symbolic variable, though we should be careful to distinguish his from the fullfledged variables of algebra. Old manuscripts of the Elements use segments to denote numbers in things like the Euclidean algorithm, and modern editions of Euclid follow this pattern. The advantage of segments, rather than collections of dots, is that it provides a way to indicate something definite (a specific length) but not too definite (an exact number of dots).

13.3 Problems In an earlier essay, I explained that A Brief Quadrivium asks students to complete “exercises” rather than “problems.” The word “problem” slips into our language in this chapter, when we say that a linear Diophantine equation is a kind of problem, so this essay takes up the theme again. In ordinary life “problems” are things we hope to dismiss expeditiously. Life on earth is called a vale of tears, which reminds us that these problems will keep popping up, like the moles whacked in the arcade game. In the quotidian sense, then, problems are temporal things. They come into being, and we hope that our efforts make them pass away. The “problems” that Diophantus and his successors have investigated are unchanging, because they are concerned with numbers. These problems are simply features of numbered things, clear in themselves but sometimes obscure to us due to our intellectual limits. The change that comes about while we wrestle with and ultimately solve these problems is in our minds, not in something outside of us. When we investigate mathematical problems, we can use tangible things to make the problem more familiar; we talk of measuring out water using cups, for instance. At this point, teachers can face a practical problem, in the non-mathematical sense of the word. Students sometimes fail to see the situation as a (real-life) problem. Things appear too contrived, and students cannot imagine dealing with such problems. If we ordinarily have ready access to a vari-

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ety of measuring devices in the kitchen, for example, it is difficult to see how the puzzle about measuring water poses any genuine challenge. There are a couple of ways to address this sort of objection, if it is posed in good faith. One is to imagine playing a game. In a game, we deliberately constrain ourselves in a way that corresponds to no real need. We play by the rules, we enter imaginatively into the frame delineated by the game. Mathematics can be approached in this way too. There is something artificial about it; it is game-like. Another way to think about the meaning of the artificial problems of mathematics is to consider other areas, like sports and musical performance, in which young people often strive for excellence. In those domains we often must go through certain techniques— footwork drills for the running back, scales for the pianist—that are not themselves the real activity. The preparatory drills are artificial and do not directly correspond to the real thing. The coach or teacher, though, knows that these rote and dusty drills prepare their charges for the improvisation that exhibits mastery. The word “problem” directly corresponds to a Greek word, “problema,” that occurs in many settings. One use of of the term occurs in a passage of Plato’s Republic we have already discussed, in Book VII, where Socrates discusses education in mathematics. Socrates says that there is a danger for those who study mathematics deeply: they might never rise to consider “problems.” What he means by problems here is something more fundamental than mathematics; instead of talking about relationships between numbers, the scholar should instead go further and ponder how it is that numbers have properties and relationships at all. Socrates says the student must rise higher. This is something we see in miniature in our study of linear Diophantine equations. The particular “problems” of measuring water or weighing an object have a common generalization, the linear Diophantine equation. In order to understand both of those more tangible situations mathematically, we must strip away the irrelevant details. Doing this gives us insight into the real nature of things. Plato would seem to say that we must do this again, at a higher level, not with particular mathematical problems but with problems and numbers themselves.

14 Numbers in Themselves, Revisited 14.1

Week 14 Plan

Overview: This chapter examines patterns that occur in special numbers, and proves that these patterns really do persist in all the special numbers that are examined, not just the smaller ones we happen to look at. The tool for proving is called “mathematical induction.” Looking Ahead: Next week is the final week of arithmetic. Plan now for the unit assessment, and remind students to begin preparing now. Notes 1. To become comfortable with using mathematical induction, students must learn by doing. The first section, which explains the general principle, will be much clearer after it is used in specific cases. 2. This chapter presents the “formulas” in an older way, using words rather than symbols. You might see something like n ( n + 1) 2 if you look in other sources (that is the result for triangular numbers). I avoid that because students get confused about what n means and treat it as fixed. We use the term “the side number” (i.e., number of objects on a side) which clearly applies to an arbitrary triangular configuration. 1+2+...+n =

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Day 1: 1. Read Section 14.1 and 14.2. 2. Complete Exercises 2 and 3. 3. Exercise 3 is related to the formula given in the second note for this chapter. Day 2: 1. Reread Section 14.2. 2. Read Sections 14.3 and 14.4. 3. Complete Exercises 4 and 8. 4. Attempt Exercise 1. Day 3: 1. Reread Section 14.3 and 14.4. 2. Read Section 14.5. 3. The proof of Theorem 86 (IX.36) is more difficult. Be sure to give it time. 4. Euclid’s theorem was the state of the art in perfect numbers for millennia. He showed that if a certain number is prime, then a related number (which is necessarily even) is perfect. Euler extended Euclid’s result by showing that all even perfect numbers are of the form given by Euclid (i.e., there are no other even perfect numbers). At the time of writing, no one knows if there are any odd perfect numbers, and no one knows if there are infinitely many perfect numbers. 5. The special primes used here are called Mersenne primes. Day 4: 1. Reread Section 14.5. 2. Students should strive to understand the proof of Theorem 86 fully, and check whether they can give it independently from memory.

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3. Complete Exercises 12–14. Day 5: 1. Review the entire chapter. 2. Complete Exercises 5–7 and 15–19. 3. Attempt Exercises 9, 10, 11, 20, and 21. Assessment: • Give a brief, general description of how proof by mathematical induction works. • Prove one of Propositions 83 and 84. • (optional) Prove Theorem 86 (IX.36) on the generation of perfect numbers.

14.2 Beginning Mathematical induction, the method we use to give proofs of the various patterns we started to see in Chapter 10, is not a part of the original sources of the quadrivium. It fits well with both the original mathematics as well as more general logical principles, so it makes a worthwhile supplement. To go from a more intuitive sense of number patterns to properly demonstrative knowledge, we must enunciate a clear principle. This principle is what we call mathematical induction. It is a kind of axiom. (Whether it is a single axiom or something more complicated depends on the exact foundation given. Technically speaking, it depends on whether you can “quantify over predicates.”) In order to use mathematical induction, we need a statement that depends on a natural number. An example is “the sum of the natural numbers from 1 to n is . . . .” This can also be thought of as an infinite collection of definite statements: “the sum 1 + 2 is . . . ,” “the sum 1 + 2 + 3 is . . . ,” etc. Applying mathematical induction means doing two things. One of these is general: we show that when the statement is true for one natural number, it is also true of the successor. The other thing is specific: we show that the statement holds for some small natural number, usually 1.

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Why is mathematical induction a good way to reason? Suppose that our statement depending on a natural number were false for at least one natural number, and maybe more. There might be many instances in which it is false, but these can be put in order, since they are natural numbers. In this collection of natural numbers, one of them must be the smallest, call it N. Since it is the smallest point at which the statement fails, we know that the statement is true for the number N − 1. Supposing, though, that the truth of the statement for one number implies that of the successor, we see that the statement must also be true for N, since that is the successor of N − 1. This is a contradiction. Thus, there must not have been some least number at which the statement failed, and so there must be no number at which the statement fails. It is useful to compare reasoning by mathematical induction with some statements that Aristotle makes in his logical works. We must keep in mind that the word “induction” is being used in a different way by Aristotle. Still, we can compare his statements to our experience of proving using mathematical induction. In his Prior Analytics, Aristotle says that induction involves enumerating all of the cases. This seems to be something like the way that we can figure out the pattern that we wish to show with mathematical induction. We look at many particular cases, and with enough of them in hand we hope to discern something that is common to all of them. In practice, we do not really look at “all” of the cases (in numbers, of course, that would take forever). We just look at enough to get the idea. There seems to be room in Aristotle’s notion of induction for this circumscription, too. Enumerating the cases need not be an exhaustive and fully explicit listing. In his Posterior Analytics, Aristotle talks about induction in a different way. He explains how we arrive at the principles that give us expertise in any area, not just mathematics. We begin with bodily sensation, we collect the sensations in memory, the memories are coordinated to yield experience, and experience finally leads to skill (in practical matters) or understanding (in theoretical matters). This whole sequence proceeds, at each stage, from greater concreteness and multiplicity towards greater abstraction and unity. When seeking patterns among numbers, by comparison, we begin by looking simply at specific numbers, and perhaps nothing appears to unite

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the cases. Over time we come to see these many instances by way of a single light. Deduction, rather than induction, is Aristotle’s main concern in both the works just cited. His remarks on the strength and the limits of deduction illustrate the way that we prove using mathematical induction, reminding us of the need for two parts. There are two kinds of extreme views about first principles that Aristotle rejects while giving his theory of deduction. One of these views is that nothing is ever deduced. On this account, any principles we ever set down will themselves need to be deduced, and this will go on forever, meaning that nothing is ever deduced. The other extreme view once again emphasizes deduction, but this time in the other direction. Proponents of this perspective claim that everything must be deduced, and that everything we know is indeed deduced, i.e., we can even reason circularly. Aristotle tackles both objections by affirming that deduction is not the only way that we know. We arrive at first principles from which we will deduce, but this movement is not itself deductive. There is something of this kind of distinction in proof by mathematical induction. While we must show that “true for n implies true for n + 1” (the analogue of deduction), this is not enough. We cannot remain at the abstract level. We must also examine a particular starting number, and show that the statement holds in that case. That step is not deductive, it is a straightforward computation. Since mathematical induction is not in Euclid, Boethius, and the like, when did it arise? It seems that Francesco Maurolico gave the first proof by mathematical induction in 1575. One of the things he showed with his new technique is that summing odd numbers yields square numbers, as in 1 + 3 + 5 + 7 = 16 = (4)2 . About 75 years later Blaise Pascal would also use mathematical induction; it is possible that he got the idea from Maurolico.

14.3

Formula

The relations that we prove using mathematical induction might be called “formulas.” You have seen the word “formula” in mathematical and scientific settings, in things like the “quadratic formula.” The word “formula” is attested in English in the late 16th cen-

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tury, and its first meaning is: set words used perhaps ceremonially by which something is defined or declared. The word seems to have acquired a pejorative sense a bit later through Thomas Carlyle, who used it to mean a rule applied mindlessly. Carlyle had excelled in mathematics and studied for a time under John Playfair, whose eponymous axiom was mentioned in an essay for Chapter 5. The word is genuinely Latin. It is the diminutive of forma, the source of many words like “form” and “formal.” Ovid uses the word in a poem that takes the form of a letter from Acontius, a young man, to Cydippe, a maiden. Nec mihi credideris—recitetur formula pacti; Neu falsam dicas esse, fac ipsa legat!1

Acontius had played a trick upon Cydippe, the beautiful girl he sought to win. While she sat in the temple of Artemis, Acontius threw a golden apple upon which he had written some words. Cydippe, there in the temple, picked up the apple and mindlessly read aloud the text, which stated “I swear by Artemis that I will marry Acontius.” She had thereby bound herself to him by a solemn vow in the holy place of the goddess. Acontius’ job was not finished, though. Cydippe was reluctant to marry him, and instead was betrothed a number of times to other suitors. Each time illness prevented the nuptials. In the passage cited above, Acontius addresses one of his competitors. He says “Don’t believe me, have her repeat the formula that she said.” In other words, he says “It was her, not me.” Later in the poem, Acontius presses the matter with Cydippe herself. He observes that even if she is happy to ignore the “vow” for now, and marry another man, she will one day be in labor pangs, and at that time will need to call upon the goddess. If she makes a new vow to Artemis while in labor, why should the patroness of childbirth trust her, if Cydippe has already been false to an earlier promise? First consider “formula” simply as a word. It does a couple of things for the poet. As a dactyl, it fills the meter in a way that “forma” would not. And being a diminutive, it captures a humor1 Do

not believe me. Let the vow’s formula be recited. Do not say it is false. Make her read it.

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ous aspect of Acontius’s claim. “She said just this little thing.” Now let us step back from the words to the the things they name. Cydippe resists Acontius, at least for a time, because she denies that she could give her assent accidentally. The words we speak need to correspond to reality in order for them to be true. Acontius’s trick is an amusing one, and might ultimately win Cydippe’s affection, but the words he deceived her into saying were not yet really her words. This story, dealing with youthful passion, is far from mathematics, and yet it provides an excellent image of mathematical “formulas” and the challenges that they present to the learner. Before the two sides of the formula are united as one, they must be known independently, just as Acontius and Cydippe must each separately assent to a union to make it real. One problem that plagues students of mathematics is that they play the part of Acontius. They hope that the mere utterance of words and symbols in a certain form makes the thing true. The symbol that stands for “equality” becomes twisted into a procedure, a tool that “makes” one side into the other. I attempt to address this difficulty in A Brief Quadrivium by emphasizing that the two things united by a “formula,” the left-hand and right-hand sides of an equation, have independent intelligibility. The formula does not have us “do” something to one side to make it into the other. Instead, each side of the formula is its own entity, and our proof requires us to show that they are indeed the same. In the end Cydippe did marry Acontius—but only after her father had given his consent.

15 Relations Between Numbers 15.1

Week 15 Plan

Overview: This week describes how to classify various ratios of numbers, and shows how all of these ratios arise in an orderly way from a single principle. Looking Ahead: Be sure that you have prepared your monochord for next week. Be sure that astronomical observations are underway, or at least planned. (They are in Chapter 22.) Next week is light, and you can use some of that time for logistical matters. Notes: 1. The terms like “superparticular” that are introduced here are present in the old sources, but not in common mathematical use now. They are useful when discussing musical ratios. 2. Figure 15.1 can be compared to the final propositions of Euclid’s Elements, and specifically to Proposition XIII.18. There, Euclid presents in a single figure the side lengths of various regular polyhedra (the Platonic solids) when they are inscribed in a sphere of a fixed radius. 3. Figure 15.1 gives, informally, a pair of inequalities. The arithmetic mean is greater than the geometric mean, and the geometric mean is greater than the harmonic mean. These inequalities can be proven formally.

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4. Exercises 8, 9, 11, and 12 can be assigned later in the week as time allows. Day 1: 1. Read Section 15.1 and 15.2. 2. Complete Exercises 1, 3, and 4. Day 2: 1. Reread Sections 15.1 and 15.2. 2. Complete Exercises 2, 5, 6, and 7. Day 3: 1. Read Section 15.3 from 15.3.1 through 15.3.3 (inclusive). 2. Complete Exercises 14 and 15. Day 4: 1. Reread Section 15.3.1–15.3.3. 2. Read Section 15.3.4. 3. Complete Exercises 16, 17, and 20. Day 5: 1. Read Section 15.4. 2. Review the rest of the chapter. 3. Complete Exercises 10, 13, 18, and 19. Assessment: • Complete Exercise 4 (or the equivalent, with a different collection of numbers). • Complete Exercise 15 (or the equivalent, with a different collection of numbers).

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15.2

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Names

A meter is just a bit longer than a yard. How did that happen? In 1791 the French Academy of Sciences recommended that a new unit of length be adopted. This length, the meter, was defined to correspond to the earth in a natural way. Take the distance from the north pole to the equator, and divide it by ten million. That is a meter. This definition of a meter leads to an important, more general point about mathematics. Sometimes we can define a thing but cannot really get our hands on it. I can say, for example, “the smallest prime greater than 21000 .” This phrase does define a number, but it does not really tell me anything about it. The same thing happened with the meter. In order to know what a meter is, we need to measure the earth. While Eratosthenes had already given a good estimate of this more than two thousand years ago, more accurate measurements required demanding, multi-year expeditions in the early 19th century. The definition of the meter arises from something natural. The yard and the foot do as well, since they use human dimensions. Even plants have a role to play. One old definition of the inch is as “three barleycorn.” The barleycorn is even still in use today; it is the basis of shoe sizes in the United States and the United Kingdom. An increase in one shoe size corresponds to an increase by a third of an inch. When we say how long something is, we do so by choosing a unit. Our units have many names: “yard,” “meter,” “metre,” “barleycorn.” Rational numbers (fractions) also involve the choice of a unit. The word “denominator” contains within itself the root for “name.” The denominator is that by which the thing is named. No matter how “natural” our choice of unit, it is still a choice. A ratio is something more fundamental, an intelligible relationship between quantities that requires no choice on our part. This is what makes the arithmetic of fractions somewhat tricky when we first learn it. Why can we add fractions when denominators are the same, but not when numerators are the same? This is not an arbitrary rule. Thinking along the lines of “name,” we see that different “denominators” mean that different things are being named.

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Ratios themselves can be given names. This is one feature of the current chapter, and it is the most obscure part of quadrivial arithmetic. You will not encounter names like “superparticular” in a modern mathematics text, unless perhaps when dealing with nuances of musical tunings (we will see superparticular ratios again in our study of music). One reason that it is helpful to include this funny, forgotten terminology is that it directs us to think carefully about what ratios are. Here is one way to say what a superparticular ratio is: “a ratio that comes from consecutive natural numbers once it is given in least terms.” Note that this is not the definition given in the text. Describing superparticular ratios by consecutive numbers is helpful as an alternative that helps us wrap our minds around superparticularity, but it is not the best definition. By talking about “least terms” we force something accidental, something involving human activity, into the definition. It is more elegant, mathematically speaking, to avoid all such things. A definition like this—“the difference divides the lesser term”—gives the most general statement, the one that captures the essence. Our work on arithmetic follows the Roman Boethius, who was himself following Nicomachus, from Roman Syria (now Jordan). In addition to his mathematical writings, Boethius also wrote on philosophy and theology. One of his works was on the Christian doctrine of the Trinity, and Thomas Aquinas wrote a commentary on this work. Boethius and Aquinas both go beyond strictly theological themes, discussing the ways that people know and disciplines are divided. At one point in his commentary, Aquinas examines the traditional division of the seven liberal arts. He addresses the question of why these “sciences” (ways of knowing) are also “arts” (productive disciplines). Thomas says that these each involve not only knowledge; they also involve work, examples of which include producing a speech (rhetoric), composing a melody (music), and counting/numbering (arithmetic). That last one is the most important for us now. Aquinas is observing that numbering things is a human work, it is not simply “seeing” a number that is already there, just as when we write an essay we see some thing that we wish to convey while also striving to assemble words in a creative and artful

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manner. The “denomination” of a ratio really is a naming, and not simply a recognition. Names and necessity, meters and philosophy come up again in the 20th century. In 1970 Saul Kripke gave a series of lectures that led to his 1980 book Naming and Necessity. The work examines the nature of names, and does so in part by looking at how we talk about “possible worlds.” He gives an example with Aristotle and Plato. We might say that “Aristotle” refers to “Plato’s best student.” But we can imagine a world in which the man named Aristotle did not study with Plato, though he was still Aristotle. So whatever “Aristotle” means, it cannot be simply convertible with the description “Plato’s best student.” Kripke looks at the meter, too. At its first definition, the meter came from the circumference of the earth. After careful terrestrial measurements, scientists produced a rod that was exactly one meter long, which became the new standard. Kripke makes an interesting observation. He notes that even though “meter” means “the length of this rod,” we can still imagine a world in which “that” rod is shorter or longer than a meter (i.e., we can mentally apply our system of measurement to a variant possible world). Somehow, what we mean by meter is not exactly the rod, even when it is defined by way of the rod. We can imagine a world where the meter is shorter than a meter. Things kept going after the standard meter rod. As light became an absolute standard for velocity in the early 20th century, the meter 1 was redefined yet again, as the distance traveled by light in 299792452 second. This gives us a new problem; what is a second? We’ll return to the question of time in our study of astronomy, but the modern answer is of interest here. A second is defined as the time required for 9192631770 vibrations (that is a non-technical description) of a cesium atom. This reduces it all to arithmetic.

15.3

Adrastus

In the small Michigan city where my father was born, you can find a Euclid Street. There is a Euclid Street in the capital of the United States, too. Cincinnati does not have a Euclid Street. It does, though, have a Euclid Avenue.

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Ptolemy Avenues seem to be less common. As the Mississippi nears the Gulf of Mexico, it makes some sharp bends in New Orleans. On the right bank, the southern bank, not too far from the river, lies a short street called Ptolemy Street. Names like Euclid and Ptolemy are still relatively familiar to us, despite their strange spellings, thousands of years after those men lived. Other thinkers also contributed to the quadrivium, perhaps in smaller ways. This essay introduces a couple of those figures. While their names are not widely known, these authors are not anonymous. A thinker named Theon wrote a book called Mathematics Useful for the Understanding of Plato around 100 AD, almost 500 years after Plato himself wrote. Theon’s book is not a direct commentary on any of Plato’s works. Instead, it compiles general mathematical knowledge, following the quadrivial division of disciplines. One general point from Plato that Theon discusses is that the Greek word logos has multiple senses. Logos can mean an unspoken mental act, expressed reasons, an explanation of the elements of the universe, and mathematical ratio. We have seen three of these four thus far in our study of the quadrivium. The pre-verbal assent that we give to the reasonability of things like a Euclidean geometric postulate is a kind of “unspoken mental act.” The proofs that we have given, in geometry and in arithmetic, are “expressed reasons.” Ratio itself was our study in Chapters 7, 8, and now 15. In the next two parts of the quadrivium, music and astronomy, we will encounter the fourth sense of logos, as we investigate the order that lies within things we hear and see. It is through Theon’s book that we learn that Adrastus discovered the rule that is presented here under his name. There are a number of figures named Adrastus in the ancient world; our Adrastus is the one from Aphrodisias. The name Aphrodisias is itself ambiguous, though it seems that the one relevant to our discussion is in modern day Turkey. We do not know much about Adrastus. Theon was interested in him for his mathematical work in relation to Plato; centuries later, Simplicius quoted Adrastus when commenting on the works of Aristotle. Simplicius, who wrote around 550 AD, seems to be the first author to make the notion of “saving the phenomena” explicit.

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Coarsely speaking, this means that the goal of scientific activity is to give accurate models more that it is to say what things are. Adrastus does not seem to have any streets named for him. I have nowhere found a Theon Avenue, but a bit north of Austin you will find a Theon, Texas. Theon himself came from Smyrna, now called Izmir, in Turkey, though not Turkey, Texas. With his rule, Adrastus showed that the apparent multiplicity of all kinds of ratios could be explained through a single, originating principle. His discovery marks the close of our course on arithmetic. As we end the study of this first of sciences, we can recall the words in the Book of Revelation addressed to Theon’s hometown: “These are the words of the first and the last.”

Part III

Music

Aperiam in psalterio propositionem meam.

16 Sound 16.1

Week 16 Plan

Overview: This week involves listening to sounds and beginning to describe their relations mathematically. It involves two chapters, so part of the week’s work is given in the next chapter. Looking Ahead: Chapter 16 (regarding sounds) is analogous to Chapter 22 (regarding the sight of the sun, moon, and stars). Be sure that astronomical observations are underway. Notes: 1. The concepts of Chapter 16 are not difficult. Still, it is good to spend a bit of time bringing them to mind. 2. Rhythm is one aspect of sound that we omit here. It is about how sounds go together. We are working with more elemental, individual sounds. Day 1: 1. Read Chapter 16. 2. Complete Exercises 1–3. 3. Exercise 3 will take repetition. Some drops might seem to make no sound. Others might make a more audible sound. Students should take time exploring the range of sounds that are produced, and paying careful attention to the specific character of the sounds.

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4. Read Exercises 4 and 5, and complete them over the course of the next day. The rest of Week 16 is in the next chapter.

16.2 Rhythm At a young age, babies can already distinguish voices. Even penguins recognize each other this way. The simple sounds that we produce using our vocal chords are immensely rich. The first chapter of our study of music introduces volume, pitch, and timbre. Volume requires little explanation. Timbre is complex, and goes a great part of the way in explaining how it is we identify our friends and family through vocal sound. Pitch, finally, is the topic that will most concern us. We will investigate the intelligibility of pitch, of high and low sounds and their relations with each other. Rhythm does not appear in the list of aspects of sound given here. It makes an important contribution to music, but we will omit it. Setting rhythm apart is somewhat like a grammatical difference, the distinction between letters and syllables. When we learn to read and write, we first learn letters. Letters are the most basic elements of written words. When we speak and listen, though, we utter and hear not mere letters, but syllables. The word “consonant” itself reveals this, since these letters do not occur alone but instead “sound with” a vowel. Our pure pitches are like letters, while timed pitches—which are necessarily rhythmic, and the only ones we can make in practice—are like syllables. Our focus on pitch reflects most of the classical musical sources. One significant exception is Augustine’s De musica. Shortly after the conversion he famously describes in his Confessions, Augustine retired to seclusion. He wished to write a series of texts on the liberal arts. He did not complete this task, but did write a work on music, which has a finished quality, even though he intended to extend its scope. Augustine’s De musica consists of six “books” (i.e., parts). The first five books focus on rhythm and meter in poetry. Augustine explores how ratios of times (long and short) go together to constitute the smallest bits of poetry, and how these small parts are assem-

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bled into larger elements (lines, couplets). In the sixth and final book, Augustine rises from more technical, metrical considerations to philosophical questions about such topics as memory, time, and beauty. The first five books involve specific technical analyses of Latin poetry. They are, perhaps, less accessible to a casual reader than the sixth book, which concerns larger questions. Taliaferro, who translated Augustine’s work on music to English in the middle part of the 20th century, makes this remark: It is usually dangerous procedure to ignore the technical details a thinker uses to test or suggest his general and more seductive theories. It is too easy to overlook the first five books and to concentrate on the sixth.1

Taliaferro’s comment reflects the order of learning that I recommend for students of the quadrivium. It is important that students first learn the technical details of geometry, arithmetic, music, and astronomy. When those subjects have been grasped, students are then capable of encountering and understanding the philosophical uses that ancient and medieval authors made of these disciplines.

1 Augustine,

“De musica,” trans. Robert Catesby Taliaferro, in The Fathers of the Church, vol. 4. Washington, DC: The Catholic University of America Press, 1947.

17 The Monochord 17.1

Week 16 Plan, Continued

Overview: This chapter introduces the use of the monochord to investigate the relationship between musical consonance and mathematical ratio. Looking Ahead: The procedure we use here for determining specific lengths of the monochord string will be treated more formally and generally in Chapter 20. Notes: 1. Procedure 17.2 relies on ratio. Recall that we have specific technical familiarity with ratio from Chapters 7 and 8, and not a merely vague and procedural familiarity. 2. Depending on the size of your monochord, you might need to get large sheets of paper. You can also tape small sheets together to make a larger one. 3. It is best to avoid making permanent marks on the monochord, which would be like frets on a guitar. Instead, the monochord should be like a violin or cello. Have your students make marks lightly, with pencil or chalk. 4. Students should be able to identify the octave, the fifth, and the fourth by ear, by the end of the week. For some students with prior musical training, this will be easy. Others will require focused practice.

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Day 2: 1. Read Sections 17.1–17.3. 2. Practice the division procedure of Section 17.2. 3. Mark the half length of the monochord, and listen to various octaves. Adjust the string tension so that students hear both higher notes and lower notes, playing the octave each time. Day 3: 1. Reread Sections 17.1–17.3. 2. Read Section 17.4. 3. Pay careful attention to the way that pitches and intervals are depicted using numbers. The numbers always refer implicitly to “parts of the whole string.” 4. Complete Exercises 1 and 3. Day 4: 1. Reread Section 17.4. 2. Complete Exercises 4 and 5. Day 5: 1. Review the Chapter. 2. Students can freely play notes on the monochord, measure, and make conjectures. 3. (optional) Complete Exercise 2. Assessment: • Divide a segment into seven equal parts, using a compass and straightedge. • State the ratios that correspond to the octave, the fifth, and the fourth. • Identify an octave, a fifth, and a fourth by ear.

the monochord

17.2

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Endurance

This essay offers brief word of encouragement as you reach the halfway point in A Brief Quadrivium. Persevere in the good work that you and your students have begun. First, let us consider where we are. Half of the course has been spent on “pure” mathematics—geometry and arithmetic. Some of the ideas involved tangible things—compass and straightedge, water poured from cups—but tangible things were not at the heart of what we did. Now, as we turn to “applied” mathematics, we involve ourselves a bit more in physical things. Dealing more with physical things involves some challenges. You need to prepare a monochord for students to complete this part on music, and will also need to use larger pieces of paper, and perhaps find a longer ruler. When stated in words, these are minor hurdles. Yet they are real hurdles, I do not wish to discount them. Keep moving forward, even if getting things planned and organized takes a bit of work. We will see the same thing with the astronomical observations. It is much easier to read something once in a book than it is to go outside, at night, repeatedly, over a long period of time. If you have made it this far, be confident that ordinary effort and diligence will pay off. Do not shortchange or ignore the preparatory tasks, and do not let practical challenges slow you excessively. Second, let us look more closely at where we have been. Examining our progress can give confidence and offer an incentive to keep going. Just fifteen weeks ago, we were not yet doing proofs. Now, only a few months later, your students are able confidently to give proofs of substantial mathematical statements. This shows real mathematical sophistication. Some of the proofs your students have learned are high points of scientific history: the infinitude of the primes, the Pythagorean theorem, the generation of perfect numbers. In other instances, your students are able to give proofs of propositions that they have never seen before; in other words, they are able to do genuinely creative mathematical work. Finally, let us take a look at where we are going. We have begun our study of music, which centers on the notion that pleasant combinations of sounds correspond to ratios of natural numbers. We

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will conclude our study of music by examining Gregorian chant, and learning to classify these chants according to the basic pitches that structure the pieces. With music done, we will move on to astronomy. In that final quadrivial discipline, we will see how basic principles of geometry allow us to calculate details about the motions of the sun and the moon.

17.3

2001

Stanley Kubrick’s 2001: A Space Odyssey opens with a striking image. The darkness of outer space is suddenly pierced by a brilliant ray, as the sun rises from behind the earth. When the sun’s light appears, trumpets play a dramatic, rising series of notes. Once the sun is fully revealed, the film’s perspective passes from space to the earth. The sun rises in a dry, rocky, occasionally mountainous landscape with scattered, low vegetation. The ensuing terrestrial segment, titled “The Dawn of Man,” shows ape-like creatures dwelling alongside tapirs, which look like primitive pigs. The apes fight for their lives, avoiding the predation of a leopard, scuffling over edible grasses with the tapirs, and contesting the water and territory with other apes. Then the apes sleep. At dawn they wake to find a strange object in their presence. A large, dark, rectangular slab, a “monolith,” stands upright among the rocks. The apes gather around it, cautiously exploring the mysterious thing as the new day begins. The soundtrack at this point offers the haunting Kyrie of György Ligeti’s Requiem. Mournful voices flit about in every direction like leaves in the wind. We see, perhaps from the perspective of an ape, the sun and crescent moon hanging together over the monolith. The apes go about their day. One is foraging near the skeleton of a long-dead tapir. We return briefly to the image of the sun and moon aligned with the monolith. The ape seems to ponder for a moment, and then picks up a bone and begins to use it as a tool, a club. At this moment, in which the proto-human discovers tool use, the same music from the very beginning of the film plays again: three triumphant, rising notes from the trumpet. The three rising notes that mark high points—the revelation of the sun’s light, the invention of the tool—involve the most funda-

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mental musical intervals. The first interval is a fifth, and the second is a fourth, so that the whole span of the three notes is exactly an octave. The smaller intervals correspond to the ratios 3 : 2 and 4 : 3, and the whole octave is the double ratio, 2 : 1. Both the sights and the sounds of Kubrick’s masterpiece offer images by which we can understand and even relive the ancient Pythagorean discovery that consonant sounds correspond to natural number ratios. That discovery led the ancient sect to proclaim that “All is number.” The absolute sphericity of the heavenly bodies and the perfect planarity of the rectangular monolith are a jarring contrast to the coarse desert landscapes and hairy, mortal animal bodies. Ligeti’s complex layers of wavering vocal sound are far removed from the large, bright intervals played by a single brass instrument. Pythagoras and his school found something clean, pure, simple, and rational amidst the great multiplicity and messiness of our sensory impressions. The ideas are clean and pure, but the tools might not be. The apes soon use their clubs to drive another band away from a small pond. They kill one of their adversaries. A victor joyfully tosses his bone club in the air. We see the club rise, flip, and begin to fall. Kubrick then cuts to the sight of a spaceship orbiting the earth. The ship’s proportions are the same as those of the bone. Isaac Newton would one day see that the moon falls just like the lowly objects of earth. A waltz plays. The repeated theme of three rising notes, an interval of a fifth followed by an interval of a fourth, comes from Richard Strauss’s Also Sprach Zarathustra, whose title and inspiration come from the novel with the same name by Friedrich Nietzsche. The waltz that accompanies the orbiting spacecraft is by a different Strauss, Johann Strauss II, and it is his well-known The Blue Danube. Here too we see something of the contrast between raw matter and rational consonance. The Zarathustra of Nietzsche’s novel dwells alone for a decade in the mountains, returning to human society to proclaim the death of God and the coming of the Übermensch. The waltz, in contrast, involves the careful, elegant, synchronous movements of couples, all coordinated in the motion of the larger group. After studying the mathematical character of the fifth, the fourth,

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and the octave, consider showing your students the first twenty minutes of 2001: A Space Odyssey. As you look at the pure lines of the monolith in the dusty scrabble of the desert, and as you hear the pure tones of the trumpet arrayed from low to high, bring to mind once again the intelligible order that so struck Pythagoras.

18 The Tone 18.1

Week 17 Plan

Overview: This week involves a study of how the octave breaks up into elemental parts, and accustoms students to careful description of musical intervals using ratio. It covers two chapters. Looking Ahead: It is not necessary to make hundreds of copies of a segment in order to produce a semitone. Chapter 20 will explain how tones and semitones can be produced efficiently using ratio. For now, it is fine to go by ear (compare with a musical instrument if you wish) while paying careful attention to the mathematical details. Notes: 1. It would be good to take a look back at Chapter 7. Review how we made careful arguments about ratio. 2. We need to know both what it means to say that ratios are the same, and what it means to say that one is greater than the other. Day 1: 1. Read Sections 18.1 and 18.2. 2. Students should patiently redo the computations of the textbook, by hand (no calculator). The volume of material is deliberately light. There is plenty of time to carry out the computations, which involve only basic multiplication. It is not enough that

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students “see” how it is done; this “seeing” must be habituated through action. 3. Complete Exercises 1, 2, 3, and 4. Day 2: 1. Reread Sections 18.1 and 18.2. 2. Read Section 18.3. 3. Complete Exercises 5, 6, and 7. The rest of Week 17 is in the next chapter.

18.2 Canon The words “canon” and “canonical” are rich with musical and mathematical meanings. The purpose of this essay is to explore them and show their relations. Euclid is famous for his Elements, but he wrote other books as well. An earlier essay briefly mentioned his Data, which examines the ways in which things in mathematical statements can be “given.” Euclid also wrote on optics. Finally, relevant to the present discussion, there is a work that has been attributed to Euclid called The Division of the Canon. It is possible, even likely, that Euclid himself did not write this musical book. Still, it offers a reasonable representation of the music theory that he would have studied. The title of the work directs us to a first sense of the word “canon.” The ancient author used the Greek kanon to refer to a measuring rod. Greek also admits the word in a metaphorical sense, as a standard or rule. The “division of a canon” in the musical sense means producing the marks by which a string can play suitable notes, those that belong to the chosen method of tuning. Chapter 20 of A Brief Quadrivium could be titled “The Division of the Canon.” A second sense of the word “canon” is curricular, and arises from the metaphorical sense of a canon as a standard. Canonical texts are those that are essential, central, enduring, representative. They are “the great books.” The use of quotation marks in the preceding

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sentence is not any sign of skepticism. I mean simply to capture a collection of words that are often uttered together, yielding a term. Canonicity has been the subject of numerous educational disputes. Is there a canon at all? Which books belong? Which are excluded? How variable is this canon over time? One specific sense of textual “canonicity” is for religious writings, as when we speak of canonical Scriptures. A third sense of canon, adjectivally in the word “canonical,” arises in contemporary mathematics. It is often the case that there is more than one choice for a mathematical object in a given setting. Even though there are many options, one of them might be most natural. This most natural, most immediate choice is called the “canonical” one. Before mathematical examples, here is one from ordinary life. If you are at a specific point in Washington, DC, there are a number of ways that you can specify your location. You can say “I am 1 mile north-northeast of the Air and Space Museum” or “I am four blocks west of the westernmost point of Rock Creek Park” or something like that. The most natural thing to do, though, is often to say something like “I’m at 5th and H St.” Using the nearest intersection is a “canonical” (in a loose sense) way of giving your location. The streets in Washington are, at least for a while, in a perfect grid, with lettered streets running east and west and numbered streets running north and south. The alphabet and the numbers both start at the US Capitol. Giving a location like “5th and H” is effectively like saying “I’m at the point (5, 8)” in the Cartesian plane. (The quadrant—NE, NW, SE, SW—must also be specified, since we do not use negative street numbers or a “negative alphabet.”) The numbers and letters are just like coordinates. In the Cartesian plane example, the first coordinate and the second coordinate are “canonical coordinates” for points in the plane. Here are more sophisticated examples from mathematics. Recall from the essay on Categories that there is a map from a vector space to the dual space of its dual space. This map is said to be “canonical.” While there are many different linear maps between the two vector spaces, the “canonical” one is the natural one that comes “for free.” Another example comes from mathematics used in physics. It is often useful to given an account of the way that

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a particle or group of particles behave using their positions and momenta. Formally, this involves something called a “cotangent bundle” or “symplectic manifold.” In this setting, the choice of coordinates to describe the position of the particle immediately yields a natural account of momentum as well. These momentum coordinates are “conjugate” or “canonical.” Let us conclude by drawing together these senses of canonicity to see how they relate to our study of music. The relevance of the first is clear, since we are “dividing the canon” just as contemporaries of Euclid would have done. There are a couple of ways in which the second sense of canon applies. First, looking at the broader outlines of A Brief Quadrivium, we presume that Euclid, Ptolemy, and Boethius are canonical authors, that they deserve an enduring place in mathematical and liberal education. More specifically, within music, we treat the Pythagorean tuning as canonical. There are many ways to divide an octave into smaller intervals using ratio. Our study examines only one, the Pythagorean one, giving it a privileged place owing to its simplicity. Finally, what about the canonicity of mathematical naturalness? The connection here is more remote, but there are a couple of points of connection. There is something of the “canonical” in the fact that we divide the tetrachord in a symmetrical manner: tone, semitone, tone. While the asymmetrical divisions require some further specifying information (where is the semitone?) the symmetrical division is, in that single word, completely determined. The monochord itself, as a physical instrument, has something of the naturally canonical in it too. It is physically possible to make a huge instrument, or a tiny one, to make an instrument that plays notes that are extremely high or extremely low. There is, despite this, a kind of natural scale; a useful monochord will tend to be of a size that a single person can easily work with it, and will play notes that are approximately in the range of the human voice. Some standards do change over time. Setting aside both lackadaisical slackness and unrealizable dreams of absolute rigidity, we can tune—and retune—the string.

19 Approximation 19.1

Week 17 Plan, Continued

Overview: This chapter explains how ratios that involve large numbers can be understood through ratios of smaller numbers. It also explores some ratios that cannot be realized by numbers. Looking Ahead: In Chapter 21, in a couple of weeks, we will explore Gregorian chant. Take a bit of time now to find a good online resource where you can listen to these chants. Notes: 1. It is good to connect what we do now to ratio as studied in Chapter 7. Then, when we found specific numbers used for repeating things like segments, we were comparing the ratios of segments to ratios of natural numbers. Now we make similar comparisons, but all of the entities involved are natural numbers, rather than things like segments and plane regions. We specifically look for “small” numbers (what that means depends on context). 2. Be sure that students carry out the computations detailed in the text. They do not need to do anything creative, and it might seem to be enough that they understand how the computations are done. Still, they should do them. Repetition will help to build the habit of thinking about ratio, pitch, and string length in an orderly way.

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3. The remarks in Exercise 4 about avoiding unnecessary computation can apply to other computations that students do as well. An example of the utility of this way of proceeding comes in the proof of Proposition 95. Day 3: 1. Read Sections 19.1 and 19.2. 2. Be sure to carry out the computations given in the book. Students must do them by hand. 3. Complete Exercises 1, 2, 3, and 4. Day 4: 1. Reread Sections 19.1 and 19.2. 2. Read Sections 19.3 and 19.4. 3. Complete Exercises 5, 7, 8, and 9. Day 5: 1. Read Section 19.5. 2. Review the rest of Chapter 19. 3. Complete Exercise 10. 4. (optional) Complete Exercise 11. Assessment: • Describe mathematically the relationship between two semitones and one tone. • (optional) Show that there is no interval composed of (possibly many) fifths that is exactly equal to an interval of (possibly many) octaves.

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19.2

135

Rational

It takes reason, rationality, to say “the quotient of the circumference of a circle and its diameter is an irrational number.” (That number is π.) And on the other hand, as Chesterton has it, to go mad does not mean losing your reason—it means losing everything but your reason. Rationality and irrationality are near relatives, then, whose genealogy is a bit of a puzzle. Let us begin strictly within the domain of the rational. One feature of Chapter 19 stands out as somewhat unusual. We talk about the rational approximation of rational numbers. An example of this is when we say that the semitone, corresponding to the ratio 256 : 243, lies between the ratios 19 : 18 and 20 : 19. If we already have the ratio as a ratio of numbers, why bother approximating by different ratios of numbers? One reason for this is that while all numbers are exact, fully specified, and thus “reasonable,” only fairly small numbers are fully reasonable to us, due to the limits of our minds. We can talk about a thing being “reasonable in itself” or “reasonable to us” and these two are not necessarily the same. This is at the heart of theodicy. See Boethius and his Consolation of Philosophy. Even quantities that are “irrational” can be approximated using rational ones. Students got a taste of this in Chapter 7. They were asked to show that the ratio of the diagonal of a square to the side of the square is larger than the ratio 7 : 5. In the end, the careful definition of “real numbers” involves just this sort of approximation: a real number is identified with the collection of all of its rational approximations. The Euclidean algorithm for the greatest common divisor turns out to be a source for rational approximations. It is easiest to understand this through an example. Consider the ratio 81 : 64, the Pythagorean major third (it is close to the purer third 5 : 4). Use the Euclidean algorithm to compute the greatest common divisor of the two numbers. (We know it is 1, since 64 is a power of 2 and 81 is odd, but still carry out the procedure.) 81 − 1 × 64 = 17 64 − 3 × 17 = 13

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17 − 1 × 13 = 4 13 − 3 × 4 = 1 4−4×1 = 0 Now take the number of times that the small number went into the larger number in each case; this is the sequence 1, 3, 1, 3, 4. Omitting some details (it is a good exercise to work them out for yourself), that sequence gives us the collection of fractions below. 1+ 1+

1+

1+

1 3

1 3+

1 1

1 3+

1 1+ 13

1 3+

1 1+

1 3+ 1 4

An expression like this, with fractions inside denominators, is called a continued fraction. Let us look at the series of fractions, written in a more standard 4 form. Each of them is an approximation of 81 64 . The first one is 3 . 5 This is the ratio of a fourth. The next one simplifies to 4 , the “pure” major third. Putting these first two approximations together, this is saying that our ratio ( 81 64 ) is close to the pure third, and slightly larger since it is also well-approximated by the (larger) fourth. The 19 . Finally, if next ratio, with a bit of arithmetic, shows itself to be 15 you are persistent, you will see that the final expression is just a convoluted way of writing 81 64 . In other words, the last approximation of the thing is the thing itself. While we only used the Euclidean algorithm for natural numbers, in which case the procedure always terminates in a finite number of steps, it is also applicable to irrational quantities, in which case the approximations keep going indefinitely. Perhaps the most appealing of all of these is the continued fraction for the golden ratio.

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1+

137

1 1+

1 1+

1+

1+

1

1

1+

1 1 1+...

It turns out that the rational approximations that arise through this procedure are the best rational approximations. What is meant by “best?” Suppose that we want to produce a rational approximation of a real number. Reasoning intuitively, we get the sense that we can get “as close as we want” by using really large numbers. The rational approximations we get from the Euclidean algorithm in fact get close to the true result without “cheating” (i.e., using an especially large numerator and denominator relative to the standard of accuracy). Geometry provides us with another good example of the in√ terplay of rationality and irrationality. While 2 is not a rational number, it is still the sort of thing we can get at in Euclidean geometry. It is the ratio of the diagonal of a square and the side. Since we √ can construct squares with compass and straightedge, we see 2 as “rational” even though it is not a ratio of natural numbers. By “rational” we now are meaning the more general “intelligible relative to some definitely posited thing.” Euclid himself enlarges the scope of the term like this, in his notoriously intricate Book X. Though the square root of 2 can be known by way of compass and straightedge, it turns out that the cube root of 2 cannot be known in this way. The cube root of 2 can be thought of very concretely. Given a cube, with a specific side length, find the length of the side of the cube that is exactly twice as large (in volume) as the first cube. This was sometimes called the “Delian problem,” as it legendarily arose in Delos. The Delphic oracle said that the people of Delos needed to double the size of Apollo’s altar. His altar was cubical. Plato explained that the oracle was in fact admonishing them to dedicate themselves to mathematics. Ancient mathematicians explored the Delian problem by introducing tools beyond the compass and straightedge, including auxiliary curves. In his Geometry, Descartes explores the ways that some things can be known when certain curves are taken as given. In particular, Descartes shows how to extract a cube root if a parabola (one of the so-called “conic sections”) is assumed. Descartes’s ge-

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ometry contributed to the rapid development of algebra, and it was through algebraic means that compass-and-straightedge solutions to the Delian problem were eventually shown to be impossible. Descartes freely and briskly assumes complicated mathematical objects (plane curves, in his case) as given, and reasons relative to that hypothesis. This kind of “relativization” has an enduring place in mathematics, from Euclid’s Book X until the present day. A final example comes from the theoretical study of computation. In the 1930s, Alan Turing developed a universal theoretical computer. These “Turing machines” are mostly for abstract purposes, for thinking about computers, although they can also be built in real life. They are sufficiently general that they cover every reasonable meaning of “computer” that you can come up with. Consider the following two “computer programs.” Program 1: Find a prime number bigger than 101000000 . Program 2: Find a pair of twin primes with each prime bigger than 101000000 . Each of those programs can be made into a formal program suitable for a Turing machine. Let us consider what our study of arithmetic tells us about the first one. Is there a prime bigger than 101000000 ? Yes, because the primes keep going indefinitely. That means that Program 1 will eventually come to a stop, it will succeed in finding a big prime number (after a very long time). We know, from looking at the “code” for Program 1, that it will definitely stop. Program 2 is a different story. That program might find a pair of twin primes. We do not know that it ever will, though. (Recall that the infinitude of twin primes is a conjecture, not a theorem.) At the time of writing, the largest known pair of twin primes are smaller than that. What if that pair is the last pair? Then Program 2 is doomed to fail in its search, and will run on endlessly, hopelessly. We now rise to a new level of abstraction and ask this question: is there a computer program that can “look at” another computer program and determine if it will ever stop? This is called the “Halting Problem.” The question is not whether some particular program that someone has already written down can make this determination; the question is whether such a program could ever be writ-

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ten down. The answer turns out to be no, and it was Turing who showed this. Mathematicians are not content with such a dead end. Instead, they develop a new line of thought. Suppose that we did have a solution to the Halting Problem? What could we do then? They explore the difficulty of other problems relative to the solution of the Halting Problem. With that solution in hand, is some other problem now solvable? This sort of thinking leads to a hierarchy of so-called “Turing degrees” of relative difficulty. These are vertiginous heights of abstraction; let us descend. In one of Chesterton’s mystery stories, The Hammer of God, Fr. Brown makes this remark: I think there is something rather dangerous about standing on these high places even to pray. Heights were made to be looked at, not to be looked from. . . . Humility is the mother of giants. One sees great things from the valley; only small things from the peak.

20 The Diatonic Genus 20.1

Week 18 Plan

Overview: Last week, we considered fundamental musical intervals through number; this week, we examine how to realize those numerical ratios practically, using geometry, in order to divide a real monochord string. Looking Ahead: Three weeks remain until the start of the final quadrivial discipline, astronomy. Wrap up any observations that are incomplete. If your students made their observations a while ago, start to track down the notebooks in which they recorded what they saw. Notes: 1. The division given by the universal tetrachord is a minimal solution. It is possible (see Exercises 5 and 6) to produce a more structured solution. The reason to give a minimal solution is that it highlights the essential ideas and omits redundancy. 2. Gregorian chant moves within a range larger than a single octave. As a result, it is useful in Section 20.3 to start with only a part of the monochord string considered as “whole,” leaving room for additional notes below. 3. Students who are attentive to detail might particularly enjoy Exercises 8 and 9. As the marginal note indicates, these exercises reveal the power of our reasoning, which is based on ratio and

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not simply counting. There is time to work on these and other exercises on Days 4 and 5. Day 1: 1. Read Section 20.1 and Section 20.2.1. 2. Attempt to carry out the instructions for producing a universal tetrachord. 3. Be sure that the work is completed neatly. Day 2: 1. Reread Section 20.1 and Section 20.2.1. 2. Read Section 20.2. 3. Produce a universal tetrachord again. 4. Carry out the two tasks of Section 20.2.2 with the student’s monochord string length. Day 3: 1. Reread through Section 20.2.2. 2. Read Section 20.3. 3. Carry out the procedure of Section 20.3. Day 4: 1. Reread Chapter 20. 2. Attempt to produce a divided octave on a bare line completely from memory, using only a compass and straightedge. Day 5: 1. Review the whole chapter. 2. Complete any of the exercises. Assessment: • Produce a universal tetrachord. • Explain how to use a universal tetrachord to divide a monochord string.

the diatonic genus

20.2

143

Scales

There are 150 Psalms. Within the whole collection, fifteen are called “songs of ascents.” We can find some symbolism in these numbers. Fifteen is a tenth part of one hundred fifty, and we have ten fingers, or digits. On one hand, we have five fingers, and fifteen is the triangular number whose side is five. Augustine plays such number games when commenting on Sacred Scripture. He also explains why a psalm would be called “of ascents,” a canticum graduum in the Latin of the Vulgate. In his Expositions on the Psalms, while introducing Psalm 120, the first song of ascents, Augustine says that the “grades” or “degrees” of ascent refer to the steps by which we rise to understand spiritual things. Let us return briefly to our current, concrete musical task. Given a string, we wish to mark out certain distances so that we can produce sounds that go well together. In the end, we arrive at an octave divided in tones and semitones. This octave is related to, but somewhat different than, the octave of what is called the “major scale.” The major scale is familiar to you, even if the name is not. It is what you get when you sing “do re mi fa sol la ti do” or when you play the white notes on a piano starting at middle C and moving upwards. The name makes clear that a “scale” involves direction; mountaineers scale a peak. It is not just a collection of notes without order; a scale is a collection of notes seen in sequence leading upward. If you sing just “do re me fa sol la” and then stop, it will feel incomplete. This incompleteness is in part due to custom; we are used to hearing the whole scale. The custom is not arbitrary, though. There really is something natural about going through the whole octave, which corresponds numerically to the first multiple ratio. Augustine and many other thinkers have spoken about a natural directedness in the liberal arts, and of the quadrivium specifically. In one of Augustine’s early works, De ordine (“on order”), he inquires into the right order of learning. The way he comes to the question is worth noting. Augustine is lying awake at night when he hears a change in the sound of the water flowing through a nearby stream. He talks with a friend about this, and the friend proposes that leaves have accumulated at a narrowing, temporar-

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ily blocking and then releasing the water’s flow. Augustine and company then move on to a more general discussion; how is it that things that seem random, like the falling of leaves into a stream, happen according to reasonable causes? There is not room here to review the whole discussion. It leads ultimately to a consideration of divine providence. Why do people who wish to have children have none, while others have children they do not want? Why is someone who would be generous left in penury, while a stingy man accumulates vast wealth? Augustine says that those who have been trained to see reason even in obscure matters—like the irregular sounds of a brook—will be disposed to recognize the possibility of a deeper order within things that are, at first glance, disordered. Augustine goes on to review the trivium, the arts of grammar, dialectic (or logic), and rhetoric. These are important, he says, but they are not enough. With the study of the trivium, the soul begins to yearn for “divine things,” but it does not go directly. Instead, he says, it takes a way of steps. The Latin here is “gradus,” just as in the Psalm title. The steps that are to follow come from the structure of our senses. Augustine observes that two of the senses—hearing and sight—are particularly suited to conveying reasonability to us. These senses correspond to two of the quadrivial arts, music and astronomy, each of which also has a “pure” aspect, arithmetic and geometry, respectively. The structure of study that he proposes arises, then, from the basic ways that humans find intelligibility through the senses. Centuries later, a different thinker would also investigate the way that specific technical studies relate to the pursuit of wisdom. Bonaventure wrote two works on this theme: Itinerarium Mentis ad Deum (The mind’s journey to God) and De Reductio Artium ad Theologiam (Retracing the arts to theology). Bonaventure’s Itinerarium introduces a six-fold division of learning. We can consider external things, inward things, and things above us. Each of these three is divided further into two parts: as God is seen through them, or as God is seen in them. Bonaventure discusses mathematics in the second and third of the six steps. The second, God seen in external things, involves recognizing the divine

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presence in the orderliness, the “numerosity,” of external things. In the third step, God seen through inward things, Bonaventure uses the relationship of mathematical arts to human knoweldge as a whole as a way to explain the Blessed Trinity. Bonaventure’s Reductio also presents mathematics in a Trinitarian light. Bonaventure was roughly contemporary with Thomas Aquinas. In his commentary on Boethius’s De Trinitate, mentioned already in an earlier essay, Aquinas joins Bonaventure and Augustine in affirming a seven-fold division of liberal arts and, like them, suggests that these arts are paths by which the mind is led to consider higher things. Aquinas cites Hugh of St. Victor, whose Didascalion, written over a century earlier, reflected and contributed to the medieval sense of the division and order of human knowledge. Having ascended through all the notes of a scale, we arrive at a note that is both the same as the first and different from it. Let us conclude by returning to the songs of ascents. Augustine and other writers gave “gradus” a spiritual or metaphorical sense. It also has a literal sense. It seems that these psalms were recited by pilgrims as they approached Jerusalem, ascending from the surrounding valleys. The third psalm of ascents begins “I was glad when they said to me: we will go into the house of the Lord.” This text has been used for coronations of British monarchs since the 17th century. One notable setting is by Hubert Parry, composed for the coronation of Edward VII in 1902. British coronations involve ecclesiastical and civil politics, things that are hardly trivial. Parry’s I was glad deserves your listening. Finally, set aside kings, queens, rule, and strife. Listen to Parry’s setting (“Repton”) of Whittier’s Dear Lord and Father of Mankind. “O Sabbath rest by Galilee! O calm of hills above.”

20.3

Universality

Mathematicians prize elegance. At times, that elegance approaches the brevity that is said to be the soul of wit. At other times, an elegant route is more roundabout. The longer path often involves the search for some kind of universality. The “universal tetrachord” of this chapter is called “universal”

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because it solves all problems of dividing a monochord string at once. The solution is independent of any specific monochord parameter. Once you have a universal tetrachord, you can divide any monochord, big or small. These sorts of universal solutions pervade mathematics, and are of interest in some of the discipline’s most technically nuanced areas. To explain why, let us start with something concrete, the Möbius band. You have likely seen one before. To make one, take a strip of paper and tape the two ends together after introducing a single twist. (Without a twist you would make a short, fat cylinder.) The Möbius band presents an interesting feature. While it is possible in a small region to talk about “this side” and “that side” of the piece of paper, this is not true of the shape as a whole. By continually moving along “this side” you arrive, without any tunneling or any dramatic leap, at “the other side.” A cylinder and a Möbius band have a common description. Each is a collection of lines that are “parameterized by” a circle. Understanding this concretely involves marking your strip of paper. Start again with a long, narrow strip. Draw a single line (in red, if you have color) along the middle of the strip in the long direction. Do this on both sides of the paper. Then draw many other lines (in blue or black) that are perpendicular to this, that run in the narrow direction. Make these on both sides, too. Now take your marked strip and consider it in two ways. First attach the two ends to each other without any twist. You will see that the central (red) line now makes a circle. There is a short blue/black line “attached to” each point on the circle. Now consider the Möbius band configuration, in which you attach the two ends with a single twist. Once again you see that the central (red) line makes a circle, and each point on this circle has a line attached to it. We see from these constructions with the marked paper that both the cylinder and the Möbius band are “families of lines parameterized by a circle.” Another way to say this is that each is a “line bundle over a circle.” Lines and circles are very simple. Lines are the simplest instances of “vector spaces” (see the earlier essay on Categories) and circles are simple shapes that can be replaced by spheres, donuts, and even higher-dimensional shapes that are difficult to visualize. In the greatest generality, mathematicians are

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interested in “vector bundles over topological spaces.” This route is starting to get long; what is the purpose? There are many different kinds of topological spaces, and over any given space there are, in general, many different kinds of vector bundles. (We already saw that the circle has two different kinds of line bundles.) Despite this great multiplicity, a dramatic unification lies open to us. It turns out that there is a very complicated space, a “classifying space.” Over that complicated space there is something called a “universal bundle.” It is a complicated vector bundle. Why is the bundle called “universal?” It turns out that every vector bundle, over every topological space, “comes from” the universal one. More specifically, each vector bundle over each base space can be understood to correspond to a relationship between the given base space and the classifying space. In the analogy with our universal tetrachord, the line AC is the classifying space, and every time we pick a point in the classifying space we get a specific tetrachord. This is a lot of abstraction, but it is for an important purpose. The key is that many things have been reduced to one thing. At first it looks like vector bundles are wildly various. Ultimately, they are all descendants of this single, universal one. The great takeaway from this is that mathematical properties of the universal bundle become, suitably reinterpreted, properties of every vector bundle. The more technical term here is “characteristic class.” The “cohomology” of the classifying space yields “invariants of vector bundles” over arbitrary bases. One mathematician who made use of characteristic classes is Friedrich Hirzebruch. Hirzebruch, the son of a math teacher, studied in his native Germany and in Switzerland immediately after the Second World War. After briefer stints in the 1950s at the Institute for Advanced Study and Princeton University, Hirzebruch returned to Germany, to Bonn, where he remained for the rest of his career. One of Hirzebruch’s important results from the 1950s is called the Signature Theorem. We can only sketch it imprecisely. Suppose you have a special kind of space, called a “manifold.” Suppose further that this manifold is “oriented” (i.e., not like the Möbius band). Then there is a special number called the “signature” of the manifold. It is just an ordinary natural number. On the other hand, a

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manifold also comes along with a special kind of vector bundle, its “tangent bundle.” That vector bundle leads to special characteristic classes (called “Pontryagin classes”) that ultimately lead to a number. In a conjunction that mathematicians find remarkable, it turns out that the number that comes from those Pontryagin classes is the same as the signature of the manifold. This is the Hirzebruch Signature Theorem. Hirzebruch was more than a pure mathematician. He led the renewal of mathematics in Germany in the years after World War II, inviting and hosting researchers from many nations to regular Arbeitstagungen and ultimately founding the Max Planck Institute for Mathematics in Bonn in 1980. Those administrative duties must sometimes have come at some cost to his own research, yet Hirzebruch had an outstanding mathematical career personally at the same time as he was building up a world-leading institution. This universality has a particular elegance.

21 Gregorian Modes 21.1

Week 19 Plan

Overview: This is the first of two weeks devoted to understanding the basic order of a body of music, using the mathematical foundation we have developed. Looking Ahead: Seek out additional chant scores for students with special talent and interests. They might wish for more next week. Consider finding images of old illuminated manuscripts, and perhaps letting students attempt to produce a copy next week. This could be coordinated with an art class. You should also prepare for the music unit assessment. Notes: 1. Use ecclesiastical Latin pronunciation when saying and singing the words. 2. Students should play each piece slowly on the monochord, and also try to sing them. Find recordings so that students can also listen, and if possible find someone who can sing some of the chants for the students. Day 1: 1. Read from the beginning of the chapter through Section 21.4, stopping before Mode I. 2. Complete Exercises 1, 2, 3, 4, and 5.

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Day 2: 1. Reread from the beginning of the chapter through Section 21.4. 2. Study Modes I and II. Day 3: 1. Study Modes III and IV. Day 4: 1. Study Modes V and VI. Day 5: 1. Study Modes VII and VIII. Assessment: • Given a staff, a clef, and a mode, mark the tonic and dominant. • Describe the difference between odd and even modes.

21.2 Week 20 Plan Overview: This is the final week of the study of music, in which students go deeper into Gregorian modality. Looking Ahead: Be sure that students are ready to share their astronomical observations next week. They will need to dig out journals or notes that they made. Your specific observational schedule will have varied, but the first prompting in the text came at Week 8. Notes: 1. Supplemental exercises are practically infinite. Find any chant— they are freely available—and analyze it in the way taught in this chapter. Day 1: 1. Read Sections 21.5 and 21.6. 2. Review Modes I–IV.

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3. Complete Exercises 7–9. Day 2: 1. Read Section 21.7. 2. Review Modes V–VIII. 3. Complete Exercises 10 and 11. Day 3: 1. Complete Exercises 12, 13, and 14. Day 4: 1. Complete Exercises 15, 16, and 17. Day 5: 1. Review the chapter. 2. Listen to, play, and sing the chants. Assessment: • Ask students to discuss the modality of a chant from the text, or find one that they have not previously seen.

21.3 Life Gregorian chant arose after the original quadrivial disciplines were firmly established. Despite this asynchrony, chant belongs to a contemporary presentation of the quadrivium. One reason it goes well is that Gregorian chant developed as the classical quadrivium was being assimilated in the Christian West. Another reason is that we can still hope to hear it. Chant is not simply a now-outdated relic lingering in the foundation of Western music; it is a living tradition that is sung today. It is alive. Teaching math and science with a historical perspective can lead to a challenge: the sense of pretending or role-play might become dominant. “If I had been X I would have thought Y.” The vitality of chant mitigates this problem. Even now we can go into churches (or,

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if need be, concert halls) to hear Gregorian chant. As we listen, we experience beautiful melodic features. Some of those features can be understood. The basic study of music that we are now completing gives us a mathematical language by which to speak of them. Your students should sing the chants of this chapter and not merely listen to them. You and they might sing poorly. This is okay. Let’s see why. In his study of leadership and knowledge work, The Effective Executive, Peter Drucker examines how executives can help organizations succeed through good management and direction of personel who engage chiefly in intellectual activity. Drucker notes that people are usually very good in, at most, one area. There is of course the old phrase “jack of all trades, master of none.” Drucker takes this further, inquiring into the needs and abilities of people and modern corporations and how they interact. Drucker brings up the great German poet Johann Wolfgang von Goethe. Goethe rose to prominence with The Sorrows of Young Werther and later produced his Faust, perhaps the greatest work of German literature. His mind was immensely fertile and productive; in addition to his literary works Goethe also wrote on botany and color theory, and he even participated in the government of Weimar and its surroundings. In the face of such excellence, what does Drucker say? He observes that if we did not have Goethe’s poetry, we would not care about the rest. Even such a great genius was not able to be a world leader in multiple, widely different areas. Drucker is concerned with running a good company, with producing. In that light, activities like Goethe’s botany and our imperfect singing can seem peripheral, amateurish, dilettantish, superfluous. There is another side to the story, though. Granting that there would be no memory of Goethe’s color theory without Goethe’s poetry, it is also the case that there would likely have been none of Goethe’s poetry if there had been no Goethe the scientist. The man who made the poetry came to be through everything that he did. We would not have had him by any other route. The Victorian artist and art critic John Ruskin makes some helpful remarks along these lines, emphasizing the significance of producing art in order to appreciate art. Ruskin had a good education

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in his youth through frequent travel, culminating in high honors at Oxford. While his earlier works are on art and architecture, Ruskin later was concerned with social matters. He wrote Unto This Last, a work of economics that influenced Mahatma Gandhi. He also supported the Working Men’s College, which aimed to provide adult artisans with liberal education. In Ruskin’s The Elements of Drawing, he argues that drawing should not be taught for the sake of producing immediately pleasing results. Instead, he prescribes a demanding (but tractable) series of technical exercises that train the power to see. The purpose of learning to draw, he says, is not that we make something that looks nice. The purpose is instead that we learn to see. Only with this new sight, trained through study, do we become able to appreciate the true greatness of a Leonardo or a Raphael. Ruskin loved the Alps, and painted in them. Some of his most beautiful, striking works are of mere rocks, not grand mountain vistas or fertile valley pastures. In his masterful hand these lowly objects—still, sober, silent—burst with life.

21.4 Classes Classifying chants by modes gives us an opportunity to talk a bit about ancient logic and its transmission. Aristotle’s logical works together go by the name Organon. They make for demanding reading, so for many centuries Porphyry’s Introduction (sometimes called Isagoge, from the Greek) helped students to get started. Porphyry was himself a student of Plotinus, the founding Neoplatonist mentioned earlier. Porphyry identifies five basic logical terms—genus, species, difference, property, and accident. Let us consider the first two, genus and species. Their contemporary biological sense is related to their original, broader sense, in which they are used to explain how we speak about and classify things generally. Both a genus (such as animal) and a species (such as man) are ways of answering the question “What is it?” Both a genus and a species are said of many things. Here, though, there is a difference. The many things that are named by a species differ only in number—Socrates and Plato are two individuals who fall under

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the species man—while the multiplicity that falls under a genus is not simply a multitude of individuals, but instead can also contain distinct species. So man is an animal and tiger is an animal, and yet neither man nor tiger is an individual. Porphyry makes some etymological remarks about the word “genus.” While philosophers use the term in the strict sense outlined above, the word has a couple other, more concrete senses. People can be called a genus in virtue of their relation to some common thing (“the people descended from Hercules”) or by their origin (“from Athens”). These two—belonging to, and coming from—are not quite the same. Together, they give the word an immediate, concrete meaning to make it useful for technical, philosophical use. These distinctions can be enriched by analogy with Gregorian chant. It seems that the Gregorian modes were first descriptive rather than prescriptive. There was an existing body of chant, and it was found helpful to divide the chants up into groups in order to associate them with psalm verses. Should we call this a division by relation to something (matching a pre-existing complex melody to a simple psalm tone), or by origin (arriving at the melody by elaborating on a psalm tone)? It is a challenging question, one that we cannot investigate here. Stepping back from a specific question in the history of music; should we call a chant a “thing?” When we talk about tigers and squirrels as species of animal, there seems to be little risk of confusion. The highest Porphyrean genus is “substance.” Individual tigers and squirrels are substances. A chant, on the other hand, does not seem to be a substance. Is it even possible, then, to find a use for the terms “genus” and “species” in describing our modal classification of chant? At this point we are getting near questions that Porphyry suggests are best avoided by a beginner. He specifically lists questions that he will not address—things like in what sense, if any, a genus “subsists”—preferring instead to stick to simple matters, suitable for the beginner. Much of A Brief Quadrivium is written in the same spirit. Students must begin by acquiring a real grasp of the basic terms, even though this initial groping is necessarily incomplete.

Part IV

Astronomy

Dilexi decorem domus tuae.

22 Observation 22.1

Week 21 Plan

Overview: We begin by turning our observations into verbal principles by which we can reason. Looking Ahead: The next chapter is perhaps the most difficult of the whole book. Notes: 1. The principles of Section 22.2 are like postulates in geometry. It is essential that students perceive the principles as reflecting what they have themselves seen. Day 1: 1. Collect and review student observations. 2. Students should present their own observations and comment on those of others. 3. Read Section 22.2. Day 2: 1. Read all of Chapter 22. 2. Complete Additional Exercises 7 and 8.

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Reference

We can be too aware of ourselves and, simultaneously, entirely ignorant of ourselves. Youthful zeal, in particular, often involves undue self-consciousness along with the trespassing of proper boundaries. The Delphic oracle said “Know thyself.” Self-knowledge need not lead us to self-worship. It will probably free us from it. It also makes us more capable of real, healthy, whole relationships with others. The seemingly tender heart that claims to see and respond to the needs of another can turn out in fact to be all too aggressive, all too presumptuous. How does one grow beyond this? Time does its part. Scrapes, trials, and humiliations remind us of our own wants and limitations. In this way we become more attuned to the needs and difficulties of others, while also recognizing our ignorance, and how little we can ever hope to “solve.” “A man who has not been a socialist before 25 has no heart. If he remains one after 25 he has no head.” Many versions of this statement have circulated, with various political labels and attributions. This one was ascribed to King Oscar II of Sweden by the Wall Street Journal in 1923. What do politics and developmental psychology have to do with the technical study of elementary astronomy? Self-knowledge and knowledge of others are closely related to something that we will do now. As we conclude the quadrivium by following Ptolemy, our classical source, we proceed “geocentrically.” The use of observations from the earth, and models made in reference to the earth, can be called “geocentrism.” Geocentrism in astronomy is, in its own way, like self-knowledge in the beginning of wisdom. Mathematical accounts of motion, apparent or otherwise, require us to establish a frame of reference. The establishment of the frame is not a purely deductive act; it is one governed by prudence or art. Many mathematical problems are largely solved simply by choosing the right frame of reference. Since Einstein, the relative character of reference frames has risen to prominence even in popular discourse. Einstein’s great achievements in physics were not, however, the first instances of relativity in accounts of the material world. Galileo has his relativity, and

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even Ptolemy introduces multiple reference frames, as students of A Brief Quadrivium will discover soon. We must begin from somewhere. We should not presume that we see so far, or so clearly, that we will arrive at the most universal, most comprehensive account in a single leap. Geocentric astronomy, in the way we study it here, allows students to begin with their own direct knowledge, the sight of the sun and moon in their courses. A firm grasp of elementary accounts of these motions is the only good basis for a clear understanding of more complicated mathematical models and more remote reference frames. Self-knowledge can start us on the path to wisdom, but it is not the end of the way. Geocentric astronomical models will be tempered and augmented as students go further in their studies. Accepting a lowly place, in humility, leaves free the possibility that we will be called up higher.

23 Plane and Spherical Trigonometry 23.1

Week 21 Plan, Continued

Overview: We are now developing general-purpose tools for reasoning about motion in the heavens. Looking Ahead: Stronger students might be capable of comparing next week’s chord table with standard trigonometric functions. Consider it in advance. While this will be good for some students, you do not want to confuse others with too many ideas at once. Notes: 1. The Pythagorean theorem is of fundamental importance. Whenever we know two sides of a right triangle, we can approximate the length of the third. Day 3: 1. Read Section 23.1. 2. Complete Exercise 1. 3. Copy and discuss Proposition 103. Day 4: 1. Reread Section 23.1. 2. Read Section 23.2.1. 3. Copy and discuss Propositions 104–106.

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4. Complete Exercise 2. Day 5: 1. Reread Section 23.1 and 23.2.1. 2. Read Section 23.2.2. 3. Complete Exercise 10. Assessment: • Define the chord of an angle. • Prove one of the Propositions 104–106.

23.2 Week 22 Plan Overview: The chord table gives us numbers—the chords of specific angles. The theorems of Menelaus gives us a way to reason using those numbers, inferring one chord from others that are given.. Looking Ahead: This is a technically demanding week, and you might feel a bit strained. While next week is also full, there is some extra space in Weeks 24 and 25 in case you need to make some adjustments with timing. Notes: 1. Strong students might wish to refine the chord table. This can be done in a couple of ways. One is to refine a given entry by computing a more accurate approximation of a square root. Another is to insert new entries by using the half-angle formala to find the chord of 1.5◦ , then using Ptolemy’s Theorem 108 as indicated to find the chord of sums and differences of angles. 2. The exercise pairs 3–4, 5–6, and 7–8 help make the computations concrete. It is good to separate the individual numerical calculations from the way that they are all put together to arrive at a conclusion. 3. The theorem of Menelaus is somewhat complicated. It is acceptable if students do not memorize the statement. They must, though, be able to interpret it.

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4. To understand the theorem of Menalaus about arcs on the sphere, emphasize the analogy to the corresponding planar statements, Propositions 109 and 110. Day 1: 1. Reread Section 23.2.2. 2. Read Section 23.3. 3. Complete Exercises 11 and 12. Day 2: 1. Reread Section 23.3. 2. Complete Exercises 5, 6, 7, and 8. Day 3: 1. Read Section 23.4. 2. Complete Exercise 15. Day 4: 1. Read Section 23.5. 2. Complete Exercise 17. Day 5: 1. Reread Section 23.5.2. 2. Complete Exercises 18 and 19. 3. (optional) Complete Exercises 13, 14, and 16. Assessment: • Discuss the chord table and the theorem of Menelaus.

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Computation

What the hammer? What the chain? In what furnace was thy brain?

Electricity has made many jobs easier. Some people still choose, though, to work wood by hand. Why? Each woodworker will have his own reasons. We can imagine some reasons by simply looking on. The shop is quiet, and what noise there is does not pierce the silence shrilly. The product is the work of one artificer; the thing is commensurate with its maker. We should not be naïve. Whence the steel? Whence the glue? Whence, indeed, the lumber? To wrestle with a single log is to grow astonished that anyone has made anything out of wood, ever. A master makes his own tools. He thereby knows best what he gives and what he receives. The plane, the marking gauge, the mallet: these are his products but not his goal. And yet the sort of work he does to make the tools is the same as the sort of work he does with his tools. The purpose of the present chapter is computation. Our long course of study should by now have convinced you and your students that we do not make or alter mathematical things. They are not subject to time. Something does change, though, as we compute. We come to know, and our knowledge is new to us. The change is in our mind. In order to speak about order in heavenly motions, we require tools. The computations of plane and spherical trigonometry are the knowing by which we will make our knowing. Doing everything from scratch is not always best. You do not wish to bring your child to a hospital made by hand. What hand work does is this: when we see that everything machine-made is ultimately hand-made, we are made duly grateful. Aristotle would have us live this way: The investigation of the truth is in one way hard, in another easy. An indication of this is found in the fact that no one is able to attain the truth adequately, while, on the other hand, we do not collectively fail, but every one says something true about the nature of things, and while individually we contribute little or nothing to the truth, by the union of all a considerable amount is amassed.1 1 Aristotle,

Metaphysics, trans. W. D. Ross. Oxford: Clarendon Press, 1908.

24 Principal Solar Events 24.1

Week 23 Plan

Overview: We will understand day length and the location of the sun’s rising and setting by applying the tools of the previous chapter. Looking Ahead: The next two weeks are lighter, so it is okay if you are behind. Notes: 1. Our tools let us work with dates other than solstices, too. This is a good challenge for stronger students. The difference is that we cannot use simply the obliquity of the ecliptic, but instead must do additional work to determine the displacement of the sun from the celestial equator. Day 1: 1. Read the introduction and Section 24.1.1. 2. Go outside and look east. Think about locations for the sun’s rising at various times of year, and disuss. Be sure that students have a concrete grasp of the way that Figure 24.1 corresponds to your surroundings. You might have them replicate it, using the things that you really see nearby. Day 2: 1. Reread Section 24.1.1.

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2. Read Section 24.1.2. 3. Complete Exercise 1 and 5. Day 3: 1. Reread Section 24.1, and read Section 24.2. 2. Complete Exercises 2, 4, and 6. 3. (optional) Make a meridian line. Day 4: 1. Read Sections 23.3 and 23.4. 2. Complete Exercises 7–11. Day 5: 1. Reread the chapter. 2. Attempt to reproduce the computations of Sections 24.1.1 and 24.1.2 using only the statement of Menelaus’ Theorem. 3. (optional) Complete Exercise 3. Assessment: • Given a partial computation of day length or the sun’s location on the horizon, complete the computation. (You can take an example from the text and simply omit portions of the reasoning.) • Offer an account in words of the sun’s behavior at extreme latitudes (the poles and the equator).

24.2

Repetition

Generations have trod, have trod, have trod

Each day is a new day. It is also yet another day. Gregorian chant is monophonic but not monotonic; our lives, in contrast, can seem both cacaphonous and tediously routine. There is somehow, still, a “dearest freshness deep down things,” as Gerard Manley Hopkins

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suggests in God’s Grandeur. Strikingly, repetition itself can be for us a perennial font of new joy. Aristotle makes the famous observation that “All men by nature desire to know.” He notes that animals other than humans have sensation and memory. What we have, and they do not, is experience. Experience arises when “several memories of the same thing produce finally a single experience.” Beyond experience lies knowledge. Speaking of knowledge, Aristotle says that wonder is the origin of the philosophical search. About what did people first begin to wonder? The philosopher had himself begun as a natural scientist, and he points us to the moon, the sun, and the stars, the very things that students of A Brief Quadrivium themselves contemplate. Repetition spurs Augustine to wonder. In his Confessions, after examining memory and sensation, he ponders the nature of time. When Augustine remembers sounds, they are no longer present, and yet when he compares the duration of sounds (as one does in poetry) he somehow “measures” things that “are not.” The answer to his conundrum—though answer is the wrong word—comes at the very end of Book XI. Prior to pronouncing the words of a memorized poem, the speaker has the whole in view, at the outset. The parts, seemingly disparate and perhaps even incommensurable, come later. The main exercises of this chapter involve repetition. Ptolemy himself would easily have computed the solstitial rising point of the sun at the latitude of 39◦ or any other latitude. It has happened over and over, for all these years. This means that there is no novelty, no discovery in what students do when they compute this. There is no prediction. Each year the location will be the same; each generation of students will arrive at the same result. The repetitive “wholes” that comprise elementary solar motion are the basis by which we can ever hope to encounter something “new” in the sky. This perspective extends beyond material things. In Metalogicon, his defense and exposition of the trivial arts, John of Salisbury notes that “The employment of tropes, just as the use of schemata, is the exclusive privilege of the very learned.”1 Genuinely 1 John

of Salisbury, The Metalogicon of John of Salisbury, trans. Daniel D. McGarry. Berkeley and Los Angeles: University of California Press, 1955.

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ornamental variations in words receive their brilliance from the regularity of most verbal expression. The repetitive year is ornamented by the changing seasons. In his work on music, Boethius says that “what winter confines, spring releases, summer heats, and autumn ripens.”2 Some bear fruit and others assist in the bearing. We have joined Martha for a time in the kitchen. The seasons are themselves punctuated by days. A certain peace descends with the setting sun. The newborn child is shushed and rocked; we rise and we rest with a longer but still regular rhythm. Rabindranath Tagore finds, in the evening, peace and even prayer. The translation from the Bengali is his own: If the day is done, if birds sing no more, if the wind has flagged tired, then draw the veil of darkness thick upon me, even as thou hast wrapt the earth with the coverlet of sleep and tenderly closed the petals of the drooping lotus at dusk. From the traveller, whose sack of provisions is empty before the voyage is ended, whose garment is torn and dustladen, whose strength is exhausted, remove shame and poverty, and renew his life like a flower under the cover of thy kindly night. 3

2 Boethius, Fundamentals of Music, trans. Calvin M. Bower. New Haven: Yale University Press, 1989. 3 Rabindranath Tagore, Gitanjali (Song Offerings). New York: Macmillan, 1915.

25 A Refined Solar Model 25.1

Week 24 Plan

Overview: We see that there is subtle non-uniformity in the sun’s motion. It is discernible with nothing more than a calendar. Looking Ahead: This non-uniformity is one part of the equation of time, which we will see in Week 26 (Chapter 26). Notes: 1. It will be helpful to make a model of an epicycle. Take a large paper plate and attach a small disk (perhaps a smaller paper plate) to its periphery, in a way such that the small disk can rotate about its center. Then rotate the paper plate about its center while simultaneously rotating the small disk the other way at the same rate. 2. If you make a model, remember that the observer is at the center, and that the daily motion of the whole celestial sphere is from the left to the right when seen from the north pole. Day 1: 1. Read Chapter 25 up to (but not including) Proposition 114. 2. Complete Exercise 1. Day 2: 1. Reread up to Proposition 114, and now read that proposition, too.

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2. Note that Figure 25.4 is not to scale. The epicycle radius is lengthened for clarity. 3. Complete Exercise 3. Day 3: 1. Study Proposition 114. 2. Complete Exercises 5 and 6. Day 4: 1. Study the computation of the angle after Proposition 114. 2. Complete Exercise 2. Day 5: 1. Review through Section 25.2. 2. Complete Exercises 4, 7, and 9. Assessment: • Prove Proposition 114, given chords of relevant angles as well as dates of solstices and equinoxes. This is challenging, so students should be given some structure (i.e., fill in the blanks).

25.2 Week 25 Plan Overview: We see that there can be two models—distinct but equivalent—of the same thing. Looking Ahead: Extra time this week can be used to begin preparing for future review and a final exam, now about a month away. Small amounts of repetition spaced over time will be more fruitful than intense work at the very end. Notes: 1. Carefully produced tables and graphs of season lengths and other calendar data are available from Dr. Irv Bromberg at his website.1 1 http://individual.utoronto.ca/kalendis/index.html

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Day 1: 1. Review the epicyclic model of Section 25.2, then read Section 25.3. 2. Complete Exercise 8. Day 2: 1. Reread Section 25.3. 2. Attempt to reproduce Figure 25.10 from memory and explain its significance. 3. Complete Exercise 8 again. Day 3: 1. Read Sections 25.4 and 25.5. Days 4 and 5: 1. Make a general review of each of the first three disciplines we studied: Geometry, Arithmetic, and Music. Assessment: • Explain the equivalence of the two models for the sun. • State a significant proposition, definition, or computation from each of the first three parts of the book: Geometry, Arithmetic, and Music.

25.3 Certitude How do we know something for certain? Do we ever? These questions have been pondered from the time of Plato to the present day. Here are the remarks of a few thinkers and some connections to our study of astronomy. Aristotle observes amusingly in his Metaphysics that “accuracy has something of this character. . . that as in trade so in argument some people think it mean.” He then notes that “[t]he minute accuracy of mathematics is not to be demanded in all cases.”2 These are 2 Aristotle,

Metaphysics, trans. W. D. Ross. Oxford: Clarendon Press, 1908.

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good reminders for us as we go forth with our students and their newly formed mathematical muscles, turning outward and above to grasp feebly at grand and cosmic structures. Thomas Aquinas uses mathematical and astronomical examples as he inquires into the way that humans can know about the Blessed Trinity, the central Christian mystery. Augustine, whom we have often encountered in our study, had famously found images of the Trinity in the human mind, in the elements of the soul and the act of perception. Thomas must confront the question: does this mean that there is a rational demonstration—a proof—of the Trinitarian doctrine, this most sacred and seemingly hidden teaching? Aquinas uses an important distinction. On the one hand, sometimes we proceed by the light of reason to conviction about a principle. On the other hand, sometimes we use reason to see that one principle is consonant with other principles, or other experience. He illustrates both of these cases astronomically. Our observations convince us that the rotation of the celestial sphere (or in contemporary terms, of the earth on its axis) is regular, uniform, consistent. That regularity is something reliable, upon which we can base other reasoning. Our use of a specific mathematical model such as our epicyclic one, on the other hand, is different. We can only hope to show that it is congruous with the things we observe, not that it is the only possible model around. The theological import of the Thomistic distinction is that any sort of reasoning we do about the Trinity is only about congruence, like epicycles and the motion of the sun. We can find correspondences between revealed and reasoned truths, but those correspondences are not of themselves causal. Instead, they are like the hints and allusions that please us in stories and poems and music. The story of Galileo and the Inquisition provides a suitable complement to the measured and even light-hearted views of Aristotle and Aquinas. Passing over the details, let us go directly to a pellucid footnote in Paul Feyerabend’s Against Method. He writes “what Galileo suggested was no less than to regard as true a theory which had only analogies in its favor. . . .”3 Such a perspective stands in contrast to the more temperate notions already cited. 3 Paul

Feyerabend, Against Method. 4th ed. London and New York: Verso, 2010.

26 Terms of Time 26.1

Week 26 Plan

Overview: We discover something that might surprise you and your students. It is remarkably difficult to define specific periods of time, such as days, in a precise way, compatibly with motions we perceive as uniform. Looking Ahead: Next week involves eclipses. Find out the most recent lunar and solar eclipses that have been visible in your area, and investigate whether any others will be visible soon, nearby. Notes: 1. We have not discussed graphs of functions, but they might be familiar to you and your students. Find a graph of the equation of time (presenting the time difference as a function of the day of the year) and compare the values you see there to the ones that we compute in this chapter. 2. While daily effects are small, the cumulative effects of the varying day lengths are significant. The difference between real and mean solar noon is around 15 minutes at some points in the year. 3. Some sundials and old clocks display the equation of time. You can find images of these and show them to students. 4. The equation of time is an instance in which geocentric accounts are better than heliocentric ones. Since it involves the sun’s ap-

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parent location when seen from a given place on earth, it is simplest to work with a reference frame centered at the earth. Day 1: 1. Read Sections 26.1–26.3. 2. Complete Exercises 1, 2, and 3. Day 2: 1. Reread Sections 26.1–26.3. 2. Read Section 26.4.1. 3. Complete Exercises 4, 5, and 6. Day 3: 1. Reread Section 26.4.1. 2. Complete Exercises 7 and 8. Day 4: 1. Read Section 26.4.2. 2. Complete Exercise 11. Day 5: 1. Reread Section 26.4.2. 2. Complete Exercises 9 and 12. 3. (optional) Complete Exercises 10 and 13. Assessment: • Prove Propositions 122 and 123. • (optional) Prove Proposition 124.

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26.2

175

Grammar

Grammar is the first of the liberal arts. In this chapter we engage in a grammatical act—defining. In geometry and arithmetic, students practiced definition in specific fields, and in the union of those two they came to know definition as such. In the juxtaposition of the Pythagorean and just major thirds they found that ordered speech can be ambiguous without thereby degenerating into sophistry. Having acquired the capacity to speak mathematically, and witnessed the sky over a period of months, students are now ready to make and analyze definitions involving real subtlety. Priscian’s Institutes of Grammar shaped the medieval notion and practice of the subject. One of his minor works is called De figuris numerorum. Why is V the Roman numeral for the number 5? Priscian explains in this work that it is the fifth of the vowels. (Think of old inscriptions, in which U is written as V.) Priscian includes simple lists of number words. What might seem to be a tedious mire turns out to yield rich questions. For example, what relation does grammatical gender bear to number words? unus una unum, duo duae duo, hi et hae tres et haec tria, hi et hae et haec quattuor

While the number one has three forms, the numbers two and three have two—in one case the feminine stands out, and in another the neuter. The numbers four and beyond have a single form, being differentiated only in their pronouns. Perhaps four, then, is really the first number. Priscian’s text also instructs us about ordering with numbers. The first two ordering words are special, “primus et secundus.”1 After that, they follow the number form: “tertius, quartus, . . . .”2 Numbering years is tricky. The list of adjectives starts with unity— “annuus, biennis, triennis”3 —while the list of nouns starts instead with two—“biennium, triennium, . . . .”4 1 First

and second. fourth, . . . . 3 Annual, biannual, triannual. 4 Biennium and triennium can be used in English to refer to two-year and threeyear periods, respectively. Their use is somewhat rare. The word millennium, on the other hand, meaning a period of one thousand years, is quite common, and takes the same form. 2 Third,

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These verbal details have practical consequences. Today, the computer programmer sees them when indexing an array. Long ago, when Julius Caesar reformed the calendar in 46 BC, he took from the Roman priests the privilege of intercalation, the declaration of specific periods of time outside the ordinary calendar, by which civic dates and astronomical motions could be kept in alignment. Despite the theoretical accuracy of the Julian calendar, it immediately went awry. Caesar died before a single four year period elapsed. “Every four years” was misunderstood as “in the fourth year,” i.e., what we would call every three years. Corrections were made after forty years of wandering.

27 Elements of Lunar Astronomy 27.1

Week 27 Plan

Overview: We come to tell where the moon is by comparing it to the sun, at the rare moments of eclipses, rather than by direct observations of the moon in its ordinary course. Looking Ahead: We will go next week from relative considerations to absolute ones. We will then see how far the moon is from us. Notes: 1. The epicycle was useful when we wanted to model the annual motion of the sun. We continue to use an epicycle here, but there is a new complication. The rotation of the epicycle and the rotation of the deferent are not exactly the same. The result is that the moon’s motion is not modeled as a simple circle. Instead, it is a complicated curve called an epitrochoid. The drawing toy Spirograph makes such curves. 2. It is possible to do the arithmetic of this chapter by hand. If possible, avoid using calculators. It is good for students to see that they can understand this independently. They do not need accuracy beyond the hundredths place. Day 1: 1. Read Sections 27.1–27.3. 2. Complete Exercises 1, 2, and 3.

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Day 2: 1. Reread Sections 27.1–27.3. 2. Read the introductory portion of Section 27.4, and then read Sections 27.4.1 and 27.4.2. 3. Complete Exercises 4 and 6. Day 3: 1. Reread Section 27.4.2. 2. Take time to be sure that students understand the way that Figure 27.3 arises from Figure 27.2. Using a physical model (like the one you might have made earlier, using a paper plate) can help. Earth, the location of the observers, is point E. Figure 27.3 shows the three eclipses relative to the earth, when the line from the earth to the center of the epicycle is taken as vertical each time. 3. Complete Exercise 9. Day 4: 1. Read Sections 27.4.3 and 27.4.4. 2. Complete Exercises 5, 8, and 19. Day 5: 1. Reread Section 27.4.4. 2. Complete Exercises 20 and 21. Assessment: • Given a month of the year and a phase of the moon, explain whether the moon will be fairly high or low in the sky, and the time at which it will rise. • Explain in words the strategy for determining the moon’s epicycle. What role do eclipses play? How many eclipses do we need? Why? • (optional) Why is it important for determining the moon’s motion accurately that the eclipses we use happen close together?

elements of lunar astronomy

27.2

179

Week 28 Plan

Overview: We find the distance to the moon. Looking Ahead: Preparations should be underway for the final examination. Be sure that students know what is expected of them. Notes: 1. The material here might require more time than the three days allotted. You can extend it without too much disruption, since the final weeks have extra time built in for review. 2. In his Principia, when he argues that the moon is subject to the same gravity as objects on earth, Isaac Newton cites Ptolemy (among other scientists) for the distance from the earth to the moon. This datum is an important part of Newton’s argument, which is Proposition 4 in his Book 3, called The System of the World. You might show this to students. Day 1: 1. Reread Sections 27.4.2–27.4.4. 2. Complete Exercises 23 and 24. Day 2: 1. Read Sections 27.5–25.7. 2. Complete Exercises 13–16 and 25. Day 3: 1. Reread Section 27.7. 2. Complete Exercises 18, 26, and 27. 3. (optional) Complete Exercises 17 and 28.

27.3

Maculate

Ours is a maculate world. Augustine and Boethius encourage us to pursue mathematical study lest this spottedness cause us to lose

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our conviction that order governs all things. Somehow, though, the disorder itself points us towards the order. Our eyes offer a sign of this. Each has a blind spot. To find yours, do the following. 1. Put two dots on a sheet of paper. Make them three or four inches apart, and slightly smaller than a pencil eraser. 2. Hold the paper in front of you, so that the two dots determine a horizontal line, at the level of your eyes. 3. Close your left eye, and look (with your right eye) at the dot on the left, which should be directly in front of your right eye. 4. Start with the paper at arm’s length, and move the paper towards your face while continually looking at the dot on the left. Pay attention to the dot on the right, but look directly at the dot on the left. Explore a range of distances. What you will find is that, at the right distance, the dot on the right disappears. You cannot see it. The anatomical explanation is that the small region on the back of your eye where the optic nerve meets the retina is devoid of those cells by which light is sensed. It is occupied instead by the bundle of nerves that concentrate there before continuing to the brain. With one eye we see surface; with two we see depth. Binocularity, seeing from more than one place, gives us parallax, the thing that makes tracking down the moon so difficult. How do we track it down? In a marvelous way, we see it by not seeing it. The eclipses of the moon are the foundation for our insight into the regularity of lunar motion. The sun is cooly distant. When we speak of its motion, our location on the surface of the earth is irrelevant. The moon, on the other hand, is coyly distant. Closer, and it would ever be eclipsed; further, and never. In either case we would not have a good lunar model. The tantalizing infrequency of those passing shadows seems calculated to stoke our ardor and keep us in pursuit.

28 Stars, Fixed and Moving 28.1

Week 28 Plan, Continued

Overview: In studying the motions of the sun and moon, we presumed uniformity within the celestial sphere. That uniformity is only approximate. We now see two kinds of irregularity. One is that some objects that look like stars move within the celestial sphere. These objects are the planets. More subtly, the ecliptic, which determines the seasons, moves with respect to the celestial sphere. We had earlier taken it to be fixed. Day 4: 1. Read Section 28.1. 2. Complete Exercises 1–5, and if possible, try Exercise 6. Day 5: 1. Read Section 28.2. 2. Complete Exercises 7–10. Assessment: • Fill in the blanks to complete the parallax computation of the moon’s distance. • Describe the precession of the equinoxes. • Determine the period between conjunctions of Jupiter and Saturn, given their synodic periods.

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Quirks

Around the time that Galileo drew the attention of the Roman Inquisition, Miguel de Cervantes was publishing Don Quixote. There is something quixotic about the study of Ptolemaic astronomy today. Have not Galileo and those who followed him freed us from a foolish enchantment, and revealed that only an eccentric would study an epicycle? A more recent Quixote meets us in John Kennedy Toole’s novel A Confederacy of Dunces. Ignatius Reilly, the protagonist, seeks—or perhaps has already found—a proper theology and geometry. He fills notebooks with what he imagines to be sophisticated historical analyses; unemployed, he lives, at thirty, at home with his mother. Reilly’s concern for purity of thought seems to preclude attention to ordinary human hygiene, and his mother must deal with his filth. Like us, Ignatius follows Boethius. It would be a good exercise, now that your course in the quadrivium is wrapping up, to have a look at Toole’s novel. If we are going to hold the “modern” scientific stance at a critical distance, or propose that Boethius be read again, we should take care. Will we end up like Ignatius Reilly? Have we become absurd? A novelist like Toole can help us to laugh at ourselves, which is a good measure of continued sanity. We are not the only ones who need caution. The contemporary scientist stands at just as great a risk of becoming Reilly-like. Striving to give an account of the world’s structure can quickly become the presumption to possess an absolute account, in human terms, of the same. Toole, and Plato, and many in between, indicate that this lies beyond us. Working long before Ptolemy, Hipparchus had already discovered the precession of the equinoxes. Our careful computations involving the celestial sphere are inexact not simply because they involve square roots, which we only approximate with numbers, but because the sphere and the ecliptic are not fixed in relation to each other. Their relation changes over time. This is an early hint that approximation might mark all mathematical descriptions of the world’s phenomena. Ignatius Reilly often cries out to “Fortuna,” and he feels bound

stars, fixed and moving

by the turning of fate’s wheel. G.K. Chesterton’s The Ballad of the White Horse offers another perspective. Centuries ago, as Danes under Guthrum threaten Wessex, King Alfred the Great protects his lands. The defenders are not sure of victory, but they do not suppose that certainty is found in human things: The men of the East may search the scrolls For sure fates and fame, But the men that drink the blood of God Go singing to their shame.1

1 G.

K. Chesterton, The Ballad of the White Horse. Mineola: Dover, 2010.

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Part V

Beyond the Quadrivium

Multiplicati sunt super numerum.

29 Looking Back and Looking Ahead 29.1

Week 29 Plan

Overview: The final chapters relate our study of the quadrivium to other things that students have learned and will learn. Looking Ahead: If possible, arrange for an oral examination of some kind at the conclusion of the course. It is good if such an event can be public. Students should be asked to present material that they understand well. Notes: • Students can and should supplement the textbook with independent reading according to their interests. They can go in many directions. One option is to read additional material from Euclid. The notes of Sir Thomas Heath (whose standard translation is now in the public domain) are clear and voluminous. Other mathematical options are Descartes’ Geometry or Newton’s Principia. Students will only be able to sample a small portion of those texts. William Dunham’s Journey Through Genius presents significant mathematical theorems along with fascinating history and biography. Dava Sobel’s Longitude tells an exciting story about keeping time. Among classic philosophical texts, Plato’s Meno uses a geomet-

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rical example, Book I of Aristotle’s Metaphysics treats views attributed to the Pythagoreans, and Thomas Aquinas’s Summa Theologiae uses noteworthy astronomical reasoning in Part I, Question 32, article 1, in the response to the second objection. Day 1: 1. Read and discuss Chapter 29. Day 2: 1. Read and discuss Chapter 30. Day 3: 1. Read and discuss Chapter 31. Days 4 and 5 and Assessment: • These days and the weekly assessment are open to instructor discretion.

29.2 Week 30 Plan Overview: We complete the course. Looking ahead: Make notes for yourself on how to teach the course the next time. Notes: • See the note on independent reading given for last week. Day 1: 1. Study the Pythagorean theorem, its proof and its uses. 2. Review Proposition 14 (I.5) and Proposition 15 (I.6), and discuss the notion of converse statements as well as the method of proof by contradiction. Day 2: 1. Study Theorem 84 on solutions of linear Diophantine equations. 2. (optional) Review how to use the Euclidean algorithm in reverse to solve linear Diophantine equations.

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3. Review the general notion of mathematical induction along with Proposition 86, which gives an inductive proof of the recipe for triangular numbers. Day 3: 1. Study the numerical ratios of musical intervals, especially the tone and semitone. 2. Determine the mode of a chant and discuss how the melody relates to the mode. Day 4: 1. Explain how the sun’s non-uniform motion can be found by looking at a calendar. 2. Study the solstice computations of daylight length and rising location in Section 24.1. 3. Discuss the importance of lunar eclipses in producing a lunar model. Day 5: 1. Review Chapter 7, on ratio. 2. Discuss the ways that ratio is used in each of the four quadrivial disciplines.

29.3 Exsultavit The wrestler Dan Gable had a legendary work ethic. He won the gold medal at the 1971 World Championships in Sofia, Bulgaria. Early the next morning, the team bus was loading for departure, but Gable was missing. Had he overslept after a night of celebration? No one knew where he was. He turned up after a couple of hours. The newly-crowned victor had been out on the road, running. He was already training again. The Olympics were next. For the champion, the joy is in the striving. There is something of this in our mathematical activity. Before we give the proof, we

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state the proposition. We already know where we will end up. The starting line and the finish coincide. Between the start and the finish we might hope to encounter beauty. We will certainly find our limits. In his poem To an Athlete Dying Young, A.E. Housman tells of how quickly humans fade, and the memories of their achievements. Smart lad, to slip betimes away From fields where glory does not stay, And early though the laurel grows It withers quicker than the rose.

Gable kept training for the 1972 Games in Munich. He earned Olympic gold, and in a spectacular fashion—he did not concede a single point to any opponent. He then retired from wrestling at the age of 23, moving on to a similarly dominant career as coach at the University of Iowa. We have encountered Thomas Aquinas at various points in our study. He too retired young. One day he stopped dictating his great Summa Theologiae, telling his assistant that it was “like straw” in comparison with what he had been given in a mystical vision. In a study of the work of thinking and writing called The Intellectual Life, A.G. Sertillanges says this of St. Thomas’s silence: “Let us not have the presumption to wish that this lofty despair should come to us too soon.”1 Our words will fall short, and yet it is our responsibility to give them. A better example of our own silence comes not from the life of a saint, but from the story of a man named Olav Audunsson, the protagonist of a novel by Sigrid Undset set in medieval Norway. Audunsson kills a man in youth. The sin burdens him for many years, and he refuses—the reasons are complex—to confess. Late in life, he grows ready to seek absolution, when a new tragedy befalls him. A stroke takes his ability to speak. Finally willing to confess, he is now unable. Sleepless one night, and wracked with pain, Olav gets out of bed before dawn and wanders down to the fjord. He wants to see the water again. As he goes, a song that he once knew comes back to 1 A.

G. Sertillanges, O.P. The Intellectual Life, trans. Mary Ryan. Washington, DC: The Catholic University of America Press, 1998

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191

his mind, the psalm that says that all the trees of the forest will rejoice before the Lord, who comes with judgment. Perhaps the trees will rejoice, but Olav will not. Standing on the earth that he had reaped and sown in the strength of his manhood, Olav is brought low, considering how poor a steward he has been. The sun rises, and Olav looks. He gazes at what becomes for him a sign, a kind of writing. Somehow, the fiery orb burns not in judgment but with purification. Man, mute, has been heard. The summary means little unless you read the whole novel, all four parts. There is no abbreviating it. Words about the quadrivium will be tepid, too, unless we have the courage to run the whole course. Our words—Euclidean and Ptolemaic and Boethian and otherwise— have begun to fail; let us turn to the sun. Adam of St. Victor wrote in the 12th century, and he knew Hugh, whose Didascalion, mentioned earlier, handed on the traditional division of liberal learning. In Adam’s poem beginning Potestate non natura, he hints at the structure of the quadrivium, while seeing the sun as a sign. Hic est gigas currens fortis, Qui, destructa lege mortis, Ad amoena primae sortis Ovem fert in humerum. Vivit, regnat Deus-homo, Trahens orco lapsum pomo; Caelo tractus gaudet homo Denum complens numerum.2

We see astronomy in the turning heavens. Arithmetic is here too, in the tenfold hierarchy, the number of our fingers, a triangular number. There are hints of geometry elsewhere in the poem. The word meta occurs twice, once meaning a boundary line, and once indicating a goal, a conical turning post for a race—a point. All that remains, for the final lines, is music. Cantemus alleluia. Amen dicant omnia.3 2 This

is the giant running strongly, who, having destroyed the law of death, in order to remedy our first fate, carries the sheep on his shoulder. The God-man lives and reigns, drawing up from the abyss the one who fell by the fruit; man drawn heavenward rejoices, filling its tenfold order. 3 Let us sing alleluia. Let all say Amen.

Index

Adam of St. Victor, 191 Adrastus, 113–115 Al-Khwarizmi, 91 Alexandria, 65–66 algebra, xiii, 91, 97–99, 138 algorithm, 91 Almagest, 66 analogy, 39–41 approximation, 135–139, 172, 182–183 Aquinas, 112, 145, 172, 190 Archimedes, 43 area, 41–43 Aristotle, 65–66, 77, 80, 88, 91, 104–105, 113, 114, 153, 164, 167, 171 Audunsson, Olav, 190 Augustine, 120–121, 143, 167, 172, 179 axiom, 15, 31

beauty, 6, 55–57 Blake, William, 164 Boethius, 25, 77, 105, 112, 132, 135, 145, 168, 179 Bonaventure, 144–145

canon, 130–132 Cantor, Georg, 88 category, 49–51, 131 Cervantes, 182 characteristic class, 147 Chesterton, G.K., 135–139, 183 classifying space, 147 cohomology, 147 content, 38 continued fraction, 136 Copernicus, xiii Dante, 36, 49 definition, 13, 15, 81, 98, 175 Dehn, Max, 41–43 Delian problem, 137 Delphi, 158 Descartes, 137 diagram, 34–36 Didascalion, 145, 191 Diophantus, 99 Dodgson, Charles, 32–34 Drucker, Peter, 152 eccentricity, 182–183 Eilenberg, Samuel, 51 Einstein, Albert, 11, 92, 158

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Eliot, T.S., 34 epicycle, 172, 182 equality, 39–41, 73, 105–107 Erasmus, 90–91 Euclid, 10, 26, 34, 35, 40, 43–44, 65, 78, 87, 91, 99, 105, 113, 130, 137 Euclidean algorithm, 91, 99, 135 existence (mathematical), 17–18, 41 Fermi, Enrico, 63 Feyerabend, Paul, 172 formula, 105–107 function, 98 functor, 50 Gable, Dan, 189 Galileo, 88, 158, 172, 182 Garfield, James, 43 Gauss, Carl Friedrich, 57–58 genus, 153 geocentrism, 158–159 goal, 81–83 Goethe, Johann Wolfgang von, 152 grammar, 16, 144, 175–176 gratitude, 164 Gregorian chant, 151–154, 166 Halting Problem, 138 Hilbert, David, 42 Hipparchus, 66, 182 Hirzebruch, Friedrich, 147–148 Hopkins, Gerard Manley, 166 Horace, 15

Housman, A.E., 190 Hugh of St. Victor, 145, 191 humility, 139, 158–159 Iamblichus, 69–71 induction, 103–105 infinite, 87–89 irrational, 48, 135–139, 167, 179–180 Isagoge, 153 Jerusalem, 145 John of Salisbury, 167 John of the Cross, 83 Julius Caesar, 65–66, 176 Kepler, xiii Kripke, Saul, 113 Kubrick, Stanley, 126–128 lecture, 23–24 liberal arts, 4–5, 144, 191 Ligeti, György, 126 Lincoln, Abraham, 87 logic, 32, 103–105, 144, 153–154 Möbius band, 146 Mac Lane, Saunders, 51 Maurolico, Francesco, 105 memory, 48–49, 89–91 Meno, 48–49 meter, 111–113 Millay, Edna St. Vincent, 55–57 moon, 180 morphism, 50 mystery, 69–71

INDEX

names, 111–113 natural transformation, 49–51 Newton, Isaac, xiii, 127, 179 Nicomachus, 112 Nietzsche, Friedrich, 127 number words, 175–176 number, prime, 87–89, 93, 125 number, triangular, 143 object (of a category), 50 Organon, 153 Ovid, 106–107 parallax, 179–180 Parry, Hubert, 145 Pascal, 92, 105 patience, 5–6, 8, 125–126 Peano arithmetic, 78, 81 Peano, Giuseppe, 78 Percy, Walker, 70 Plato, 12, 25, 42, 48, 63, 100, 114, 137, 171, 182 Playfair’s axiom, 32 Plotinus, 69, 153 Plutarch, 89 Porphyry, 153–154 postulate, 6, 15, 17–18, 31–34 priority, 77–79, 103–105 Priscian, 175 Proclus, 69 proof, 7, 22–23, 31, 73, 80–82 prudence, 24–27 psalm, 143 Ptolemy, 27, 65, 114, 132, 158 Pythagoras, xiii, 25, 63, 69, 126–128, 132 Pythagorean theorem, 29, 37, 43–44, 69, 125

195

quadrivium, xiii, 4–5, 8, 25, 114, 191 Quintillian, 16 Quixote, 182 ratio, 26, 44, 48–57, 112, 114, 135–139 reference frame, 158 Reilly, Ignatius, 182–183 relativization, 138 Republic, 12, 25, 100 rhetoric, 31, 144 rhythm, 120–121 Romulus, 89 Ruskin, John, 152 scale, 143 Sertillanges, A.G., 190 Shakespeare, 56 silence, 55–57, 190–191 Simplicius, 114 solfege, 143 species, 153 speech, 22–23 Strauss, Johann II, 127 Strauss, Richard, 127 sun, 191 symbol, 69–71, 191 Tagore, Rabindranath, 168 tetrahedron, 42 Theon, 113–115 theorem, 36 Timaeus, 41–43, 64 topology, 146 trivium, 4, 144, 167 Turing degrees, 139 Turing machine, 138

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Turing, Alan, 138 Undset, Sigrid, 190 unit, 41, 111–113 utility, xiii, 10–11, 32, 152, 164–165 variable, 78, 97–99

vector space, 50, 132, 147 volume, 42 von Neumann, John, 92 Weil, Simone, 82 Whittier, John Greenleaf, 145 Zarathustra, 127