The Large Egyptian Pyramids: Modelling a Complex Engineering Project 9781407305462, 9781407335872

Table of contents :
53-54.pdf
Pages from 9781407305462.t
Pages from 9781407305462.t-2
Pages from 9781407305462.t-3
Front Cover
Title Page
Copyright
Dedication
About the Author
PREFACE
ABSTRACT
Table of Contents
1. INTRODUCTION
2. THE MODEL
3. THE SIZE OF THE PROJECT
4. THE EFFICIENCY OF HUMAN LABOUR
5. HOW MANY MAN-HOURS?
6. WORKFORCE AND WORK SCHEDULE
7. COMBINATION OF A RAMP WITH LEVERING
8. AREA PER BUILDER AND HAULING TRAINS
9. NUMERICAL EXAMPLE
10. DISCUSSION OF BASIC DATA
11. PLACING THE PYRAMIDION
12. THE BUILDING MATERIALS
13. LITERATURE REVIEW
14. CONCLUSIONS
APPENDICES
BIBLIOGRAPHY
ILLUSTRATION SOURCES
TABLES
FIGURES
AUTHOR INDEX
SUBJECT INDEX
SYMBOLS AND UNITS USED

Citation preview

BAR S2057 2010

The Large Egyptian Pyramids Modelling a Complex Engineering Project

DE HAAN

H. J. de Haan

THE LARGE EGYPTIAN PYRAMIDS

B A R

BAR International Series 2057 2010

The Large Egyptian Pyramids Modelling a Complex Engineering Project

H. J. de Haan

BAR International Series 2057 2010

ISBN 9781407305462 paperback ISBN 9781407335872 e-format DOI https://doi.org/10.30861/9781407305462 A catalogue record for this book is available from the British Library

BAR

PUBLISHING

To Henna

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

About the author Hendrik Jan de Haan received an MSc degree in physics from the Delft University of Technology. After a career with Shell International Petroleum Company he was appointed Professor of Petroleum Engineering at this university. He is now retired.

e-mail: [email protected]

iii

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

PREFACE When standing in awe in front of the great pyramids in Giza, I wondered, as so many people before me, how many man-years it must have taken to build these gigantic structures. At the same time, as a physicist, I felt that it ought not to be too difficult to make a rough estimate of this number by applying the well-known laws of mechanics. Just calculate the average amount of energy per day needed to build a pyramid in, say 20 years, divide this by the energy a man can deliver in a day and you have the answer. However, on closer examination the problem appeared to be a bit more complex than that. With all the attention spent on the construction of the Egyptian pyramids, I expected that this problem would have received exhaustive treatment. To my surprise, a search in the well-stocked library of the NINO (Netherlands Institute for the Near East) revealed that a systematic quantitative analysis was still lacking. So I decided to study the problem more closely myself. Gradually, I began to realise that a solution could only be reached by means of a well-founded mathematical model, tying together the various activities and quantifying the effort corresponding with the various building methods proposed in the literature. With interruptions this kept me busy for the next few years. No doubt the model presented still suffers from imperfections. However, I feel that it has now reached a stage where the most important aspects have been taken into account. I thank Prof. Ir. Jaap Oosterhoff who encouraged me at an early stage to publish this study and provided me with some of his own studies on ancient architecture. I am indebted to Prof. Dr. Ir Koen Weber for his active interest in the subject of this book, which led him not only to make critical and constructive remarks, but also to contribute two interesting chapters and last but not least, a large number of drawings that are a welcome compensation for the somewhat abstract subject matter. I also thank Stuart Evans Bsc Msc who kindly checked my English. Finally, I consider myself fortunate that BAR Publishing accepted this rather unusual treatment of an archaeological subject for publication. The general reader may wish to limit himself to the more descriptive parts of the study, notably Chapters 1 to 6, the last part of Chapter 7, Chapters 10 to 14 and Appendix 9. I sincerely hope that this work will help to bridge the gap between the egyptologist and the scientist, if only by showing that mathematics is an indispensable tool to describe this complex building process.

Wonder en is gheen wonder

I trust the famous Dutch scientist Simon Stevin (1548-1620) would have forgiven me for borrowing his motto shown above which seems pertinent to our problem: “Magic is no magic”, illustrated by his imaginative proof of the vector law. Wassenaar, August 2009

iv

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

ABSTRACT The building process of the Egyptian pyramids has been the subject of many publications. However, a thorough review of this literature reveals that only certain aspects of this process have been studied in isolation, without taking into account the interaction between various activities involved, such as quarrying, transportation and building and without a sound quantitative basis. The present study aims at filling this gap by means of an integrated mathematical model. We focus our attention on the largest pyramid, the one built by Cheops. The model simulates an efficient project co-ordination by balancing supply and demand of the building material, with all the activities related to the growth of the pyramid and by assuming a constant total workforce. It enables us to determine the effect of for instance, different building methods and of the productivity of the workers, on the workforce required for the various tasks. Three building methods have been studied, successively making use of a linear ramp, of a spiral ramp and of levers. These methods are compared in terms of the number of men and man-years required. Calculations have been carried out for two sets of input data, indicated as base case and maximum case. A gently sloping linear ramp reaching to the top of the pyramid would surpass the actual pyramid in size and require an excessive amount of material and manpower and thus appears an unrealistic undertaking. By comparison, the levering method requires little manpower, because no ramp is required. As the manpower for the case of a linear ramp increases steeply during the latter phase of the project, a considerable reduction in manpower and workforce can be achieved by limiting the height of the ramp and constructing the last part of the pyramid by means of levering. This combination therefore may well have been the method adopted by the pyramid builders. Assuming a constant total workforce and a project duration of 20 years, with the work continuing throughout the year, the workforce for this combination method is estimated to range from 5000 to 14000 men, depending on the basic data used. These numbers include only those who are directly involved in the building of the pyramid and the ramp and the supply of the necessary material. The number of workers required for the various activities changes as the project proceeds. Although the pyramid building rate decreases, the number of workers employed in the building of the pyramid and the ramp increases with time and the number of those allocated to the other activities, such as quarrying and hauling the building material to the building site, decreases accordingly. On completion of the ramp a rather drastic re-allocation of the workers is required so as to take maximum advantage of the total available workforce. To find out if the large number of workers might cause overcrowding on the surface of the ramp or the pyramid, the area available per member of the building team has been determined. It appears that congestion problems would occur for the spiral ramp. For this and for other reasons this building method is not considered practical. The main source of uncertainty in the basic data is the amount of limestone that could be quarried per man-hour. It is estimated that relatively little manpower was required for the excavation of a canal and for the shipping of the Tura stone used as casing for the pyramid. By testing a number of building theories presented in the literature against this model it was found that in all cases essential features of the building process, have been ignored.

v

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

PREFACE

IV

ABSTRACT

V

CONTENTS

VI

Chapters 1. Introduction

1

2. The model

2

• Activities and building methods

2

• The building process

3

3. The size of the project

5

• Pyramid and casing

5

• Linear ramp

5

• Spiral ramp

7

• Other ramps

7

4. The efficiency of manual labour

7

• Definitions of unit performance

7

• Quarrying

8

• Excavation of a canal

13

• Brick production

14

• Transportation

15

5. How many man-hours?

26

6. Workforce and work schedule

29

• Total workforce

29

• Time

30

• Building rate

31

• Workforce for the various activities

33

7. Combination of a ramp with levering

36

• Conditions and summary of results

36

• Transition between successive phases

38

• The levering phase

39

• Hindsight versus foresight

43

8. Area per builder and hauling trains

44

• Minimum area per builder

44

• Available areas

44

• Number of hauling trains

46 vi

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

9. Numerical example

47

10. Discussion of basic data

49

11. Placing the pyramidion (by K. J. Weber)

50

12. The building materials (by K. J. Weber)

53

• The Giza limestone

53

• The Tura limestone

59

• Granite

59

13. Literature review

59

14. Conclusions

78

Appendices 1. Volumes of pyramid and casing

80

2. Volume of linear ramp

81

3. Volume of spiral ramp

84

4. Manpower

87

5. Workforce, time and rate

92

6. Combination of a ramp with levering

98

7. Area per builder

102

8. Transportation

104

9. How does it work for a mini-pyramid?

110

Bibliography

112

Illustration sources

114

Tables attached I.

Unit performance for transportation – literature data

115

II. Basic data

116

III. Some results

117

Figures attached A.2.1 Exterior ramp - cross-section

118

A.2.2 Exterior ramp - base areas

118

A.2.3 Interior ramp - cross-section

118

A.2.4 Staircase - cross-section

118

A.3.1 Spiral ramp schematically

119

A.3.2 Spiral ramp - cross-section

119 vii

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

A.3.3 Spiral ramp – calculation

119

A.7.1 Available area for levering on side of the pyramid

120

A.8.1 Force diagram

120

A.8.2 Number of blocks arriving per unit time

120

A.8.3 Stepwise displacement by levering

120

Author Index

121

Subject Index

122

List of symbols and units used

124

Tables in text 4.1 Hauling experiment

19

6.1 Total workforce for the various building methods

29

7.1 Results for the combination method

37

7.2 Exterior ramp, base case - Workforce by activity vs height

41

7.3 Interior ramp, base case - Workforce by activity vs height

41

8.1 Numerical values for the for the levering method - Volume equations

47

8.2 Numerical values for the for the levering method - Manpower equations

48

12.1 Layer thickness of Sphinx and Pyramid

55

13.1 Literature review

61

13.2 Workforce corresponding with pyramid building in series

66

13.3 Comparison with Romer’s workforce numbers

69

13.4 Matching Romer's volume curve

69

13.5 Characteristics of some model curves

71

13.6 Stone production

72

Tables in Appendices 4-1 Volume equations for pyramid, casing and ramps

88

4-2 Manpower equations for the various activities

91

9-1 Calculations for mini-pyramid

110

viii

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

Figures in text 1.1 Map of ancient Egypt

X

1.2 The Cheops pyramid

1

2.1 Project schedule of the pyramid construction

2

2.2 Exterior and interior linear ramp

3

2.3 Spiral ramp

3

2.4 Material flow

4

3.1 Calculation of volume linear ramp

6

3.2 Cross-section of linear ramp for successive stages

6

3.3 Ratio of ramp/pyramid volumes vs relative height

6

3.4 Ratio of interior/exterior ramp volumes vs relative total height

6

4.1 Giza plateau with possible orientation of linear ramp

9

4.2 Quarrying limestone

10

4.3 North face of Meidum pyramid showing undressed inner stones

11

4.4 Unfinished obelisk at Aswan

12

4.5 Granite quarrying squatting

12

4.6 Granite quarrying standing

12

4.7 Transport of Djehutihotep monument

16

4.8 Hauling a block up the ramp

17

4.9 Levering

24

5.1 Calculation of the average horizontal distance

27

5.2 Total manpower versus fraction of pyramid completed

28

5.3 Fractional manpower for the various activities

28

6.1 Calculation scheme

` 30

6.2 Fraction of pyramid completed versus time

30

6.3 Block travel time

30

6.4 Effective building rate versus fraction of pyramid completed

31

6.5 Adjusted building rate versus fraction of pyramid completed

31

6.6 Simultaneous ramp and pyramid building

32

6.7 Number of builders as a fraction of total workforce, vs fraction of pyramid completed 33 6.8 Workforce vs time for exterior ramp of 133 m

34

6.9 Workforce vs time for spiral ramp of 133 m

34

6.10 Workforce vs time for levering

34

6.11 Shipping workforce versus fraction of pyramid completed

33

7.1 Workforce vs time for exterior ramp of 82 m

40

7.2 Workforce vs time for interior ramp of 82 m

40

7.3 Workforce vs time for spiral ramp of 82 m

40 ix

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

7.4 Height of pyramid in cubits versus thickness layers in metres

42

8.1 Area per hauling team

44

8.2 Horizontal area per builder versus fraction of pyramid completed

45

8.3 Inclined area per builder versus fraction of pyramid completed

45

8.4 Minimum number of hauling trains vs time for linear ramp

46

8.5 Minimum number of hauling trains vs time for spiral ramp

46

11.1 Seagoing vessel dating from about 2500 B.C.

51

11.2 H-shaped hoist

51

11.3 Bearing stones of the fourth dynasty

52

11.4 Placing the topmost layers

52

12.1 Section of a nummulites limestone

53

12.2 Detail of a nummulites limestone

54

12.3 Chephren quarry showing the trenches around the removed building blocks

54

12.4 Side view of the Sphinx showing the constant thickness of the layers

55

12.5 The Sphinx prior to the extensive restoration

56

12.6 Cross-section of the Mokattam formation

60

12.7 Estimated contour map of the quarry area prior to the excavation

57

12.8 Estimated contour map of the quarry area on completion of the excavation

57

12.9 Thin section of Tura limestone

60

13.1 Stone production for the construction of pyramids

65

13.2 Matching Romer’ building schedule

70

13.3 Romer’s building schedule vs model curves

71

13.4 Effect of early start of quarrying on volume to be stored

76

A.9.1 Mini-pyramid

110

A.9.2 Workforce for mini-pyramid vs time in hours

111

x

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

Fig. 1.1 Map of Ancient Egypt

xi

1. INTRODUCTION Which methods were used and how many people were needed to build the large Egyptian pyramids is still a matter of debate. A variety of building methods has been suggested in the literature. From drawings and reliefs it is evident that the Egyptians of the Old Kingdom made use of ramps and levers. Traces of small ramps still exist, also in the Giza area. However they did not discover the pulley or the winch until much later. Although much has been written on the building of the great pyramids, it appears that so far, little quantitative work has been done on the logistics of the building process. A critical review reveals that in practically all cases these calculations lack a proper physical basis. A notable exception is the study by Wier, but this paper treats the problem in an oversimplified way. The purpose of this study therefore is to fill this gap by developing a systematic and quantitative approach that takes into account the interdependence of the more important activities and that makes it possible to evaluate the effect of all the variables involved. In this way we can compare different building methods in terms of manpower required and identify crucial factors and possible limitations.1 Moreover, the model may serve as a tool to analyse building concepts presented in the literature. We focus our attention on the largest pyramid, the one built by Cheops or Khufu who reigned from 2551 to 2528 B.C.2 (Fig. 1.2).

Fig. 1.2 The Cheops pyramid

Admittedly, there is a margin of uncertainty in the basic data. However, a number of the activities involved can be – and in some cases have been – reconstructed, based on every-day experience with the possibilities of manual labour and by applying well established physical principles. In this way the uncertainty in the final results, notably the total workforce required, can be narrowed down considerably. To assess the sensitivity of the final results to the uncertainty in the basic data two sets of input data have been used, denoted as base case and maximum case. In fact, the main purpose of this work is to relate the results to the basic data in a systematic, verifiable way, recognising that precise answers are beyond reach.

1

For a simpler version of the model which minimises the mathematics by describing only the levering method, I refer to my paper mentioned under Bibliography. 2 According to Baines and Malek p. 36

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

Chapter 2 summarises the approach followed, Chapter 3 and Appendices 1 to 3 deal with the volumetric aspects. In Chapter 4 the performance efficiencies for the various activities are evaluated, based on data from the literature in combination with relationships developed in the Appendices. Chapters 5 to 8 and Appendices 4 to 8 are concerned with the dynamic aspects of the problem. Chapter 9 contains a numerical example and Chapter 10 a discussion of the basic data used. In Chapter 11 a special construction method is proposed for the uppermost levels of the pyramid. In Chapter 12 some geological aspects of the building materials are highlighted. Building methods suggested in the literature are reviewed in Chapter 13. The conclusions are summarised in Chapter 14. The details of the calculations are presented in the Appendices. The data used and results obtained are presented in the Tables and Figures. Fig. 1.1 shows the places mentioned in this study. The calculation procedure is illustrated for the simple case of a threelayer mini-pyramid in Appendix 9. 2. THE MODEL Activities and building methods In total we distinguish 8 different activities (Fig. 2.1): 1. 2. 3. 4. 5.

building the pyramid building one or more ramps quarrying and dressing of the building blocks production of the bricks used to reinforce the ramps transporting the blocks from the quarry and the bricks from the brickyard to the building site 6. shipping the casing material from the Tura quarry to the building site 7. excavating a canal for the transportation of building material from the Nile bank to the Giza plateau 8. demolishing the ramps

construction pyramid (1) construction ramp (2) quarrying/dressing (3) excavation canal (7)

brick production (4)

demolition ramp ( 8)

material transport by hauling to building site (5) shipping Tura blocks ( 6)

T7

T1 = T2 = T3 = T4 = T5 = T6 =T- T7 -T8

T8

T = total project duration

Fig. 2.1 Project schedule of the pyramid construction

Activities 1 to 6 were supposed to be carried out simultaneously with a constant total workforce. For obvious reasons, activities 7 and 8 had to be performed before the start and after completion of the building respectively, but – for the sake of consistency – were supposed to be carried out with the same workforce. Although shipping and the excavation of a canal turn out to absorb 2

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

only a small fraction of the total manpower, these activities have been retained because they illustrate special aspects of the model. Three different building methods were investigated: 1. a linear ramp approaching the pyramid from one or more sides (Fig. 2.2) 2. a spiral ramp around the pyramid built against its side walls (Fig. 2.3) 3. levering, essentially jacking up the blocks from one step of the pyramid to the next (Fig. 4.9) The building process The approach followed consists essentially of adding the total number of man-hours required which we shall denote as manpower - for simultaneous activities and determining the corresponding workforce by dividing this combined manpower by the duration. The workforce thus equals the manpower delivered per unit time. This concept is useful for our purpose because the manpower depends only on the effort required for an activity, not on its duration which is not the same for all activities. The manpower depends on the volumes produced or displaced. For the construction of the pyramid it is determined layer by layer by combining the layer volume with the lifting height. As the pyramid was built by human labour, the progress of the construction was related to the power – i.e. the energy per unit time - that can be developed per man (Chapter 5 and Appendix 4).

Fig. 2.2 Exterior and interior linear ramp

Fig. 2.3 Spiral ramp

An important aspect of the building of these large pyramids is the high building rate. The pyramids are generally associated with the reign of a pharaoh (Baines and Malek p. 36 and 140). For the Cheops pyramid we therefore assume a total project life of 20 years. With all activities continuing throughout the year, one free day during the Egyptian ten-day week and 8 effective working hours per day (excluding rest) this adds up to 328x8=2624 working hours per year. With a pyramid volume of about 2.6 million m3, the average building rate then is 2600000/(20x2624)=49.5 m3/hour or, for an average block size of 1 m3, nearly one block every minute. This means that the rate of supply of the building material from the various sources must be tuned to the building rate in order to prevent major problems with accumulation or shortfall of building material (Fig. 2.4). In our analysis the progress of all activities has therefore been related to the growth of the pyramid. Another important assumption made is that the total workforce is kept constant, so as to ensure an efficient utilisation of the available workforce. This means that the size of the teams in charge of the various activities must be 3

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

source

transport

Giza quarry

hauling

brick factory

shipping* & hauling

Tura quarry

shipping & hauling**

Aswan quarry

shipping & hauling***

quantity

building site

Notes: *shipping neglected, **hauling included in pyramid, ***manpower neglected

Fig. 2.4 Material flow

adjusted regularly. In view of the long project period concerned, this was probably not a serious constraint, as there must have been enough time for training. In this way I am modelling a perfectly coordinated project, thus determining a minimum workforce. This may not be far from the truth, for when the Egyptians built the Cheops pyramid they could draw on the experience with similar projects of many generations before them, in fact since Imhotep built the step pyramid for king Djoser. The sub-division of the workforce in “phyles” or work gangs (Hawass p. 30-31) is a clear indication that indeed the project was being well organised. In this connection it is interesting to mention a more recent example of what experience can do. Whereas it took Dutch shipbuilders not more than 8 months between 1627 and 1628 to build the “Batavia”, a ship meant for the East India trade, it took a workforce numbering a maximum of 1140 men 10 years between 1985 and 1995 to build its replica with better tools, but without the experience of their ancestors. It is an even more sobering thought that the ancient Egyptians could have completed several of these great pyramids during the time it has taken generations of scholars to find out how they might have done it. It stands to reason that in practice co-ordination must have been more difficult and as discussed in Chapter 7, the building process must have been less efficient than according to the ideal model presented here. On the other hand, as explained in Chapter 10, there are a number of reasons why the input data used for the model and thus the results obtained, may be pessimistic. In view of the size of the project a large workforce must have been required to complete it in the relatively short time available. To ascertain if and when this would have led to overcrowding of the pyramid and the ramps, the area available per member of the building team has been determined in Chapter 8 and Appendix 7. It is noteworthy that this problem lends itself to a systematic approach because of its relative simplicity, namely: (a) the simple, well-defined shape and relative homogeneity of the pyramid and the ramps, so that their volumes can be described by simple functions of the height reached and (b) relatively simple, well-defined manual activities which allow the definition of the unit performances for the productivity of the workers, as discussed in Chapter 4. In other words, a rather unique combination of circumstances that is not easily found elsewhere. 4

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

3. THE SIZE OF THE PROJECT To determine how the manpower required for the various activities depends on the progress of the project, we must know the relationships between the pyramid and ramp volumes and the height reached. As shown in Appendix 1 to 3 these relationships are described by simple quadratic equations. Pyramid and casing With a base of 230x230 m2 and an original height of 146 m, the pyramid had a volume of 2.57 million m3. It was finished with an outer layer of higher quality limestone obtained from the Tura quarry located on the other side of the Nile and shipped from there across the river. With a thickness of one metre this layer comprised some 3 % of the total volume (Table III). In Appendix 1 equations are derived for the volumes of the pyramid and the casing as functions of the height. For this purpose, following Stadelmann (p. 109) and Arnold (p. 159) and based on the outward appearance of the pyramid (Fig. 1.2), I assume that it consisted of horizontal layers.3 Moreover, I assume that the casing was put in place concurrent with the construction of the pyramid (see also Chapter 6). The fact that the existing rock outcrop4 fills part of the pyramid volume has been ignored. I consider the pyramid as uniform and thus disregard the detailed internal structure of corridors and chambers. Oosterhoff (p. 6) points out that for a density of 2500 kg/m3 the maximum pressure at the bottom of the pyramid is 9.81x2500x146=3.6x106 N/m2. In his view such a high pressure must have caused crushing of the blocks at the contact points and possibly even fracturing and subsidence. Linear ramp The linear ramp is depicted in detail in Fig. 3.1 and discussed in Appendix 2. There is evidence that ramps consisted at least partly of loose material, presumably the building debris from the quarry (Arnold p. 80, 81). Therefore it had to be reinforced by supporting side walls. A drawing found in the tomb of Rekhmira indeed shows a ramp that is supported by brick walls (Arnold p.97, Fig. 3.52). Ramps have also been observed at Meidum, consisting of debris supported by brick sidewalls (Arnold p.81). A ramp found at Giza, near the Cheops pyramid, consists of two walls carefully built of rough stone (Arnold p.83 and Fig. 3.33). Lehner (in Hawass p. 40) points out that no remainders of bricks have been found in Giza, but that large amounts of limestone and gypsum are present in the quarries south of the pyramids. In his view this must have been the construction material for the ramps. Presumably the production of this material would have required less manpower than bricks. We assume a top surface with a constant width of 20 m, with a gradient of 15 % (corresponding with 8.60). As the ramp became higher, its base had to be widened to ensure its stability (Fig. 3.2). This implies that the flanks of the ramp must have consisted of solid walls, whereas the central part, underneath the top surface, was possibly filled with debris. To remain on the conservative side it has been assumed that the ramp consisted entirely of bricks. In Appendix 2 the ramp volume has been determined for the general case in which the ramp walls slope at an arbitrary angle κ. However, unless otherwise stated, in this study the slope of the flanks is supposed to be parallel to those of the pyramid (κ = β). The ramp reaches its maximum height when the width of the pyramid becomes equal to that of the top surface of the ramp (Fig. 3.2). In our case that corresponds with a height of 133 m. The ramp will then almost completely

3 For a pyramid built up of buttress walls essentially the same results will be obtained, because the workforce is determined by the total power required and the power per man (Chapter 5). 4 Estimated at 74419 m3 by Romer (Appendix 2, p.451).

5

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

hmax

Fig. 3.2 Cross-section linear ramp for successive stages H

κ β

h

L

α

Fig. 3.1 Linear Ramp

b

2.0

1.0

spiral ramp exterior ramp

1.5

0.8 0.6

1.0 0.4

0.5

0.2

0.0

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 3.3 Ratio of ramp/pyramid volumes vs relative height

0

0.2

0.4

0.6

0.8

1

Fig. 3.4 Ratio of interior/exterior ramp volumes vs relative height

cover one side of the pyramid, resulting in a volume almost twice that of the pyramid volume (Table III and Fig. 3.3). Obviously, such a ramp would have been impractical, as its construction would be a more challenging undertaking than that of the pyramid proper. It follows from equation (2-7) and Fig. 3.3 that by limiting the height of the ramp, its volume can be reduced drastically. As explained in Chapter 7, by reducing the ramp height to 82 m its volume is reduced to 47 % of the pyramid volume, whereas this height is still sufficient to complete more than 90 % of the pyramid, with the last 10 % being constructed by levering. In Appendix 2 it is shown that by means of an interior ramp (Fig. 2.2) as proposed by Arnold (p. 101) the ramp volume can be reduced for the same total height or the height increased for the

6

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

same ramp volume. However, as shown in Fig. 3.4, based on equations (2-11) and (2-12), with λramp=0.5, this advantage decreases with increasing height. Spiral ramp The spiral ramp is illustrated in Fig. 2.3. We assume that it was built against the pyramid. With every course added its base area increases, thus at the same time lengthening the courses. This interdependence complicates the calculation of the total number of courses required (Appendix 3). With a 15 % slope and 10 m wide courses 8.5 courses or a little more than 2 complete windings are needed to reach a height of 133 m, equal to the maximum height of a linear ramp. The volume of the ramp, though smaller than the linear ramp, then still amounts to nearly 35 % of the pyramid volume (Table III). However, also in this case we shall consider the alternative of a lower ramp, combined with levering. The spiral ramp then has the advantage of a smaller volume, namely only 13 % of the pyramid volume for a height of 82 m, compared to 47 % for the linear ramp. On the other hand, the spiral ramp entirely covers the pyramid surface and thus hampers the outlining of the pyramid. This is much less of a problem with the linear ramp which covers only 17 % of the total pyramid surface for the same height (see also Chapter 7 and Table 7.1). In Fig. 3.3 the ramp volumes are plotted against the relative height. Other ramps In the literature several other types of ramp have been considered, more or less comparable with the spiral ramp, notably (1) a ramp supported by the steps of the pyramid and thus with a constant base area and (2) a reversing ramp with courses built in a zig-zag pattern against one side of the pyramid, rather than around it. Obviously, the first type of spiral ramp requires relatively little material and was found to require about the same amount of manpower as the levering method. However, as emphasised by Hodges (p. 13), this ramp suffers from the fact that its side walls which are already rather steep from the beginning, would become vertical somewhere below the top of the pyramid, the reason being that the vertical distance between the courses becomes smaller with increasing height. For a given width of the ramp this results in steepening of the side walls. In other words, this ramp presents a stability problem, in particular as it would be exposed to the effects of constant and long-term transportation of heavy loads. The reversing ramp, also built against the pyramid, is similar to the corresponding spiral ramp in terms of manpower and material required. Its only advantage compared to the spiral ramp is that it obscures only one side of the pyramid. All three ramps suffer from the problem that the sledges must be hauled around sharp corners. 4. THE EFFICIENCY OF HUMAN LABOUR Definitions of unit performance The duration of the project and the size of the pyramid determine how much material had to be lifted and thus also how much energy had to be spent per day, in other words how much power was required on average. To determine how many men were needed we must therefore know how much power one man can deliver. As is shown in this Chapter and in Appendix 8, this approach can indeed be followed for the transportation activities overland (activities 1, 2, 5 and 8), by making use of the laws of mechanics and of historical data. For other activities, such as quarrying, brick production and shipping, we have to find other ways to measure the performance per man. An obvious measure in the case of quarrying, brick production and the excavation of a canal is the volume produced or removed per man-hour: uq, so that the number of man-hours required to produce a volume V equals: 7

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

Mq =

V uq

It is then convenient to express the unit performance for transportation also in terms of volume, rather than mass. For this purpose we therefore introduce the volume that can be displaced at a velocity of one m/hour by one man (or the distance L over which a unit volume can be moved in one hour by one man). If na men are required to move a volume V at a speed v=L/T, the corresponding unit performance is: ut =

Vv VL = na naT

The manpower required to displace a volume V over a distance L then is: M t = naT =

VL ut

The power is defined as the force exerted on the object times its speed. For example, the force that must be overcome to hoist an object with volume V and density ρ vertically, is: F = ρ gV

The power needed to hoist it at a velocity v then is: P = F v = ρ gV v = ρ gV

L T

If the maximum power that can be exerted by one man is Pm, this takes na=P/Pm men, so that according to the definition of the unit performance: Pm = ρ g V v / n a = ρ g u t

Hence manpower and power per man should not be confused. Manpower has the dimension of man-hour, whereas power per man has the dimension of force x distance per man-hour. As shown above, the power per man is closely related to the unit performance for transportation, in fact equal to the unit performance times the density of the object being moved. For the cases considered here, namely transportation along a ramp or by levering, additional factors and terms must be included, as shown in Appendix 8. The values of these unit performances have been determined as far as possible from whatever data could be found in the literature on similar activities in ancient Egypt or on model experiments. For all activities we assume a work schedule of 2624 working hours per year, as discussed in Chapter 2. Basic data used and results derived have been summarised in Tables I and II, both for the base case and the maximum case (where applicable). The results are summarised in Table III. Below we shall discuss the unit performance for the various activities. Quarrying (activity 3) Limestone The limestone for the core stones was obtained from a quarry a few hundred metres south of the pyramid (Fig. 4.1). The availability of sufficient building material must have been an important reason why this location was chosen. In the NOVA experiment, set up to simulate working 8

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

Fig. 4.1 Giza plateau with possible orientation of linear ramp

conditions in ancient Egypt, 186 blocks of limestone were produced by 12 men in 22 days. Based on this experience Lehner (p. 206) estimates that under conditions in ancient Egypt production of a daily amount of about 322 m3 would have required 1212 men. The size of the stones is not mentioned, but according to these data must have been: 32 x 322 x 22 /( 1212 x 186) = 1.006 m3 The corresponding NOVA unit performance for an 8-hour working day is: 186 x 1.006 / (22 x 12 x 8) = 0.0886 m3/man-hour For the same number of working hours per day, the unit performance of the Egyptian quarrymen then would have been: 322/ (1212 x 8) = 0.033 m3/man-hour The additional amount of rock removed was relatively large, as wide trenches were made around the blocks to facilitate the quarrying (Figures 4.2 and 12.3). Although the ancient Egyptians had a variety of copper or bronze tools at their disposal, such as chisels, saws, drilling tools and wedges, these could not of course, compete with the iron tools used in the NOVA experiment. On the other hand, the Egyptians were well trained in this type of work. Moreover, 9

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

Fig. 4.2 The limestone quarry

The core stones are known to be less well cut. A hole made in the South face of the pyramid reveals that these stones are of an irregular shape and that the open spaces are filled with grout and rubble (Lehner in Hawass, p. 38). This can also be observed at the Meidum pyramid (Fig. 4.3). This must have speeded up the quarrying and the dressing considerably. For our base case we assume: u3g = 0.03 m3/man-hour and for our maximum case: u3g = 0.02 m3/man-hour Obviously, the Tura stone, used as casing, had to be dressed more carefully than the material from the Giza quarry which was to be used for the inner part of the pyramid. However, as the casing volume comprises only about 3 % of the total pyramid volume, this aspect has been neglected. Illig (p. 26) is convinced that the blocks could be easily dislodged by utilising the layered structure of the limestone and that it was not necessary to make trenches around the blocks. He 10

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

estimates that 8 men, including helpers, working 8 hours per day, could produce at least 2 to 3 blocks per hour. For a block size of 1 m3 this corresponds with a unit performance of 0.25 to 0.4 m3/man-hour, in other words 8 to 13 times our base case value. Indeed limestone formations are often broken up in blocks by horizontal layering and vertical fissures which would have greatly facilitated the task of the quarrymen (see also Chapter 12).

Fig. 4.3 North face of the Meidum pyramid, showing undressed inner stones

Granite In Cheops’ pyramid granite was used only for the King’s and the pressure relieving chambers. This is only a small quantity, but because of its hardness it required a relatively large quarrying effort. We shall therefore try to get an impression of the corresponding unit performance from the production process of queen Hatshepsut’s obelisks. These two obelisks, 30 m high and each weighing 323 tons were reportedly produced and transported in 7 months (Arnold p. 40). The unfinished obelisk at Aswan (Fig. 4.4) gives a clear impression of the size of the trenches excavated around it and thus not only of the amount of material that had to be removed but also of the number of men these trenches could accommodate (Fig. 4.5, inspired by Arnold p. 36-40 and Goyon p. 164-166). As pointed out in Chapter 12 and illustrated by Fig. 4.6, it would seem more efficient to operate the hammer in a standing position, but the space required per man would have been about the same. We shall assume that the quarrymen needed a 60 cm wide working space. This implies that about 50 quarrymen were active on each side of the obelisk at any one moment. Supposing that out of the 7 months available one month was used for transport and assuming two 8 hour shifts with one free day per 10 days, this corresponds with a total of 2 x 2 x50 x 6 x 30 x 8 x 0.9 = 259200 man-hours

11

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

Fig. 4.4 The unfinished obelisk at Aswan

Fig. 4.5 Granite quarrying squatting

Fig. 4.6 Granite quarrying standing

With a rock density of 2.9 tons/m3, the volume of the obelisks becomes 323/2.9 = 111.4 m3, resulting in a unit performance in terms of useful volume produced of: 111.4/259200 = 0.00043 m3/man-hour Presumably, this figure includes the dressing effort. The fact that it is about 100 times smaller than that for limestone is not surprising, as granite, because of its greater hardness, had to be quarried by pounding with dolerite balls, rather than with copper tools as these were too soft. However, for our purpose the effort required per unit volume of rock removed is more relevant than the effort per unit of ‘useful product’. The average width of the obelisks is: √ (111.4/30)= 1.97 m 12

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

Assuming a 60 cm wide trench around the obelisks and a 60 cm high undertunnelling, the amount of material removed then is roughly: (1.927+0.6)*(30+2 x 0.6)*(1.927+2 x 0.6)-111.4=135.1 m3 In terms of unit performance: 135.1/259200 = 0.00052 m3/(man-hour) The actual rate of removal must have been higher as part of the time was required for dressing and polishing. Moreover, removal of the top layer has been ignored. Engelbach (p. 48), in an attempt to determine a value experimentally, found that he could remove a granite volume of 563 cm3 in one hour (see also Goyon, p. 164). Lehner (p. 207) managed to remove 30 x 30 x 2 cm by pounding in 5 hours. This corresponds with a unit performance of 0.00056 and 0.00036 m3/man-hour respectively, in reasonable agreement with the value derived above. The ratio of volume removed to the volume of the obelisk is 135.1/111.4 = 1.21. The roofing blocks above the burial crypt in Cheops’s pyramid were smaller and this ratio was therefore larger. According to Arnold (p. 60), the dimensions of these blocks are 1.3 x 1.8 x 8 = 18.7 m3, corresponding with a weight of 18.7 x 2.9 = 54.3 tons. Let us assume that on average trenches 60 cm wide were made on 3 sides around blocks (e.g. on right side, behind and underneath), then the ratio of the volume to be removed to the volume of the block is: 1.9 x 2.4 x 8.6/18.7-1 = 2.10 Assuming the same ratio for all the 56 blocks, the unit performance in terms of useful volume produced per man-hour then is: 0.00052/2.10 = 0.000244 m3/man-hour We shall assume: u3a = 0.00024 m3/man-hour In spite of this extremely low unit performance the manpower required for the quarrying turns out to be not more than a few percent of the total manpower, due to the small volume involved.5 This aspect has therefore been neglected. Here again Illig (p.144) is much more optimistic. He assumed that the Egyptians had iron tools at their disposal and were able to produce one block “of normal size” per day with a team of 20. For a block of 1 m3 and an 8-hour work day this corresponds with a unit performance of 0.00625 m3/man-hour. Excavation of a canal (activity 7) No data are available that are representative for the conditions in ancient Egypt. However, the information about the excavation of canals in the 19th century may serve as a guideline, because at that time only manual labour was available. I was fortunate to receive some data on the excavation in The Netherlands of the “Overijssels” canal, dug in the period 1852-1858, 25 km long, 5 m wide, with a number of workers varying from 80 to 700 (sources: “Stichting Apeldoorns kanaal” and “Maatschappij der Overijsselse kanalen”).

5 As granite quarrying took place only during the first phase of the project, the effect on the workforce allocation must have been of importance only during this period.

13

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

During one 17 months period 213000 m3 was excavated by 221 men. Assuming 8 effective working hours per day and 6 days per week, this corresponds with: 213000/(221x17x30x(6/7)x8)=0.276 m3/man-hour During a 2.5 months period 41160 m3 was excavated by a maximum of 320 men. We round the number of workers off to 300 and find a unit performance of: 41160 /(300x75x(6/7)x8)=0.267 m3/man-hour Another relevant piece of information is that the workers earned 12 cents/m3 and 40 cents/day. This would mean that they were able to dig 40/(12x8)=0.417 m3/man-hour. The average of these three numbers is 0.32 m3/man-hour. According to Arnold (p.65) and Goyon (p. 189) at least 15 tons or 6 m3 could be transported by boat.6 To limit the draft to 0.5 m, a raft or boat loaded with 15 tons would need a surface area of at least 30 m2, a size well within the capability of ancient Egyptian ship builders, bearing in mind the size of the “sun-boat” exhibited near the pyramid of Cheops which measures 43 x 5.6 m2. To be on the safe side and to take into account irregularities of the terrain, we assume an average canal depth of 2 m. In view of this shallow depth we assume that the canals were excavated in relatively soft rock. On the other hand, we must take into account that the ancient Egyptians had no steel spades and no wheelbarrow.7 We shall therefore scale the above average value down to 0.25 m3/man-hour. As excavation of a canal plays a very minor role in terms of manpower, variation of this unit performance has little effect on the estimation of the total workforce. Brick production (activity 4) Manual fabrication of bricks was still common in the first half of the 20th century. One of my friends who made his career in the building industry recollects that a brickmaker, assisted by a helper who hands him the clay, could fill a mould with clay sufficient for six bricks, each sized 24x12x6=1728 cm3, in not more than five minutes. Thereafter the bricks must be fired – or in ancient Egypt dried in the sun – and collected. These later stages are more a matter of mass production and thus take much less time per brick, say another minute for the six bricks, hence on average one minute per brick. This then results in a unit performance of 6x1728x10-6x60/(6x2)=0.052 m3/man-hour A ramp found by Petrie in the limestone quarries of Antaeopolis was built with bricks sized 12 x 18 x 37 cm3 = 7992 cm3 stamped with the name of Amenhotep III (Arnold p.93). To match his modern colleague the Egyptian brickmaker, also assisted by a helper, would have to produce one of these larger bricks every 7992/1728=4.6 minutes which seems rather generous. This means that it would take 2x4.6/(7992x10-6x60)= 19.2 man-hours to produce 1 m3 of these bricks. To this we must add the time needed to supply the material. Presumably this consisted mainly of an excavation activity. As discussed above, this would take an additional 1/0.25 = 4 man-hours. Thus we arrive at a unit performance of 1/23.2 =0.043 m3/man-

6 Shipment of Hatsjepsoets’obelisks, weighing 323 tons (admittedly during a much later period), indicates that this is not necessarily a maximum load. 7 Curiously, this relatively simple vehicle was first mentioned in Chinese documents dating back to the 2d century AD and did not even appear in Europe until the Middle Ages (Cotterell and Kamminga p. 214).

14

H. J. de Haan - The large Egyptian pyramids – Modelling a complex engineering project

hour. Here also we must take into account that the ancient brickmakers did not have the same facilities, for instance no wheelbarrows and therefore we assume u4 =0.035 m3/man-hour Like their colleagues in more recent times, the ancient Egyptians would produce the bricks near the river where there must have been clay in abundance. The bricks then probably were transported by boat to the Giza plateau and from there on sledges to the building site. In view of the large unit performance for shipping (see below under transportation) and the short distance to be covered this means that the manpower involved in this shipment can be neglected. According to Baines and Malek (p. 15) in ancient times the course of the Nile was further West, so that the distance between the river and the plateau was probably not more than one kilometer. We shall assume that the remaining distance to be covered by overland transport ranged from 500 to 1000 m. Transportation (activities 1, 2, 5, 6, 8) Friction, force and power As explained above, transportation is controlled by the maximum power (force x speed), rather than by the maximum force per man Fm as assumed in most publications. Of course, the force exerted must be large enough to overcome the minimum force F1 to move an object, for instance a building block. Once this condition is met, a variation of the force exerted per man in the range F1/na