Part 1

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_With the respectful compliments of the author._

Address, Mr. R. BALLARD, Malvern, ENGLAND.

Illustration: Plumb Line over Pyramids

THE SOLUTION OF THE PYRAMID PROBLEM OR, PYRAMID DISCOVERIES.

WITH A NEW THEORY AS TO THEIR ANCIENT USE.

BY ROBERT BALLARD, M. INST. C.E., ENGLAND; M. AMER, SOC. C.E. CHIEF ENGINEER OF THE CENTRAL AND NORTHERN RAILWAY DIVISION OF THE COLONY OF QUEENSLAND, AUSTRALIA.

NEW YORK: JOHN WILEY & SONS. 1882.

Copyright, 1882, By JOHN WILEY & SONS.

PRESS OF J. J. LITTLE & CO., NOS. 10 TO 20 ASTOR PLACE, NEW YORK.

NOTE.

In preparing this work for publication I have received valuable help from the following friends in Queensland:--

E. A. Delisser, L.S. and C.E., Bogantungan, who assisted me in my calculations, and furnished many useful suggestions.

J. Brunton Stephens, Brisbane, who persuaded me to publish my theory, and who also undertook the work of correction for the press.

J. A. Clarke, Artist, Brisbane, who contributed to the Illustrations.

Lyne Brown, Emerald,--(photographs).

F. Rothery, Emerald,--(models).

and--A. W. Voysey, Emerald,--(maps and diagrams).

CONTENTS.

PAGE

§1. The Ground Plan of the Gïzeh Group 9

Plan Ratios connected into Natural Numbers 13

§2. The Original Cubit Measure of the Gïzeh Group 15

§3. The Exact Measure of the Bases of the Pyramids 19

§4. The Slopes, Ratios and Angles of the Three Principal Pyramids of the Gïzeh Group 20

§5. The Exact Dimensions of the Pyramids 23

§6. Geometrical Peculiarities of the Pyramids 28

§6A. The Casing Stones of the Pyramids 32

§7. Peculiarities of the Triangles 3, 4, 5 and 20, 21, 29 35

§8. General Observations 38

§9. The Pyramids of Egypt the Theodolites of the Egyptian Land Surveyors 41

§10. How the Pyramids were made use of 44

§11. Description of the ancient Portable Survey Instrument 51

Table to explain Figure 60 64

§12. Primary Triangles and their Satellites;--or the Ancient System of Right-angled Trigonometry unfolded by a Study of the Plan of the Pyramids of Gïzeh 64 Table of some "Primary Triangles" and their Satellites 74

§13. The Size and Shape of the Pyramids indicated by the Plan 77

§14. A Simple Instrument for laying off "Primary Triangles" 79

§14A. General Observations 80

§15. Primary Triangulation 82

§16. The Pentangle or Five-pointed Star the Geometric Symbol of the Great Pyramid 91

Table showing the comparative Measures of Lines 96

§17. The manner in which the Slope Ratios of the Pyramids were arrived at 104

LIST OF WORKS CONSULTED.

_Penny Cyclopædia. (Knight, London. 1833.)_

_Sharpe's Egypt._

_"Our Inheritance in the Great Pyramid." Piazzi Smyth._

_"The Pyramids of Egypt." R. A. Proctor. (Article in Gentleman's Magazine. Feb. 1880.)_

_"Traite de la Grandeur et de la Figure de la Terre." Cassini. (Amsterdam. 1723.)_

_"Pyramid Facts and Fancies." J. Bonwick._

Illustration: General Plan of Gïzeh Group

THE SOLUTION OF THE PYRAMID PROBLEM.

With the firm conviction that the Pyramids of Egypt were built and employed, among other purposes, for one special, main, and important purpose of the greatest utility and convenience, I find it necessary before I can establish the theory I advance, to endeavor to determine the proportions and measures of one of the principal groups. I take that of Gïzeh as being the one affording most data, and as being probably one of the most important groups.

I shall first try to set forth the results of my investigations into the peculiarities of construction of the Gïzeh Group, and afterwards show how the Pyramids were applied to the national work for which I believe they were designed.

§ 1. THE GROUND PLAN OF THE GIZEH GROUP.

I find that the Pyramid Cheops is situated on the acute angle of a right-angled triangle--sometimes called the Pythagorean, or Egyptian triangle--of which base, perpendicular, and hypotenuse are to each other as 3, 4, and 5. The Pyramid called Mycerinus, is situate on the greater angle of this triangle, and the base of the triangle, measuring _three_, is a line due east from Mycerinus, and joining perpendicular at a point due south of Cheops. (_See Figure 1._)

Fig. 1. Cheops, Mycerinus

I find that the Pyramid Cheops is also situate at the acute angle of a right-angled triangle more beautiful than the so-called triangle of Pythagoras, because more practically useful. I have named it the 20, 21, 29 triangle. Base, perpendicular, and hypotenuse are to each other as twenty, twenty-one, and twenty-nine.

The Pyramid Cephren is situate on the greater angle of this triangle, and base and perpendicular are as before described in the Pythagorean triangle upon which Mycerinus is built. (_See Fig. 2._)

Fig. 2. Cheops, Cephren

Fig. 3. Cheops, Cephren, Mycerinus

_Figure 3_ represents the combination,--A being Cheops, F Cephren, and D Mycerinus.

Lines DC, CA, and AD are to each other as 3, 4, and 5; and lines FB, BA, and AF are to each other as 20, 21, and 29.

The line CB is to BA, as 8 to 7; the line FH is to DH, as 96 to 55; and the line FB is to BC, as 5 to 6.

The Ratios of the first triangle multiplied by forty-five, of the second multiplied by four, and the other three sets by twelve, one, and sixteen respectively, produce the following connected lengths in natural numbers for all the lines.

DC 135 CA 180 AD 225 ----------- FB 80 BA 84 AF 116 ----------- CB 96 BA 84 ----------- FH 96 DH 55 ----------- FB 80 BC 96

Figure 4 connects another pyramid of the group--it is the one to the southward and eastward of Cheops.

In this connection, A Y Z A is a 3, 4, 5 triangle, and B Y Z O B is a square.

Lines YA to CA are as 1 to 5 CY to YZ as 3 to 1 FO to ZO as 8 to 3 and DA to AZ as 15 to 4.

I may also point out on the same plan that calling the line FA radius, and the lines BA and FB sine and co-sine, then is YA equal in length to versed sine of angle AFB.

This connects the 20, 21, 29 triangle FAB with the 3, 4, 5 triangle AZY.

I have not sufficient data at my disposal to enable me to connect the remaining eleven small pyramids to my satisfaction, and I consider the four are sufficient for my purpose.

_Fig. 4._

_At level of_ _At level of_ _Own base_ _Cephren's base_

_Of these natural_ } _Cheops_ 56½ 52½ _numbers the bases_ } = _Cephren_ 52½ 52½ _of the pyramids are_} _Mycerinus_ 26¼ 27¼ _as follows._ }

I now establish the following list of measurements of the plan in connected natural numbers. (_See Figure 4._)

Plan Ratios connected into Natural Numbers.

+------------+------------+------------+-------------+ |BY 1} 48|BC 6} 96|DC 45} 135|FB 5} 80| | } 48 | } 16 | }3 | } 16 | |YZ 1} 48|FB 5} 80|BC 32} 96|BY 3} 48| +------------+------------+------------+-------------+ |DN 61} 183|DN 61} 183|CY 3} 144|FH 96} 96| | } 3 | } 3 | } 48| } 1 | |NR 60} 180|NZ 48} 144|BC 2} 96|DH 55} 55| +------------+------------+------------+-------------+ |CY 16} 144|PN 61} 146·4|JE 3} 72|YX 7} 63| | } 9 | } 2·4 | } 24 | } 9 | |DC 15} 135|PA 48} 115·2|EX 2} 48|AY 4} 36| +------------+------------+------------+-------------+ |BA 21} 84|CA 4} 180|BC 32} 96|EA 7} 105| | } 4 | } 45 | } 3 | } 15 | |FB 20} 80|DC 3} 135|EB 21} 63|AZ 4} 60| +------------+------------+------------+-------------+ |CB 8} 96|YZ 4} 48|FO 32} 128|AB 7} 84| | } 12 | } 12 | } 4 | } 12 | |BA 7} 84| AY 3} 36|OR 21} 84|BO 4} 48| +------------+------------+------------+-------------+ |ED 8} 120|BA 4} 84|FT 84} 84|BC 8} 96| | } 15 | } 21 | } 1 | } 12 | |AE 7} 105|EB 3} 63|ST 55} 55|AC 15} 180| +------------+------------+------------+-------------+ |VW 55} 55| GE 4} 96|VW 55} 55|ND 61} 183| | } 1 | } 24 | } 1 | } 3 | |FW 48} 48|DG 3} 72|SV 36} 36|NO 32} 96| +------------+------------+------------+-------------+ |SJ 7} 84|HN 4} 128|BJ 45} 135|PA 48} 115·2| | } 12 | } 32 | } 3 | } 2·4 | |SU 6} 72|FH 3} 96|AB 28} 84|AZ 25} 60| +------------+------------+------------+-------------+ |GX 2} 144|GU 5} 180|EO 37} 111|SR 61} 183| | } 72 | } 36 | } 3 | } 3 | |DG 1} 72|DG 2} 72|AY 12} 36|RZ 12} 36| +------------+------------+------------+-------------+ |SU 2} 72|HW 144} 144|HT 36} 180|FH 96} 96| | } 36 | } 1 | } 5 | } 1 | |SV 1} 36|DH 55} 55|DH 11} 55|FE 17} 17| +------------+------------+------------+-------------+ |TW 36} 36|FO 8} 128|DA 15} 225|EA 105} 105| | } 1 | } 16 | } 15 | } 1 | |TU 17} 17|OZ 3} 48|AZ 4} 60|EF 17} 17| +------------+------------+------------+-------------+ |SR 61} 183|JB 45} 135|AC 15} 180|WH 144} 144| | } 3 | } 3 | } 12 | } 1 | |RO 28} 84|BY 16} 48|CN 4} 48|HG 17} 17| +------------+------------+------------+-------------+ |YW 20} 80|FW 48} 48|YV 15} 135|TH 180} 180| | } 4 | } 1 | } 9 | } 1 | |AY 9} 36|FE 17} 17|AY 4} 36|HG 17} 17| +------------+------------+------------+-------------+ |MY 9} 108|AC 20} 180|VZ 61} 183| | | } 12 | } 9 | } 3 | | |ZY 4} 48|CG 7} 63|ZO 16} 48| | +------------+------------+------------+-------------+ |AC 9} 180|EA 35} 105|EU 84} 84| | | } 20 | } 3 | } 1 | | |CH 4} 80|AY 12} 36|FE 17} 17| | +------------+------------+------------+-------------+ |NZ 12} 144|CY 3} 144|CA 5} 180| | | } 12 | } 48 | } 36 | | |ZA 5} 60|YZ 1} 48|AY 1} 36| | +------------+------------+------------+-------------+

The above connected natural numbers multiplied by eight become R.B. cubits. R.B.C.

(Thus, BY 48 × 8 = 384 GX 144 × 8 = 1152).

§ 2. THE ORIGINAL CUBIT MEASURE OF THE GIZEH GROUP.

Mr. J. J. Wild, in his letter to Lord Brougham written in 1850, called the base of Cephren seven seconds. I estimate the base of Cephren to be just seven thirtieths of the line DA. The line DA is therefore thirty seconds of the Earth's Polar circumference. The line DA is therefore 3033·118625 British feet, and the base of Cephren 707·727 British feet.

I applied a variety of Cubits but found none to work in _without fractions_ on the beautiful set of natural dimensions which I had worked out for my plan. (_See table of connected natural numbers._)

I ultimately arrived at a cubit as the ancient measure which I have called the R.B. cubit, because it closely resembles the Royal Babylonian Cubit of ·5131 metre, or 1·683399 British feet. The difference is 1/600 of a foot.

I arrived at the R.B. cubit in the following manner.

Taking the polar axis of the earth at five hundred million geometric inches, thirty seconds of the circumference will be 36361·02608--geometric inches, or 36397·4235 British inches, at nine hundred and ninety-nine to the thousand--and 3030·0855 geometric feet, or 3033·118625 British feet. Now dividing a second into sixty parts, there are 1800 R.B. cubits in the line DA; and the line DA being thirty seconds, measures 36397·4235 British inches, which divided by 1800 makes one of my cubits 20·2207908 British inches, or 1·685066 British feet. Similarly, 36361·02608 geometric inches divided by 1800 makes my cubit 20·20057 geometric inches in length. I have therefore defined this cubit as follows:--One R.B. cubit is equal to 20.2006 geo. inches, 20·2208 Brit. inches, and 1·685 Brit. feet.

I now construct the following table of measures.

+---------+----------+--------+--------+--------+ | R. B. |PLETHRA OR| | | | | CUBITS. | SECONDS. |STADIA. |MINUTES.|DEGREES.| +---------+----------+--------+--------+--------+ | 60 | 1 | | | | | 360 | 6 | 1 | | | | 3600 | 60 | 10 | 1 | | | 216000 | 3600 | 600 | 60 | 1 | |77760000 | 1296000 | 216000 | 21600 | 360 | +---------+----------+--------+--------+--------+

Thus there are seventy-seven million, seven hundred and sixty thousand R.B. cubits, or two hundred and sixteen thousand stadia, to the Polar circumference of the earth.

Thus we have obtained a perfect set of natural and convenient measures which fits the plan, and fits the circumference of the earth.

And I claim for the R.B. cubit that it is the most perfect ancient measure yet discovered, being the measure of the plan of the Pyramids of Gïzeh.

The same forgotten wisdom which divided the circle into three hundred and sixty degrees, the degree into sixty minutes, and the minute into sixty seconds, subdivided those seconds, for earth measurements, into the sixty parts represented by sixty R.B. cubits.

We are aware that thirds and fourths were used in ancient astronomical calculations.

* * * * *

The reader will now observe that the cubit measures of the main Pythagorean triangle of the plan are obtained by multiplying the original 3, 4 and 5 by 360; and that the entire dimensions are obtained in R.B. cubits by multiplying the last column of connected natural numbers in the table by eight,--thus--

R. B. CUBITS. DC 3 × 360 = 1080 CA 4 × 360 = 1440 DA 5 × 360 = 1800

or,

NATURAL R. B. NUMBERS. CUBITS. DC 135 × 8 = 1080 CA 180 × 8 = 1440 DA 225 × 8 = 1800 &c., &c.

(_See Figure 5, p. 18._)

According to Cassini, a degree was 600 stadia, a minute 10 stadia; and a modern Italian mile, in the year 1723, was equal to one and a quarter ancient Roman miles; and one and a quarter ancient Roman miles were equal to ten stadia or one minute. (_Cassini, Traite de la grandeur et de la Figure de la Terre. Amsterdam, 1723._)

_Fig. 5_

_R.B.Cubits._

FB = 640 EB = 504 BA = 672 BA = 672 AF = 928 AE = 840

DC = 1080 NO = 768 CA = 1440 OF = 1024 AD = 1800 FN = 1280

DG = 576 AY = 288 GE = 768 ZY = 384 ED = 960 ZA = 480

_R.B. Cub._

_At level of Cephren's_ } {_Cheops' Base_ 420 _Base which is the plane_} {_Cephren's Base_ 420 _or level of the plan_.--} {_Mycerinus' Base_ 218

Dufeu also made a stadium the six hundredth part of a degree. He made the degree 110827·68 metres, which multiplied by 3·280841 gives 363607·996+ British feet; and 363607·996+ divided by 600 equals 606·013327 feet to his stadium.

I make the stadium 606·62376 British feet.

There being 360 cubits to a stadium, Dufeu's stadium divided by 360, gives 1·6833 British feet, which is the exact measure given for a Royal Babylonian Cubit, if reduced to metres, viz.: 0·5131 of a metre, and therefore probably the origin of the measure called the Royal Babylonian cubit. According to this measure, the Gïzeh plan would be about 1/1011 smaller than if measured by R.B. cubits.

§ 3. THE EXACT MEASURE OF THE BASES OF THE PYRAMIDS.

A stadium being 360 R.B. cubits, or six seconds--and a plethron 60 R.B. cubits, or one second, the base of the Pyramid Cephren is seven plethra, or a stadium and a plethron, equal to seven seconds, or four hundred and twenty R.B. cubits.

Mycerinus' base is acknowledged to be half the base of Cephren.

Piazzi Smyth makes the base of the Pyramid Cheops 9131·05 pyramid (or geometric) inches, which divided by 20·2006 gives 452·01 R.B. cubits. I call it 452 cubits, and accept it as the measure which exactly fits the plan.

I have not sufficient data to determine the exact base of the other and smaller pyramid which I have marked on my plan.

The bases, then, of Mycerinus, Cephren, and Cheops, are 210, 420 and 452 cubits, respectively.

But in plan the bases should be reduced to one level. I have therefore drawn my plan, or horizontal section, at the level or plane of the base of Cephren, at which level or plane the bases or horizontal sections of the pyramids are--Mycerinus, 218 cubits, Cephren, 420 cubits, and Cheops, 420 cubits. I shall show how I arrive at this by-and-by, and shall also show that the horizontal section of Cheops, corresponding to the horizontal section of Cephren at the level of Cephren's base, occurs, as it should do, at the level of one of the courses of masonry, viz.--the top of the tenth course.

§ 4. THE SLOPES, RATIOS, AND ANGLES OF THE THREE PRINCIPAL PYRAMIDS OF THE GIZEH GROUP.

Before entering on the description of the exact slopes and angles of the three principal pyramids, I must premise that I was guided to my conclusions by making full use of the combined evolutions of the two wonderful right-angled triangles, 3, 4, 5, and 20, 21, 29, which seem to run through the whole design as a sort of dominant.

From the first I was firmly convinced that in such skilful workmanship some very simple and easily applied templates must have been employed, and so it turned out. Builders do not mark a dimension on a plan which they cannot measure, nor have a hidden measure of any importance without some clear outer way of establishing it.

This made me "go straight" for the slant ratios. When the Pyramids were cased from top to bottom with polished marble, there were only two feasible measures, the bases and the apothems;[1] and for that reason I conjectured that these would be the definite plan ratios.

Footnote 1: The "_Apothem_ is a perpendicular from the vertex of a pyramid on a side of the base."--_Chambers' Practical Mathematics, p. 156._

* * * * *

Figures 6, 7 and 8 show the _exact_ slope ratios of Cheops, Cephren, and Mycerinus, measured as shown on the diagrams--viz., Cheops, 21 to 34, Cephren, 20 to 33, and Mycerinus, 20 to 32--that is, half base to apothem.

Fig. 6 Cheops. _Note. The Ratios of Bases to Altitudes are very Fig. 7 Cephren. nearly as follows, viz_:--

Fig. 8 Mycerinus. _Cheops 33 to 21 or 330 to 210_ _Cephren 32 to 21 or 320 to 210_ _Mycerinus 32 to 20 or 336 to 210_

The ratios of base to altitude are, Cheops, 33 to 21, Cephren, 32 to 21, and Mycerinus, 32 to 20: not exactly, but near enough for all practical purposes. For the sake of comparison, it will be well to call these ratios 330 to 210, 320 to 210, and 336 to 210, respectively.

* * * * *

Fig. 9. Cheops. Fig. 10. Cheops.

Figures 9 and 10 are meridional and diagonal sections, showing ratios of Cheops, viz., half base to apothem, 21 to 34 _exactly_; half base to altitude, 5½ to 7 nearly, and 183 to 233, nearer still (being the ratio of Piazzi Smyth). The ratio of Sir F. James, half diagonal 10 to altitude 9 is also very nearly correct.

My altitude for Cheops is 484·887 British feet, and the half base 380·81 British feet.

The ratio of 7 to 5½ gives 484·66, and the ratio of 233 to 183 gives 484·85 for the altitude.

My half diagonal is 538·5465, and ratio 10 to 9, gives 484·69 British feet for the altitude.

I have mentioned the above to show how very nearly these ratios agree with my exact ratio of 21 to 34 half base to apothem.

* * * * *

Fig. 11. Cephren. Fig. 12. Cephren.

Figures 11 and 12 show the ratios of Cephren, viz., half base to apothem, _20 to 33 exactly_, and half base, altitude, and apothem respectively, as 80, 105, and 132, very nearly.

Also half diagonal, altitude, and edge, practically as 431, 400, and 588.

* * * * *

Fig. 13. Mycerinus. Fig. 14. Mycerinus.

Figures 13 and 14 show the ratios of Mycerinus, viz., half base to apothem, 20 _to_ 32 _exactly_, and half base, altitude, and apothem respectively, as 20, 25, and 32 very nearly.

Also full diagonal to edge as 297 to 198, nearly. A peculiarity of this pyramid is, that base is to altitude as apothem is to half base. Thus, 40 : 25 :: 32 : 20; that is, half base is a fourth proportional to base, apothem, and altitude.

§ 5. THE EXACT DIMENSIONS OF THE PYRAMIDS.

R.B. Cub. Brit. Ft. Fig. 15. Cheops. 452 = 761·62 287·767 = 484·887 Fig. 16. Cheops. 365·9047 = 616·549 430·058 = 724·647 639·2244 = 1077·093

Figures 15 to 20 inclusive, show the linear dimensions of the three pyramids, also their angles. The base angles are, Cheops, 51° 51′ 20"; Cephren, 52° 41′ 41″; and Mycerinus, 51° 19′ 4″.

R.B. Cub. Brit. Ft. Fig. 17. Cephren. 420 = 707·70 275·61 = 464·40 Fig. 18. Cephren. 346·50 = 583·85 405·16 = 682·69 593·97 = 1000·84

R.B. Cub. Brit. Ft.

Fig. 19. Mycerinus. 210 = 353.85 168 = 283.08 Fig. 20. Mycerinus: 131.14 = 220.97 198.10 = 333·7985 296·9848 = 500·42

In Cheops, my dimensions agree with Piazzi Smyth--in the base of Cephren, with Vyse and Perring--in the height of Cephren, with Sir Gardner Wilkinson, nearly--in the base of Mycerinus, they agree with the usually accepted measures, and in the height of Mycerinus, they exceed Jas. J. Wild's measure, by not quite one of my cubits.

In my angles I agree very nearly with Piazzi Smyth, for Cheops, and with Agnew, for Cephren, differing about half a degree from Agnew, for Mycerinus, who took this pyramid to represent the same relation of [Pi] that P. Smyth ascribes to Cheops (viz.: 51° 51′ 14·″3), while he gave Cheops about the same angle which I ascribe to Mycerinus.

I shall now show how I make Cephren and Cheops of equal bases of 420 R.B. cubits at the same level, viz.--that of Cephren's base.

John James Wild made the bases of Cheops, Cephren, and Mycerinus, respectively, 80, 100, and 104·90 cubits above some point that he called Nile Level.