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With the respectful compliments of the author.
Address,
Mr. R. BALLARD,
Malvern,
ENGLAND.
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.
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.
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.)
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.)
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.
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.
I may also point out on the same plan that calling the line FA radius, and the lines BA
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.
Cephren
Mycerīnus 56½
52½
26¼ 52½
52½
27¼
I now establish the following list of measurements of the plan in connected natural numbers. (See Figure 4.)
Plan Ratios connected into Natural Numbers.
YZ 1
1 } 48 48
48 BC
FB 6
5 } 16 96
80 DC
BC 45
32 } 3 135
96 FB
BY 5
3 } 16 80
48 DN
NR 61
60 } 3 183
180 DN
NZ 61
48 } 3 183
144 CY
BC 3
2 } 48 144
96 FH
DH 96
55 } 1 96
55 CY
DC 16
25 } 9 144
135 PN
PA 61
48 } 2.4 146.4
115.2 JE
EX 3
2 } 24 72
48 YX
AY 7
4 } 9 63
36 BA
FB 21
20 } 4 84
80 CA
DC 4
3 } 45 180
135 BC
EB 32
21 } 3 96
63 EA
AZ 7
4 } 15 105
60 CB
BA 8
7 } 12 96
84 YZ
AY 4
3 } 12 48
36 FO
OR 32
21 } 4 128
84 AB
BO 7
4 } 12 84
48 ED
AE 8
7 } 15 120
105 BA
EB 4
3 } 21 84
63 FT
ST 84
55 } 1 84
55 BC
AC 8
15 } 12 96
180 VW
FW
48 } 1 55
48 GE
DG 4
3 } 24 96
72 VW
SV 55
36 } 1 55
36 ND
NO 61
32 } 3 183
96 SJ
SU 7
6 } 12 84
72 HN
FH 4
3 } 32 128
96 BJ
AB 45
28 } 3 135
84 PA
AZ 48
25 } 2.4 115.2
60 GX
DG 2
1 } 72 144
72 GU
DG 5
2 } 36 180
72 EO
AY 37
12 } 3 111
36 SR
RZ 61
12 } 3 183
36 SU
SV 2
1 } 36 72
36 HW
DH 144
55 } 1 144
55 HT
DH 36
11 } 5 180
55 FH
FE 96
17 } 1 96
17 TW
TU 36
17 } 1 36
17 FO
OZ 8
3 } 16 128
48 DA
AZ 15
4 } 15 225
60 EA
EF 105
17 } 1 105
17 SR
RO 61
28 } 3 183
84 JB
BY 45
16 } 3 135
48 AC
CN 15
4 } 12 180
48 WH
HG 144
17 } 1 144
17 YW
AY 20
9 } 4 80
36 FW
FE 48
17 } 1 48
17 YV
AY 15
4 } 9 135
36 TH
HG 180
17 } 1 180
17 MY
ZY 9
4 } 12 108
48 AC
CG 20
7 } 9 180
63 VZ
ZO 61
16 }
48 AC
CH 9
4 } 20 180
80 EA
AY 35
12 } 3 105
36 EU
FE 84
17 } 1 84
17 NZ
ZA 12
5 } 12 144
60 CY
YZ 3
1 } 48 144
48 CA
AY 5
1 } 36 180
36
The above connected natural numbers multiplied by eight become
§ 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.
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—
CUBITS. DC 3 × 360 = 1080 CA 4 × 360 = 1440 DA 5 × 360 = 1800
or,
NUMBERS. R. B.
CUBITS. DC 135 × 8 = 1080 CA 80 × 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.)
Cubits. 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
the plane or level of the plan.— } { Cheops' Base
Cephren's Base
Mycerīnus' Base 420
420
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.