Produced by Adrian Mastronardi, Hazel Batey and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive/American Libraries.)
_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.
His cubit was, I believe, the Memphis, or Nilometric cubit--but at any
rate, he made the base of Cephren 412 of them.
I therefore divided the recognized base of Cephren--viz., 707·75 British
feet--by 412, and got a result of 1·7178 British feet for his cubit.
Therefore, his measures multiplied by 1·7178 and divided by 1·685 will
turn his cubits into R.B. cubits.
I thus make Cheops, Cephren, and Mycerinus, respectively, 81·56,
101·93, and 106·93 R.B. cubits above the datum that J. J. Wild calls
Nile Level. According to Bonwick's "Facts and Fancies," p. 31, high
water Nile would be 138½ ft. below base of Cheops (or 82·19 R.B.
cubits).
Piazzi Smyth makes the pavement of Cheops 1752 British inches (or 86·64
R.B. cubits) above _average Nile Level_, but, by scaling his map, his
_high Nile Level_ appears to agree nearly with Wild.
It is the _relative levels_ of the Pyramids, however, that I require, no
matter how much above Nile Level.
Cephren's base of 420 cubits being 101·93 cubits, and Cheops' base of
452 cubits being 81·56 cubits above Wild's datum, the difference in
level of their bases is, 20·37 cubits.
The ratio of base to altitude of Cheops being 330 to 210, therefore
20·37 cubits divided by 210 and multiplied by 330 equals 32 cubits; and
452 cubits minus 32 cubits, equals 420.
Similarly, the base of Mycerinus is 5 cubits _above_ the base of
Cephren, and the ratio of base to altitude 32 to 20; therefore, 5 cubits
divided by 20 and multiplied by 32 equals 8 cubits to be _added_ to the
210 cubit base of Mycerinus, making it 218 cubits in breadth at the
level of Cephren's base.
Thus, a horizontal section or plan at the level of Cephren's base would
meet the slopes of the Pyramids so that they would on plan appear as
squares with sides equal to 218, 420, and 420 R.B. cubits, for
Mycerinus, Cephren, and Cheops, respectively.
Fig. 21.
R.B. Cub.
Apex of Cephren above Base Cheops 295·98
Apex of Cheops above Base Cheops 287·77
Apex of Mycerinus above Base Cheops 156·51
Base Cephren above base of Cheops 20·37
Base Mycerinus above base of Cheops 25·37
Piazzi Smyth makes the top of the tenth course of Cheops 414 pyramid
inches above the pavement; and 414 divided by 20·2006 equals 20·49 R.B.
cubits.
But I have already proved that Cheops' 420 cubit base measure occurs at
a level of 20·37 cubits above pavement; therefore is this level the
level of the top of the tenth course, for the difference is only 0·12
R.B. cubits, or 2½ inches.
* * * * *
I wish here to note as a matter of interest, but not as affecting my
theory, the following measures of Piazzi Smyth, turned into R.B. cubits,
viz.:--
PYR. INCHES. R.B. CUBITS.
King's Chamber floor, above pavement 1702· = 84·25
Cheops' Base, as before stated 9131·05 = 452·01
King's Chamber, "True Length," 412·132 = 20·40
" " "True First Height," 230·389 = 11·40
" " "True Breadth," 206·066 = 10·20
He makes the present summit platform of Cheops 5445 pyramid inches above
pavement. My calculation of 269·80 R.B. cub. (See Fig. 21) is equal to
5450 pyramid inches--this is about 18 cubits below the theoretical apex.
* * * * *
Figure 21 represents the comparative levels and dimensions of Mycerinus,
Cephren, and Cheops.
The following peculiarities are noticeable:--That Cheops and Cephren are
of equal bases at the level of Cephren's base;--that, at the level of
Cheops' base, the latter is only half a cubit larger;--that, from the
level of Mycerinus' base, Cheops is just double the height of
Mycerinus;--and that from the level of Cephren's base, Cephren is just
double the height of Mycerinus; measuring in the latter case, however,
only up to the level platform at the summit of Cephren, which is said to
be about eight feet wide.
The present summit of Cephren is 23·07 cubits above the present summit
of Cheops, and the completed apex of Cephren would be 8·21 cubits above
the completed apex of Cheops.
In the summit platforms I have been guided by P. Smyth's estimate of
_height deficient_, 363 pyr. inches, for Cheops, and I have taken 8 feet
base for Cephren's summit platform.
§ 6. GEOMETRICAL PECULIARITIES OF THE PYRAMIDS.
_In any pyramid, the apothem is to half the base as the area of the four
sides is to the area of the base._
Thus--Ratio apothem to half base Mycerinus 32 to 20
" " " " Cephren 33 to 20
" " " " Cheops 34 to 21
AREA OF THE FOUR SIDES. AREA OF THE BASE.
Mycerinus 70560· 44100
Cephren 291060· 176400
Cheops 330777·90 204304
All in R.B. cubits.
Therefore--32 : 20 : : 70560· : 44100
33 : 20 : : 291060· : 176400
34 : 21 : : 330777·90 : 204304
[2]Herodotus states that "_the area of each of the four faces of Cheops
was equal to the area of a square whose base was the altitude of a
Pyramid;_" or, in other words, that altitude was a mean proportional to
apothem and half base; thus--area of one face equals the fourth of
330777·90 or 82694·475 R.B. cubits, and the square root of 82694·475 is
287·56. But the correct altitude is 287·77, so the error is 0·21, or
4¼ British inches. I have therefore the authority of Herodotus to
support the theory which I shall subsequently set forth, that this
pyramid was the exponent of lines divided in mean and extreme ratio.
By taking the dimensions of the Pyramid from what I may call its
_working level_, that is, the level of the base of Cephren, this
peculiarity shows more clearly, as also others to which I shall refer.
Thus--base of Cheops at working level, 420 cubits, and apothem 340
cubits; base area is, therefore, 176400 cubits, and area of one face is
(420 cubits, multiplied by half apothem, or 170 cubits) 71400 cubits.
Now the square root of 71400 would give altitude, or side of square
equal to altitude, 267·207784 cubits: but the real altitude is
√(340²-210²) = √71500 = 267·394839. So that the error of Herodotus's
proposition is the difference between √714 and √715.
Footnote 2: Proctor is responsible for this statement, as I am
quoting from an essay of his in the _Gentleman's Magazine_. R. B.
This leads to a consideration of the properties of the angle formed by
the ratio _apothem_ 34 to _half base_ 21, peculiar to the pyramid
Cheops. (_See Figure 22._)
Fig. 22. Diagram illustrating relations of ratios of the pyramid Cheops.
Calling apothem 34, _radius_; and half base 21, _sine_--I find that--
Radius is the square root of 1156
Sine 441
Co-sine 715
Tangent 713
Secant 1869
and Co-versed-sine 169
So it follows that the area of one of the faces, 714, is a mean between
the square of the altitude or co-sine, 715, and the square of the
tangent, 713.
Thus the reader will notice that the peculiarities of the Pyramid
Cheops lie in the regular relations of the _squares_ of its various
lines; while the peculiarities of the other two pyramids lie in the
relations of the lines themselves.
Mycerinus and Cephren, born, as one may say, of those two noble
triangles 3, 4, 5, and 20, 21, 29, exhibit in their lineal developments
ratios so nearly perfect that, for all practical purposes, they may be
called correct.
Thus--Mycerinus, [3]20² + 25² = 1025, and 32² = 1024.
and Cephren, [4]80² + 105² = 17425, and 132² = 17424.
or [5]400² + 431² = 345761, and 588² = 345744.
See diagrams, Figures 11 to 14 inclusive.
In the Pyramid Cheops, altitude is _very nearly_ a mean proportional
between apothem and half base. Apothem being 34, and half base 21, then
altitude would be √(34²-21²) = √715 = 26·7394839, and--
21 : 26·7394839 :: 26·7394839 : 34, nearly.
Here, of course, the same difference comes in as occurred in
considering the assumption of Herodotus, viz., the difference
between √715 and √714; because if the altitude were √714, then would
it be _exactly_ a mean proportional between the half base and the
apothem; (thus, 21 : 26·72077 :: 26·72077 :: 34.)
Footnote 3: Half base to altitude.
Footnote 4: Half base to altitude.
Footnote 5: Half diagonal of base to altitude.
In Cheops, the ratios of apothem, half base and edge are, 34, 21, and
40, very nearly, thus, 34² + 21² = 1597, and 40² = 1600.
The dimensions of Cheops (from the level of the base of Cephren) to be
what Piazzi Smyth calls a [Pi] pyramid, would be--
Half base 210 R.B. cubits.
Altitude 267·380304, &c. "
Apothem 339·988573, &c. "
Altitude being to perimeter of base, as radius of a circle to
circumference.
My dimensions of the pyramid therefore in which--
Half base = 210 R.B. cubits.
Altitude = 267·394839 &c. "
Apothem = 340 "
come about as near to the ratio of [Pi] as it is possible to come, and
provide simple lines and templates to the workmen in constructing the
building; and I entertain no doubt that on the simple lines and
templates that my ratios provide, were these three pyramids built.
§ 6A. THE CASING STONES OF THE PYRAMIDS.
Figures 23, 24, and 25, represent ordinary casing stones of the three
pyramids, and Figures 26, 27, and 28, represent angle or quoin casing
stones of the same.
The casing stone of Cheops, found by Colonel Vyse, is represented in
Bonwick's "Pyramid Facts and Fancies," page 16, as measuring four feet
three inches at the top, eight feet three inches at the bottom, four
feet eleven inches at the back, and six feet three inches at the
front. Taking four feet eleven inches as _Radius_, and six feet three
inches as _Secant_, then the _Tangent_ is three feet ten inches and
three tenths.
Fig. 23. Cheops Casing Stone.
Fig. 24. Cephren Casing Stone.
Fig. 25. Mycerinus Casing Stone.
Fig. 26. Cheops Angle or Quoin Stone.
Fig. 27. Cephren Angle or Quoin Stone.
Fig. 28. Mycerinus Angle or Quoin Stone.
Thus, in inches (√(75²-59²)) = 46·30 inches; therefore the inclination
of the stone must have been--slant height 75 inches to 46·30 inches
horizontal. Now, 46·30 is to 75, as 21 is to 34. Therefore, Col. Vyse's
casing stone agrees exactly with my ratio for the Pyramid Cheops, viz.,
21 to 34. (_See Figure 29._)
Fig. 29. =Col. Vyse's Casing Stone.= 75 : 46·3 :: 34 : 21
This stone must have been out of plumb at the back an inch and seven
tenths; perhaps to give room for grouting the back joint of the marble
casing stone to the limestone body of the work: or, because, as it is
not a necessity in good masonry that the back of a stone should be
exactly plumb, so long as the error is on the right side, the builders
might not have been particular in that respect.
Fig. 59. (Temple of Cheops, standing at angle of wall.)
Figure 59 represents such a template as the masons would have used in
building Cheops, both for dressing and setting the stones. (The courses
are drawn out of proportion to the template.) The other pyramids must
have been built by the aid of similar templates.
Such large blocks of stone as were used in the casing of these pyramids
could not have been completely dressed before setting; the back and
ends, and the top and bottom beds were probably dressed off truly, and
the face roughly scabbled off; but the true slope angle could not have
been dressed off until the stone had been truly set and bedded,
otherwise there would have been great danger to the sharp arises.
* * * * *
I shall now record the peculiarities of the 3, 4, 5 or Pythagorean
triangle, and the right-angled triangle 20, 21, 29.
§ 7. PECULIARITIES OF THE TRIANGLES 3, 4, 5, AND 20, 21, 29.
Fig. 30 to 35. PECULIARITIES OF THE TRIANGLES
The 3, 4, 5 triangle contains 36° 52′ 11·65″ and the complement or
greater angle 53° 7′ 48·35″
Radius 5 = 60 whole numbers.[6]
Co-sine 4 = 48"
Sine 3 = 36"
Versed sine 1 = 12"
Co-versed sine 2 = 24"
Tangent 3¾ = 45"
Secant 6¼ = 75"
Co-tangent 6⅔ = 80"
Co-secant 8⅓ = 100"
Tangent + Secant = Diameter or 2 Radius
Co-tan + Co-sec = 3 Radius
Sine : Versed-sine :: 3 : 1
Co-sine : Co-versed sine :: 2 : 1
Figure 30 illustrates the preceding description. Figure 31 shows the 3·1
triangle, and the 2·1 triangle built up on the sine and co-sine of the
3, 4, 5 triangle.
The 3·1 triangle contains 18° 26′ 5·82″ and the 2·1 triangle
26° 33′ 54·19″; the latter has been frequently noticed as a
pyramid angle in the gallery inclinations.
Figure 32 shows these two triangles combined with the 3, 4, 5 triangle,
on the circumference of a circle.
Footnote 6: 60 = 3 × 4 × 5
The 20, 21, 29 triangle contains 43° 36′ 10·15″ and the
complement, 46° 23′ 49·85″.
Expressed in whole numbers--
Radius 29 = 12180[7]
Sine 20 = 8400
Co-sine 21 = 8820
Versed sine 8 = 3360
Co-versed sine 9 = 3780
Tangent = 11600
Co-tangent = 12789
Secant = 16820
Co-sec = 17661
Tangent + Secant = 2⅓ radius
Co-tan + Co-sec = 2½ radius
Sine : Versed sine :: 5 : 2
Co-sine : Co-versed sine :: 7 : 3
Footnote 7: 12180 = 20 × 21 × 29
It is noticeable that while the multiplier required to bring radius 5
and the rest into whole numbers, for the 3, 4, 5 triangle is twelve, in
the 20, 21, 29 triangle it is 420, the key measure for the bases of the
two main pyramids in R.B. cubits.[8]
Footnote 8: 12 = 3 × 4, and 420 = 20 × 21
I am led to believe from study of the plan, and consideration of the
whole numbers in this 20, 21, 29 triangle, that the R.B. cubit, like the
Memphis cubit, was divided into 280 parts.
The whole numbers of radius, sine, and co-sine divided by 280, give a
very pretty measure and series in R.B. cubits, viz., 43½, 30, and
31½, or 87, 60, and 63, or 174, 120 and 126;--all exceedingly useful
in right-angled measurements. Notice that the right-angled triangle 174,
120, 126, in the sum of its sides _amounts to_ 420.
Figure 33 illustrates the 20, 21, 29 triangle. Figure 34 shows the 5·2
and 7·3 triangles built up on the sine and co-sine of the 20, 21, 29
triangle.
The 5·2 triangle contains 21° 48′ 5·08″ and the 7·3 triangle
23° 11′ 54·98″.
Figure 35 shows how these two triangles are combined with the 20, 21, 29
triangle on the circumference, and Figure 36 gives a general view and
identification of these six triangles which occupied an important
position in the trigonometry of a people who did all their work by right
angles and proportional lines.
Fig. 36. Ratios of Leading Triangles.
§ 8. GENERAL OBSERVATIONS.
It must be admitted that in the details of the building of the Pyramids
of Gïzeh there are traces of other measures than R. B. cubits, but that
the original cubit of the plan was 1·685 British feet I feel no doubt.
It is a perfect and beautiful measure, fit for such a noble design, and,
representing as it does the sixtieth part of a second of the Earth's
polar circumference, it is and was a measure for all time.
It may be objected that these ancient geometricians could not have been
aware of the measure of the Earth's circumference; and wisely so, were
it not for two distinct answers that arise. The first being, that since
I think I have shown that Pythagoras never discovered the Pythagorean
triangle, but that it must have been known and practically employed
thousands of years before his era, in the Egyptian Colleges where he
obtained his M.A. degree, so in the same way it is probable that
Eratosthenes, when he went to work to prove that the earth's
circumference was fifty times the distance from Syene to Alexandria, may
have obtained the idea from his ready access to the ill-fated
Alexandrian Library, in which perhaps some record of the learning of the
builders of the Pyramids was stored. And therefore I claim that there is
no reason why the pyramid builders should not have known as much about
the circumference of the earth as the modern world that has calmly stood
by in its ignorance and permitted those magnificent and, as I shall
prove, useful edifices to be stripped of their beautiful garments of
polished marble.
My second answer is that the correct cubit measure may have been got by
its inventors in a variety of other ways; for instance, by observations
of shadows of heavenly bodies, without any knowledge even that the earth
was round; or it may have been evolved like the British inch, which Sir
John Herschel tells us is within a thousandth part of being one five
hundred millionth of the earth's polar axis. I doubt if the
circumference of the earth was considered by the inventor of the British
inch.
It was a peculiarity of the Hindoo mathematicians that they tried to
make out that all they knew was _very old_. Modern savants appear to
take the opposite stand for any little information they happen to
possess.
The cubit which is called the Royal Babylonian cubit and stated to
measure O·5131 metre, differs so slightly from my cubit, only the
six-hundredth part of a foot, that it may fairly be said to be the same
cubit, and it will be for antiquaries to trace the connection, as this
may throw some light on the identity of the builders of the Pyramids of
Gïzeh. Few good English two-foot rules agree better than these two
cubits do.
While I was groping about in the dark searching for this bright needle,
I tried on the plan many likely ancient measures.
For a long time I worked in Memphis or Nilometric cubits, which I made
1·7126 British feet; they seem to vary from 1·70 to 1·72, and although I
made good use of them in identifying other people's measures, still they
were evidently not in accordance with the design; but the R.B. cubit of
1·685 British feet works as truly into the plan of the Pyramids _without
fractions_ as it does into the circumference of the earth.
Here I might, to prevent others from falling into one of my errors,
point out a rock on which I was aground for a long time. I took the base
of the Pyramid Cheops, determined by Piazzi Smyth, from Bonwick's
"Pyramid Facts and Fancies" (a valuable little reference book), as
763.81 British feet, and the altitude as 486.2567; and then from Piazzi
Smyth's "Inheritance," page 27, I confirmed these figures, and so worked
on them for a long time, but found always a great flaw in my work, and
at last adopted a fresh base for Cheops, feeling sure that Mr. Smyth's
base was wrong: for I was absolutely grounded in my conviction that at a
certain level, Cheops' and Cephren's measures bore certain relations to
each other. I subsequently found in another part of Mr. Smyth's book,
that the correct measures were 761.65 and 484.91 British feet for base
and altitude, which were exactly what I wanted, and enabled me to be in
accordance with him in that pyramid which he appears to have made his
particular study.
For the information of those who may wish to compare my measures, which
are the results of an even or regular circumference without fractions,
with Mr. Smyth's measures, which are the results of an even or regular
diameter without fractions, it may be well to state that there are just
about 99 R.B. cubits in 80 of Piazzi Smyth's cubits of 25 pyramid inches
each.
§ 9. THE PYRAMIDS OF EGYPT, THE THEODOLITES OF THE EGYPTIAN LAND
SURVEYORS.
About twenty-three years ago, on my road to Australia, I was crossing
from Alexandria to Cairo, and saw the pyramids of Gïzeh.
I watched them carefully as the train passed along, noticed their clear
cut lines against the sky, and their constantly changing relative
position.
I then felt a strong conviction that they were built for at least one
useful purpose, and that purpose was the survey of the country. I said,
"Here be the Theodolites of the Egyptians."
Built by scientific men, well versed in geometry, but unacquainted with
the use of glass lenses, these great stone monuments are so suited in
shape for the purposes of land surveying, that the practical engineer or
surveyor must, after consideration, admit that they may have been built
mainly for that purpose.
Not only might the country have been surveyed by these great
instruments, and the land allotted at periodical times to the people;
but they, remaining always in one position, were there to correct and
readjust boundaries destroyed or confused by the annual inundations of
the Nile.
The Pyramids of Egypt may be considered as a great system of landmarks
for the establishment and easy readjustment at any time of the
boundaries of the holdings of the people.
The Pyramids of Gïzeh appear to have been main marks; and those of
Abousir, Sakkârah, Dashow, Lisht, Meydoun, &c., with the great pyramids
in Lake Moeris, subordinate marks, in this system, which was probably
extended from Chaldea through Egypt into Ethiopia.
The pyramid builders may perhaps have made the entombment of their Kings
one of their exoteric objects, playing on the morbid vanity of their
rulers to induce them to the work, but in the minds of the builders
before ever they built must have been planted the intention to make use
of the structures for the purposes of land surveying.
The land of Egypt was valuable and maintained a dense population; every
year it was mostly submerged, and the boundaries destroyed or confused.
Every soldier had six to twelve acres of land; the priests had their
slice of the land too; after every war a reallotment of the lands must
have taken place, perhaps every year.
While the water was lying on the land, it so softened the ground that
the stone boundary marks must have required frequent readjustment, as
they would have been likely to fall on one side.
By the aid of their great stone theodolites, the surveyors, who
belonged to the priestly order, were able to readjust the boundaries
with great precision. That all science was comprised in their secret
mysteries may be one reason why no hieroglyphic record of the scientific
uses of the pyramids remains. It is possible that at the time of
Diodorus and Herodotus, (and even when Pythagoras visited Egypt,)
theology may have so smothered science, that the uses of the pyramids
may have been forgotten by the very priests to whom in former times the
knowledge belonged; but "a respectful reticence" which has been noticed
in some of these old writers on pyramid and other priestly matters would
rather lead us to believe that an initiation into the mysteries may have
sealed their lips on subjects about which they might otherwise have been
more explicit.
The "_closing_" of one pyramid over another in bringing any of their
many lines into true order, must even now be very perfect;--but now we
can only imagine the beauties of these great instrumental wonders of the
world when the casing stones were on them. We can picture the rosy
lights of one, and the bright white lights of others; their clear cut
lines against the sky, true as the hairs of a theodolite; and the sombre
darkness of the contrasting shades, bringing out their angles with
startling distinctness. Under the influence of the Eastern sun, the
faces must have been a very blaze of light, and could have been seen at
enormous distances like great mirrors.
I declare that the pyramids of Gïzeh in all their polished glory,
before the destroyer stripped them of their beautiful garments, were in
every respect adapted to flash around clearly defined lines of sight,
upon which the lands of the nation could be accurately threaded. The
very thought of these mighty theodolites of the old Egyptians fills me
with wonder and reverence. What perfect and beautiful instruments they
were! never out of adjustment, always correct, always ready; no magnetic
deviation to allow for. No wonder they took the trouble they did to
build them so correctly in their so marvellously suitable positions.
If Astronomers agree that observations of a pole star could have been
accurately made by peering up a small gallery on the north side of one
of the pyramids only a few hundred feet in length, I feel that I shall
have little difficulty in satisfying them that accurate measurements to
points only _miles_ away could have been made from angular observations
of the whole group.
§ 10. HOW THE PYRAMIDS WERE MADE USE OF.
It appears from what I have already set forth that the plan of the
Pyramids under consideration is geometrically exact, a perfect set of
measures.
I shall now show how these edifices were applied to a thoroughly
geometrical purpose in the true meaning of the word--to measure the
Earth.
I shall show how true straight lines could be extended from the Pyramids
in given directions useful in right-angled trigonometry, by direct
observation of the buildings, and without the aid of other instruments.
And I shall show how by the aid of a simple instrument angles could be
exactly observed from any point.
This Survey theory does not stand or fall on the merits of my theory of
the Gïzeh plan. Let it be proved that this group is not built on the
exact system of triangulation set forth by me, it is still a fact that
its plan is in a similar shape, and any such shape would enable a
surveyor acquainted with the plan to lay down accurate surveys by
observations of the group even should it not occupy the precise lines
assumed by me.
And here I must state that although the lines of the plan as laid down
herein agree nearly with the lines as laid down in Piazzi Smyth's book,
in the Penny Cyclopædia, and in an essay of Proctor's in the
_Gentleman's Magazine_, still I find that they do not agree at all
satisfactorily with a map of the Pyramids in Sharp's "Egypt," said to be
copied from Wilkinson's map.
We will, however, for the time, and to explain my survey theory, suppose
the plan theory to be correct, as I firmly believe it is.
And then, supposing it may be proved that the respective positions of
the pyramids are slightly different to those that I have allotted to
them on my plan, it will only make a similar slight difference to the
lines and angles which I shall here show could be laid out by their aid.
Let us in the first place comprehend clearly the shape of the land of
Egypt.
A sector or fan, with a long handle--the fan or sector, the delta; and
the handle of the fan, the Nile Valley, running nearly due south.
The Pyramids of Gïzeh are situate at the angle of the sector, on a rocky
eminence whence they can all be seen for many miles. The summits of the
two high ones can be seen from the delta, and from the Nile Valley to a
very great distance; how far, I am unable to say; but I should think
that while the group could be made general use of for a radius of
fifteen miles, the summits of Cephren and Cheops could be made use of
for a distance of thirty miles; taking into consideration the general
fall of the country.
It must be admitted that if meridian observations of the star Alpha of
the Dragon could be made with accuracy by peeping up a small hole in one
of the pyramids, then surely might the surveyors have carried true north
and south lines up the Nile Valley as far as the summit of Cheops was
visible, by "_plumbing in_" the star and the apex of the pyramid by the
aid of a string and a stone.
True east and west lines could have been made to intersect such north
and south lines from the various groups of pyramids along the river
banks, by whose aid also such lines would be prolonged.
Next, supposing that their astronomers had been aware of the latitude of
Cheops, and the annual northing and southing of the sun, straight lines
could have been laid out in various sectoral directions to the
north-eastward and north-westward of Cheops, across the delta, as far as
the extreme apex of the pyramid was visible, by observations of the sun,
rising or setting over his summit. (That the Dog-star was observed in
this manner from the north-west, I have little doubt.)
Fig. 37. Sun above Pyramid and sinking behind Pyramid
For this purpose, surveyors would be stationed at suitable distances
apart with their strings and their stones, ready to catch the sun
simultaneously, and at the very moment he became transfixed upon the
apex of the pyramid, and was, as it were, "swallowed by it." (_See
Figure 37_.) The knowledge of the pyramid slope angle from different
points of view would enable the surveyor to place himself in readiness
nearly on the line.
Surely such lines as these would be as true and as perfect as we could
lay out nowadays with all our modern instrumental appliances. A string
and a stone here, a clean-cut point of stone twenty miles away, and a
great ball of fire behind that point at a distance of ninety odd million
miles. The error in such a line would be very trifling.
Such observations as last mentioned would have been probably extended
from Cephren for long lines, as being the higher pyramid above the
earth's surface, and may have been made from the moon or stars.
In those days was the sun the intimate friend of man. The moon and stars
were his hand-maidens.