Part 1
The Solution of the Pyramid Problem.
by Robert Ballard.
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 princ.i.p.al groups. I take that of Gzeh 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 Gzeh 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_ 56 52 _numbers the bases_ } = _Cephren_ 52 52 _of the pyramids are_} _Mycerinus_ 26 27 _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} 1464
JE 3} 72
YX 7} 63
} 9
} 24
} 24
} 9
DC 15} 135
PA 48} 1152
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} 1152
} 12
} 32
} 3
} 24
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 circ.u.mference. The line DA is therefore 3033118625 British feet, and the base of Cephren 707727 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 1683399 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 circ.u.mference will be 3636102608--geometric inches, or 363974235 British inches, at nine hundred and ninety-nine to the thousand--and 30300855 geometric feet, or 3033118625 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 363974235 British inches, which divided by 1800 makes one of my cubits 202207908 British inches, or 1685066 British feet. Similarly, 3636102608 geometric inches divided by 1800 makes my cubit 2020057 geometric inches in length.
I have therefore defined this cubit as follows:--One R.B. cubit is equal to 20.2006 geo. inches, 202208 Brit. inches, and 1685 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 circ.u.mference of the earth.
Thus we have obtained a perfect set of natural and convenient measures which fits the plan, and fits the circ.u.mference 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 Gzeh.
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.
