The Second Pyramid

1. The Second Pyramid is almost as high as the Great Pyramid, but because it was built on higher ground it appears higher. The height of the Great Pyramid was 279.53 cubits=146.515 meters, whereas the height of the Second Pyramid was 274.867=144.302 meters. If the pyramids had been built merely for the purpose of erecting impressive monuments, as many believe, it would have been easy for the builders of the Second Pyramid to outdo the height of the Great Pyramid. But there were mathematical reasons why the Second Pyramid could not be higher.

The base of the Second Pyramid is smaller than that of the Great Pyramid: 410 cubits on each side against 439 ½. In spite of this, the Second Pyramid was the product of a tremendous construction enterprise: its volume is only about 1/7 less than that of the Great Pyramid. The volume of the Second Pyramid was 15,402,777.78 cubic cubits, whereas the volume of the Great Pyramid was 18,000,000 cubic cubits. However, the cubit of the Second Pyramid is the standard cubit of 525 mm. whereas the cubit of the Great Pyramid is the lesser cubit of 524.1483 mm. The cube of the second cubit has a volume of 144 liters, whereas the cube of the first cubit has a volume of 144.703125 liters. In cubic cubits of the Great Pyramid the Second Pyramid had a volume of 15,477,986.65 cubic cubits.

As a construction project the main inferiority of the Second Pyramid is the lack of elaborate inner chambers such as occur inside the Great Pyramid.

Even though the slope of the Second Pyramid is only about 2 ½ degrees sharper than that of the Great Pyramid, the Second Pyramid impresses the observer by its steepness. This is in contrast with the balance between horizontal and vertical thrusts in the Great Pyramid: this balance may be the result of the use of the Golden Section in the proportions of this pyramid. In the Second Pyramid the triangularity of the faces appears strongly emphasized. This emphasis could be related to the fact that in all dimensions of the Second Pyramid, the number 3 is the key to the figures; even the volume, as we shall see, was calculated as 3,327,000 cubic nebiu; this figure was obtained by multiplying 1,109,000 by 3,

809 1/60 atur

12,135.250

6,371,006.25

1,108,998,656

3,326,995,967

From Petrie’s survey I have concluded that the volume of the pyramid was 3,326,995 modified by increasing the slope to 53° 17’ 00”. Hence the meridian triangle came to be:

Petrie reported that the best empirical datum about the slope of the Second Pyramid was provided by the face of the casing block, which he found to be 53° 14’ ± 05’

The slope was increased to 53° 17’ not only to obtain the right volume, but also to obtain an apothem of 600 feet = 180.00 meters. In Egyptian feet the dimensions of the meridian triangle are:

Greek authors in describing the pyramids of Gizah, stress that these pyramids are stadiaai, ”of stadium size,” meaning that the apothem is a stadium long. The concept of stadium embraced several slightly different entities: the most important one was a unit of 600 feet; the second was the length of 1/10 of minute of latitude. The apothem of the Great Pyramid, as I shall have occasion to explain at length, was calculated as 1/10 of minute of degree. The apothem of the Second Pyramid on the contrary, was stadium-long in the sense of being 600 Egyptian feet of 300 mm. The apothem of the Great Pyramid is about 1/24 longer than that of the Second Pyramid.

From the report of Petrie one gathers that the base of the Second Pyramid was 246 nebiu in the direction East-West but was increased over 246 nebiu by one finger (1/28 of cubit) in the direction North-South. This increased the surface of the base by a small amount in order to arrive at a pyramid the volume of which was exactly 3,327,000 cubit nebiu.

By raising the slope to 53° 17’, the volume of the pyramid come to be 3,326,706.136 cubic nebiu. But the builders wanted to arrive at volume closer to the intended figure of 3,327,000 cubic nebiu. Hence, they increased the surface of the base. The dimension of the base was left at 123 nebiu = 205 cubits in the direction East-West, but was increased by one finger (1/28 of cubit) in the direction North-South.

By this small increase the surface of the base came to be 60521.27143 square nebiu instead of 246² = 60,516 square nebiu. As a result of this small increase of about 5 squarenebiu, the volume of the pyramid came to be 3,326,995.918 cubic nebiu.

As I have said the intended figure was 3,327,000 cubic nebiu. That there was the mentioned small incrase in the surface of the base can be gathered from Petrie’s survey of the base.

I have mentioned the figures 3,327,000 cubic nebiu, because this is a figure that the builders had in mind. But they aimed at extreme precision beginning from an average radius of the northern hemisphere of 809 1/60 atur. Counting from this figure the volume of the hemisphere is 1,108,998,656 cubic atur and hence the volume of the pyramid should be 3,326,995.967 cubic nebiu.

From Petrie’s survey I have concluded that the volume of the pyramid was 3,326,995.918 cubic nebiu.

One could hardly ask for more exquisite precision in calculation.

If the volume of the pyramid is 3,327,000 cubic nebiu and the pyramid was intended to correspond to the volume of the northern hemisphere according to a scale of 3 cubic nebiu for each 1000 cubic atur, the volume of the northern hemisphere is

1,109,000,000 cubic atur.

This means that the northern hemisphere was has assumed to have an average radius of

809.00169936 atur

= 12,135,254.9 cubits

 

= 6,371,008.825 meters

Since the scale is 3 cubic nebiu for 1000 cubic atur of volume, scale is 3 5³/3³: 15,000³ x 1000 = 1:243,000,000,000,000.

But the scale of geographic representation was calculated by the Egyptians, as it is by us, according to linear units.

The pyramid had a volume of 3,327,000 cubic nebiu; this is the volume of a hemisphere with a radius of

116.6804411 nebiu

194.4674019 cubits

In linear units the scale is

1:³Ö 243 x 10¹² = 1: 62,402.51469

This means that if a hemisphere equal in volume to the pyramid has a radius of 194.4674019 cubits, the northern hemisphere is understood to be 194.4674019 x 62,402.51469 = 12,135,254. cubits = 6,371,008.794 meters.

As I will have occasion to explain, Petrie obtained not too reliable results in measuring the base of the Great Pyramid, but was much more successful in measuring the Second Pyramid. One of the reasons is that the bottom layer of the casing of the Second Pyramid is of granite, a hardy material, whereas that of the Great Pyramid is of limestone and badly eroded.

Petrie’s data about the sides of the Second Pyramid are the following:

West

8475.5 inches

= 215,277.6 mm.

North

8471.9

= 215,185.9

East

8475.2

= 215,269.7

South

8476.9

= 215,312.9

Petrie reported also the following data about the orientation of the faces;

West

4’21” north of true east

North

5’31” west of true north

East

6’13” north of true east

South

5°40” west of true north

Petrie made no attempt interpret his own finding. In interpreting them one must remember that he was much more successful measuring linear distances, than in measuring angles.

In my opinion the figures indicate that the pyramid was rotated slightly to the west of true north. The North side was shorter than the other sides, as is the case in the Great Pyramid.

My understanding is that the base of the Second Pyramid had the shape of a regular trapeze. The North side and South side were parallel to each other. The West and the East sides were rotated at the middle by ½ minute toward the center of the North side. I assume that the standard length of the sides was 246 nebiu = 410 cubits = 215,250 mm. The rotation of West and East sides would cause a shortening of the North side and a lengthening of the South side by 62.6 mm.

I would expect the North side to be 215,187.4 mm. and the South side to be 215,312,6 mm. My tentative conclusions agree almost perfectly with Petrie’s findings.

But the distance between the North side and South side, which in my opinion were parallel each other, is reported by Petrie as 215,277.4 or 215,269.7 mm. My understanding is that the distance between the North and South sides was increased by 3/120 of nebiu = 21.875 mm.

The Third Pyramid was calculated by the meridian triangle

The Second Pyramid was calculated by the sexagesimal meridian triangle

modified into

The scale of these two pyramids was suggested by the circumstance that the average radius of the northern hemisphere was taken as:

12,135,250 cubits

809 1/60 atur

= 6,371006.25 meters

This suggested the scale 1:120,000 for the Third Pyramid. For the same reason the scale of the Second Pyramid was 1;60,000, adjusted in the final reckoning to 1:62,402.5.