I found 'Colonel H. Vyse volume 2 Operations at Gizeh' on line free in google books earlier today, and also the dimensions of the burial chamber of Khafre's pyramid in which it appears to me that the gable height of 38 inches is uncertain which is just as well because I had remembered the height incorrectly as 36 inches from Petrie in earlier posts:

PROPOSED NEW MODEL IS 410 CUBITS AS SIDE LENGTH OF BASE AND 273 PLUS 1/3 CUBITS AS HEIGHT

A set of say 10 master stones with a length of 10 cubits could have been used to mark out the sides of the base square, and the same set could have been used to check the dimensions of the burial chamber excavated from the bedrock. The proposed design length of the chamber is 27 cubits which should be a reliable estimate of the length of the cubit used in the construction of Khafre’s pyramid because it is a whole number of cubits straightforward to measure as a horizontal length.

It would have been difficult to achieve a precise length from an excavation in the bedrock, but Petrie noted that the walls of the chamber had been made good with stone let in as required. It is noted here that the chamber has probably retained its original dimensions because it is cut out of the bedrock.

The apparent length of the cubit can be calculated from Petrie’s measurements of the long walls of the chamber:
North Wall: 557.9 inches (14.171 metres) = 27 cubits x 20.66 inches (524.8 millimetres)
South Wall: 557.4 inches (14.156 metres) = 27 cubits x 20.64 inches (524.3 millimetres)
Mean length: 557.65 inches (14.164 metres) = 27 cubits x 20.65 inches (524.5 millimetres)

Petrie determined the mean side length of the base as 8474.9 inches (215.262 metres), so if the intended length was 410 cubits, then the apparent length of cubit is 20.67 inches (525.0 millimetres), which is very close indeed to the apparent length of 20.65 inches from the chamber.

There may be a clue in the internal architecture relating to the design height of the pyramid. In 1840 Colonel H. Vyse reported the height of the peak of the burial chamber and also the height of the long walls, which Petrie noted as having a difference of 38 inches.

This difference of 38 inches may be incorrect because the heights had been measured to the base of the walls not to the floor level, with no assurance that the base of the chamber hewn out of the bedrock is level, and each of the determinations may have a significant error of measurement as judged from comparison to Petrie's measurements where both have measured the same length.

Direct measurement of the gables for the height of the triangular section of the end walls would have been difficult, and the measurement was not attempted by Petrie either directly or indirectly, so Petrie's height of the ridge of the roof relies on the report of Vyse.

The design area of the roof can be calculated as approximately 275 square cubits if the intended slope was the seked of 18 palms (a gable height of 38.1 inches), or very close to 273 plus 1/3 square cubits if the intended slope was the seked of 19 palms (36.1 inches) for which the height of the gable would be 1.75 cubits (7 x 1/4 cubits) and the hypotenuse of the triangle would be ‘the square root of 410 square cubits’ x 1/4 from the horizontal of 4.75 cubits (19 x 1/4 cubits).

It is interesting that the right-angled triangle for the seked of 19 palms has an area relationship to ‘410’ albeit as a coincidence in relation to the proposition based on roof area:
(19 x 19 square palms) + (7 x 7 square palms) = 410 square palms (square on hypotenuse)
Roof area = 2 x 27 cubits x (square root of 410 square cubits) x 1/4
In this model the roof area is precisely 273 plus 1/3 square cubits

In this model side length of base = 3/2 x 273 plus 1/3 cubits = 410 cubits as a round number of cubits.

Khafre's pyramid has sloping faces with a rise of '4' for a run of '3' so a hypotenuse of '5'.

Therefore the projected height of the pyramid is the half side length of 205 cubits x 4/3.

The shape of the pyramid is the same as a model pyramid in the RMP with a height of '8' and base of '12' so a ratio of 2 to 3.

2/3 x 410 = 273 + 1/3

The height of a pyramid arises from the side length of the base and the slope of the faces, so the irregular height would not have been a problem for the surveyors or builders as it would not have been measured directly.

The proposed model requires a survey of the gables of the burial chamber in order to determine if the model is tenable.

The model predicts the height of the gables so it can be 'falsified' if incorrect.

If the model proves to be tenable then perhaps the ancient Egyptians were able to calculate the hypotenuse from the square of the hypotenuse
(3 x 3) + (4 x 4) = 5 x 5 = 25
(7 x 7) + (19 x 19) = 49 + 361 = 410 = square root of 410 x square root of 410

The square root of 410 could have been measured very precisely (or calculated from Egyptian methods, which I can detail if anybody is interested in square roots).

Mark



Edited 3 time(s). Last edit at 05/16/2021 02:17PM by Mark Heaton.