Hi Pistol,
The proposed model of the internal area equal to the area of a circle with a diameter of 2.5 cubits can be tested by ratios which yield the same answer irrespective of the conversion factor to inches or metres because the model requires an internal depth of 1.66 cubits so the internal volume corresponds to its sphere, and the proposed internal diagonal is 4 cubits.
Ratios are multiplied by 1000 to aid comparison:
D9/D3 = Internal diagonal / Internal depth
D9/D3 = 4 cubits / 1.66 cubits
D9/D3 x 1,000 = 2,400 (model), 2399.3 (Smyth), 2,397.9 (Petrie)
It is conspicuous that Smyth's measurements are extremely close to the theoretical model. He had been a professor of astronomy for over twenty years so his ability to measure lengths and angles was undoubtedly at least as good as Petrie, and apparently better from this analysis, but the most important point is that the intended internal diagonal was certainly 4 cubits and the intended internal depth was certainly 1.667 cubits on conversion from a fraction to a decimal to 3 decimal places. Some now disregard Smyth's measurements, but he had calibrated his instruments against very precise standards and had access to the Warden of British Standards. Some of Smyth's instruments were old and very expensive, and others he had made for the task in hand such as the rod to measure the vertical height of the Grand Gallery.
D1/D2 = Internal length / Internal width (irregular dimensions governed by D9)
D1/D2 x 1,000 = 2,916 (22/7 model), 2,916 (Smyth), 2,911 (Petrie)
D1/D2 x 1,000 = 2,886 (8/9 model), 2,916 (Smyth), 2,911 (Petrie)
These ratios are independent of the conversion factor from cubits to inches or metres.
If we imagine that the sarcophagus was made close to perfection then Smyth recovered the ratio of D9/D3 better than Petrie.
Egyptologist fondly imagine Petrie's measurements were superior to Smyth's measurements, so we shall suppose Smyth was just lucky to get the 2916 ratio at a time when he had no idea of what the internal depth might prove to be in relation to a theoretical model. Petrie knew full well what the model was, but discredited Smyth's theory rather effectively by relegating Smyth's principal theory to a single sentence in his first edition and no mention in his second edition.
Let's take Petrie's ratio of 2911. Is it closer to 2916 or 2886?
The 8/9 ratio is a fair approximation, but the use of ratios indicates the model was based on a pi approximation close to the true value of Pi, but not necessarily the 22/7 ratio. It could be, for example, 314 and 1/4 digits / 100 digits which converts to 3.1425 which is slightly closer to the true value of pi, but so little different from 22/7 that differentiation is not possible.
In fairness to Petrie, he thought that the dimensions of the King's Chamber indicated knowledge of a closer approximation to Pi than 22/7 having examined Smyth's theory that the perimeter of the long walls was intended to be equal to the circumference of a virtual circle with a diameter equal to the length of the chamber.
Petrie had to disprove Smyth's model of squaring the area of a circle, otherwise a circle with a radius of 10 cubits is equal to the area of a square with a side length of 365.25 pyramid inches of 1.001 inches, but it was actually also equal to 365 x 34/25 digits and Petrie measured the lean-in of the east wall of the Queen's Chamber as precisely 1 inch to a very small fraction of an inch.
For simplicity, and to avoid digression, the pi ratio apparent from the sacophagus is certainly not the 8/9 ratio for squaring the area of a circle.
Mark
Edited 3 time(s). Last edit at 11/20/2019 08:02PM by Mark Heaton.