THE HIGHT OF THE GREAT PYRAMID
Even Borchardt agreed that the line marked off center on the N side of the Pyramid, marked the position of the North-South axis, but he never even concerned why this axis should have been displaced to the West, It is not the matter of a corrects measurement, as Burchardt possibly would have agreed if he had been compelled to answer the question, since the Petrie survey show that the East West axis is at distance of exactly Petrie 3/4 cubits from the N side. Lucky Petrie could measure from the N line of the Pyramid through the ascenders passages to east of the or of the Queen's chamber and the face of the Great Steps of the position of the East-West axis.
In my opinion the reason for the dispocement to the West of the North-South axis, is that the height of the Pyramid have to be calculated according to two different principles; the factor 9 and two factor _, Up to now those who have been interpreted the dimensions of the Pyramid have been divided into two factions; those who give that the height was calculated by the factor p (such as Taylor., Piazzi Smyth, and Petrie) and those who any we that it was calculated by the factor ( which is the one mentioned by Herodotos). In my opinion both grops are correct, except that the factor was given greater importance (one of the measures being derived from that of 4), the two mast basic sides of the Pyramid doe the N and the W sides, the calculation of the Pyramid beyond from the marking of the angle which is the only one which is a perfect right angle.
In any pyramid the most important side is two N side, Hence the N side was calculated by the factor and the W side was calculated by the factor .
As I have instaled, the N side was at the exact distance of 219 3/± cubits from the East-West axis of the Pyramid, Hence, if the height of the Pyramid had been calculated by the exact value of , it would have been 219 ¾ x = 279. 526318 cubits = 146,513.2384 cubits. But the factor was calculated in an easy manner by the relation 610/377 of the Fibonocci series by which the height is 279.526590 cubits = 146, 513,3809 mm. The difference between the two values of the height is for less than 1/672 cubits = 0.77998 mm., which was the menimun unit of lineal unit of measurement employed by the builders.
The inclination of the N face according to the exact value of would have been cos¯¹ = 51° 49 38.25; it come to be cos ¯¹ 610/377 = 51° 49 38. 57.
If the height of the Pyramid had been calculated by the exact value of , the inclination of W face should have been 51° 51 14. 30. This would have coused the apex of the Pyramid to be too heigh. Hence, the base of the W meridian triangle was shortened, by shifting the North-South axis to the West. the inclination of the face, the W face, was calculated by the round figure 51° 51 00 = 51°85 (which makes = 3,142041). Possibly this slope was chosen so as to provide an exsily measurable angle for the actual builders; 51° 51 is equal to 51°, 17/20. Even through the human eye con discern differences of a minute of degree, without the aid of optical tools if one counts from the normal with the of the pupil, it has great difficulty in keeping apeart two points that are a distance of less than 3 minutes from each other, We perceive as one heigh two stars that are at a distance of less than 3 minutes Tycho Brahe, whose observations were essential for the development of the Scientific Revolution, with the huge instruments which he had set up at Uraniburg (instruments, which did not use lenses or mirrors) was able to establish celestial position with an accuracy of 3 minutes, which then was a great improvement over earlier observations. They were accurate enough to provide Kepler with the data by which he established his laws of planetary movements.
If the slope of the W face is 51° 51 and the base of the W meridian triangle is 219 4/7, became of the shift of the North-South axis, the apex of the Pyramid is at height of 279, 52 711 cubits = 146, 513.6541.
The most important return in the dimensions of the pyramid was the height which was set ¹/434,000 of polar radius of the earth.
As indicated by the Book of the Dead the Egyptian datum for the polar radius was:
12,128,182 royal cubits
= 6,356,965,9 meters
This figure had been originally calculated as 34,422 geographic stadia = 6,356,966.1 m.
The height of the pyramid should have been 146,473.86 mm. This measurement was so important that although the rest of the pyramid was calculated by sixteenth of figure, in the height was referred to ¼ of sixteenth (1/1536 of cubit. This will be proved by the calculations that follow.
The height of the pyramid was set as
Hence, the height of the pyramid was exact that one could derive from it the length of the polar radius with the precision of a meter.
If the meridian section of the pyramid had been calculated by the exact value of it would have been.
But it was decided that the West Side of the pyramid should have a slope of 5150, either to indicate that this is the earent round fogure to the angle which cuted by the factor * or because it was necessary for the side of construction to have at the slope of one side reckoned by a figure rounded to the minute of an.
In order to give to the West forced slope of 5105000 the north side was shortened so as to move teh north south axis closer to the base of the West side.The meridian section of the west side is
The sides are calculated by
If the pyramid had been calculated by the exact already of cosecont and tangent = secont = _*
The meridian sector fo the North side turned to be the following
The figures indicate what the resourse why the west sise way shortened by 4 siateenths, so as to move the east -west axis 2 sixteenth tothe north was to obtain anaceptance of north more expressed by the rounal figure 355.5 cubits
The slope of westside is something more than the ideal slope 510 49039, whereas the slope of the north side is something less.
West side 510 50 00
* slope 510 290 39 21
North side 510 490 3925
Resuming the various steps, The builders beyond by a general outhine indicated by Herodotus
Them the figures were changed to
355,5 279,5 cubits
The basic overage length of the semi sides was kept at 219,75 cubits
The slaret height was keept at 355.5 on the most side. On the west side here was introduce the slope 51050 as the basic slope of the pyramid.
The initial plan of the Pyramid was drawn according to the terms 55 and 89 of the Feboracci series,
89 x 4=356
55 x 4=220
But the relation 89/55 =
1.16181818.., given only a rough approximation to * = 1.618033898. Hence, the builder calculated the final plan by the later terms of the Fibonacci series, the proportions of the Pyramid were calculated by *= 610/377 = 1.618037135. This value differs from the exact value of * by about 1/50,00.
The Pyramid could be conceived as heaving the following meridian section
The angle of the slope is
510.82737997 = 510 49 38,5679
According the exact value of * it would be
510. 82729237 = 510 49 38. 2525
The difference is less them a second of degree and hence could be dissergarded eventh the unusually accounts computation of the Pyramid.
Accoracy to the therm of Pytagores the other side of the meridian triangle, the height, would be 479.5529160
But the relation between height and base should be _/4. The figures just qouted would emply _ = 3.144596.
The meridian triangle could be conceived as being roughly
_ con be calculate to good accuracy as _ = 3x 16/113 =3 x 1/12
355/113 = 3.14159292
which is, correct to the sixth decim figure. This value would cause an error of slightly more them a meter is the distance west the two poles of the earth.
_/4 is 355/452 ; hence, the height of the Pyramid con be calculated
as 355/452 = 377/480.0112676
Now, 377/480 = 0, 785 91666...;
This 5 wuold imply _ = 3.141666... This value is much more occurent by _ = 3.142857 which the builders began their plan.
The relation 377/480 implies an angle of the slope
510. 85331835 = 51051 111946
According to _/9 , the slope is 51085 397401 = 51051 14304, in theterminal plan; 377 goegraphic feet are 116, 038. 9677 mm.
was used merely to calculate the proportions of the Pyramid However, it happened that if one assumed that the figures refered to geographic feet, they would be almost right, being valy slightly in excess of the actual dimension of the Pyramid.
The sembase of the Pyramid in the initial plan was 220 royal en bit = 115,312. 62 mm. , and become 219.75 royal cubits= 115,181. 58 mm.