[2]Herodotus states that "the area of each of the four faces of Cheops was equal to the area of a square whose base was the altitude of a Pyramid;" or, in other words, that altitude was a mean proportional to apothem and half base; thus—area of one face equals the fourth of 330777·90 or 82694·475 R.B. cubits, and the square root of 82694·475 is 287·56. But the correct altitude is 287·77, so the error is 0·21, or 4¼ British inches. I have therefore the authority of Herodotus to support the theory which I shall subsequently set forth, that this pyramid was the exponent of lines divided in mean and extreme ratio.
By taking the dimensions of the Pyramid from what I may call its working level, that is, the level of the base of Cephren, this peculiarity shows more clearly, as also others to which I shall refer. Thus—base of Cheops at working level, 420 cubits, and apothem 340 cubits; base area is, therefore, 176400 cubits, and area of one face is (420 cubits, multiplied by half apothem, or 170 cubits) 71400 cubits. Now the square root of 71400 would give altitude, or side of square equal to altitude, 267·207784 cubits: but the real altitude is √(340²-210²) = √71500 = 267·394839. So that the error of Herodotus's proposition is the difference between √714 and √715.
[2] Proctor is responsible for this statement, as I am quoting from an essay of his in the Gentleman's Magazine. R. B.
This leads to a consideration of the properties of the angle formed by the ratio apothem 34 to half base 21, peculiar to the pyramid Cheops. (See Figure 22.)
Fig. 22. Diagram illustrating relations of ratios of the pyramid Cheops.
Calling apothem 34, radius; and half base 21, sine—I find that—
| Radius is the square root of | 1156 | |
| Sine | 441 | |
| Co-sine | 715 | |
| Tangent | 713 | |
| Secant | 1869 | |
| and | Co-versed-sine | 169 |
So it follows that the area of one of the faces, 714, is a mean between the square of the altitude or co-sine, 715, and the square of the tangent, 713.
Thus the reader will notice that the peculiarities of the Pyramid Cheops lie in the regular relations of the squares of its various lines; while the peculiarities of the other two pyramids lie in the relations of the lines themselves.