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. Footnote 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. Mycerinus and Cephren, born, as one may say, of those two noble triangles 3, 4, 5, and 20, 21, 29, exhibit in their lineal developments ratios so nearly perfect that, for all practical purposes, they may be called correct. Thus--Mycerinus, [3]20² + 25² = 1025, and 32² = 1024. and Cephren, [4]80² + 105² = 17425, and 132² = 17424. or [5]400² + 431² = 345761, and 588² = 345744. See diagrams, Figures 11 to 14 inclusive. In the Pyramid Cheops, altitude is _very nearly_ a mean proportional between apothem and half base. Apothem being 34, and half base 21, then altitude would be √(34²-21²) = √715 = 26·7394839, and-- 21 : 26·7394839 :: 26·7394839 : 34, nearly. Here, of course, the same difference comes in as occurred in considering the assumption of Herodotus, viz., the difference between √715 and √714; because if the altitude were √714, then would it be _exactly_ a mean proportional between the half base and the apothem; (thus, 21 : 26·72077 :: 26·72077 :: 34.) Footnote 3: Half base to altitude. Footnote 4: Half base to altitude. Footnote 5: Half diagonal of base to altitude.