The Solution of the Pyramid Problem; or, Pyramid Discoveries / With a New Theory as to their Ancient Use

by Robert Ballard

§ 6. GEOMETRICAL PECULIARITIES OF THE PYRAMIDS.

In any pyramid, the apothem is to half the base as the area of the four sides is to the area of the base.

Thus—Ratio apothem to half base Mycerinus	32 to 20
Thus—Ratio apothem to half base Cephren	33 to 20
Thus—Ratio apothem to half base Cheops	34 to 21

	AREA OF THE FOUR SIDES.	AREA OF THE BASE.
Mycerinus	70560·	44100
Cephren	291060·	176400
Cheops	330777·90	204304

All in R.B. cubits.

Therefore—	32	:	20	: :	70560·	:	44100
	33	:	20	: :	291060·	:	176400
	34	:	21	: :	330777·90	:	204304

[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.

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.)

[3] Half base to altitude.

[4] Half base to altitude.

[5] Half diagonal of base to altitude.

In Cheops, the ratios of apothem, half base and edge are, 34, 21, and 40, very nearly, thus, 34² + 21² = 1597, and 40² = 1600.

The dimensions of Cheops (from the level of the base of Cephren) to be what Piazzi Smyth calls a Π pyramid, would be—

Half base	210	R.B. cubits.
Altitude	267·380304, &c.	"
Apothem	339·988573, &c.	"

Altitude being to perimeter of base, as radius of a circle to circumference.

My dimensions of the pyramid therefore in which—

Half base	=	210	R.B. cubits.
Altitude	=	267·394839 &c.	"
Apothem	=	340	"

come about as near to the ratio of Π as it is possible to come, and provide simple lines and templates to the workmen in constructing the building; and I entertain no doubt that on the simple lines and templates that my ratios provide, were these three pyramids built.