Within a circle describe a pentangle, around the interior pentagon of the star describe a circle, around the circle describe a square; then will the square represent the base of Cheops.
Draw two diameters of the outer circle passing through the centre square at right angles to each other, and each diameter parallel to sides of the square; then will the parts of these diameters between the square and the outer circle represent the four apothems of the four slant sides of the pyramid. Connect the angles of the square with the circumference of the outer circle by lines at the four points indicated by the diameters, and the star of the pyramid is formed, which, when closed as a solid, will be a correct model of Cheops.
| Calling apothem of Cheops, | MH = 34 |
| and half base, | HC = 21 |
| as per Figure 6. Then— MH + | MC = 55 |
and 55 : 34 :: 34 : 21·018, being only in error a few inches in the pyramid itself, if carried into actual measures.
The ratio, therefore, of apothem to half-base, 34 to 21, which I ascribe to Cheops, is as near as stone and mortar can be got to illustrate the above proportions.
Correctly stated arithmetically let MH = 2.
| Then | HC | = √5 - 1 | ||
| MC | = √5 + 1 | |||
| and altitude of Cheops | = √(MH × MC) | |||
Let us now compare the construction of the two stars:—
| Fig. 69. | Fig. 70. |
| TO CONSTRUCT THE STAR PENTALPHA FIG. 69. | TO CONSTRUCT THE STAR CHEOPS, FIG. 70. |
| Describe a circle. Draw diameter MCE. Divide MC in mean and extreme ratio at H. Lay off half MH from C, to D. Draw chord ADB, at right angles to diameter ECM. Draw chord BHN, through H. Draw chord AHO, through H. Connect NE. Connect EO. | Describe a circle. Draw diameter MCE. Divide MC in mean and extreme ratio, at H. Describe an inner circle with radius CH, and around it describe the square a, b, c, d. Draw diameter ACB, at right angles to diameter ECM. Draw Aa, aE, Eb, bB, Bd, dM, Mc, and cA. |
The question now arises, does this pyramid Cheops set forth by the relations of its altitude to perimeter of base the ratio of diameter to circumference; or, does it set forth mean proportional, and extreme and mean ratio, by the proportions of its apothem, altitude, and half-base? The answer is—from the practical impossibility of such extreme accuracy in such a mass of masonry, that it points alike to all, and may as fairly be considered the exponent of the one as of the others. Piazzi Smyth makes Cheops 761·65 feet base, and 484·91 feet altitude, which is very nearly what he calls a Π pyramid, for which I reckon the altitude would be about 484·87 feet with the same base: and for a pyramid of extreme and mean ratio the altitude would be 484·34 feet.