§ 4. THE SLOPES, RATIOS, AND ANGLES OF THE THREE PRINCIPAL PYRAMIDS OF THE GIZEH GROUP.

Before entering on the description of the exact slopes and angles of the three principal pyramids, I must premise that I was guided to my conclusions by making full use of the combined evolutions of the two wonderful right-angled triangles, 3, 4, 5, and 20, 21, 29, which seem to run through the whole design as a sort of dominant.

From the first I was firmly convinced that in such skilful workmanship some very simple and easily applied templates must have been employed, and so it turned out. Builders do not mark a dimension on a plan which they cannot measure, nor have a hidden measure of any importance without some clear outer way of establishing it.

This made me "go straight" for the slant ratios. When the Pyramids were cased from top to bottom with polished marble, there were only two feasible measures, the bases and the apothems;[1] and for that reason I conjectured that these would be the definite plan ratios.

[1] The "Apothem is a perpendicular from the vertex of a pyramid on a side of the base."—Chambers' Practical Mathematics, p. 156.

Figures 6, 7 and 8 show the exact slope ratios of Cheops, Cephren, and Mycerinus, measured as shown on the diagrams—viz., Cheops, 21 to 34, Cephren, 20 to 33, and Mycerinus, 20 to 32—that is, half base to apothem.

Fig. 6 Cheops.

Note. The Ratios of Bases to Altitudes are very nearly as follows, viz:—
Cheops33 to 21or 330 to 210
Cephren 32 to 21 or 320 to 210
Mycerīnus32 to 20 or 336 to 210

Fig. 7 Cephren.

Fig. 8 Mycerīnus.

The ratios of base to altitude are, Cheops, 33 to 21, Cephren, 32 to 21, and Mycerinus, 32 to 20: not exactly, but near enough for all practical purposes. For the sake of comparison, it will be well to call these ratios 330 to 210, 320 to 210, and 336 to 210, respectively.

Fig. 9. Cheops. Fig. 10. Cheops.

Figures 9 and 10 are meridional and diagonal sections, showing ratios of Cheops, viz., half base to apothem, 21 to 34 exactly; half base to altitude, 5½ to 7 nearly, and 183 to 233, nearer still (being the ratio of Piazzi Smyth). The ratio of Sir F. James, half diagonal 10 to altitude 9 is also very nearly correct.

My altitude for Cheops is 484·887 British feet, and the half base 380·81 British feet.

The ratio of 7 to 5½ gives 484·66, and the ratio of 233 to 183 gives 484·85 for the altitude.

My half diagonal is 538·5465, and ratio 10 to 9, gives 484·69 British feet for the altitude.

I have mentioned the above to show how very nearly these ratios agree with my exact ratio of 21 to 34 half base to apothem.

Fig. 11. Cephren. Fig. 12. Cephren.

Figures 11 and 12 show the ratios of Cephren, viz., half base to apothem, 20 to 33 exactly, and half base, altitude, and apothem respectively, as 80, 105, and 132, very nearly.

Also half diagonal, altitude, and edge, practically as 431, 400, and 588.

Fig. 13. Mycerīnus. Fig. 14. Mycerīnus.

Figures 13 and 14 show the ratios of Mycerinus, viz., half base to apothem, 20 to 32 exactly, and half base, altitude, and apothem respectively, as 20, 25, and 32 very nearly.

Also full diagonal to edge as 297 to 198, nearly. A peculiarity of this pyramid is, that base is to altitude as apothem is to half base. Thus, 40 : 25 :: 32 : 20; that is, half base is a fourth proportional to base, apothem, and altitude.