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. Footnote 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 Fig. 7 Cephren. nearly as follows, viz_:-- Fig. 8 Mycerinus. _Cheops 33 to 21 or 330 to 210_ _Cephren 32 to 21 or 320 to 210_ _Mycerinus 32 to 20 or 336 to 210_ 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. Mycerinus. Fig. 14. Mycerinus. 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.