4 165″ --- = ·8 = Tan. < ADC = 38° 39′ 35--- 5 477 312″ ∴ < DAC = 51° 20′ 24--- 477 but as it is probable that the pyramid was built to the ratio of 32 to 20, I have shown its base angle in Figure 19, as 51° 19′ 4″. Figure 72 shows how the slopes of _Cephren_ were arrived at. _Fig. 72. (Cephren)_ LHNM represents the base of the pyramid. On the half-base AC, I described a 3, 4, 5 triangle ABC. I then projected the line CF (ratio 21 to BC 20), thus erecting the 20, 21, 29 triangle BCF. From this datum, I arrived at the triangles BEA, ADC, and GKH; GK, EA and AD each representing apothem; CF and CD, altitude; and HK, edge. The lengths of the lines AD, HK and CH being got at as in the pyramid Mycerinus. These measures reduced to cubits, calling AC = ratio 16 = 210 cubits (half-base of pyramid) give the following result. R. B. BRITISH CUBITS. FEET. Half-base 210·00 353·85 = LA Apothem 346·50 583·85 = EA Edge 405·16 682·69 = HK Altitude 275·625 464·43 = CD Half-diag. of base 296·985 500·42 = CH thus I get the ratios of--Apothem : Half-Base :: 33 : 20, &c. The planes in the diagram are placed in their correct positions, as directed for Figure 71. The angle at the base of Cephren, if built to the ratio of 16 to 21 (half-base to altitude), and not to the practical ratio of 33 to 20 (apothem to half-base), would be the complement of < ADC, thus-- 16 16″ -- = ·761904 = Tan. < ADC = 37° 18′ 14-- 21 46 30″ ∴ < DAC = 52° 41′ 45-- 46 but as it is probable that the pyramid was built to the ratio of 33 to 20, I have marked the base angle in Fig. 17, as 52° 41′ 41″. I took _Cheops_ out, first as a [Pi] pyramid, and made his lines to a base of 420 cubits, as follows-- Half-base 210 Altitude 267·380304 Apothem 339·988573 (_See Fig_. 73.) _Fig. 73. (Cheops) _ But to produce the building ratio of 34 to 21, as per diagram Figure 6 or 9, I had to alter it to-- Half-base 210 Altitude 267·394839 Apothem 340° Thus the theoretical angle of Cheops is 51° 51′ 14·3″, and the probable angle at which it was built, is 51° 51′ 20″, as per figure 15. Cheops is therefore the mean or centre of a system--the slopes of Mycerinus being a little flatter, and those of Cephren a little steeper, Cheops coming fairly between the two, within about 10 minutes; and thus the connection between the ground plan of the group and the slopes of the three pyramids is exactly as one might expect after examination of Figure 3, 4 or 5.