* * * * * Figure 21 represents the comparative levels and dimensions of Mycerinus, Cephren, and Cheops. The following peculiarities are noticeable:--That Cheops and Cephren are of equal bases at the level of Cephren's base;--that, at the level of Cheops' base, the latter is only half a cubit larger;--that, from the level of Mycerinus' base, Cheops is just double the height of Mycerinus;--and that from the level of Cephren's base, Cephren is just double the height of Mycerinus; measuring in the latter case, however, only up to the level platform at the summit of Cephren, which is said to be about eight feet wide. The present summit of Cephren is 23·07 cubits above the present summit of Cheops, and the completed apex of Cephren would be 8·21 cubits above the completed apex of Cheops. In the summit platforms I have been guided by P. Smyth's estimate of _height deficient_, 363 pyr. inches, for Cheops, and I have taken 8 feet base for Cephren's summit platform. § 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 " " " " Cephren 33 to 20 " " " " 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.