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ARITHMETIC AND GEOMETRY 27

Khufu, the variation of a course of casing is .04 in. on 20 ft., and .00 to a course of the core 40 ft. distant. The sides of the pyramid varied 2.3 ins. on an average in a length of 9069.4 ins., or I in 4,000. If this were laid out with copper measuring rods, this error would result from I5° C. difference of temperature. The angular accuracy of building is also a test; on the same pyramid (where the diagonals could not be laid out, owing to rock) the average error is 1' 12" from a right angle, or 2 ins. on the length.

Considering these limits of accuracy, which must have needed extraordinary care to carry them out in a gigantic mass of masonry, it is reasonable to look for accuracy of proportion. It so happens that five different relations of the parts would each give the same angle as that of the Khufu pyramid, within 2' or 3'. A help in judging which of these is likely is given by the Meydum pyramid of Sneferu, next in age before that of Khufu, and of the. same angle. If supposed numbers of proportion will not result in a recognised unit, when applied to both these pyramids, there is little probability that they were intended. There remain two possible designs, (1) the base circuit equal to a circle struck by the height (or the modification of this in height to base as 7 : 11); (2) the face area = height squared (or height a mean proportional between half base and slope). The first design is strongly supported by the dimensions in cubits being multiples of 7 and 11; and this is made almost certain by the Meydum pyramid dimensions also being multiples of 7 and 11. Thus :

Pyramid of Sneferu 25 X 7 high, 25 X 11 base in cubits.
Pyramid of Khufu 40 X 7 high, 40 X 11 base in cubits.
The angle of 7 on 11 base is 51' 50' (strict pi ratio 51' 50; observed at Meydum pyramid 51° 48' +/- 7' and Khufu's pyramid 5I° 52' +/- 2'. We conclude there-

28 WISDOM OF THE EGYPTIANS

fore that the approximation of 7 to 22 as the ratio of diameter to circumference was recognised.