Mr. Petrie gives a synopsis of the various theories connected with the Great Pyramid, which applies not only to the outside form but to the several chambers and passages in the interior. “There are three great lines of theory,” he says,[35] “throughout the Pyramid, each of which must stand or fall as a whole, they are scarcely contradictory, and may almost subsist together;” these are (1) the Egyptian cubit (20·62 in.) theory; (2) the π proportion or radius and circumference theory; (3) the theory of areas, squares of lengths and diagonals. Of the two first, and applying these only to the exterior by the cubit theory, the outside form of pyramid is 280 cubits high and 440 cubits length of side, or 7 in height to 11 of width. This is confirmed by the π theory, where we get the very common proportion that the height is to the circumference as the radius is to the circumference of a circle inscribed within its base; thus taking the mean height of 481 ft. 4 in., we have 481·33 × 2 × 3·1416 = 3024, whilst the side 755·75 × 4 = 3023, so near a coincidence that it can hardly be accidental, and if it was intended, all the other external proportions follow as a matter of course. Even if this theory should not be accepted as the true one, it has at least the merit of being nearer the truth than any other yet proposed. I confess it appears to me so likely that I would hardly care to go further, especially as all the astronomical theories have signally failed, and it seems as if it were only to some numerical fancy that we must look for a solution of the puzzle. Be this as it may, the small residuum we get from all these pyramid discussions is, that they were built by the kings of the early dynasties of the old kingdom of Egypt as their tombs. The leading idea that governed their forms was that of durability—a quasi-eternity of duration is what they aimed at. The entrances were meant to be concealed, and the angle of the passages was the limit of rest at which heavy bodies could be moved while obtaining the necessary strength where they opened at the outside, and the necessary difficulty for protection inside, without trenching on impossibility. By concealment of the entrance, the difficulties of the passages, and the complicated but most ingenious arrangement of portcullises, these ancient kings hoped to be allowed to rest in undisturbed security for at least 3000 years. Perhaps they were successful, though their tombs have since been so shamefully profaned.