The origins of geometry are even more obscure than those of arithmetic. Not only is geometry as highly developed as arithmetic when it first appears in occidental civilization, but, in addition, the problems of primitive peoples seem to have been such that they have developed no geometrical formulae striking enough to be recorded by investigators, so far as I have been able to discover. But just as the commercial life of the Phoenicians early forced them self-consciously to develop arithmetical calculation, so environmental conditions seem to have forced upon the Egyptians a need for geometrical considerations. It is almost platitudinous to quote Herodotus' remark that the invention of geometry was necessary because of the floods of the Nile, which washed away the boundaries and changed the contours of the fields. And as Proclus Diadochus adds (_Procli Diadochi, in primum Euclidis elementorum librum commentarii_--quoted Cantor, I, p. 125): "It is not surprising that the discovery of this as well as other sciences has sprung from need, because everything in the process of beginning proceeds from the incomplete to the complete. There takes place a suitable transition from sensible perception to thoughtful consideration and rational knowledge. Just as with the Phoenicians, for the sake of business and commerce, an exact knowledge of numbers had its beginning, so with the Egyptians, for the above-mentioned reasons, was geometry contrived." The earliest Egyptian mathematical writing that we know is that of Ahmes (2000 B. C.), but long before this the mural decorations of the temple wall involved many figures, the construction of which involved a certain amount of working knowledge of such operations as may be performed with the aid of a ruler and compass. The fact that these operations did not earlier lead to geometry, as ruler and compass work seems to have done in Japan in the nineteenth century (Smith and Mikami, index, "Geometry"), is probably due to the stage at which the development of Egyptian intelligence had arrived, feebly advanced on the road to higher abstract thinking. It is everywhere characteristic of Egyptian genius that little purely intellectual curiosity is shown. Even astronomical knowledge was limited to those determinations which had religious or magically practical significance, and its arithmetic and geometry never escaped these bounds as with the more imaginative Pythagoreans, where mystical interpretation seems to have been a consequence of rather than a stimulus to investigation. An old Egyptian treatise reads (Cantor, p. 63): "I hold the wooden pin (Nebi) and the handle of the mallet (semes), I hold the line in concurrence with the Goddess S[a.]fech. My glance follows the course of the stars. When my eye comes to the constellation of the great bear and the time of the number of the hour determined by me is fulfilled, I place the corner of the temple." This incantation