of {41} securing a right angle between two very long lines was to stretch round three pegs a rope measured in three portions which were to one another in the ratio 3:4:5. The triangle thus formed is right-angled. Further the operation of rope stretching is mentioned in Egypt, without explanation, at an extremely early time (Amenemhat I). If this be the correct explanation of it, then the Egyptians were acquainted 2,000 years B.C., with a particular case of the proposition now known as the Pythagorean theorem." This may not seem to mean very much. Yet what it illustrates is just this. These men wanted a certain development of mathematics. They needed it for the work that they were engaged at. They set themselves to the solution of certain problems and in doing so evolved a theorem in pure mathematics and an application of it which greatly simplified construction and gave an impetus to mechanics. In so doing they anticipated the work of a long after time. This is what I would insist is always true with regard to man. When he needs some intellectual development he makes it. When he requires an application of it he succeeds in working it out. Later ages may go farther, but had he needed further developments he evidently had the power to make them and probably would have made them. The old Greeks had a much better opportunity to study Egyptian remains than we have, and especially was this true after the foundation of {42} Alexandria. There must have been a lively interest in things Egyptian aroused in the Greek minds by this Greek settlement in old Egypt. It is not surprising, then, to find some magnificent compliments to the old Egyptians in the mouths of some of the writers about the time of the foundation of Alexandria. Eudemus, for instance, the pupil of Aristotle, wrote the history of Geometry in which he traces its invention to the Egyptians, and states that the reason for its invention was its necessity in the remeasurement of land demanded after the removal of landmarks by the annual rise of the Nile. Always does one find this, that when there is a serious demand for an invention in theory or practice men make it. It is not a change or development in man that brings about inventions, but a change in his environment which causes new necessities to arise, and then he proceeds with an ability always the same to respond properly to those necessities. Eudemus says: "Geometry is said by many to have been invented among the Egyptians, its origin being due to the measurement of plots of land. This was necessary there because of the rising of the Nile, which obliterated the boundaries appertaining to separate owners. Nor is it marvellous that the discovery of this and other sciences should have arisen from such an occasion, since everything which moves in development will advance from the imperfect to the {43} perfect. From mere sense-perception to calculation, and from this to reasoning, is a natural transition."