of the shadow thrown on the ground (this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids).

After Thales come the Pythagoreans. We are told that the Pythagoreans were the first to use the term μαθήματα (literally “subjects of instruction”) in the specialised sense of “mathematics”; they, too, first advanced mathematics as a study pursued for its own sake and made it a part of a liberal education. Pythagoras, son of Mnesarchus, was born in Samos about 572 B.C., and died at a great age (75 or 80) at Metapontum. His interests were as various as those of Thales; his travels, all undertaken in pursuit of knowledge, were probably even more extended. Like Thales, and perhaps at his suggestion, he visited Egypt and studied there for a long period (22 years, some say).

It is difficult to disentangle from the body of Pythagorean doctrines the portions which are due to Pythagoras himself because of the habit which the members of the school had of attributing everything to the Master (αὐτὸς ἔφα, ipse dixit). In astronomy two things at least may safely be attributed to him; he held

that the earth is spherical in shape, and he recognised that the sun, moon and planets have an independent motion of their own in a direction contrary to that of the daily rotation; he seems, however, to have adhered to the geocentric view of the universe, and it was his successors who evolved the theory that the earth does not remain at the centre but revolves, like the other planets and the sun and moon, about the “central fire”.

Perhaps his most remarkable discovery was the dependence of the musical intervals on the lengths of vibrating strings, the proportion for the octave being 2 : 1, for the fifth 3 : 2 and for the fourth 4 : 3.

In arithmetic he was the first to expound the theory of