44. The history of ὰριθμητική, or the scientific study of numbers in the abstract, begins in Greece with Pythagoras(cir.b.c. 530), whose example determined for many centuries its symbolism, its nomenclature and the limits of its subject-matter. How Pythagoras came to be interested in such inquiries is not at all clear. It cannot be doubted that he lived a considerable time in Egypt: it is said also, though on far inferior authority, that he visited Babylon. In the first country, he would at least have found calculation brought to a very considerable development, far superior to that which he can have known among his own people: he would have also found a rudimentary geometry, such as was entirely unknown to the Western Greeks. At Babylon, if he ever went there, he might have learnt a strange notation (the sexagesimal) in arithmetic and a great number of astronomical observations, recorded with such numerical precision as was possible at that time. But Pythagoras was not the first to be initiated into this foreign learning, for the Asiatic Greeks had certainly, before his time, acquired a good deal of Chaldaean astronomy and had even improved upon Egyptian geometry. Nor was the bent of his mind altogether singular in his time. Among the Greeks everywhere, a new speculative spirit was abroad and they were burning to discover some principle of homogeneity in the universe. Some fundamental unity was surely to be discerned either in the matter or the structure of things. The Ionic philosophers chose the former field: Pythagoras took the latter. But the difficulty is to determine whether mathematical studies led him to a philosophy of structure or vice versa.

147. IF the materials for a history of Greek geometry in the second century B. C. are scanty, they become still more so for the next 250 years. Only a few works, and those not of a very valuable character, survive from this period.

About 70 b.c. lived Geminus of Rhodes who seems to have been the freedman of a wealthy Roman and who wrote, beside the astronomical work εἰσαγωγἠ εἰς τὰ ϕαvóμενα, still extant, a book on the Arrangement of Mathematics, περὶ τῆς τῶν μαθη μάτων τάξεως, which, without being expressly historical, con tained abundant notices of the early history of Greek mathematics and from which Proclus and Eutocius derived much of their most correct and valuable information on that subject. A book of this kind, written not long after the classical age by a competent geometer, would, if preserved, have cleared up a hundred difficulties which do not now admit of solution.

148. Probably near to the time of Geminus lived Theodosius (? of Tripolis), who is mentioned by Strabo and Vitruvius and must therefore be a pre-Christian writer, though Suidas attributes to him a commentary on one Theudas of Trajan's time. He is the author of Sphaerica, a very complete treatise on the geometry of the sphere, in three books. It was remarked above, however, on the subject of Euclid's Phaenomena, that both that and the treatise of Theodosius are evidently founded on some earlier work on Spherics, perhaps by Eudoxus. The work of Theodosius contains no trigonometry (a spherical triangle is not mentioned) and there is nothing particularly interesting either in his style or in his discoveries, if indeed he made any.