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GROWTH OF MATHEMATICAL SCIENCE 17 Thales presented his geometrical teaching. We infer, how- ever, from Proclus that it consisted of a number of isolated propositions which were not arranged in a logical sequence, but that the proofs were deductive, so that the theorems were not a mere statement of an induction from a large number of special instances, as probably was the case with the Egyptian geo- metricians. The deductive character which he thus gave to the science is his chief claim to distinction. Pythagoras (born about 580 B.C.) changed geometry into the form of an abstract science, regarding its principles in a purely abstract manner, and investigated its theorems from the immaterial and intel- lectual point of view. Among the successors of these men, the best known are Archytas of Tarentum (428-347 B.C.), Plato (429-348 B.C.), Hippocrates of Chios (born about 470 B.C.), Meuaechmus (about 375-325 B.C.), Euclid (about 330-275 B.C.), Archimedes (287-212 B.C.), and Apollonius (260-200 B.C.). The only geometry known to the Egyptian priests was that of surfaces, together with a sketch of that of solids, a geometry consisting of the knowledge of the areas contained by some simple plane and solid figures, which they had obtained by actual trial. Thales introduced the ideal of establishing by exact reasoning the relations between the different parts of a figure, so that some of them could be found by means of others in a manner strictly rigorous. This was a phenomenon quite new in the world, and due, in fact, to the abstract spirit of the Greeks. In connection with the new impulse given to geometry, there arose with Thales, moreover, scientific astro- nomy, also an abstract science, and undoubtedly a Greek creation. The astronomy of the Greeks differs from that of the Orientals in this respect : the astronomy of the latter, which is altogether concrete and empirical, consisted merely in determining the duration of some periods or in indicating, by means of a mechanical process, the motions of the sun and planets, whilst the astronomy of the Greeks aimed at the discovery of the geometrical laws of the motions of the heavenly bodies. Let us consider a simple case. The area of a right-angled field of length 80 yards and breadth 50 yards is 4000 square yards. Other fields which are not rectangular can be approxi- mately measured by mentally dissecting them a process which
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