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.

The materials for a history of Greek geometry after Apollonius are both scanty in quantity and most unsatisfactory in quality. We know the names of many geometers who lived during the next three centuries, but very few indeed of their works have come down to us, and we are compelled to rely for the most part on such scraps of information as the later scholiasts, Pappus, Proclus, Eutocius and the like, have incidentally preserved. But this information, again, generally affords little clue to the date of the geometer in question. Thus, though we have abundant evidence that mathematics remained a chief constituent of the Greek liberal curriculum, we cannot tell with any accuracy what subjects were most in vogue or what mathematicians were most generally regarded at any particular time. It is certain, however, that during the whole period between Apollonius and Ptolemy only two mathematicians of real genius, Hipparchus and Heron, appeared, that both of these lived about the same time (120 b.c.), and that neither was interested in mathematics per se, for Hipparchus was above all things an astronomer, Heron above all things a surveyor and engineer. The result might have been different if some new methods had been introduced. The force of nature could go no further in the same direction than the ingenious applications of exhaustion by Archimedes and the portentous sentences in which Apollonius enunciates a proposition in conies. A briefer symbolism, an analytical geometry, an infinitesimal calculus were wanted, but against these there stood the tremendous authority of the Platonic and Euclidean tradition, and no discoveries were made in physics or astronomy which rendered them imperatively necessary.

In the book of Problemata, attributed to Aristotle, the following question is asked (xv. 3): “Why do all men, both barbarians and Hellenes, count up to 10 and not to some other number?” It is suggested, among several answers of great absurdity, that the true reason may be that all men have ten fingers: “using these, then, as symbols of their proper number (viz. 10), they count everything else by this scale.” The writer then adds “Alone among men, a certain tribe of Thracians count up to 4, because, like children, they cannot remember a long sum neither have they any need for a great quantity of anything.”

It is natural to regret that an author who at so early a date was capable of writing this passage, was not induced to ask himself more questions and to collect more facts on the same and similar subjects. Had he done so, he might have anticipated, by some two thousand years, the modern method of research into prehistoric times and might have attempted, with every chance of success, a hundred problems which cannot now be satisfactorily treated. In the fourth century B. c. and for long after, half the Aryan peoples were still barbarous and there must still have survived, even among Greeks and Italians, countless relics of primitive manners, forming a sure tradition of the past. Nearly all these materials, so abundant in Aristotle's day, are irretrievably lost to us and the primeval history of Aryan culture depends now chiefly on the evidence supplied by comparative philology. It is so with the art of calculation.

It has been already pointed out that the conditions of life in Athens were unfavourable to the growth of any “natural” science. Her practical men were absorbed in politics, her philosophers in metaphysical speculation. Neither of these classes objected to deductive science, for deduction is the chief instrument of rhetoric and is also the most interesting part of logic: but the patient and unrewarded industry, which leads to inductive science, was not to the Athenian taste. The practical men thought it profane, the philosophers vulgar. The schools of inductive science remained therefore far away from the turmoil of Athens: the observatories of the astronomers were at Cyzicus on the Hellespont or at Cnidus on the south coast of Asia Minor: the school of medicine was maintained by one illustrious family in the island of Cos. If it be objected that Aristotle lived in Athens, the answer is that Aristotle was the son of a physician, was not born or bred in Athens, never became an Athenian citizen, disliked Athens and left it, and was not able to command in Athens an audience for anything but metaphysics. The Peripatetic school was as unscientific as the Platonic. There was not yet a “university, ” to which all the world might come and learn all the knowledge that was in existence. Alexandria was the first city to deserve that name. Athens might have won it, but when Athenian politics were no more and the field was free for other pursuits, Alexandria had forestalled her.