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problem of the quadrature of the circle was reducible to that of the lune on the side of the inscribed hexagon; and what was stated conditionally may have been taken up by Hippocrates as unconditional;103

3. The further attempt which Hippocrates made to solve the problem by squaring a lune and circle together (see p. 74);

4. The obscurity and deficiency in the construction given in p. 72; and the dependence of that construction on a problem which we know was Pythagorean (see p. 24 (e), and note 26); 104

5. The passage in Iamblichus, see p. 58 (ƒ); and, generally, the unfavourable opinion entertained by the ancients of Hippocrates.

This conjecture gains additional strength from the fact that the publication of the Pythagorean doctrines was first made by Philolaus, who was a contemporary of Socrates, and, therefore, somewhat junior to Hippocrates: Philolaus may have thought that it was full time to make this publication, notwithstanding the Pythagorean precept to the contrary.

The view which I have taken of the form of the

103 In reference to this paralogism of Hippocrates, Bretschneider (Geom. vor Eukl., p. 122) says, 'It is difficult to assume so gross a mistake on the part of such a good geometer,' and he ascribes the supposed error to a complete misunderstanding. He then gives an explanation similar to that given above, with this difference, that he supposes Hippocrates to have stated the matter correctly, and that Aristotle took it up erroneously: it seems to me more probable that Hippocrates took up wrongly what he had heard at lecture than that Aristotle did so on reading the work of Hippocrates. Further, we see from the quotation in p. 98, from Anal. Prior., that Aristotle fully understood the conditions of the question.

104 Referring to the application of areas, Mr. Charles Taylor, An Introduction to the Ancient and Modern Geometry of Conics, Prolegomena, p. xxv., says, 'Although it has not been made out wherein consisted the importance of the discovery in the hands of the Pythagoreans, we shall see that it played a great part in the system of Apollonius, and that he was led to designate the three conic sections by the Pythagorean terms Parabola, Hyperbola, Ellipse.'

I may notice that we have an instance of these problems in the construction referred to above: for other applications of the method see Ch. II., pp. 41, 43.

demonstrations in geometry at this period differs altogether from that put forward by Bretschneider and Hankel, and agrees better not only with what Simplicius tells us of the summary manner of Eudemus, who, according to archaic custom, gives concise proofs' (see p. 69), but also with what we know of the origin, development, and transmission of geometry: as to the last, what room would there be for the silent meditation on difficult questions which was enjoined on the pupils in the Pythagorean schools, if the steps were minute, and if laboured proofs were given of the simplest theorems ?

The need of a change in the method of proof was brought about at this very time, and was in great measure due to the action of the Sophists, who questioned everything.

Flaws, no doubt, were found in many demonstrations which had hitherto passed current; new conceptions arose, while others, which had been secret, became generally known, and gave rise to unexpected difficulties; new problems, whose solution could not be effected by the old methods, came to the front, and attracted general attention. It became necessary then on the one hand to recast the old methods, and on the other to invent new methods, which would enable geometers to solve the new problems.

I have already indicated the men who were equal to this task, and I propose in the following chapters to examine their work.

CHAPTER IV.*

ARCHYTAS.

State of Hellas during the last generation of the fifth century B.C.-Magna Graecia again became flourishing.-Archytas of Tarentum.-His life, eminence as a Statesman and noble character.-Notices of his Geometrical work. Was there a Roman Agrimensor named Architas?-The problem 'to find two Mean Proportionals between two Given Lines' was first solved by Archytas.-His Solution.-Theorems which occur in it.-Inferences from it as to Archytas's knowledge of Geometry.-The conception of Geometrical Loci involved in this Solution.-Different opinions as to its importance.-Construction of Archytas's Solution.-Was Plato the inventor of the method of Geometrical Analysis ?-Passage in the 'Republic' of Plato, in which the backward state of Solid Geometry is noticed.-Yet Archytas had, for the period, a profound knowledge of Geometry of Three Dimensions; and Stereometry was founded in Plato's lifetime by Eudoxus.

DURING the last thirty years of the fifth century before the Christian era no progress was made in geometry at Athens, owing to the Peloponnesian war, which having broken out between the two principal States of Greece, gradually spread to the other States, and for the space of a generation involved almost the whole of Hellas. Although it was at Syracuse that the issue was really decided, yet the Hellenic cities of Italy kept aloof from the contest,' and Magna Graecia enjoyed at this time a

* In the preparation of this and the following Chapters I have again made use of the works of Bretschneider and Hankel, and have derived much advantage from the great work of Cantor-Vorlesungen über Geschichte der Mathematik. I have also constantly used the Index Graecitatis appended by Hultsch to vol. III. of his edition of Pappus; which, indeed, I have found invaluable.

1 At the time of the Athenian expedition to Sicily they were not received into any of the Italian cities, nor were they allowed any market, but had only the liberty of anchorage and water-and even that was denied them at Tarentum and Locri. At Rhegium, however, though the Athenians were not received into the city, they were allowed a market without the walls; they then made proposals to the Rhegians, begging them, as Chalcideans, to aid the Leontines. To which was answered, that they would take part with neither, but whatever should seem fitting to the rest of the Italians that they also would do.' Thucyd. VI. 44.

period of comparative rest, and again became flourishing. This proved to be an event of the highest importance; for, some years before the commencement of the Peloponnesian war, the disorder which had long prevailed in the cities of Magna Graecia had been allayed through the intervention of the Achaeans; party feeling, which had run so high, had been soothed, and the banished Pythagoreans allowed to return. The foundation of Thurii (443 B C.), under the auspices of Pericles, in which the different Hellenic races joined, and which seems not to have incurred any opposition from the native tribes, may be regarded as an indication of the improved state of affairs, and as a pledge for the future. It is probable that the pacification was effected by the Achaeans on condition

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2The political creed and peculiar form of government now mentioned also existed among the Achaeans in former times. This is clear from many other facts, but one or two selected proofs will suffice, for the present, to make the thing believed. At the time when the Senate-houses (ovvédpia) of the Pythagoreans were burnt in the parts about Italy then called Magna Graecia, and a universal change of the form of government was subsequently made (as was likely when all the most eminent men in each State had been so unexpectedly cut off), it came to pass that the Grecian cities in those parts were inundated with bloodshed, sedition, and every kind of disorder. And when embassies came from very many parts of Greece with a view to effect a cessation of differences in the various States, the latter agreed in employing the Achaeans, and their wellknown integrity, for the removal of existing evils. Not only at this time did they adopt the system of the Achaeans, but, some time after, they set about imitating their form of government in a complete and thorough manner. For the people of Crotona, Sybaris, and Caulon, sent for them by common consent; and first of all they established a common temple dedicated to Zeus, 'the Giver of Concord,' and a place in which they held their meetings and deliberations: in the second place, they took the customs and laws of the Achaeans, and applied themselves to their use, and to the management of their public affairs in accordance with them. But some time after, being hindered by the overbearing power of Dionysius of Syracuse, and also by the encroachments made upon them by the neigbouring natives of the country, they renounced them, not voluntarily, but of necessity.' Polybius, II., 39. Polybius uses σvvédpiov for the senate at Rome: there would be one in each Graeco-Italian State—a point which, as will be seen, has not been sufficiently noted.

3 The foundation of Thurii, near the site of Sybaris, seems to have been regarded as an event of high importance; Herodotus was amongst the first citizens, and Empedocles visited Thurii soon after it was founded. The names of the tribes of Thurii show the pan-Hellenic character of the foundation,

that, on the one hand, the banished Pythagoreans should be allowed to return to their homes, and, on the other, that they should give up all organised political action.* Whether this be so or not, many Pythagoreans returned to Italy, and the Brotherhood ceased for ever to exist as a political association. Pythagoreanism, thus purified, continued as a religious society and as a philosophic School; further, owing to this purification and to the members being thus enabled to give their undivided atten

4 Chaignet, Pythagore et la Philosophie Pythagoricienne, I., p. 93, says so, but does not give his authority; the passage in Polybius, II. 39, to which he refers, does not contain this statement.

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5 There are so many conflicting accounts of the events referred to here that it is impossible to reconcile them (see p. 53). The view which I have adopted seems to me to fit best with the contemporary history, with the history of geometry, and with the balance of the authorities. Zeller, on the other hand, thinks that the most probable account is that the first public outbreak must have taken place after the death of Pythagoras, though an opposition to him and his friends may perhaps have arisen during his lifetime, and caused his migration to Metapontum. The party struggles with the Pythagoreans, thus begun, may have repeated themselves at different times in the cities of Magna Graecia, and the variations in the statements may be partially accounted for as recollections of these different facts. The burning of the assembled Pythagoreans in Crotona, and the general assault upon the Pythagorean party, most likely did not take place until the middle of the fifth century; and lastly, Pythagoras may have spent the last portion of his life unmolested at Metapontum.' (Zeller, PreSocratic Philosophy, vol. 1., p. 360, E. T.).

Ueberweg takes a similar view :—

'But the persecutions were also several times renewed. In Crotona, as it appears, the partisans of Pythagoras and the 'Cylonians' were for a long time after the death of Pythagoras living in opposition as political parties, till at length, about a century later, the Pythagoreans were surprised by their opponents, while engaged in a deliberation in the 'house of Milo' (who himself had died long before), and the house being set on fire and surrounded, all perished with the exception of Archippus and Lysis of Tarentum. (According to other accounts, the burning of the house, in which the Pythagoreans were assembled, took place on the occasion of the first reaction against the Society, in the lifetime of Pythagoras.) Lysis went to Thebes, and was there (soon after 400 B.C.) a teacher of the youthful Epaminondas.' (Ueberweg, History of Philosophy, vol. I., p. 46, E. T.)

Zeller, in a note on the passage quoted above, gives the reasons on which his suppositions are chiefly based. Chaignet, Pyth. et la Phil. Pyth. vol. I., p. 88, and note, states Zeller's opinion, and, while admitting that the reasons advanced by him do not want force, says that they are not strong enough to convince him: he then gives his objections. Chaignet, further on, p. 94, n., referring to the name Italian, by which the Pythagorean philosophy is known, says: 'C'est même ce qui

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