CHAPTER II.

THE IONIAN AND PYTHAGOREAN SCHOOLS.1 CIRC. 600 B.C.-400 B.C.

WITH the foundation of the Ionian and Pythagorean schools we emerge from the region of antiquarian research and conjecture into the light of history. The materials at our disposal for estimating the knowledge of the philosophers of these schools previous to about the year 430 B.C. are, however, very scanty, Not only have all but fragments of the different mathematical treatises then written been lost, but we possess no copy of the history of mathematics written about 325 B.C. by Eudemus (who was a pupil of Aristotle). Luckily Proclus, who about 450 a.d. wrote a commentary on the earlier part of Euclid's Elements, was familiar with Eudemus's work, and freely utilised it in his historical references. We have also a fragment of the General View of Mathematics written by Geminus about 50 B.C., in which the methods of proof used by the early Greek geometricians are compared with those current at a later date. In addition to these general statements we have biographies of a few of the

1 The history of these schools has been discussed by G. Loria in his Le Scienze Esatte nell' Antica Grecia, Modena, 1893-1900; by Cantor, chaps. v-viii; by G. J. Allman in his Greek Geometry from Thales to Euclid, Dublin, 1889; by J. Gow, in his Greek Mathematics, Cambridge, 1884; by C. A. Bretschneider in his Die Geometrie und die Geometer vor Eukleides, Leipzig, 1870; and partially by H. Hankel in his posthumous Geschichte der Mathematik, Leipzig, 1874.

leading mathematicians, and some scattered notes in various writers in which allusions are made to the lives and works of others. The original authorities are criticised and discussed at length in the works mentioned in the footnote to the heading of the chapter.

Thales.1

The Ionian School.

The founder of the earliest Greek school of mathematics and philosophy was Thales, one of the seven sages of Greece, who was born about 640 B.C. at Miletus, and died in the same town about 550 B.C. The materials for an account of his life consist of little more than a few anecdotes which have been handed down by tradition.

During the early part of his life Thales was engaged partly in commerce and partly in public affairs; and to judge by two stories that have been preserved, he was then as distinguished for shrewdness in business and readiness in resource as he was subsequently celebrated in science. It is said that once when transporting some salt which was loaded on mules, one of the animals slipping in a stream got its load wet and so caused some of the salt to be dissolved, and finding its burden thus lightened it rolled over at the next ford to which it came; to break it of this trick Thales loaded it with rags and sponges which, by absorbing the water, made the load heavier and soon effectually cured it of its troublesome habit. At another time, according to Aristotle, when there was a prospect of an unusually abundant crop of olives Thales got possession of all the olive-presses of the district; and, having thus "cornered " them, he was able to make his own terms for lending them out, or buying the olives, and thus realized a large sum. These tales may be apocryphal, but it is certain that he must have had considerable reputation as a man of affairs and as a good engineer, since he was employed to construct an embankment so as to divert the river Halys in such a way as to permit of the construction of a ford.

1 See Loria, book 1, chap. ii; Cantor, chap. v; Allman, chap. i:

Probably it was as a merchant that Thales first went to Egypt, but during his leisure there he studied astronomy and geometry. He was middle-aged when he returned to Miletus; he seems then to have abandoned business and public life, and to have devoted himself to the study of philosophy and science subjects which in the Ionian, Pythagorean, and perhaps also the Athenian schools, were closely connected: his views on philosophy do not here concern us. He continued

to live at Miletus till his death circ. 550 B.C.

We cannot form any exact idea as to how Thales presented his geometrical teaching. We infer, however, 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 geometricians. The deductive character which he thus gave to the science is his chief claim to distinction.

The following comprise the chief propositions that can now with reasonable probability be attributed to him; they are concerned with the geometry of angles and straight lines.

(i) The angles at the base of an isosceles triangle are equal (Euc. 1, 5). Proclus seems to imply that this was proved by taking another exactly equal isosceles triangle, turning it over, and then superposing it on the first-a sort of experimental demonstration.

(ii) If two straight lines cut one another, the vertically opposite angles are equal (Euc. 1, 15). Thales may have regarded this as obvious, for Proclus adds that Euclid was the first to give a strict proof of it.

(iii) A triangle is determined if its base and base angles be given (cf. Euc. 1, 26). Apparently this was applied to find the distance of a ship at sea— -the base being a tower, and the base angles being obtained by observation.

(iv) The sides of equiangular triangles are proportionals (Euc. vi, 4, or perhaps rather Euc. VI, 2). This is said to

have been used by Thales when in Egypt to find the height of a pyramid. In a dialogue given by Plutarch, the speaker, addressing Thales, says, "Placing your stick at the end of the shadow of the pyramid, you made by the sun's rays two triangles, and so proved that the [height of the] pyramid was to the [length of the] stick as the shadow of the pyramid to the shadow of the stick." It would seem that the theorem was unknown to the Egyptians, and we are told that the king Amasis, who was present, was astonished at this application of abstract science.

(v) A circle is bisected by any diameter. This may have been enunciated by Thales, but it must have been recognised as an obvious fact from the earliest times.

(vi) The angle subtended by a diameter of a circle at any point in the circumference is a right angle (Euc. III, 31). This appears to have been regarded as the most remarkable of the geometrical achievements of Thales, and it is stated that on inscribing a right-angled triangle in a circle he sacrificed an ox to the immortal gods. It is supposed that he proved the proposition by joining the centre of the circle to the apex of the right angle, thus splitting the triangle into two isosceles triangles, and then applied the proposition (i) above: if this be the correct account of his proof, he must have been aware that the sum of the angles of a right-angled triangle is equal to two right angles.

It has been ingeniously suggested that the shape of the tiles used in paving floors may have afforded an experimental demonstration of the latter result, namely, that the sum of the angles of a triangle is equal to two right angles. We know from Eudemus that the first geometers proved the general property separately for three species of triangles, and it is not unlikely that they proved it thus. The area about a point can be filled by the angles of six equilateral triangles or tiles, hence the proposition is true for an equilateral triangle. Again, any two equal right-angled triangles can be placed in juxtaposition so as to form a rectangle, the sum of whose angles is four right

angles; hence the proposition is true for a right-angled triangle ; and it will be noticed that tiles of such a shape would give an ocular demonstration of this case. It would appear that this proof was given at first only in the case of isosceles right-angled triangles, but probably it was extended later so as to cover any right-angled triangle. Lastly, any triangle can be split into the sum of two right-angled triangles by drawing a perpendicular from the biggest angle on the opposite side, and therefore again the proposition is true. The first of these proofs is evidently included in the last, but there is nothing improbable in the suggestion that the early Greek geometers continued to teach the first proposition in the form above given.

Thales wrote on astronomy, and among his contemporaries was more famous as an astronomer than as a geometrician. A story runs that one night, when walking out, he was looking so intently at the stars that he tumbled into a ditch, on which an old woman exclaimed, "How can you tell what is going on in the sky when you can't see what is lying at your own feet?” -an anecdote which was often quoted to illustrate the unpractical character of philosophers.

Without going into astronomical details, it may be mentioned that he taught that a year contained about 365 days, and not (as is said to have been previously reckoned) twelve months of thirty days each. It is said that his predecessors occasionally intercalated a month to keep the seasons in their customary places, and if so they must have realized that the year contains, on the average, more than 360 days. There is some reason to think that he believed the earth to be a disc-like body floating on water. He predicted a solar eclipse which took place at or about the time he foretold; the actual date was either May 28, 585 B.C., or September 30, 609 B.C. But though this prophecy and its fulfilment gave extraordinary prestige to his teaching, and secured him the name of one of the seven sages of Greece, it is most likely that he only made use of one of the Egyptian or Chaldaean registers which stated that solar eclipses recur at intervals of about 18 years 11 days.

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