Early Greek Philosophy by John Burnet, with Burnet's notes
48. Triangular, Square and Oblong Numbers 50. Incommensurability

From Chapter II., Science and Religion

49. Geometry and Harmonics
It is easy to see how this way of representing numbers would suggest problems of a geometrical nature. The dots which stand for the pebbles are regularly called "boundary-stones" (ὅροι, termini, "terms"), and the area they mark out is the "field " (χώρα).83 This is evidently an early way of speaking, and may be referred to Pythagoras himself. Now it must have struck him that "fields" could be compared as well as numbers,84 and it is likely that he knew the rough methods of doing this traditional in Egypt, though certainly these would fail to satisfy him. Once more the tradition is helpful in suggesting the direction his thoughts must have taken. He knew, of course, the use of the triangle 3, 4, 5 in constructing right angles. We have seen (p. 20) that it was familiar in the East from a very early date, and that Thales introduced it to the Hellenes, if they did not know it already. In later writers it is actually called the "Pythagorean triangle." Now the Pythagorean proposition par excellence is just that, in a right-angled triangle, the square on the hypotenuse is equal to the squares on the other two sides, and the so-called Pythagorean triangle is the application of its converse to a particular case. The very name "hypotenuse" (ὑποτείνουσα) affords strong confirmation of the intimate connexion between the two things. It means literally "the cord stretching over against," and this is surely just the rope of the "arpedonapt." It is, therefore, quite possible that this proposition was really discovered by Pythagoras, though we cannot be sure of that, and though the demonstration of it which Euclid gives is certainly not his.85



Burnet's Notes

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83. Proclus, in Eucl. I. p. 136, 8, ἔστι δὲ τὸ ὄνομα (sc. ὅρος) οἰκεῖον τῇ ἐξ ἀρχῆς γεωμετρίᾳ, καθ' ἣν τὰ χωρία ἐμέτρουν καὶ τοὺς ὅρους αὐτῶν ἐφύλαττον ἀσυγχύτους. We have ὅροι of a series (ἔκθεσις), then of a proportion, and in later times of a syllogism. The signs :, ::, .·. seem to be derived from this. The term χώρα is often used by the later Pythagoreans, though Attic usage required χωρίον for a rectangle. The spaces between the γραμμαί of the abacus and the chess-board were also called χῶραι.

84. In his commentary on Euclid i. 44, Proclus tells us on the authority of Eudemos that the παραβολή, ἔλλειψις and ὑπερβολή of χωρία were Pythagorean inventions. For these and the later application of the terms in Conic Sections, see Milhaud, Philosophes géomètres, pp. 81 sqq.

85. See Proclus's commentary on Euclid i. 47.




















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