49. 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 which they occupy, or rather mark out, is the “field” (χώρα).[240] This is evidently a very early way of speaking, and may therefore be referred to Pythagoras himself. Now it must have struck him that “fields” could be compared as well as numbers,[241] and it is even likely that he knew the rough methods of doing this which were traditional in Egypt, though certainly these would fail to satisfy him. Once more the tradition is singularly helpful in suggesting the direction that 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. 24) 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” 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 “harpedonapt.”[242] An early tradition says that Pythagoras sacrificed an ox when he discovered the proof of this proposition, and indeed it was the real foundation of scientific mathematics.[243] Footnote 240: We have ὅροι of a series (ἔκθεσις), then of a proportion, and in later times of a syllogism. The signs :, ::, and ∴ are a survival of the original use. 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. Footnote 241: 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 an account of these and the subsequent application of the terms in Conic Sections, see Milhaud, _Philosophes géomètres_, pp. 81 sqq. Footnote 242: The verb ὑποτείνειν is, of course, used intransitively. The explanation suggested in the text seems to me much simpler than that of Max C. P. Schmidt (_Kulturhistorische Beiträge_, Heft i. pp. 64 sqq.). He explains the hypotenuse as the longest string in a triangular harp; but my view seems more in accordance with analogy. So ἡ κάθετος is, literally, a plumb-line. Footnote 243: The statement comes from Eudemos; for it is found in Proclus’s commentary on Euclid i. 47. Whether historical or not, it is no Neopythagorean fancy.