done by dividing half the diagonal of the base by the "ridge," and it is obvious that such a method might quite well be discovered empirically. It seems an anachronism to speak of elementary trigonometry in connexion with a rule like this, and there is nothing to suggest that the Egyptians went any further.[1] That the Greeks learnt as much from them is highly probable, though we shall see also that, from the very first, they generalised it so as to make it of use in measuring the distances of inaccessible objects, such as ships at sea. It was probably this generalisation that suggested the idea of a science of geometry, which was really the creation of the Pythagoreans, and we can see how far the Greeks soon surpassed their teachers from a remark attributed to Demokritos. It runs (fr. 299): "I have listened to many learned men, but no one has yet surpassed me in the construction of figures out of lines accompanied by demonstration, not even the Egyptian arpedonapts, as they call them."[2] Now the word ἀρπεδονάπτης is not Egyptian but Greek. It means "cord-fastener,"[3]and it is a striking coincidence that the oldest Indian geometrical treatise is called the Śulvasūtras or "rules of the cord." These things point to the use of the triangle of which the sides are as 3, 4, 5, and which has always a right angle. We know that this was used from an early date among the Chinese and the Hindus, who doubtless got it from Babylon, and we shall see that Thales probably learnt the use of it in Egypt.[4] There is no reason for
- ↑ For a fuller account of this method see Gow, Short History of Greek Mathematics, pp. 127 sqq.; and Milhaud, Science grecque, p. 99.
- ↑ R. P. 188. It should be stated that Diels now considers this fragment spurious (Vors.³ ii. p. 124). He regards it, in fact, as from an Alexandrian forgery intended to show the derivative character of Greek science, while insisting on its superiority. However that may be the word ἀρπεδονάπται is no doubt a real one, and the inference drawn from it in the text is justified.
- ↑ The real meaning of ἀρπεδονάτης was first pointed out by Cantor. The gardener laying out a flower-bed is the true modern representative of the "arpedonapts."
- ↑ See Milhaud, Science grecque, p. 103.
