The geometry of the Rhind papyrus is of a similarly utilitarian character, and Herodotos, who tells us that Egyptian geometry arose from the necessity of measuring the land afresh after the inundations, is obviously far nearer the mark than Aristotle, who says that it grew out of the leisure enjoyed by the priestly caste.[33] We find, accordingly, that the rules given for calculating areas are only exact when these are rectangular. As fields are usually more or less rectangular, this would be sufficient for practical purposes. The rule for finding what is called the _seqt_ of a pyramid is, however, on a rather higher level, as we should expect; for the angles of the Egyptian pyramids really are equal, and there must have been some method for obtaining this result. It comes to this. Given the “length across the sole of the foot,” that is, the diagonal of the base, and that of the _piremus_ or “ridge,” to find a number which represents the ratio between them. This is 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.[34] That the Greeks learnt as much from them, we shall see to be highly probable, though we shall see also that, from a comparatively early period, 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 of Demokritos which has been preserved. He says (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 _harpedonapts_, as they call them.”[35] Now the word ἁρπεδονάπτης is not Egyptian but Greek. It means “cord-fastener,”[36] and it is a striking coincidence that the oldest Indian geometrical treatise is called the _Çulvasutras_ or “rules of the cord.” These things point to the use of the triangle of which the sides are 3, 4, 5, and which has always a right angle. We know that this triangle 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.[37] There is no reason whatever for supposing that any of these peoples had in any degree troubled themselves to give a theoretical demonstration of its properties, though Demokritos would certainly have been able to do so. Finally, we must note the highly significant fact that all mathematical terms are of purely Greek origin.[38] Footnote 33: Herod. ii. 109; Arist. _Met._ Α, 1. 981 b 23. Footnote 34: