


From Burnet's IntroductionXI. Egyptian Mathematics The Rhind papyrus in the British Museum^{46} gives us a glimpse of arithmetic and geometry as they were understood on the banks of the Nile. It is the work of one Aahmes, and contains rules for calculations both of an arithmetical and a geometrical character. The arithmetical problems mostly concern measures of corn and fruit, and deal particularly with such questions as the division of a number of measures among a given number of persons, the number of loaves or jars of beer that certain measures will yield, and the wages due to the workmen for a certain piece of work. It corresponds exactly, in fact, to the description of Egyptian arithmetic Plato gives us in the Laws, where he tells us that children learnt along with their letters to solve problems in the distribution of apples and wreaths to greater or smaller numbers of people, the pairing of boxers and wrestlers, and so forth.^{47} This is clearly the origin of the art which the Greeks called λογιστική, and they probably borrowed that from Egypt, where it was highly developed; but there is trace of what the Greeks called ἀριθμητική, the scientific study of numbers. The geometry of the Rhind papyrus is of a similar character, and Herodotos, who tells us that Egyptian geometry arose from the necessity of measuring the land afresh after the inundations, is clearly far nearer the mark than Aristotle, who says it grew out of the leisure enjoyed by the priestly caste.^{48} 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. It is even assumed that a rightangled triangle can be equilateral. The rule for finding what is called the seqt of a pyramid is, however, on a rather higher level, as we should expect. It comes to this. Given the "length across the sole of the foot," that as, 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.^{49} 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."^{50} Now the word ἀρπεδονάπτης is not Egyptian but Greek. It means "cordfastener,"^{51} and it is a striking coincidence that the oldest Indian geometrical treatise is called the Sulvasutras 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.^{52} There is no reason for supposing that any of these peoples had troubled themselves to give a theoretical demonstration of its properties, though Demokritos would certainly have been able to do so. As we shall see, however, there is no real evidence that Thales had any mathematical knowledge which went beyond the Rhind papyrus, and we must conclude that mathematics in the strict sense arose in Greece after his time. It is significant in this connexion that all mathematical terms are purely Greek in their origin.^{53} Burnet's Notes46. I am indebted for most of the information which follows to Cantor's Vorlesungen über Geschichte der Mathematik, vol. i. pp. 4663. See also Gow's Short History of Greek Mathematics, §§ 7380; and Milhaud, La Science grecque, pp. 91 sqq. The discussion in the lastnamed work is of special value because it is based on M. Rodet's paper in the Bulletin de la Société Mathématique, vol. vi., which in some important respects supplements the interpretation of Eisenlohr, on which the earlier accounts depend. 47. Plato, Laws, 819 b 4 μήλων τέ τινων διανομαὶ καὶ στεφάνων πλείοσιν ἅμα καὶ ἐλάττοσιν ἁρμοττόντων ἀριθμῶν τῶν αὐτῶν, καὶ πυκτῶν καὶ παλαιστῶν ἐφεδρείας τε καὶ συλλήξεως ἐν μέρει καὶ ἐφεξῆς καὶ ὡς πεφύκασι γίγνεσθαι. καὶ δὴ καὶ παίζοντες, φιάλας ἅμα χρυσοῦ καὶ χαλκοῦ καὶ ἀργύρου καὶ τοιούτων τινῶν ἄλλων κεραννύντες, οἱ δὲ καὶ ὅλας πως διαδιδόντες. 48. Herod ii. 109; Arist Met. A, 1. 981 b 23. 49. For a fuller account of this method see Gow, Short History of Greek Mathematics, pp. 127 sqq.; and Milhaud, Science grecque, p. 99. 50. 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. 51. The real meaning of ἀρπεδονάτης was first pointed out by Cantor. The gardener laying out a flowerbed is the true modern representative of the "arpedonapts." 52. See Milhaud, Science grecque, p. 103. 53. Cf. e.g. κύκλος, κύλινδρος. Very often these terms are derived from
the names of tools, e.g. γνώμων, which is the carpenter's
square, and τομεύς, "sector," which is a cobbler's knife. The
word πυραμίς is sometimes supposed to be an exception and has
been derived from the term piremus used in the Rhind papyrus,
which, however, does not mean "pyramid" (p. 19); but it too is
Greek. Πυραμίς (or πυραμοῦς) means a "wheatcake," and is formed
from πυροί on the analogy of σησαμίς (or σησαμοῦς). The
Greeks had a tendency to give jocular names to things
Egyptian. Cf. κροκόδειλος, ὀβελίσκος, στρουθός, καταράκτης (lit. "sluice"). We seem to hear an echo
of the slang of the mercenaries who cut their names on the
colossus at AbuSimbel.


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