and so of every other quality. Number, he inferred, is quantity and quantity is form and form is quality1. The genesis of the Pythagorean philosophy here suggested has strong historical warrant. It is certain that the Egyptian geometry was such as I have described it: the empirical knowledge of the land-surveyor, not the generalised deductions of the mathematician. If not certain, it is at least undeniable that Pythagoras lived in Egypt and there learnt such geometry as was known. It is certain that Pythagoras considered number to be the basis of creation: that he looked to arithmetic for his definitions of all abstract terms and his explanation of all natural laws: but that his arithmetical inquiries went hand in hand with geometrical and that he tried always to find arithmetical formulae for geometrical facts and vice versa3. 45. But the details of his doctrines are now hopelessly lost. For a hundred years they remained the secrets of his school in Italy and when at last a Pythagorean philosophy was published*, it was far more elaborate than the teaching of its founder. Even the tenets of this later school come to us only by hearsay. Of Pythagoreans we know something from Plato and Aristotle 1 It was Pythagoras who discovered that the 5th and the octave of a note could be produced on the same string by stopping at and of its length respectively. Harmony therefore depends on a numerical proportion. It was this discovery, according to Hankel, which led Pythagoras to his phi losophy of number. It is probable at least that the name harmonical proportion was due to it, since 1:1(1): (} − 1). Iamblichus says that this proportion was called vπevavría originally and that Archytas and Hippasus first called it harmonic. Nicomachus gives another reason for the name: viz. that a cube, being of 3 equal dimensions, was the pattern åpμovía: and having 12 edges, 8 corners, 6 faces, it gave its name to harmonic proportion, since 126 128:8-6. Vide Cantor, Vorles. p. 152. Nessel- 3 See Diog. Laert. vIII. 12 and 14. In the second passage Pythagoras is said, on the authority of Aristoxenus, to have introduced weights and measures into Greece. 4 By Philolaus. See Diog. Laert. VIII. 15. 85. The silence of Pythagoras was proverbial. On this and the facts stated in the text cf. Ritter and Preller, Hist. Phil. §§ 96, 97, 102-128. and the historians of philosophy, but hardly anything remains which is attributed, by any writer of respectable authority, to Pythagoras himself1. He is probably responsible for some of the fantastic metaphysics of his followers. Aristotle expressly says that he referred the virtues to numbers and perhaps he agreed with Philolaus that 5 is the cause of colour, 6 of cold, 7 of mind and health and light, 8 of love and friendship and invention. Plutarch says that he held that the earth was the product of the cube, fire of the pyramid, air of the octahedron, water of the eicosahedron, and the sphere of the universe of the dodecahedron2. But doctrines of this kind, though they imply an interest in mathematics, are not themselves contributions to mathematical knowledge and do not require to be discussed in this place. For our present purpose, it is sufficient only to consider what advances in arithmetic are due to Pythagoras or his school, without speculating on the mode or order in which they were obtained or their place in the Pythagorean philosophy. The following discoveries, at any rate, with the accompanying nomenclature, are as old as Plato's time. All numbers were classified as odd or even (ἄρτιοι or περισσοί). Of these the odd numbers were gnomons (yváμoves) and the sum of the series of gnomons from 1 to 2n+1 was a square (TETρáywvos). The root of a square number was called its side (Tλeupá). Some compound numbers have no square roots. These latter were oblongs (ἑτερομήκεις οι προμήκεις). Products of two numbers were plane (èπíπedοi), of three solid (σTepeoí). A number multiplied twice into itself was a cube (kiBos). Some more classifications 1 Porphyrius, a Syrian, late in the 3rd century after Christ, and Iamblichus both wrote a 'life of Pythagoras.' 2 See Ritter and Preller, pp. 72, 79, §§ 116, 117, 127. 3 Aristotle, Phys. III. 4. The gnomon is properly a carpenter's instrument, a T square with only one arm. The name was afterwards used in other senses. 4 See Plato Theaet. 147 D-148 B. A surd was probably at this early time called inexpressible or irrational (äßin Tos or aλoyos), but this is not certain. 5 Cf. Aristotle on Plato in Pol. v. 12. 8. 6 Plato, Rep. VIII. p. 246. The same passage invites one or two other little remarks. ἀριθμὸς ἀπὸ in later Greek writers means 'the square of': åp‹Ðμòs ὑπὸ means the product of. ἀπόστασις once in Plato (Timaeus 43 D) means the 'interval' between the terms of an arithmetical progression. avŋois may (like augn, Rep. VII. 528 в) mean 'mul V are given by authorities of less antiquity. Any number of the form n (n + 1) was called triangular (Tpíywvos). Perfect (TéλEO) numbers are those which are equal to the sum of all their possible factors (e.g. 28=1+2+4+7+14): for similar reasons numbers are excessive (VTEρтéλeioi) or defective (exTeis)'. Amicable (píλioi) numbers are those of which each is the sum of the factors of the other (e.g. 220 = 1 + 2 + 4+ 71+142 ; 2841 + 2 + 4 + 5 + 10 + 11+ 20 + 22 + 44 +55 +110). Beside this work in classification of single numbers, numbers were treated in groups comprised either in a series (exeous or ἀναλογία συνεχής) or a proportion (αναλογία). Each number of such a series or proportion was called a term (opos). The mean terms of a proportion were called μeoÓTηTES3. Three kinds of proportion, the arithmetical, geometrical and harmonical were certainly known. To these Iamblichus adds a fourth, the musical, which, he says, Pythagoras introduced from Babylon. It is composed between two numbers and their arithmetical a + b 2ab and harmonical means, thus a: : b (e.g. 6:9: 8:12). 2 a+b Plato knew that there was only one expressible geometrical mean between two square numbers, two between two cubes. It is a familiar fact that the geometrical proposition, Euclid 1. 47, is ascribed to Pythagoras. It follows that a right-angled triangle may be always constructed by taking sides which are to one an tiplication.' For other terms see Journal of Philology, xii. p. 92. 1 Theon Smyrnaeus (ed. Hiller) pp. 31, 45. 2 Iamblichus in Nicom. Ar. (ed, Tennulius) pp. 47, 48. 3 It will have been observed that much of our modern nomenclature (e.g. square,' 'cube,'' surd,' 'term,' 'mean') is taken from the Latin translation of the Greek expressions. 4 Philolaus in Nicomachus Introd. Ar. (ed. Hoche) p. 135, Archytas in Porphyrius, ad Ptol. Harm. cited by Gruppe, Die Fragm. des Archytas, etc. p. 94. This quotation (with one or two more) I take from Cantor, Vorl. p. 140 sqq. The statement in the text might be easily confirmed from other sources. See for instance Simplicius on Ar. de Anima 409, b. 23. Dr Allman doubts (Hermathena v. p. 204) whether these proportions were first applied to number, but see Ar. An. Post. I. 5. 74, and Hankel p. 114. 5 In Nicom. Ar. (ed. Tennulius) pp. 141-2, 168. 6 Timaeus, 32 B. Nicomachus, Introd. Ar. 11. c. 24. other in the ratios 3:4: 5, and to these numbers therefore great importance was attached in Pythagorean philosophy1. To Pythagoras himself is ascribed a mode of finding other numbers which would serve the same purpose. He took as one side an odd number (2n+1): half the square of this minus 1 is the other side (2n+2n): this last number plus 1 is the hypotenuse (2n2+2n+1). He began, it will be noticed, with an odd number. Plato invented another mode, beginning with an even number (2n): the square of half this plus 1 is the hypotenuse (n2 + 1): the same square minus 1 is the other side (n-1). 2 46. A few more details expressly alleged by, or inferred from hints of, later authors might be added to the foregoing but it is impossible to frame with them a continuous history even of the most meagre character. We cannot say precisely what Pythagoras knew or discovered, and what additions to his knowledge were successively made by Philolaus or Archytas or Plato or other inquirers who are known to have been interested in the philosophy of numbers. Proclus says that the Pythagoreans were concerned only with the questions 'how many' (тò TOσóv) and 'how great' (Tò Tηíkov) that is, with number and magnitude. Number absolute was the field of arithmetic: number applied of music: stationary magnitude of geometry, magnitude in motion of 1 This rule was known to the Egyptians, the Chinese and perhaps the Babylonians at a very remote antiquity, v. Cantor, Vorles. pp. 56, 92,153—4. The discovery is expressly attributed to Pythagoras (Vitruvius, Ix. 2). Cantor (Vorles. p. 153 sqq.) is of opinion that Pythagoras knew this empirical rule for constructing right-angled triangles before he discovered Eucl. 1. 47. 2 Proclus (ed. Friedlein), p. 428. It will be noticed that both the Pythagorean and Platonic methods apply only to cases in which the hypotenuse differs from one side by 1 or 2. They would not discover such an eligible group of side-numbers as 29: 21: 20. See Nesselmann, pp. 152-3. These are provided for by the first lemma to Euclid, x. 29. Infra, p. 81 n. 3 Plutarch, Quaest. Conv. VIII. 9. 11-13, says that Xenocrates, the pupil of Plato, discovered that the number of possible syllables was 1,002,000,000,000. This looks like a problem in combinations, but the theory of combinations does not appear in any Greek mathematician, and the number seems too round to have been scientifically obtained. (Cantor, Vorles. pp. 215, 220.) 4 Ed. Friedlein, pp. 35, 36. For the distinction between number and magnitude compare Aristotle, An. Post. 1. 7 and 10, and Cat. c. 6. spheric or astronomy. But they did not so strictly dissociate discrete from continuous quantity. An arithmetical fact had its analogue in geometry and vice versa; a musical fact had its analogue in astronomy and vice versa. Pythagorean arithmetic and geometry should therefore be treated together, but there is so little known of either, that it seemed unadvisable, for this purpose only, to alter the plan of this book. The history of Greek geometry is so much fuller and more important and proceeds by so much more regular stages than that of arithmetic, that it deserves to be kept distinct. The facts above stated are sufficient to show that, from the first, Greek apilμnτin was closely connected with geometry and that it borrowed, from the latter science, its symbolism and nomenclature. It had not yet wholly discarded the abacus2, but its aim was entirely different from that of the ordinary calculator and it was natural that the philosopher who sought in numbers to find the plan on which the Creator worked, should begin to regard with contempt the merchant who wanted only to know how many sardines, at 10 for an obol, he could buy for a talent. 47. Whensoever and by whomsoever invented, most of the known propositions of ȧpilμntiký were collected together, not much later than 300 B. C. by Euclid in his Elements. Only the seventh, eighth and ninth books are specially devoted to numbers, but it cannot be doubted that the second and the tenth, though they profess to be geometrical and to deal with 1 These four sciences became, through the Pythagorean influence of Alexandria, the quadrivium of early mediaevalism. The subjects of this fourfold education are mentioned in the familiar line "Mus canit: Ar numerat: Ge ponderat: Ast colit astra." To this, however, another trivium Rhetoric, Dialectic and Grammar, were added ("Gram loquitur: Dia vera docet: Rhet verba colorat") and these seven are the goddesses of science and art who attend at the nuptials of Philology and Mercury celebrated by Martianus Capella (cir. A. D. 400). The same seven branches of education are discussed by Cassiodorus (born about A.D. 468), De Artibus ac Disciplinis Liberalium Litterarum. 2 E.g. Plato, Legg. 737 E, 738 ▲ says that 5040 has 59 divisors including all the numbers from 1 to 10. A fact of this sort must have been discovered empirically by means of the abacus. 3 Archimedes uses one or two propositions which are not in the Elements. |