Early Greek Philosophy by John Burnet, with Burnet's notes
46. Arithmetic 48. Triangular, Square and Oblong Numbers

From Chapter II., Science and Religion

47. The Figures
One of the most remarkable statements we have about Pythagoreanism is what we are told of Eurytos on the unimpeachable authority of Archytas. Eurytos was the disciple of Philolaos, and Aristoxenos mentioned him along with Philolaos as having taught the last of the Pythagoreans, the men with whom he himself was acquainted. He therefore belongs to the beginning of the fourth century B.C., by which time the Pythagorean system was fully developed, and he was no eccentric enthusiast, but one of the foremost men in the school.72 We are told of him, then, that he used to give the number of all sorts of things, such as horses and men, and that he demonstrated these by arranging pebbles in a certain way. Moreover, Aristotle compares his procedure to that of those who bring numbers into figures (σχήματα) like the triangle and the square.73

Now these statements, and especially the remark of Aristotle last quoted, seem to imply the existence at this date, and earlier, of a numerical symbolism quite distinct from the alphabetical notation on the one hand and from the Euclidean representation of numbers by lines on the other. The former was inconvenient for arithmetical purposes, because the zero was not yet invented.74 The representation of numbers by lines was adopted to avoid the difficulties raised by the discovery of irrational quantities, and is of much later date. It seems rather that numbers were originally represented by dots arranged in symmetrical and easily recognised patterns, of which the marking of dice or dominoes gives us the best idea. And these markings are, in fact, the best proof that this is a genuinely primitive method of indicating numbers; for they are of unknown antiquity, and go back to the time when men could only count by arranging numbers in such patterns, each of which became, as it were, a fresh unit.

It is, therefore, significant that we do not find any clue to what Aristotle meant by "those who bring numbers into figures like the triangle and the square" till we come to certain late writers who called themselves Pythagoreans, and revived the study of arithmetic as a science independent of geometry. These men not only abandoned the linear symbolism of Euclid, but also regarded the alphabetical notation, which they did use, as inadequate to represent the true nature of number. Nikomachos of Gerasa says expressly that the letters used to represent numbers are purely conventional.75 The natural thing would be to represent linear or prime numbers by a row of units, polygonal numbers by units arranged so as to mark out the various plane figures, and solid numbers by units disposed in pyramids and so forth.76 We therefore find figures like this

Now it ought to be obvious that this is no innovation. Of course the employment of the letter alpha to represent the units is derived from the conventional notation; but otherwise we are clearly in presence of something which belongs to the very earliest stage of the science. We also gather that the dots were supposed to represent pebbles (ψῆφοι), and this throws light on early methods of what we still call calculation.

Burnet's Notes


72. Apart from the story in Iamblichos (V. Pyth. 148) that Eurytos heard the voice of Philolaos from the grave after he had been many years dead it is to be noticed that he is mentioned after him in the statement of Aristoxenos referred to (Diog. viii. 46; R. P. 62).

73. Arist. Met. N, 5. 1092 b 8 (R. P. 76 a). Aristotle does not quote the authority of Archytas here, but the source of his statement is made quite clear by Theophr. Met. p. vi. a 19 (Usener), τοῦτο γὰρ (sc. τὸ μὴ μέχρι του προελθόντα παύεσθαι) τελέου καὶ φρονοῦντος, ὅπερ Ἀρχύτας ποτ' ἔφη ποιεῖν Εὔρυτον διατιθέντα τινὰς ψήφους· λέγειν γὰρ ὡς ὅδε μὲν ἀνθρώπου ὁ ἀριθμός, ὅδε δὲ ἵππου, ὅδε δ' ἄλλου τινὸς τυγχάνει.

74. The notation used in Greek arithmetical treatises must have originated at a date and in a region where the Vau and the Koppa were still recognised as letters of the alphabet and retained their original position in it. That points to a Dorian state (Taras or Syracuse?), and to a date not later than the early fourth century B.C. The so-called Arabic figures are usually credited to the Indians, but M. Carra de Vaux has shown (Scientia, xxi. pp. 273 sqq.) that this idea (which only makes its appearance in the tenth century A.D.) is due to a confusion between the Arabic hindi, "Indian," and hindasi, "arithmetical." He comes to the conclusion that the "Arabic" numerals were invented by the Neopythagoreans, and brought by the Neoplatonists to Persia, whence they reached the Indians and later the Arabs. The zero, on which the value of the whole system depends, appears to be the initial letter of οὐδέν.

75. Nikomachos of Gerasa, Introd. Arithm. p. 83, 12, Hoche, Πρότερον δὲ ἐπιγνωστέον ὅτι ἕκαστον γράμμα ᾧ σημειούμεθα ἀριθμόν, οἷον τὸ ι, ᾧ τὸ δέκα, τὸ κ, ᾧ τὰ εἴκοσι, τὸ ω, ᾧ τα ὀκτακόσια, νόμῳ καὶ συνθήματι ἀνθρωπίνῳ, ἀλλ' οὐ φύσει σημαντικόν ἐστι τοῦ ἀριθμοῦ κτλ. Cf. also Iambl. in Nicom. p. 56, 27, Pistelli, ἰστέον γὰρ ὡς τὸ παλαιὸν φυσικώτερον οἱ πρόσθεν ἐσημαίνοντο τὰς τοῦ ἀριθμοῦ ποσότητας, ἀλλ' οὐχ ὥσπερ οἱ νῦν συμβολικῶς.

76. For the prime or rectilinear numbers, cf. Iambl. in Nicom. p. 26, 25, Pistelli, πρῶτος μὲν οὖν καὶ ἀσύνθετος ἀριθμός ἐστι περισσὸς ὃς ὑπὸ μόνης μονάδος πληρούντως μετρεῖται, οὐκέτι δὲ καὶ ὑπ' ἄλλου τινὸς μέρους, καὶ ἐπὶ μίαν δὲ διάστασιν προβήσεται ὁ τοιοῦτος, διὰ τοῦτο δὲ αὐτὸν καὶ εὐθυμετρικόν τινες καλοῦσι, Θυμαρίδας δὲ καὶ εὐθυγραμμικόν· ἀπλατὴς γὰρ ἐν τῇ ἐκθέσει ἐφ' ἓν μόνον διιστάμενος. It is generally recognised now that Thymaridas was an early Pythagorean (Tannery, Mém. scient. vol. i. n. 9; G. Loria, Scienze esatte, p. 807); and, if that is so, we have a complete proof that this theory goes back to the early days of the school. For the triangular, oblong, and square numbers, etc., see Theon of Smyrna, pp. 27-37, Hiller, and Nicom. loc. cit.

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