From Chapter II., Science and Religion
48. Triangular, Square and Oblong Numbers
and represented the number ten as the triangle of four. It showed at a glance that 1+2+3+4=10. Speusippos tells us of several properties which the Pythagoreans discovered in the dekad. It is, for instance, the first number that has in it an equal number of prime and composite numbers. How much of this goes back to Pythagoras himself, we cannot tell; but we are probably justified in referring to him the conclusion that it is "according to nature" that all Hellenes and barbarians count up to ten and then begin over again.
It is obvious that the tetraktys may be indefinitely extended so as to exhibit the sums of the series of successive integers in a graphic form, and these sums are accordingly called "triangular numbers."
For similar reasons, the sums of the series of successive odd numbers are called "square numbers," and those of successive even numbers "oblong." If odd numbers are added in the form of gnomons,80 the result is always a similar figure, namely a square, while, if even numbers are added, we get a series of rectangles,81 as shown by the figure:
It is clear, then, that we are entitled to refer the study of
sums of series to Pythagoras himself; but whether he went
beyond the oblong, and studied pyramidal or cubic numbers, we
77. Cf. the formula Οὐ μὰ τὸν ἁμετέρᾳ γενεᾷ παραδόντα τετρακτύν, which is all the more likely to be old that it is put into the mouth of Pythagoras by the forger of the Χρυσᾶ ἔπη,, thus making him swear by himself ! See Diels, Arch. iii. p. 457.
78. Speusippos wrote a work on the Pythagorean numbers, based chiefly on Philolaos, and a considerable fragment of it is preserved in the Theologumena Arithmetica. It will be found in Diels, Vorsokratiker, 32 A 13, and is discussed by Tannery, Science hellène, pp. 374 sqq.
79. See Theon, Expositio, pp. 93 sqq., Hiller. The τετρακτύς used in the Timaeus is the second described by Theon (Exp. p. 94, 10 sqq.).
80. In accordance with analogy (p. 21, n. i), the original meaning of the word γνώμων must have been that of the carpenter's square. From that are derived its use (1) for the instrument; (2) for the figure added to a square or rectangle to form another square or rectangle. In Euclid (ii. def. 2) this is extended to all parallelograms, and finally the γνώμων is defined by Heron (ed. Heiberg, vol. iv. def. 58) thus: καθόλου δὲ γνώμων ἐστὶν πᾶν, ὃ προσλαβὸν ὁτιοῦν, ἀριθμὸς ἢ σχῆμα, ποιεῖ τὸ ὅλον ὅμοιον ᾧ προσείληφεν These, however, are later developments; for the use of γνώμων in the sense of "perpendicular" by Oinopides of Chios shows that, in the fifth century B.C., it only applied to rectangular figures.
81. Cf. Milhaud, Philosophes géomètres, pp. 115 sqq. Aristotle puts the matter thus (Phys. Γ, 4. 203 a 13): περιτιθεμένων γὰρ τῶν γνωμόνων περὶ τὸ ἓν καὶ χωρὶς ὁτὲ μὲν ἄλλο ἀεὶ γίγνεσθαι τὸ εἶδος, ὁτὲ δὲ ἕν.. This is more clearly stated by Ps.-Plut. (Stob. i. p. 22, 16, ἔτι δὲ τῇ μονάδι τῶν ἐφεξῆς περισσῶν περιτιθεμένων ὁ γινόμενος ἀεὶ τετράγωνός ἐστι· τῶν δὲ ἀρτίων ὁμοίως περιτιθεμένων ἑτερομήκεις καὶ ἄνισοι πάντες ἀποβαίνουσιν, ἴσως δὲ ἰσάκις οὐδείς. It will be observed that Aristotle here uses εἶδος in the sense of "figure." The words καὶ χωρὶς apparently mean χωρὶς τοῦ ἑνός, i.e. starting from 2, not from 1.
82. Speusippos (cf. p. 102, n. 2) speaks of four as the first pyramidal number; but this is taken from Philolaos, so we cannot safely ascribe it to Pythagoras.
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