48. This is still further confirmed by the tradition which represents the great revelation made by Pythagoras to mankind as having been precisely a figure of this kind, namely the _tetraktys_, by which the Pythagoreans used to swear,[235] and we have no less an authority than Speusippos for holding that the whole theory which it implies was genuinely Pythagorean.[236] In later days there were many kinds of _tetraktys_,[237] but the original one, that by which the Pythagoreans swore, was the “tetraktys of the dekad.” It was a figure like this— • •   • •   •   • •   •   •   • and represented the number ten as the triangle of four. In other words, 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. Footnote 235: 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. The Doric dialect shows, however, that it belongs to the later generations of the school. Footnote 236: 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_, p. 235, 15, and is discussed by Tannery, _Science hellène_, pp. 374 sqq. Footnote 237: For these see Theon, _Expositio_, pp. 93 sqq. Hiller. The τετρακτύς used by Plato in the _Timaeus_ is the second described by Theon (_Exp._ p. 94, 10 sqq.). It is no doubt Pythagorean, but hardly as old as Pythagoras. It is obvious that the _tetraktys_ may be indefinitely extended so as to exhibit the sums of the series of successive numbers 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 to the unit in the form of _gnomons_, the result is always a similar figure, namely a square, while, if even numbers are added, we get a series of rectangles,[238] as shown by the figure:—