It is, therefore, very significant that we do not find any adequate account of what Aristotle can have 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 something conventional, and inadequate to represent the true nature of number. Nikomachos of Gerasa says expressly that the letters used to represent numbers are only significant by human usage and convention. The most natural way 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.[[234]] He therefore gives us figures like this:—

α              α α α

α      α α             ααα

α     α α                     α α             α α α

α α     α α             ααα

α α             α α α

Now it ought to be obvious that this is no innovation, but, like so many things in Neopythagoreanism, a reversion to primitive usage. 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—something, in fact, which gives the only possible clue to the meaning of Aristotle’s remark, and to what we are told of the method of Eurytos.

Triangular, square, and oblong numbers.

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—

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