Arithmetic is older than geometry, and was much more advanced in Egypt, though still in the form which the Greeks called λογιστική rather than as ἀριθμητική proper. Even Plato puts Arithmetic before Geometry in the _Republic_ in deference to the tradition. His own theory of number, however, suggested the inversion of this order which we find carried out in Euclid. 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. Footnote 234: 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 τὰ ὀκτακόσια, νόμῳ καὶ συνθήματι ἀνθρωπίνῳ, ἀλλ’ οὐ φύσει σημαντικόν, ἐστι τοῦ ἀριθμοῦ, κ.τ.λ. The same symbolism is used by Theo, _Expositio_, pp. 31 sqq. Cf. also Iambl. _Introd._ p. 56, 27, Pistelli, ἰστέον γὰρ ὡς τὸ παλαιὸν φυσικώτερον οἱ πρόσθεν ἐσημαίνοντο τὰς τοῦ ἀριθμοῦ ποσότητας, ἀλλ’ οὐχ ὥσπερ οἱ νῦν συμβολικῶς. [Sidenote: Triangular, square, and oblong numbers.]