145. In this way, then, the Odd and the Even were identified with the Limit and the Unlimited, and it is possible, though by no means certain, that Pythagoras himself had taken this step. In any case, there can be no doubt that by his Unlimited he meant something spatially extended, and we have seen that he identified it with air, night, or the void, so we are prepared to find that his followers also thought of the Unlimited as extended. Aristotle certainly regarded it so. He argues that, if the Unlimited is itself a reality, and not merely the predicate of some other reality, then every part of it must be unlimited too, just as every part of air is air.[788] The same thing is implied in his statement that the Pythagorean Unlimited was outside the heavens.[789] Further than this, it is hardly safe to go. Philolaos and his followers cannot have regarded the Unlimited in the old Pythagorean way as Air; for, as we shall see, they adopted the theory of Empedokles as to that “element,” and accounted for it otherwise. On the other hand, they can hardly have regarded it as an absolute void; for that conception was introduced by the Atomists. It is enough to say that they meant by the Unlimited the _res extensa_, without analysing that conception any further. Footnote 788: Arist. _Phys._ Γ, 4. 204 a 20 sqq., especially a 26, ἀλλὰ μὴν ὥσπερ ἀέρος ἀὴρ μέρος, οὕτω καὶ ἄπειρον ἀπείρου, εἴ γε οὐσία ἐστὶ καὶ ἀρχή. Footnote 789: See Chap. II. § 53. As the Unlimited is spatial, the Limit must be spatial too, and we should naturally expect to find that the point, the line, and the surface were regarded as all forms of the Limit. That was the later doctrine; but the characteristic feature of Pythagoreanism is just that the point was not regarded as a limit, but as the first product of the Limit and the Unlimited, and was identified with the arithmetical unit. According to this view, then, the point has one dimension, the line two, the surface three, and the solid four.[790] In other words, the Pythagorean points have magnitude, their lines breadth, and their surfaces thickness. The whole theory, in short, turns on the definition of the point as a unit “having position.”[791] It was out of such elements that it seemed possible to construct a world. Footnote 790: Cf. Speusippos in the extract preserved in the _Theologumena arithmetica_, p. 61 (Diels, _Vors._ p. 235), τὸ μὴν γὰρ ᾱ στιγμή, τὸ δὲ β̄ γραμμή, τὸ δὲ τρία τρίγωνον, τὸ δὲ δ̄ πυραμίς. We know that Speusippos is following Philolaos here. Arist. _Met._ Ζ, 11. 1036 b 12, καὶ ἀνάγουσι πάντα εἰς τοὺς ἀριθμούς, καὶ γραμμῆς τὸν λόγον τὸν τῶν δύο εἶναί φασιν. The matter is clearly put in the Scholia on Euclid (p. 78, 19, Heiberg), οἱ δὲ Πυθαγόρειοι τὸ μὲν σημεῖον ἀνάλογον ἐλάμβανον μονάδι, δυάδι δὲ τὴν γραμμήν, καὶ τριάδι τὸ ἐπίπεδον, τετράδι δὲ τὸ σῶμα. καίτοι Ἀριστοτέλης τριαδικῶς προσεληλυθέναι φησὶ τὸ σῶμα, ὡς διάστημα πρῶτον λαμβάνων τὴν γραμμήν. Footnote 791: