xii. 8; 1083 b 9. The Pythagorean standpoint has on the one hand fewer difficulties than those that have been discussed, but it has new difficulties of its own. The fact that they do not regard number as separate, removes many of the contradictions; but it is impossible that bodies should consist of numbers, and that this number should be mathematical. Nor is it true that indivisible elements have quantity; but, granted that they have this quality of indivisibility, the units have no quantity; for how can quantity be composed of indivisible elements? but arithmetical number consists of units. But these say that things are number; at least, they adapt their speculations to such bodies as consist of elements which are numbers. xiii. 3; 1090 a 20. On the other hand the Pythagoreans, because they see many qualities of numbers in bodies perceived by sense, regard objects as numbers, not as separate numbers, but as derived from numbers. And why? Because the qualities of numbers exist in harmony both in the heaven and in many other things. But for those who hold that number is mathematical only, it is impossible on the basis of their hypothesis to say any such thing; and it has already been remarked that there can be no science of these numbers. But we say, as above, that there is a science of numbers. Evidently the mathematical does not exist apart by itself, for in that case its qualities could not exist in bodies. In such a matter the Pythagoreans are restrained by nothing; when, however, they construct out of numbers physical bodies—out of numbers that have neither weight nor lightness, bodies that have weight and lightness—they seem to be speaking about another heaven and other bodies than those perceived by sense. _Eth._ i. 4; 1096 b 5. And the Pythagoreans seem to speak more persuasively about it, putting the unity in the co-ordination of good things. ii. 5; 1106 b 29. The evil partakes of the nature of the infinite, the good of the finite, as the Pythagoreans conjectured. v. 8; 1132 b 21. Reciprocity seems to some to be absolutely just, as the Pythagoreans say; for these defined the just as that which is reciprocal to another. _Mor._ i. 1; 1118 a 11. First Pythagoras attempted to speak concerning virtue, but he did not speak correctly; for bringing virtues into correspondence with numbers, he did not make any distinct. PYTHAGORAS AND THE PYTHAGOREANS: PASSAGES IN THE DOXOGRAPHISTS.