Book 1
23
When we wish to refer substances to their principles we derive linesThe lines, planes, and solids here discussed are probably the Ideal lines, etc., which are immediately posterior to the Idea-Numbers. Cf. 30, Aristot. Met. 13.6.10, Aristot. Met. 13.9.2, and see Introduction. from Long and Short, a kind of Great and Small; and the plane from Wide and Narrow, and the solid body from Deep and Shallow. But in this case how can the plane contain a line,
24
or the solid a line and a plane? for Wide and Narrow and Deep and Shallow are different genera. Nor is Number contained in these objects (because Many and Few is yet another class); and in the same way it is clear that none of the other higher genera will be contained in the lower. Nor, again, is the Broad the genus of which the Deep is a species; for then body would be a kind of plane.
25
Further, how will it be possible for figures to contain points?Lines, planes, and solids are generated from varieties of the Great and Small, but points cannot be, having no magnitude; how, then, can the latter be present in the former? Plato steadily rejected this class of objects as a geometrical fiction, but he recognized the beginning of a line, and he frequently assumed this latter class, i.e. the indivisible lines. That Plato denied the existence of the point and asserted that of indivisible lines is not directly stated elsewhere, but the same views are ascribed to Xenocrates, and were attacked in the treatise Xenocrates De lineis insecabilibus. See Ross ad loc. But these must have some limit; and so by the same argument which proves the existence of the line, the point also exists.Sc. if the point is the limit of the line.
