But how are we to differentiate the continuous, comprising as it does line, surface and solid? The line may be rated as of one dimension, the surface as of two dimensions, the solid as of three, if we are only ****** a calculation and do not suppose that we are dividing the continuous into its species; for it is an invariable rule that numbers, thus grouped as prior and posterior, cannot be brought into a common genus; there is no common basis in first, second and third dimensions.Yet there is a sense in which they would appear to be equal- namely, as pure measures of Quantity: of higher and lower dimensions, they are not however more or less quantitative.
Numbers have similarly a common property in their being numbers all; and the truth may well be, not that One creates two, and two creates three, but that all have a common source.
Suppose, however, that they are not derived from any source whatever, but merely exist; we at any rate conceive them as being derived, and so may be assumed to regard the smaller as taking priority over the greater: yet, even so, by the mere fact of their being numbers they are reducible to a single type.
What applies to numbers is equally true of magnitudes; though here we have to distinguish between line, surface and solid- the last also referred to as "body"- in the ground that, while all are magnitudes, they differ specifically.
It remains to enquire whether these species are themselves to be divided: the line into straight, circular, spiral; the surface into rectilinear and circular figures; the solid into the various solid figures- sphere and polyhedra: whether these last should be subdivided, as by the geometers, into those contained by triangular and quadrilateral planes: and whether a further division of the latter should be performed.
14.How are we to classify the straight line? Shall we deny that it is a magnitude?
The suggestion may be made that it is a qualified magnitude.May we not, then, consider straightness as a differentia of "line"? We at any rate draw on Quality for differentiae of Substance.
The straight line is, thus, a quantity plus a differentia; but it is not on that account a composite made up of straightness and line:
if it be a composite, the composite possesses a differentiae of its own.
But [if the line is a quantity] why is not the product of three lines included in Quantity? The answer is that a ******** consists not merely of three lines but of three lines in a particular disposition, a quadrilateral of four lines in a particular disposition: even the straight line involves disposition as well as quantity.
Holding that the straight line is not mere quantity, we should naturally proceed to assert that the line as limited is not mere quantity, but for the fact that the limit of a line is a point, which is in the same category, Quantity.Similarly, the limited surface will be a quantity, since lines, which have a far better right than itself to this category, constitute its limits.With the introduction of the limited surface- rectangle, hexagon, polygon- into the category of Quantity, this category will be brought to include every figure whatsoever.
If however by classing the ******** and the rectangle as qualia we propose to bring figures under Quality, we are not thereby precluded from assigning the same object to more categories than one: in so far as it is a magnitude- a magnitude of such and such a size- it will belong to Quantity; in so far as it presents a particular shape, to Quality.
It may be urged that the ******** is essentially a particular shape.Then what prevents our ranking the sphere also as a quality?
To proceed on these lines would lead us to the conclusion that geometry is concerned not with magnitudes but with Quality.But this conclusion is untenable; geometry is the study of magnitudes.The differences of magnitudes do not eliminate the existence of magnitudes as such, any more than the differences of substances annihilate the substances themselves.
Moreover, every surface is limited; it is impossible for any surface to be infinite in extent.
Again, when I find Quality bound up with Substance, I regard it as substantial quality: I am not less, but far more, disposed to see in figures or shapes [qualitative] varieties of Quantity.Besides, if we are not to regard them as varieties of magnitude, to what genus are we to assign them?
Suppose, then, that we allow differences of magnitude; we commit ourselves to a specific classification of the magnitudes so differentiated.
15.How far is it true that equality and inequality are characteristic of Quantity?
Triangles, it is significant, are said to be similar rather than equal.But we also refer to magnitudes as similar, and the accepted connotation of similarity does not exclude similarity or dissimilarity in Quantity.It may, of course, be the case that the term "similarity"has a different sense here from that understood in reference to Quality.
Furthermore, if we are told that equality and inequality are characteristic of Quantity, that is not to deny that similarity also may be predicated of certain quantities.If, on the contrary, similarity and dissimilarity are to be confined to Quality, the terms as applied to Quantity must, as we have said, bear a different meaning.
But suppose similarity to be identical in both genera; Quantity and Quality must then be expected to reveal other properties held in common.