[Footnote 217: "The infinite is distinct from the finite, and consequently from the multiplication of the finite by itself; that is, from the _indefinite_. That which is not infinite, added as many times as you please to itself, will not become infinite."--Cousin, "Hist, of Philos.," vol. ii. p. 231.] [Footnote 218: Saisset, "Modern Pantheism," vol. ii. pp. 127, 128.] The introduction of the idea of "the mathematical infinite" into metaphysical speculation, especially by Kant and Hamilton, with the design, it would seem, of transforming the idea of infinity into a sensuous conception, has generated innumerable paralogisms which disfigure the pages of their philosophical writings. This procedure is grounded in the common fallacy of supposing that _infinity_ and _quantity_ are compatible attributes, and susceptible of mathematical synthesis. This insidious and plausible error is ably refuted by a writer in the "North American Review."[219] We can not do better than transfer his argument to our pages in an abridged form. [Footnote 219: "The Conditioned and the Unconditioned," No. CCV. art. iii. (1864).] Mathematics is conversant with quantities and quantitative relations. The conception of quantity, therefore, if rigorously analyzed, will indicate _à priori_ the natural and impassable boundaries of the science; while a subsequent examination of the quantities called infinite in the mathematical sense, and of the algebraic symbol of infinity, will be seen to verify the results of this _à priori_ analysis. Quantity is that attribute of things in virtue of which they are susceptible of exact mensuration. The question _how much_, or _how many_ (_quantus_), implies the answer, _so much_, or _so many_ (_tantus_); but the answer is possible only through reference to some standard of magnitude or multitude arbitrarily assumed. Every object, therefore, of which quantity, in the mathematical sense, is predicable, must be by its essential nature _mensurable._ Now mensurability implies the existence of actual, definite limits, since without them there could be no fixed relation between the given object and the standard of measurement, and, consequently, no possibility of exact mensuration. In fact, since quantification is the object of all mathematical operations, mathematics may be not inaptly defined as _the science of the determinations of limits_. It is evident, therefore, that the terms _quantity_ and _finitude_ express the same attribute, namely, _limitation_--the former relatively, the latter absolutely; for quantity is limitation considered with relation to some standard of measurement, and finitude is limitation considered simply in itself. The sphere of quantity, therefore, is absolutely identical with the sphere of the finite; and the phrase _infinite quantity_, if strictly construed, is a contradiction in terms.