regarding the corresponding features of geometrical surfaces and solids. For, in all these objects alike, either continuous alterations, or else characters that, although matters of spatial coexistence, may be ideally expressed in terms of such continuous alterations, fell within the range of the methods of the Calculus.
The new method, however, seemed to involve, at first, the conception both of “infinitely small” quantities, and of devices whereby an “infinite number” of such quantities could be summed together, or otherwise submitted to computation. The science of the continuous, in the realm of geometrical forms, as well as in the realm of physical changes, thus seemed to depend upon the conception both of the infinitely small and of the infinitely great; and the successful application of the results of such science in the realm of physics, was sometimes used as a proof that nature contains actually infinite and actually infinitesimal collections or magnitudes. But the early methods of the Infinitesimal Calculus were not free from inexactness, and led, upon occasion, to actually false conclusions. Hence, the paradoxes apparently involved in the logical bases of the science attracted more and more critical attention, as time went on; and, as a consequence, within the present century, the whole method of the Calculus has been repeatedly and carefully revised, — with the result, to be sure, that the conceptions of the actually infinite, in the sense here in question, and the actually infinitesimal (in the older sense of the term), have been banished from the principal modern text-books of both the Differential and the Integral Calculus. The terms, “Infinite” and “Infinitesimal,” have been, indeed, very generally retained in such text-books for the sake of conciseness of expression; but with a definition that wholly avoids all the problems which our foregoing discussion has raised. The infinite and the infinitesimal of the Calculus can, therefore, no longer be cited in favor of a theory of the “actually Infinite.”
In the world of varying quantities, namely, it often happens that, by the terms of definition of a given problem,
