To disprove this doctrine, it might be sufficient to point out that there is a vast mass of mathematical science which includes the notion of infinites, and leads to a great body of propositions concerning Infinites. The whole of the infinitesimal calculus depends upon conceiving finite magnitudes divided into an infinite number of parts: these parts are infinitely small, and of these parts there are other infinitesimal parts infinitely smaller still, and so on, as far as we please to go. And even those methods which shun the term _infinite_, as Newton's method of Ultimate Ratios, the method of Indivisibles, and the method of Exhaustions of the ancient geometers, do really involve the notion of infinite; for they imply a process continued without limit. 3. But perhaps it will be more useful to point out the fallacies of the pretended proofs that we can know nothing concerning Infinity and infinite things. The argument offered is, that of infinity we have no notion but the negation of a limit, and that from this negative notion no positive result can be deduced. But to this I reply: It is not at all true that our notion of what is infinite is merely that it is _that_ which has no limit. We must ask further that _what_? that space? that time? that number?--And if that space, that what kind of space? That line? that surface? that solid space?--And if that line, that line bounded at one end, or not? If that surface, that surface bounded on one, or on two, or on three sides? or on none? However any of these questions are answered, we may still have an infinite space. Till they are answered, we can assert nothing about the space; not because we can assert nothing about infinites; but because we are not told what _kind_ of infinite we are talking of. In reality the definition of an Infinite Quantity is not negative merely, but contains a positive part as well. We assume a quantity of a certain kind which may be augmented by carrying onward its limits in one or more directions: this is a finite quantity of a given kind. We _then_--when we have thus positively determined the kind of the quantity--suppose the limit in one or more directions to be annihilated, and thus we have an infinite quantity. But in this infinite quantity there remain the positive properties from which we began, as well as the negative property, the negation of a limit; and the positive properties joined with the negative property may and do supply grounds of reasoning respecting the infinite quantity.