Note AU — PREFACE

One of the ablest mathematicians, and the most persevering Hamiltono-mastix of the day, maintains the applicability of the metaphysical notion of infinity to mathematical magnitudes; but with an assumption which unintentionally vindicates Hamilton’s position more fully than could have been done by a professed disciple. “I shall assume,” says Professor De Morgan, in a paper recently printed among the Transactions of the Cambridge Philosophical Society, “the notion of infinity and of its reciprocal infinitesimal: that a line can be conceived infinite, and therefore having points at an infinite distance. Image apart, which we cannot have, it seems to me clear that a line of infinite length, without points at an infinite distance, is a contradiction.” Now it is easy to show, by mere reasoning, without any image, that this assumption is equally a contradiction. For if space is finite, every line in space must be finite also; and if space is infinite, every point in space must have infinite space beyond it in every direction, and therefore cannot be at the greatest possible distance from another point. Or thus: Any two points in space are the extremities of the line connecting them; but an infinite line has no extremities; therefore no two points in space can be connected together by an infinite line.