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268 NEW BOOKS. necessity of the infinite and the infinitesimal in explaining curves, tan- gents, irrationals, continuity, velocity, acceleration, the infinitesimal calculus, etc. Various contradictions liable to arise in treating these subjects are discussed, and are solved by the old doctrine of orders of the infinite and the infinitesimal, together with a logico-metaphysical theory which the author calls that of Behaftungen. It is unfortunate for his views that the infinitesimal is now known not to occur in any of the problems which he discusses, being replaced everywhere by the doctrine of limits ; it is still more unfortunate that the idea of orders in the infinite and the infinitesimal has been shown to be quite inexact and vague. When two infinite series of finite numbers, whose n th terms are %n and y n respectively, are such that, given any finite number N, there exists a finite integer n such that x n , yn, and x n jy n are all greater than N, we say that the limits of the two series are infinite, and that the series of x's becomes infinite of a higher order than the series of j/'s. But as a matter of fact both series have as their limit the same number, namely, the number of finite numbers. Similarly where infinitesimals appear we have really nothing but series of finite numbers whose limit is zero. That infinite divisibility involves infinitesimals is assumed by Dr. Geissler as self-evident, although, in the case of the rational or the real numbers, the opposite is capable of formal proof. In Euclidean space, as treated by analytical geometry, although space is infinite and infinitely divisible, yet very distance is finite, i.e., has a finite ratio to every other distance ; and the apparent impossibility of such a state of things is a mere illusion dispelled by exact reasoning. Pages 297-335 are occupied in a historical review of opinions as to infinity. Only seven pages in the whole book (pp. 325-332; are devoted to Cantor, with whom the author appears to be very imperfectly ac- quainted. He discusses chiefly the more or less popular " Zur Lehre voin Transfiniten " ; it is doubtful whether he has read the " Grundlagen einer allgemeinen Mannichfaltigkeitslehre," and he appears to have never heard of the very important articles in Math. Annalen, volumes 46, 49. He mentions Cantor's sketch of a proof that there are no infinitesimal numbers, 1 which consists in showing that, if there were -an infinitesimal number , and if v were any transfinite number, how- ever great, ( v would still be infinitesimal. Dr. Geissler retorts (p. 328) : But how if instead of v we were to put a magnitude not obeying the prescriptions for the so-called transfinite numbers ? This retort is dis- posed of by the logical theory of Arithmetic, which proves the impossi- bility of our author's hypothesis. He objects also that Cantor has not established the existence (in the mathematical sense) of his transfinite numbers. On this point, it is true, the theory requires some supplement- ing; but what is necessary is easily supplied. It can be proved that every class has a number, and the finite integers form a class, but they have no finite number of terms ; consequently they have an infinite number. The doctrine of the transfinite is not merely, as Dr. Geissler is willing to allow (p. 331), one among possible theories of infinity ; it can be proved, from the general principles of Logic, to be the only possible theory. To deny this it would be necessary to deny the Syllogism, or the Law of Contradiction, or some equally elementary proposition of Logic. The last eighty pages are occupied in philosophical considerations, con- cerning chiefly the doctrine of Behaflungen, with which is connected a theory of so-called "metaphysical relativity". This theory maintains 1 Which is expanded and rendered intelligible by Peano, Rivista di Mat., vol. ii, pp. 58-62.