THIS is emphatically a text-book, deductive in method and Euclidean in arrangement; as such, it has the defects of its qualities, but its merits are undeniable. In this volume the author deals with the elementary notions of rational and irrational number, point aggregates, function, continuity, differentiation and integration. The subject last mentioned occupies pp. 333–560, so that conditions of integrability, change of order of integration, upper and lower integrals, &c., receive a proper amount of attention. It should be noted, too, that although it is confessedly incomplete, the discussion of maxima and minima of functions of two or more variables is satisfactory as far as it goes, a most unusual circumstance as things are at present. Perhaps the most valuable feature of novelty is that the author occasionally criticises arguments once thought sufficient, but now known to be fallacious, illustrating by examples the way in which the defective proofs break down. This is an excellent way of making a student feel the necessity of mastering the more refined methods of recent analysis. There is one point in which the author has not quite done justice to his authorities. After explaining Cantor's theory of irrational numbers, he gives a brief sketch of Dedekind's method of partitions, but he does not give this in its genuine form. The essence of a partition is that it divides all rational numbers (with the possible exception of one) into two classes, each element of one class being less than each of the other. After this definition it is proved that the aggregate of partitions is continuous. Prof. Pierpont (p. 82) defines a partition as dividing all real numbers into two classes; this enables him to use Dedekind's notation, when convenient, but it does not give a just idea of Dedekind's theory, and this is a pity. For bibliographical details the reader is referred to the “Encyclopädie der mathematischen Wissenschaften”this is all very well for those who have access to that work, but in the interests of the student it would be well to give a list of the most important original sources. It ought to be said that in his preface the author acknowledges his special obligation to Jordan, Stolz, and Vallée-Poussin; at the same time it is evident that he has made use of this and other matérial in an independent way.
Lectures on the Theory of Functions of Real Variables.
Vol. i. By J. Pierpont. Pp. xii + 560. (London and Boston: Ginn and Co., n.d.) Price 20s. net.
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