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Einleitung in die Theorie der Invarianten Unearer Transformationen au f Grund der Vektorenrechnung, Ers ter Teil. By E. Study. Braunschweig, Vieweg und Sobn, 1923. 268 pp. Like other writings of Professor Study the present work is interesting from several different angles. It is a development of the theory of invariants of ternary forms, based upon what he calls vectors; it is an exposition of his thesis that mathematics is the study* of "natural-(positive, integral) numbers and everything which can be based upon them, but nothing else"; it is a running commentary and criticism of many things and many mathematicians» One needs occasionally a background of other papers of his dealing with questions of a semi-philosophical character. He particularly takes to task the investigators who have developed systems of vector analysis because they have not contented themselves with utilising the theory of invari-ants, as developed by Clebsch, Aronhold, etc. He says (p. 3) "Whether one develops vector analysis for the sake of its applications or as a self-contained discipline, there must always come first, as stated, a consideration of certain topics of algebra, namely invariants of certain groups of linear transformations : all expressions considered are invari-ants of orthogonal transformations, or invariants of groups closely related to the group of orthogonal transformations. But for more than fifty years we have had a highly developed theory of the group of all linear transformations, and for more than twenty-five years at least the fundamentals of a theory of invariants of the other groups just

Book Review: Einleitung in die Theorie der Invarianten linearer Transformationen auf Grund der Vektorenrechnung