6? $ SCIENCE PROGRESS 



Prof. Conway's four lectures, which are contained in No. 3, give an historical 

 sketch of the theory of relativity down to the stage in which it was left by 

 Minkowski. From the student's point of view it is a piece of good fortune that 

 this theory is as yet so new that no teacher has yet dared to turn it into the 

 dead form of a " subject " as usually taught, so that students can still have the 

 stimulus of an historical treatment. 



No. 6 is the first text-book in English on the theory of automorphic functions. 



The fundamental concepts and theorems of the theory are thoroughly treated, and 



certain important developments of the theory — the applications to non-Euclidean 



geometry and the uniformisation of analytic functions — are treated in less detail. 



Perhaps the most valuable part of the Tract is the very full bibliography of works 



from 1881 to 1913, which will be of the utmost value to those who wish to make a 



deeper study of the theory. 



Philip E. B. Jourdain. 



Contributions to the Founding of the Theory of Transfinite Numbers. By 



GEORG CANTOR. Translated and provided with an Introduction and 

 Notes by Philip E. B. Jourdain, M.A. (Cantab). The Open Court 

 Series of Classics of Science and Philosophy, No. 1. [Pp. vii + 211.] 

 (Chicago and London: The Open Court Publishing Co., 191 5. Price 

 3 s. 6d. net.) 



This book is No. 1 of the Open Court Series of Classics of Science and Philosophy. 

 It would have been impossible to make a better choice of an initial volume. The 

 work consists of an Introduction by Mr. Jourdain consisting of 82 pages, and then 

 comes Cantor's contribution ; the volume closes with a few pages of notes and an 

 index. The work of Cantor was published in the Mathematische Annalen, 1895-7. 

 Perhaps it is the most strikingly original mathematical theory of the last twenty- 

 five years. The author begins his first article with the assertion, Hypotheses non 

 fingo ; it marks the firmness of the foundation upon which he knew he was 

 building. His assertion has been verified ; there are, it is true, points in the 

 theory upon which there is no certain knowledge, but the theory is built upon a 

 rock, and will stand as long as the human mind exercises its highest functions. 

 English students have had to wait nearly twenty years for a translation of this 

 important book into their language ; French readers in 1899 had before them 

 M. Mariotte's translation, after indeed it had been published apparently in the 

 Me"moires de la Sociite" des Sciences physiques et naiurelles de Bordeaux. And 

 now at last this edition comes from an American publisher. Unfortunately this 

 occurrence is so common as to excite no surprise ; it is part of the pleasant fiction 

 that English mathematicians are sufficiently accomplished to read mathematics in 

 any language in which the Latin or Gothic script is used. 



Mr. Jourdain has done his work of translation well, but it is in his long 

 Introduction that he has established his surest claim to our gratitude ; while 

 many could have discharged the duties of translator, few could have given such a 

 vivid historical and philosophical sketch. We have in these pages an account of 

 the great French and German analysts to whom we owe so much in modern 

 analysis ; it is not, however, a general history. The author has selected as his 

 thread the arithmetical theories which have rendered possible the theory of 

 transfinite numbers, and from Fourier and Cauchy to Weierstrass and Cantor he 

 has traced for us the true apostolic succession of modern analysts. 



C. 



