516 SCIENCE PROGRESS 



.'. by the rule just proved 



If x denote "men" andjj/ "mortal beings," the interpretation of our result is— 



" Men that are not mortal do not exist." 



In the space available for this review it is impossible to give more than a 

 glimpse of the method and scope of the work. Boole proceeds to extend his 

 method to Secondary propositions, where the symbols are taken to represent pro- 

 positions. He further applies it to an examination of the theological arguments of 

 Dr. Samuel Clarke and Spinoza, and finally to the theory of Probabilities. The 

 subject is one of absorbing interest, and well merits the attention of the student. 

 There are no difficulties which need deter him. Here, as in his mathematical 

 treatises, Boole is at great pains to make his meaning clear, even at the cost of a 

 certain amount of repetition. I must add that the present reprint is in every way 

 worthy of the work and its author. The type and paper are excellent, and, not- 

 withstanding the intricacy of some of the mathematical processes, I have not 

 discovered a single erratum. 



W. H. Winter. 



MATHEMATICS 



Historical Introduction to Mathematical Literature. By G. A. Miller, 

 Professor of Mathematics in the University of Illinois. [Pp. xiv + 302.] 

 (New York : The Macmillan Company ; London : Macmillan & Co., Ltd., 

 1916. Price 7s. net.) 



The author of this book states in the preface that his main object is to guide the 

 reader to " points from which he can overlook domains of considerable extent in 

 order that he may be able to form a somewhat independent judgment as regards 

 the regions which he might like to examine more closely" (p. v). The first 

 chapter consists of general observations on the changes brought about in the 

 nineteenth century in the teaching of mathematics, mathematical production, and 

 the place assigned to the history of mathematics ; also the author deals with 

 periods in the history of mathematics and the growth of mathematical journals, 

 and gives a sketch of mathematics in America from about 1870 onwards. The 

 second chapter deals with mathematical societies, congresses, and tables, and also 

 with periodicals, encyclopaedias, and other works of reference ; a list of books of 

 the kind last named being given in the Appendix. The third chapter deals briefly 

 with the nature and dominant concepts of mathematics, mathematical notation, 

 and mathematics as an educational subject. The fourth, fifth, and sixth chapters 

 deal with fundamental developments in arithmetic, geometry, and algebra re- 

 spectively. The seventh chapter contains short notices on the life and work of 

 twenty-five prominent deceased mathematicians. 



The book will be found far more useful as a book of hints to a teacher who 

 knows how to select and construct than to a student. There are many minute and 

 valuable details, such as a notice of errors in some collected works (p. 77), the 

 Encyclopedia Britannica (p. 281), and expositions of the theory of groups (pp. 97 

 109, 114, 263), which will be of great service to a teacher or a writer ; but the 



