€^ SCIENCE PROGRESS 



Mr. Keynes formulates a notation and bases upon it a calculus of proba- 

 bilities by means of which such problems may be discussed with great facility 

 and the nature of underlying assumptions made clear. The conception of 

 probability is critically discussed, and emphasis is laid on the fact that it is not 

 always possible to assign a numerical value to the probability of a conclusion 

 based upon a certain premiss. 



In this volume the logical basis of the subject is kept in the forefront and 

 systematically developed. The ideas which the author advances are applied 

 to such questions as the weights of arguments, laws of error, inference, in- 

 duction and analogy, and many particular problems of historical interest are 

 discussed in detail. A dif&cult subject has been handled in so clear a manner 

 that the reader, whilst being made to realise how errors have crept into much 

 of the previous work dealing with questions of probability, is left wondering 

 that these errors could ever have been made. The author has, in fact, pro- 

 vided the most logical exposition of the whole subject that has yet been given. 

 The volume is a masterpiece of clear exposition which should considerably 

 enhance the author's already high reputation. A very complete bibliography 

 on the subject of probability is given at the end of the volume. fj. S. J. 



Three Lectures on Fermat's Last Theorem. By L. J. Mordell. [Pp. v + 

 60.] (Cambridge: At the University Press, 192 1. Price45.net.) 



In this small book Mr. Mordell has given an interesting account of Fermat's 



last theorem. The theorem states that if « is a positive integer greater than 



two, the equation 



xn -if-ynssgn 



cannot be satisfied by integer values of the unknowns x, y, z unless one of them 

 is zero. It is of course well known that the equation 



possesses an infinite set of solutions in integers. 



The theorem was stated by Fermat in the first half of the seventeenth cen- 

 tury. Fermat, in the course of reading a new edition of Diophantus's work 

 which was published 1621, noted down in the margin a number of theorems. 

 It is a tantalising fact that he entered the theorem — now called Fermat's 

 last theorem — in the margin and remarked that he had found a truly wonder- 

 ful proof of it, but that the margin was too small to contain it. In spite of the 

 efiorts of the greatest mathematicians, including Euler, Legendre, Gauss, Abel, 

 Dinchlet, and Cauchy, and in spite of the stimulus of a prize of a hundred 

 thousand marks, no proof for all values of n has yet been found. 



In the course of the book, the author deals with various aspects of the prob- 

 lem and treats the special cases « = 3 and « = 4, 5 and 7. The important 

 work of Kummer, which proved the theorem for values of less than 100 (except 

 of course n = 2) is carefully explained. The several other lines of attack due 

 to Libri, Legendre, and Wendt are briefly mentioned. 



The substance of this little book was given in London in the form of lectures 

 in March 1920, at Birkbeck College. The subject is, of course, intrinsically 

 one of great interest ; but the author is to be congratulated on having seen the 

 opportunity of using the theorem as a focus for examples of different kinds of 

 mathematical reasoning and of constructing so interesting an account of its 

 historical development. D. M. Wrinch. 



An Introduction to Projective Geometry. By L. N. G. Filon, M.A., D.Sc, 



F.R.S. Third Edition. [Pp. vii + 253.] (London : Edward Arnold 



& Co. Price 75. 6d) 



The third edition of Professor Filon's well-known textbook has been seen 



through the Press by Mr. T. L. Wren, Reader in Geometry in the University 



