528 SCIENCE PROGRESS 



on mathematics — but the latter are so rare that we must warmly welcome 

 Professor Moritz's interesting work. In his preface the author explains that 

 u ten years have been devoted to its preparation, years which, if they could have 

 been more profitably, could scarcely have been more pleasantly, employed" ; and 

 the book actually contains more than two thousand analects from mathematicians 

 of distinction and others who have written on mathematics. Foreign quotations 

 have always been given in English, and the name of the author and the work 

 from which the extract is taken are appended after each. The analects are 

 classified under twenty-one subjects, as for instance, The Nature of Mathematics, 

 The Value of Mathematics, Study and Research in Mathematics, Modern Mathe- 

 matics, The Mathematician, Mathematics and Logic, The Fundamental Concepts 

 of Time and Space, and Paradoxes and Curiosities. An excellent index easily 

 enables us to locate our favourite passages. Perhaps the most interesting chapters 

 are the two devoted to Persons and Anecdotes ; and the reader would gladly 

 have seen these much extended. But the author's hope has evidently been that 

 " the present volume will prove indispensable to every teacher of mathematics, 

 and to every writer on mathematics "; and this plan requires rather a quotation 

 of opinions than an exhibition of personalities. There are many stories about 

 Newton, but the one which narrates how his dinner was eaten by some one 

 else while he was in one of his moods of abstraction is not among them. On 

 the other hand we have the famous story of the professor who wrote on his 

 blackboard that he would not " meet his classes to-day." One of the students 

 scratched out the first letter of the word "classes." Presently, however, the 

 professor returned and struck out the next letter of the same word. But Pro- 

 fessor Moritz appears to be doubtful whether the story is to be told of Lord 

 Kelvin or of Professor J. S. Blackie — and we must confess that we have heard 

 it generally quoted of the latter. Another story of J. J. Sylvester was told by 

 \V. P. Durfee, who instanced, as a case of the manner in which Sylvester forgot 

 propositions, that " I remember once submitting to Sylvester some investigations 

 that I had been engaged on, and he immediately denied my first statement, 

 saying that such a proposition had never been heard of, let alone proved. To 

 his astonishment, I showed him a paper of his own in which he had proved the 

 proposition." The chapter on mathematics as a fine art is one of the most 

 interesting, and well exhibits the aesthetic pleasure to be derived from these 

 beautiful studies. Evidently most mathematicians have been as much carried 

 away by this aesthetic pleasure as are poets and musicians by the joy of com- 

 position ; and it is true that mathematics and the highest arts are sisters who 

 live hand in hand upon the summits of Parnassus. What is there more beautiful 

 than a geometric theorem or an analytic series ? Thus J. W. A. Young remarks : 

 "It was a felicitous expression of Goethe's to call a noble cathedral 'frozen 

 music,' but it might even better be called 'petrified mathematics.'" For mathe- 

 matics and for high art there is one requirement — a sense of order. De Morgan 

 said that " the moving power of mathematical invention is not reasoning, but 

 imagination." Both are the moving powers, not only of mathematics, but of 

 art. When Plato said that "God geometrises" he merely expressed the truth 

 that this sense of order belongs to the highest part of the mind. The lower type 

 of mind does not possess it. And some one else has said, " To the small mind 

 there is nothing great, to the great mind nothing small." The author has fully 

 achieved his purpose, and, as he himself remarks, " the absence of similar English 

 works has made the author's work largely that of the pioneer." Not only 

 mathematicians, but all men of science, will thank him for his labours. 



