REVIEWS iS9 



In making the changes he seems to have acted with judgment : a brief account 

 of Dedekind's theory of numbers has been introduced ; it might indeed have 

 been extended with advantage. Again, he has given a more complete account 

 of integration which has involved several preliminary propositions, including the 

 Heine-Borel theorem. Perhaps in a future edition the author will consider the 

 advantage of proceeding even farther with the real variable, even if the complex 

 variable has to suffer. An alternative suggestion is that the course should be 

 divided into two parts, in which the real and the complex are separately dis- 

 cussed, a division which has the sanction of Goursat's example in his Cours 

 it Analyse. In English teaching there is a thoroughly unsound practice with 

 regard to complex numbers, and the only correction possible seems to be a 

 complete division between the study of the two fields of the real and the 

 complex. Such a change would no doubt involve additional space, but if it 

 effected a reform which resulted in leading students and teachers into more 

 logical methods, it would be well worth the trouble of making. 



Mr. Hardy writes well, and never shirks telling us how very little he is doing : 

 he has also a pleasant habit of interspersing matter which is not germane to his 

 subject. Such rubbing-posts are very welcome to the restless student in a 

 journey which is necessarily at times tedious. But it is doubtful whether he was 

 well advised in trotting out Mr. B. Russell's trite paradox on the value of 

 mathematics : we are sure that he was unwise in attempting to dissect it. The 

 wit of a philosopher is often worthy of admiration, but it should always be admired 

 from a distance. To base upon a paradox " highly important truths " is certainly 

 out of place in a treatise in which above all a plea is made for secure 

 foundations. 



C 



A New Analysis of Plane Geometry, Finite and Differential. With numerous 

 examples. By A. W. H. Thompson, B.A. [Pp. xvi + 120.] (Cambridge : 

 at the University Press. Price js. net.) 



The choice of a title is by no means the easiest part of an author's task. In the 

 present instance the author can hardly have been satisfied with the short title with 

 which this book is labelled. " Plane Geometry " in no way prepares the reader for 

 the scope of the book. It is true that the full title does suggest that the author is 

 making a new attack upon an old fortress, but how novel the method is can hardly 

 be explained in a title, or even in a short review. The method of treatment is so 

 entirely the author's own that the critic has to keep in mind that few things are so 

 hard to judge as the power and adaptability of a new notation ; it is only by forget- 

 ting old habits and associations of thought that he is able to estimate at all such 

 novel methods as those introduced in this book. The author too who devises new 

 attacks upon old problems does not easily comprehend the difficulties which he 

 will have in changing the habits of his critics and readers ; there is a certain 

 fascination about new methods of locomotion which unsettles the judgment of the 

 inventor and leads him to underestimate the perils of the first performance of a 

 process which he has himself learnt to execute with ease and safety. Mr. Thompson 

 has perhaps erred in the presentation of his new method inasmuch as he has not 

 sought to convince us of its charm ; he may, however, claim that if he has erred, he 

 has done so in good company, for our greatest thinkers have too often failed to 

 present their best thoughts in an attractive form. 



The elements of which the book treats are lines and points. The author defines 

 the measure of a pair of elements as the quantity determined by the pair, and 



