REVIEWS 697 



much practice they will find it impossible to reach the true attitude of mind 

 which, in geometry, is of more importance than knowledge of facts. After this 

 Mr. Mathews offers us an account, original often and brilliant always, of von 

 Staudt's theory of complex elements : then he discusses the theory of casts and 

 establishes homogeneous co-ordinates upon a projective basis. From this point 

 the author gives a little too much play to his analysis. It is always with regret 

 that one sees the domains of geometry annexed into the spreading kingdom of 

 algebra. Some readers would have been pleased if it had been possible to 

 keep up the geometrical illusion a little longer. We are not complaining that 

 geometrical methods are excluded, for this is not the case ; rather we lament 

 that the superior strength of analysis is brought so early into play. 



In the last hundred pages of the book we have discussions of such important 

 subjects as projectivities in space, quadric-surfaces, null-systems, and skew in- 

 volutions. Line geometry has a chapter to itself, while projective problems are 

 discussed in Chap. XXXII. and their solutions by geometry and algebra com- 

 pared. Throughout the later chapters the analytical theory of linear transforma- 

 tions is continually kept before the reader's attention. 



In conclusion Mr. Mathews is to be congratulated upon having written a book 

 which cannot fail to have a wide and lasting influence upon the progress of 

 geometrical knowledge in England. If the book receives the distinction of 

 translation into foreign tongues, England will pay back a portion of the heavy debt 

 which she owes to continental mathematicians who have done so much work 

 in the field of Projective Geometry. 



C. 



Plane Geometry. Part I. By G. St. L. Carson, M.A., M.Sc, and D. E. 

 Smith, Ph.D., LL.D. [Pp. iv + 266.] (Messrs. Ginn & Co. Price 2s. 6d.) 



There are signs that teachers of geometry after a long period of wandering are 

 returning to saner ways of regarding their subject. The book before us bears 

 testimony to the new attitude. Its authors are mathematicians versed in the 

 philosophy and history of the subject; they also bring to their task the ripe 

 experience of a long connection with practical teaching. The first part of the 

 geometry which lies before us treats of triangles and rectilinear figures, and is to 

 be followed by a second part in which the other portions of the school course in 

 geometry will be included ; enough is given here to allow us to judge the scheme 

 of the authors. 



The book may be divided roughly into two portions — (1) the introduction, in 

 which first notions in geometry are discussed, and (2) the formal portion, in 

 which the subject is deduced from postulates. A very wise step has been taken 

 in separating the two sections. It would have been perhaps still better to have 

 carried out the principle more completely and to have excluded some of the large 

 number of numerical examples given in the second part, which can add little to 

 what has been effected so thoroughly in the earlier portion. In the introductory 

 sections the authors give full play to their invention : here we find the map, the 

 photograph, the pantograph, hand mirrors, gothic windows, cricket-balls, doll's 

 gymnasia — in short, the delightful variety of the Christmas grotto. But throughout 

 the authors show that they know their audience, particularly by choosing as the 

 keynote of their introduction to geometry the problem of the hidden treasure. 

 How can the utility of the subject be better demonstrated than by showing its 

 intimate connection with a topic which occupies at some time or other the 

 imagination of every boy and girl ? I hope that many teachers will be encouraged 



