338 SCIENCE PROGRESS 



are preceded by a programme of the contents. Are we right in supposing that 

 these contain the lectures referred to in the preface ? Whether this conjecture 

 is correct or not, they consist of admirable statements of matters which should be 

 of the deepest interest to teachers, all presented in an attractive form by a writer 

 of keen and vigorous intellect. Here are a few of the author's obiter dicta, which 

 may give, in a fragmentary way, some idea of Prof. Nunn's scope : 



" Over a large part of the field of mathematics the fundamental idea is not 

 magnitude but order." 



"The long-delayed rational interpretation of 'imaginary numbers' appeared 

 almost simultaneously in three distinct quarters at the beginning of the nineteenth 

 century." 



"Motion is simply ' geometry plus time,' and any reason which justifies the 

 study of geometry as a branch of mathematics must justify equally the inclusion 

 of kinematics." 



" Statistics constitute at once the oldest and the newest branch of mathematics : 

 the oldest, for their practice, in some form, is one of the primary necessities of an 

 ordered social life ; the newest, for their theory is to a large extent a production 

 of the present generation. For both these reasons it is greatly to be desired that 

 an elementary study of the subject should come to be regarded as part of the 

 normal programme of secondary school mathematics." 



How strange all this sounds to an ear attuned to the frigid chants of the older 

 school of algebraists ! 



In the bare enumeration of contents we have seen the wide scope given by the 

 author to his subject : it is to include trigonometry, plane and spherical, infini- 

 tesimal calculus, and statistics. Is the author quite fair in the wide sweep which 

 he takes ? Elementary analysis might have provided a tent large enough to cover 

 such a range ; but algebra, even when it includes trigonometry, is hardly an 

 appropriate title. 



With regard to the Teaching of Algebra, Part I., and the Exercises, Part I., 

 there is little that will raise controversy. The exercises are a very interesting set, 

 interspersed with short notes and diagrams ; there is a choice from which the 

 teacher will obtain the routine examples which the average boy has to do, and also 

 the harder questions which stimulate and interest the cleverer students. 



But in the treatment of Part II. the author has deviated from the sound lines 

 on which he has developed Part I. To show the author's method we will take 

 Section VIII., Limits. Here we have on this subject in the first volume 22 pages, 

 and in the third volume about 100. It is, apparently, in the third volume that the 

 main discussion is contained, while the first volume is, as it were, a commentary 

 upon the main treatment. Somewhere perhaps in the 1,588 pages of the volumes 

 before us this is made clear ; but an arrangement in which the natural order is 

 violated constitutes a serious defect. 



The headings of the sections of Part II. have been given above ; but enumera- 

 tion hardly gives an idea of their contents. Limits, in the writer's view, means the 

 infinitesimal calculus and ordinary differential equations, and even such a subject 

 as space-filling curves is treated in some detail ; while spherical trigonometry 

 covers map-making, great-circle sailing, and some parts of astronomy. 



To sum up, the book contains a most valuable and admirable elementary 

 algebra ; but we do not expect that Prof. Nunn will find writers and teachers who 

 will adopt his course in what some writers call complementary algebra. There 

 are many w se and excellent things in this part, but they constitute a mathematical 

 miscellany, and most of it forms no part of a course in algebra. C. 



