MATHEMATICS IN ENGLISH SCHOOLS 169 



the curriculum can be disencumbered of the obsolete and the 

 unessential. No one who has taken the trouble to acquaint 

 himself with the present world-movement in education will 

 doubt that one of the tasks of the twentieth century is to find 

 a way of importing the notions of the infinitesimal calculus 

 into the ordinary school curriculum. 



The world has moved in and left school mathematics in a 

 backwater. Geometry stands a venerable monument of anti- 

 quity on which I will lay no sacrilegious hands. But mathematics 

 is applied to modern life in an analytical or algebraical not 

 a geometrical form. Newton revolutionised scientific thought 

 with a geometrical treatise but he had arrived at his results 

 by analytical means, inventing the calculus for the purpose ; 

 his successors have not thought it necessary to clothe 

 their work in a garb which respectability in those days 

 demanded. Formal and demonstrative geometry is not going 

 to help us very much on the side of outlook ; it must be taught 

 as mental training, for we can hardly break with the training 

 theory entirely, though our hold on it may be weakening. 



Now consider school arithmetic. What does it contain ? 

 It teaches in the first place ordinary ciphering, a necessary 

 art. It teaches all sorts of operations with English weights 

 and measures and coinage — mostly superfluous. It betrays its 

 commercial origin by treating of a number of commercial rules 

 which may have been practised in the City in Cocker's day. 

 But it is mainly a complete collection of methods for solving 

 all problems that have been set by all examiners since the 

 invention of printing, a snowball that still grows, a burden 

 to boyhood, a nightmare to mathematicians. " ' If this were only 

 cleared away,' they said, ' it would be grand.' " If it were 

 cleared away we might discern the true simplicity of arithmetic. 

 The invention of logarithms has left much of arithmetic on the 

 scrap-heap, a shrine where it is still worshipped. There is no 

 outlook to be got from arithmetic ; the main thing is to have 

 done with it rapidly and ever after to use it as a tool. 



Algebra is perhaps the mathematical subject which gives 

 the smallest return for the time spent on it and I shall indicate 

 the cause of this. 



It is, I am afraid, a fact that in this country the main 

 preoccupation of teachers of algebra is to impart to their 

 pupils a high degree of mechanical manipulative dexterity in 



