DISCOVERY 



61 



have the natural aptitude, but also to that greater num- 

 ber who apparently' use it as a means towards an end ? 

 The justification of the introduction of any element 

 that will have the tendency to drive away monotony 

 is apparent when we consider how progress is dependent 

 on the awakening and the maintaining of interest. 



It is naturally impossible for me to attempt a detailed 

 exposition of the recreative aspect of mathematical 

 work ; and so I must be content to give a brief survey 

 of the subject from this point of view, to make a short 

 reference to one or two of the suggestions that will have 

 been made, and to make one theme the chief illustration. 

 Mathematics is particularly fortunate in possessing a 

 wide field from which to select work of this descrip- 

 tion, the following being some of the possibilities: 

 Recreations, Arithmetical, Geometrical, and Mechani- 

 cal ; Practical Mathematics ; its uses in such spheres 

 as business, sport, or war ; its applications to other 

 sciences ; its history, not only of individuals, but of the 

 problems that have been handed down from the past ; 

 and, lastlj-, a more recondite subject, hardly suitable 

 for the immature — I mean the study of its fundamental 

 notions, a study that is necessary for obtaining a true 

 conception of its ideals and infinite possibilities. 



There are three facts in the modern history of 

 Mathematics that show a tendency in the direction of 

 which I am speaking — a tendency towards the lighten- 

 ing of the burden of the mathematical pupil. The first 

 is the use of Analytical methods, which became 

 universal in England about ninety years ago ; the 

 other two, of comparatively recent date, are the 

 superseding of Euclid and the introduction of Practical 

 Mathematics. The treatment that has replaced Euclid 

 is open to innumerable objections, but it still remains 

 a step in the right direction, and was taken on the 

 Continent long before we thought about it. The 

 deadening influence on the youthful mind of abstract 

 methods of exposition is further mitigated bj- the 

 introduction of Practical Mathematics ; and it is 

 therefore not unreasonable to suppose that this subject, 

 which is in its infancy, will in future include sections 

 not hitherto contemplated. 



This naturally leads us on to think of the interesting 

 story of the applications of Mathematics to other 

 sciences and to the ordinary business of life. Much 

 of it is known, at least vaguely, to most people, and it 

 is unnecessary here to illustrate what is common 

 knowledge. It is difficult, however, to think of the 

 subject at all without at least referring to the remark- 

 able work of that illustrious mathematician and 

 physicist, James Clerk Maxwell. But rather than give 

 positive instances of the extensive use of mathematical 

 results, I venture to describe an incident from the late 

 war to show the kind of situation that arises when the 

 application of a mathematical idea is entrusted to the 



uninitiated. In the schools of instruction established 

 for officers, many of the classes were, as is well known, 

 conducted by sergeants of the regular army. These 

 instructors were efficient in many ways, but it would 

 be an exaggeration to say that their knowledge of science 

 was profound. One of them I know of was explaining 

 to a class of Artillery officers how the angle of elevation 

 of a gun was measured, and in the course of his argument 

 made the statement that the diameter of a circle went 

 exactly three times into the circumference. One of 

 his audience was dubious of this, and asked him if that 

 was quite accurate. " Was there not a little bit left 

 over when the division was made ? " This made the 

 instructor shaky of his position, and he said he would 

 get the sergeant-major to explain. This particular 

 sergeant-major had the reputation of being omniscient ; 

 and, in any case, being a sergeant-major, was never at 

 loss for a reply. He came over to the class and said 

 he would explain how it was that the diameter went 

 into the circumference exactly three times. Taking a 

 pennj' from his pocket and holding it up, he said, 

 " Look at the little circle formed by this penny. If 

 you were to measure the diameter and then the 

 circumference, you would find that the one was three 

 times the other." " But, of course," he added dis- 

 dainfully, making a large sweeping movement with his 

 hands," if you had a 6;^ circle, anything might happen." 



Some people are above Mathematical recreations, but 

 when we remember that in most of us there is something 

 imaginative, something of the spirit of emulation, we 

 can afford to ignore their views. Some would say that 

 recreations are too trivial to deserve serious thought, 

 but what has been the study of men like Euler, Fermat, 

 and Legendre, we cannot affect to despise. Liebnitz 

 in one of his letters remarked ' that men were never 

 more ingenious than in the invention of games ; here 

 the mind was at ease ; after games that depended solely 

 on numbers came those of position ; and after those, 

 where only number and position appeared, came the 

 games that involved motion ; that one would, in fact, 

 desire to have a course of study, entirely devoted to 

 games treated mathematically. 



A typical arithmetical fallacy is one dependent on 

 the ambiguity of the root sign. For example, we may 

 express the identity 



hence {V— i)' = ( V^i)" o" multiplying across 



.•. — 1 = 1. 

 1 Quoted by W. W. Rouse Ball in Mathematical Recreations, 

 1919. (Macmillau, 12s. 6d.) 



