30 



KNOWLEDGE ♦ 



[December 1, 1887. 



USE AND BEAUTY IN MATHEMATICS. 



ilEAl)EES of Knowledge are aware that I 

 brought out in these pages, and have since 

 republished, in a single small volume, a 

 series of " Easy Lessons in the Differential 

 Calculus,' whose purpose was to encourage 

 young mathematicians to study a method 

 of calculation exceedingly useful in all 

 departments of research to which mathe- 

 matics can be applied. I did not raise the question whether 

 the study of mathematics is chiefly to be valued for tlie help 

 which mathematical methods may afford the student of 

 science, or for its effect in training the mind to exact reason- 

 ing. I simply noted that numbers w-ho have occasion to use 

 mathematical methods are deterred from the study of the 

 Differential and Integral Calculus by the sup])osition that 

 it is not a mathematical system readily available in 

 researches depending on calculation. For such studeuts 

 I wrote, proposing to show them that the Differential Cal- 

 culus is as directly availalile as an aid in scientific researches 

 and calculations as algebra or trigonometry. I tliink I 

 succeeded in showing this. In fact that I did so I can 

 safely infer from the large number of letters which reached 

 me while my " Lessons " were in progress, communications 

 to which, I may remark, the publication of the " Lessons" 

 in a volume was chiefly due. 



In the Practical Teacher an exceedingly unpractical and, 

 to say the truth, somewhat pretentious critic reviews my 

 little book as if it were intended to strike a blow at a much- 

 loved doctrine of his own, that the learner's object in study- 

 ing mathematics should be " chiefly disciplinary and only 

 subsidiarily utilitarian " (like most critics of this type, our 

 "practical teacher" writes fearful English) "to acquire 

 that finesse of mental culture which this pursuit alone can 

 impart, that nice logical perception of minute differences, 

 the lack of which constitutes the most distinguishing 

 characteristic of the ill-balanced mind, to demonstrate in 

 short the falsity of Voltaire's foolish remark, ' J'ai toujours 

 remarque que la geometrie laisse I'esprit oil elle I'a trouve.' " 

 This critic, whose modesty is as conspicuous as hLs acumen, 

 may possibly possess that nice logical perception of minute 

 diflerences to which he refers ; but he evidently possesses 

 no perception of marked and noteworthy differences. If he 

 did he would perceive (passing over the distinction between 

 a remark of Voltaire's as to what he had observed and a 

 mere opinion attributed to him) that there is all the 

 difference in the world between encouraging the stud}' of 

 a particular branch of calculation because it is useful and 

 deciding the question he raises — one way or another. The 

 study of mathematics, apart from any idea of usefully 

 employing mathematical methods, may have all the dis- 

 ciplinary and jjurifying effects the critic in the Practical 

 I'eacher attributes to it ; the men who have chiefly been 

 remarkable for developing the simph' beautiful parts of 

 mathematics may be among those whose nice logical percep- 

 tion of minute differences has chiefly charmed an admiring 

 world : or, on the contrary, the study of mathematics of 

 this particular kind (the charm of which, by the way, I 

 probably know much more about than my critic) may tend 

 to impau- the mind's powers, and especially that mental 

 common sense on which all real progress in knowledge 

 depends ; love for such study may be regarded leather as 

 an appetite to be controlled (and often resisted) than as a 

 taste to be encouraged. But whether one view or the 

 other is right, I have not raised the question in my " Easy 

 Lessons in the Differential Calculus " ; and criticism turn- 

 ing on that question is entirely out of place so far as my 

 little book is concerned. 



If, however, the general question had been raised ))y me, 

 my " practical teacher " might at least have put the ques- 

 tion properly. It does not require that nice logical percep- 

 tion of minute differences which he lauds to perceive that 

 the utilitarian side of mathematics is by no means limited 

 to the pxltry uses and classes of uses mentioned by him — 

 the application of mathematics to problems in chance, to 

 measuring the heights of mountains, to finding the way, 

 and so forth. The utilitarian value of mathematics belongs 

 to the whole domain of science, from the profoundest 

 researches of astronomers, physicists, chemists, and biolo- 

 gists, to the humblest inquii-ies of every-day observers, from 

 investigations of the infinitely great and the infinitely little 

 to the study of the most familiar objects of every-day life. 

 ISTewton and his followers, in dealing with the mutual 

 attractiijns and perturbations of the heavenly bodies, had to 

 use, and on occasion invent, mathematical methods 

 ranging, be it observed, from the simplest to the most com- 

 plex ; but one cannot investigate the movements of a reeling 

 top without mathematical methods, which also range from 

 the simplest to the most complex. I imagine, by the way, 

 that few, even among the mathematicians of Dreamland, 

 can have been more tempted than Newton to luxuriate in 

 mere mathematical reveries, such as have delighted the 

 Hamiltons, Sylvesters, and Henrys : the loving touch with 

 which he presents some quaint suggestion tending that way 

 is at least as characteristic as the cjuick return which his 

 common sense forces him to make to his actual subjects of 

 research. 



I know not that even though the mere moonshiners 

 among mathematicians included the names deservedly held 

 in most esteem among great thinkers, if they could be set 

 above the Xewtons, Laplaces, Lagranges, Herschels,* 

 Leverriers, Adamses and the rest, it would greatly affect 

 the question of the genei-al value of mathematics as a means 

 of mere mental discipline. For the number of those who 

 will take any discipline of this .sort, with or without any 

 question of the usefulness of mathematics, is very small 

 indeed. And though this may be a very trifling consideration 

 to mere dreamers, it is one which every " practical teacher " 

 ought to take seriously into account. 



Having completely missed the whole purpose of my little 

 work, a modest little manual enough, this unpractical 

 "practical teacher " naturally fails to understand the details 

 of my plan. For instance, " why Vanishing Fractions 

 should bs treated " he cannot imagine; " they do not swim 

 into the student's ken a tenth as often as do Series, the 

 whole question of which Mr. Proctor entirely ignores." 

 Now, if my critic were only a student of the actual prob- 

 lems which arise in scientific research — in all departments, 

 high and low alike — he would know that vanishing frac- 

 tions are constantly appearing to perplex the student, and 

 that the power of getting rid of their evanescent qualities, 

 of making them represent something instead of nothing, 

 which the Differential Calculus supplies, is an immense relief 

 to the student. As for Series, no student ever finds occasion 

 to deal with them until he has passed far be3-ond the stage 

 where the methods of the Differential Calculus first begin to 

 be needed. Series are pitchforked into elementary books on 

 algebra and on trigonometry, and so " swim into the student's 

 ken " quite early in his reading ; but he does not, for all 

 that, begin early to want them. 



Then my reviewer pretends to be concerned because 

 there is no mention of the Newtonian method for Maxima 

 and Minima ; but he omits to point out any reason why the 

 method should be mentioned. My book was not intended 



* William Herschel cannot be regarded as a mathematician of 

 power, bu^ John Herschel may justly be so regarded. 



