510 



TEE POPULAR SCIENCE HOXTELY.— SUPPLEMENT. 



series of expressions which, although successive- 

 ly dependent, bear no outward resemblance to 

 one another, we may read off symmetrical for- 

 mula), of almost any length, at a glance. A page 

 of such formulas becomes a picture : known forms 

 are seen in definite groupings ; their relative posi- 

 tions, or perspective as it may be called, their 

 very light and shadow, convey their meaning al- 

 most as much through the artistic faculty as 

 through any conscious ratiocinative process. 

 Few principles have been more suggestive of ex- 

 tended ideas or of new views and relation than 

 that of which I am now speaking. In order to 

 pass from questions concerning plane figures to 

 those which appertain to space, from conditions 

 having few degrees of freedom to others which 

 have many — in a word, from' more restricted to 

 less restricted problems — we have in many cases 

 merely to add lines and columns to our array of 

 letters or symbols already formed, and then read 

 off pictorially the extended theorems. 



Next as to mechanical appliances. Mr. Bab- 

 bage, when speaking of the difficulty of insuring 

 accuracy in the long numerical calculations of 

 theoretical astronomy, remarked that the science 

 which in itself is the most accurate and certain 

 of all had, through these difficulties, become in- 

 accurate and uncertain in some of its results. 

 And it was doubtless some such consideration as 

 this, coupled with his dislike of employing skilled 

 labor where unskilled labor would suffice, which led 

 him to the invention of his calculating-machines. 

 The idea of substituting mechanical for intellect- 

 ual power has not lain dormant ; for besides the 

 arithmetical machines, whose name is legion 

 (from Napier's Bones, Earl Stanhope's calculator, 

 to Schultz's and Thomas's machines now in actual 

 use), an invention has lately been designed for 

 even a more difficult task. 1 Prof. James Thom- 

 son has in fact recently constructed a machine 

 which, by means of the mere friction of a disk, a 

 cylinder, and a ball, is capable of effecting a va- 

 riety of the complicated calculations which occur 

 in the highest application of mathematics to physi- 

 cal problems. By its aid it seems that an un- 

 skilled laborer may, in a given time, perform the 

 work often skilled arithmeticians. The machine 

 is applicable alike to the calculation of tidal, of 

 magnetic, of meteorological, and perhaps also of 

 all other periodic phenomena. It will solve dif- 

 ferential equations of the second, and perhaps of 

 even higher orders. And through the same in- 

 vention the problem of finding the free motions 



1 Royal Society's " Proceedings," February 8, 1876, 

 and May 9, 1S78. 



of any number of mutually attracting particles, 

 unrestricted by any of the approximate suppo- 

 sitions required in the treatment of the lunar and 

 planetary theories, is reduced to the simple pro- 

 cess of turning a handle. 



When Faraday had completed the experi- 

 mental part of a physical problem, and desired 

 that it should thenceforward be treated mathe- 

 matically, he used irreverently to say, " Hand 

 it over to the calculators." But truth is ever 

 stranger than fiction ; and, if he had lived until 

 our day, he might with perfect propriety have 

 said, " Hand it over to the machine." 



Had time permitted, the foregoing topics 

 would have led me to point out that the mathe- 

 matician, although concerned only with abstrac- 

 tions, uses many of the same methods of research 

 as are employed in other sciences, and in the arts, 

 such as observation, experiment, induction, im- 

 agination. But this is the less necessary because 

 the subject has been already handled very ably, 

 although with greater brevity than might have 

 been wished, by Prof. Sylvester in his address to 

 Section A at our meeting at Exeter. 



In an exhaustive treatment of my subject 

 there would still remain a question which in one 

 sense lies at the bottom of all others, and which 

 through almost all time has had an attraction for 

 reflective minds, viz. : " What was the origin of 

 mathematical ideas ? Are they to be regarded as 

 independent of, or dependent upon, experience ? " 

 The question has been answered sometimes in one 

 way and sometimes in another. But the absence 

 of any satisfactory conclusion may, after all, be 

 understood as implying that no answer is possi- 

 ble in the sense in which the question is put; or 

 rather that there is no question at all in the mat- 

 ter, except as to the history of actual facts. And, 

 even if we distinguish, as we certainly should, 

 between the origin of ideas in the individual and 

 their origin in a nation or mankind, we should 

 still come to the same conclusion. If we take 

 the case of the individual, all we can do is to 

 give on account of our own experience ; how we 

 played with marbles and apples ; how we learned 

 the multiplication-table, fractions, and propor- 

 tion ; how we were afterward amused to find that 

 common things conformed to the rules of num- 

 ber ; and later still how we came to see that the 

 same laws applied to music and to mechanism, to 

 astronomy, to chemistry, and to many other sub- 

 jects. And then, on trying to analyze our own 

 mental processes, we find that mathematical ideas 

 have been imbibed in precisely the same w^iy as 

 all other ideas, viz., by learning, by experience, 



