August 15, 1878] 



NATURE 



41. 



formulse, where at best we can just make sure of their process 

 step by step, without any general survey of the path which we 

 have traversed, and still less of that which we have to pursue. 

 But almost within our own generation a new method has been 

 devised to clear this entanglement. More correctly speaking, 

 the method is not new, for it is inherent in the processes of 

 algebra itself, and instances of it, unnoticed perhaps or disre- 

 garded, are to be found cropping up throughout nearly all 

 mathematical treatises. By Lagrange, and to some extent also 

 by Gauss, among the older writers, the method of which I am 

 speaking was recognised as a principle ; but beside these per- 

 haps no others can be named until a period within our own 

 recollection. The method consists in symmetry of expression. 

 In algebraical formulae combinations of the quantities entering 

 therein occur and recur ; and by a suitable choice of these 

 quantities the various combinations may be rendered symmetrical 

 and reduced to a few well-known types. This having been 

 done, and one such combination having been calculated, the 

 remainder, together with many of their results, can often be 

 written down at once, without further calculations, by simple 

 permutations of the letters. Symmetrical expressions, more- 

 over, save as much time and trouble in reading as in writing. 

 Instead of wading laboriously through a series of expressions 

 which, although successively dependent, bear no outward resem- 

 blance to one another, we may read off symmetrical formulae, of 

 almost any length, at a glance. A page of such formulae becomes 

 a picture : known forms are seen in definite groupings ; their 

 relative positions, or perspective as it may be called, their very 

 light and shadow, convey their meaning almost as much through 

 the artistic factilty as through any conscious ratiocinative pro- 

 cess. Few principles have been more suggestive of extended 

 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. 



Mechanical Methods. 

 Next as to mechanical appliances. Mr. Babbage, 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 aU 

 had, through these difficulties, become inaccurate and uncertain 

 in some of its results. And it was doubtless some such consi- 

 deration as this, coupled with his dislike of employing skilled 

 labour where unskilled labour would suffice, which led him to 

 the invention of his calculating machines. The idea of substi- 

 tuting mechanical for intellectual power has not lain dormant ; 

 for beside 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 

 oeen designed for even a more difficult task.' Prof. James 

 Thomson has in fact recently constructed a machine which, by 

 means of the mere friction of a disc, a cylinder, and a ball, is 

 capable of effecting a variety of the complicated calculations 

 which occur in the highest application of mathematics to physical 

 problems. By its aid it seems that an unskilled labourer may, 

 in a given time, perform the work of ten 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 wiU solve differential equations of the 

 second and perhaps of even higher orders. And through the 

 same invention the problem of finding the free ..motions of any 

 number of mutually attracting particles, unrestricted by any of 

 the approximate suppositions required in the treatment of the 

 lunar and planetary theories, is reduced to the simple process of 

 turning a handle. 



When Faraday had completed the experimental part of a 

 physical problem, and desired that it should thenceforward be 

 treated mathematically, 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 pro- 

 priety have said, " Hand it over to the machine." 



Mathematics and Observation. 

 Had time permitted, the foregoing topics would have led me 

 to point out that the mathematician, although concerned only 

 Royal Society's Proceedings, February 3, 1876, and May 9, 1878. 



with abstractions, uses many of the same methods of research as 

 are employed in other sciences, and in the arts, such as observa- 

 tion, experiment, induction, imagination. But this is the less 

 necessary because the subject has been already handled veiy 

 ably, although with greater brevity than might have been wished, 

 by Prof. Sylvester in his address to Section A at our meeting at 

 Exeter. 



Origin of Mathematical Ideas. 



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 possible in the sense in which the question is 

 put ; or rather that there is no question at all in the matter, 

 except as to the history of actual facts. And, even if we dis- 

 tinguish, 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 an account of our own expe- 

 rience ; how we played with marbles and apples ; how we learnt 

 the multiplication table, fractions, and proportion ; how we 

 ^^■ere afterwards amused to find that common things conformed 

 to the rules of number ; 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 subjects. And then, on trying 

 to analyse our own mental processes, we find that mathematical 

 ideas have been imbibed in precisely the same way as all other 

 ideas, viz., by learning, by experience, and by reflection. The 

 apparent difference in the mode of first apprehending them and 

 in their ultimate cogency arises from the difference of the ideas 

 themselves, from the preponderance of quantitative over quali- 

 tative considerations in mathematics, from the notions of absolute 

 equality and identity which they imply. 



If we turn to the other question. How did the world at large 

 acquire and improve its idea of number and of figures ? How 

 can we span the interval between the savage who counted only 

 by the help of outward objects, to whom fifteen was " half the 

 hands and both the feet," and Newton or Laplace ? The answer 

 is the history of mathematics and its successive developments, 

 arithmetic, geometry, algebra, &c. The first and greatest step 

 in all this was the transition from number in the concrete to 

 number in the abstract. This was the beginning not only of 

 mathematics but of all abstract thought. The reason and mode 

 of it was the same as in the individual. There was the same 

 general influx of evidence, the same unsought for experimental 

 proof, the same recogrition of general laws running through all 

 maimer of purposes and relations of life. No wonder then if, 

 under such circumstances, mathematics, like some other subjects, 

 and perhaps with better excuse, came after a time to be clothed 

 with mysticism ; nor that, even in modern times, they shoidd 

 have been placed upon an i priori basis as in the philosophy of 

 Kant. 



Their Survival and Transition. 



Number was so soon found to be a principle common to many 

 branches of knowledge that it was readily assumed to be the key 

 to all. It gave distinctness of expression, if not clearness of 

 thought, to ideas which were floating in the untutored mind, and 

 even suggested to it new conceptions. In " the one " " the all," 

 "the many in one" (terms of purely arithmetic origin), it gave 

 the earliest utterance to men's first crude notions about God and 

 the world. In "the equal," " the solid," " the straight," and 

 " the crooked," which still survive as figures of speech among 

 ourselves, it supplied a vocabulary for the moral notions of 

 mankind, and quickened them by giving them the power of 

 expression. In this lies the great and endm-ing interest in the 

 fragments which remain to us of the Pythagorean philosophy. 



The consecutive processes of mathematics led to the con- 

 secutive processes of logic, but it was not until long after man- 

 kind had attained to abstract ideas that they attained to any 

 clear notion of their connection with one another. The leading 

 ideas of mathematics became the leading ideas of logic. The 

 "one" and the "many" passed into the "whole" and its 

 " parts ; " and thence into the "universal" and the "particular." 

 Tlie fallacies of logic, such as the well-known puzzle of Achilles 

 and the tortoise, partake of the nature of both sciences. And 

 perhaps the conception of the infinite and the infinitesimal, as 



