INAUGURAL ADDRESS BEFORE THE BRITISH ASSOCIATION. 509 



were, in these various directions from a common 

 centre ; while Cayley, Sylvester, and Clifford, in 

 this country, Klein in Germany, Lobatscheffsky 

 in Russia, Bolyai in Hungary, and Beltrami in 

 Italy, with many others, have reflected similar 

 ideas with all the modifications due to the chro- 

 matic dispersion or their individual minds. But 

 to the main question the answer must be nega- 

 tive. And, to use the words of Newton, since 

 " geometry has its foundation in mechanical prac- 

 tice," the same must be the answer until our ex- 

 perience is different from what it now is. And 

 yet, all this notwithstanding, the generalized con- 

 ceptions of space are not without their general 

 utility. The principle of representing space of 

 one kind by that of another, and figures belong- 

 ing to one by their analogues in the other, is not 

 only recognized as legitimate in pure mathemat- 

 ics, but has long ago found its application in car- 

 tography. In maps or charts, geographical posi- 

 tions, the contour of coasts, and other features 

 belonging in reality to the earth's surface, are 

 represented on the flat ; and to each mode of 

 representation, or projection, as it is called, there 

 corresponds a special correlation between the 

 spheroid and the plane. To this might perhaps 

 be added the method of descriptive geometry, and 

 all similar processes in use by engineers, both 

 military and civil. 



It has often been asked whether modern re- 

 search in the field of pure mathematics has not 

 so completely outstripped its physical applications 

 as to be practically useless ; whether the analyst 

 and the geometer might not now, and for a long 

 time to come, fairly say, " Hie artem remumque 

 repono," and turn his attention to mechanics and 

 to physics. That the Pure has outstripped the 

 Applied is largely true ; but that the former is 

 on that account useless is far from true. Its 

 utility often crops up at unexpected points : wit- 

 ness the aids to classification of physical quanti- 

 ties, furnished by the ideas (of Scalor and Vector) 

 involved in the " Calculus of Quaternions ; " or 

 the advantages which have accrued to physical 

 astronomy from Lagrange's " Equations," and 

 from Hamilton's "Principle of Varying Action;" 

 or the value of complex quantities, and the prop- 

 erties of general integrals, and of general theo- 

 rems on integration, for the theories of electricity 

 and magnetism. The utility of such researches 

 can in no case be discounted, or even imagined 

 beforehand ; who, for instance, would have sup- 

 posed that the calculus of forms or the theory of 

 substitutions would have thrown much light upon 

 ordinary equations ; or that Abolian functions 



and hyperelliptic transcendents would have told 

 us anything about the properties of curves ; or 

 that the calculus of operations would have helped 

 us in any way toward the figure of the earth ? 

 But upon such technical points I must not now 

 dwell. If, however, as I hope, it has been suffi- 

 ciently shown that any of these more extended 

 ideas enable us to combine together, and to deal 

 with as one, properties and processes which, from 

 the ordinary point of view, present marked dis- 

 tinctions, then they will have justified their own 

 existence ; and in usins; them we shall not have 

 been walking in a vain shadow, nor disquieting 

 our brains in vain. 



These extensions of mathematical ideas would, 

 however, be overwhelming, if they were not com- 

 pensated by some simplifications in the processes 

 actually employed. Of these aids to calculation 

 I will mention only two, viz., symmetry of form 

 and mechanical appliances ; or, say, mathematics 

 as a fine art, and mathematics as a handicraft. 

 And, first, as to symmetry of form. There are 

 many passages of algebra in which long processes 

 of calculation at the outset seem unavoidable. 

 Results are often obtained in the first instance 

 through a tangled maze of formulae, 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 speak- 

 ing, the method is not new, for it is inherent in 

 the processes of algebra itself, and instances of 

 it, unnoticed perhaps or disregarded, are to be 

 found cropping up throughout nearly all mathe- 

 matical treatises. By Lagrange, and to some ex- 

 tent also by Gauss, among the older writers, the 

 method of which I am speaking was recognized 

 as a principle ; but, besides these, perhaps no 

 others can be named until a period within our 

 own recollection. The method consists in sym- 

 metry 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 fur- 

 ther calculations, by simple permutations of the 

 letters. Symmetrical expressions, moreover, save 

 as much time and trouble in reading as in writ- 

 ing. Instead of wading laboriously through a 



