26 eepoet— 1878. 



I hope, it has been sufficiently 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 using 

 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 over- 

 whelming, if they were not compensated by some simplifications in the 

 processes actually employed. Of these aids to calculation I will men- 

 tion 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. Re- 

 sults are often obtained in the first instance through a tangled maze of 

 formulas, where at best we can just make sure of our 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 entangle- 

 ment. More correctly speaking, the method is not new, for it is inhe- 

 rent in the processes of algebra itself, and instances of it, unnoticed per- 

 haps or disregarded, 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 recog- 

 nised as a principle ; but beside these perhaps no others can be named 

 until a period within our own recollection. The method consists in sym- 

 metry of expression. In algebraical formulas combinations of the quan- 

 tities 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 expres- 

 sions, moreover, 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 resemblance to one 

 another, we may read off symmetrical formulas, 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 faculty as through any conscious ratioci- 

 native process. Few principles have been more suggestive of extended 

 ideas or of new views and relations than that of which I am now speak- 

 ing. 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 re- 



