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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 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 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 

 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 recognized as a 

 principle ; but beside these perhaps no others can be 

 named until a period within our own recollection. The 

 method consists in symmetry of expression. In alge- 

 braical 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 combina- 

 tion having been calculated, the remainder, together 

 with many of their results, can often be written down 

 at once, without further calculations, by simple per- 

 mutations of the letters. Symmetrical expressions, 

 moreover, save as much time and trouble in reading 

 as in writing. Instead of wading laboriously through a 

 series of expressions which, although successively depen- 

 dent, bear no outward resemblance to one another, we 

 may read off symmetrical formulae, of almost any 

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



