1878.] 
The British Association. 
509 
be overwhelming if they were not compensated 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 mass of formulae. But almost 
within our own generation a new method has been de- 
vised to clear this entanglement. By Lagrange, and to 
some extent also by Gauss, among the older writers, the 
method of which I am speaking was recognised as a prin- 
ciple ; but beside these perhaps 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 re- 
duced to a few well-known types. This having been done, 
and one such combination having been calculated, the re- 
mainder, together with many of their results, can often be 
written down at once, without further calculations, by simple 
permutations of the letters. Symmetrical expressions, 
moreover, save as much time and trouble in reading as in 
writing. 
With regard to mechanical appliances, Mr. Babbage, 
when speaking of the difficulty of ensuring accuracy in the 
long numerical calculations of theoretical astronomy, re- 
marked that the science which in itself is the most accurate 
and certain of all, had, through these difficulties, become 
inaccurate and uncertain in some of its results. And it was 
doubtless some such consideration as this, coupled with his 
dislike of employing skilled labour where unskilled would 
suffice, which led him to the invention of his calculating 
machines. Prof. James Thomson has recently constructed 
a machine which, by means of the mere friction of a disk, 
a cylinder, and a ball, is capable of effecting a variety of 
complicated calculations, which occur in the highest appli- 
cation of mathematics to physical problems. By its aid it 
seems that an unskilled labourer may, in a given time, per- 
form 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 will solve differential equations of the 
second, and perhaps of even higher orders^ And through 
