ADDRESS. ^7 



stricted 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. 



Next as to mechanical appliances. Mr. Babbage, when speaking of 

 the difficulty of ensuring accuracy in the long numerical calculations of 

 theoretical astronomy, remarked 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 inven- 

 tion of his calculating machines. The idea of substituting mechanical 

 for intellectual power has not lain dormant ; for beside the arith- 

 metical machines whose name is legion (from Napier's Bones, Earl 

 Stanhope's calculator, to Schultz and Thomas's machines now in actual 

 use) an invention has lately been 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 disk, a cyUnder, and 

 a ball, is capable of effecting a variety of the complicated calcula- 

 tions 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 mag- 

 netic, of meteorological, and perhaps also of all other periodic phe- 

 nomena. It will 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 treat- 

 ment 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 mathemati- 

 cally, 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 propriety have said, " Hand it over to the machine." 



Had time permitted, the foregoing topics would have led me to point 

 out that the mathematician, although concerned only with abstractions, 

 uses many of the same methods of research as are employed in other 

 sciences, and in the arts, such as observation, experiment, induction, 

 imagination. But this is the less necessary because the subject has 

 been already handled very ably, although with greater brevity than might 

 have been wished, by Professor Sylvester in his address to Section A. 

 at our meeting at Exeter. 



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., 



