ADDRESS. 25 



cient indeed for daily use, but after all only a rougli approximation to a 

 truer although perhaps more complicated scheme ? Traces of these ques- 

 tions are in fact to be found in the writings of some of our greatest and 

 most original mathematicians. Gauss, Riemann, and Helmholtz have 

 thrown out suggestions radiating as it were in these various directions 

 from a common centre ; while Cayley, Sylvester, and Clifford in this 

 country, Klein in Germany, Lobatcheffsky in Russia, Bolyai in Hungary, 

 and Beltrami in Italy, with many others, have reflected kindred ideas with 

 all the modifications due to the chromatic dispersion of their individual 

 minds. But to the main question the answer must be in the negative. 

 And, to use the words of Newton, since " Geometry has its foundation in 

 mechanical practice," the same must be the answer until our experience 

 is different from what it now is. And yet, all this notwithstanding, 

 generalised conceptions of space are not without their practical utility. 

 The principle of representing space of one kind by that of another, and 

 figures belonging to one by their analogues in the other, is not only 

 recognised as legitimate in pure mathematics, but has long ago found its 

 application in cartography. In maps or charts, geographical positions, 

 the contour of coasts, and other featiires, belonging in reality to the 

 Earth's surface, are represented on the flat ; and to each mode of repre- 

 sentation, or projection as it is called, there corresponds a special correla- 

 tion 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 research 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 

 Pui'e 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 ao 

 unexpected points ; witness the aids to classification of physical quanti- 

 ties, furnished by the ideas (of Scalar and Vector) involved in the Calculus 

 of Quaternions ; or the advantages which have accrued to Physical Astro- 

 nomy from Lagrange's Equations, and from Hamilton's Principle of Vary- 

 ing Action ; on the value of Complex Quantities, and the properties of 

 general Integrals, and of general theorems 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 supposed that the Calculus of Forms or the Theory of Substitu- 

 tions would have thrown much light upon ordinary equations ; or that 

 Abelian Functions and Hyperelliptic Transcendents would have told us 

 anything about the properties of curves ; or that the Calculus of Opera- 

 tions would have helped us in any way towards the figure of the Earth ? 

 But upon such technical points I must not now dwell. If, however, as 



