

45 



Eiemann, and Helmlioltz have thrown out suggestions 

 radiating as it were in these various directions from a 

 common centre; while Oayley, Sylvester, and Clifford 

 in this country, Klein in Germany, Lobatcheffsky in 

 Eussia, Bolyai in Hungary, and Beltrami in Italy, with 

 many others, have reflected similar 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 me- 

 '' chanical practice,'* the same must be the answer until 

 our experience is different from what it now is. And 

 yet, all this notwithstanding, the 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 posi- 

 tions, the contour of coasts, and other features, belong- 

 ing in reality to the Earth's surface, are represented on 

 the flat ; and to each mode of representation, or projec- 

 tion 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 Pure has out- 



