DEVELOPMENT . OF MATHEMATICAL THOUGHT. 671 



of bringing some order into the tangled web of mathe- 

 matical speculation, mainly represented by these, I shall 

 identify the name of Pliicker with the great advance 32. 



Pliicker, 



which has taken place in geometry through the change chasies, 



J ~ Cayley. 



in our ideas as to the elements of space construction and 

 the generalisation of our ideas of co-ordinates : with 

 Chasles I shall specially connect the modern habit in 

 geometry of combining figures in finite space with their 

 infinitely distant elements, and with Cayley the application 

 to geometrical science of the novel and comprehensive 

 methods of modern algebra. Let us dwell for a moment 

 on each of these three great departures. 



The elements of any science are a very different thing 

 from the elements of the special object with which that 

 science is concerned. The elements of chemistry are not 

 the chemical elements. The latter are, we suppose, 33. 



Historical 



something existing in nature, something fixed and un- and logical 



foundations 



alterable, which science aims at finding out ; the former 

 are certain conceptions from which we find it convenient 

 to start in teaching, expounding, and building up the 

 science of chemistry. The latter are artificial, the former 

 are natural. The same remark obtains in geometrical 

 science. The elements of geometry have an historical, 

 a practical beginning : the elements of space form a con- 

 ception which gradually emerges in the progress of geo- 

 metrical science. In every science there is a tendency 

 to replace the casual and artificial elements by the 

 natural or real elements, and to build up the historical 

 traditional body of doctrine anew, using the very 

 elements which Nature herself, as it were, employs in 

 producing her actual forms and objects. As the pass- 



