46 



stripped tlie Applied is largely true ; but that the former 

 is on that account useless is far from true. Its utility 

 often crops up at unexpected points ; witness the aids 

 to classification of physical quantities, furnished by the 

 ideas (of Scalar and Vector) involved in the Calculus 

 of Quaternions ; or the advantages which have accrued 

 to Physical Astronomy from Lagrange's Equations, and 

 from Hamilton's Principle of Varying 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 

 Substitutions would have thrown much light upon 

 ordinary equations ; or that Abelian Functions and 

 Hyperelliptic Transcendents would have told us any- 

 thing about the properties of curves ; or that the 

 Calculus of Operations 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 

 I hope, it has been sufficiently shown that any of these 

 more extended ideas enable us to combine together, and 

 to deal with as one, properties and processes which 

 from the ordinary point of view present marked distinc- 

 tions, then they will have justified their own existence ; 

 and in using them we shall not have been walking in 

 a vain shadow, nor disquieting our brains in vain. 



These extensions of mathematical ideas would however 

 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 



