io LECTURES AND ESSAYS [1869- 



synthetic constructions, but simply serves to guide us 

 in these constructions, and so to dispense more or less 

 completely with the tact required to find out the geo 

 metrical solution. This is true in every case, but most 

 obviously in the investigation of new curves. The 

 tracing of curves, from their equations, is a process in 

 which no man can succeed by mere rule without the 

 use of his eyes. Suppose asymptotes, cusps, concavity, 

 everything else found, the union of these features in one 

 curve will remain a synthetic process. 



Still more remarkable is the use made in analysis 

 of imaginary quantities. To the logician an imaginary 

 quantity is nonsense, but geometrically it has a real 

 interpretation. The geometrical power gained by a new 

 method like quaternions is radically distinct from that 

 gained by the solution of a new differential equation. 

 The latter is a triumph of algebra, the former is a triumph 

 of synthetic geometry the discovery of a whole class of 

 new guides to construction. 



[A brief discussion followed, in the course of which 

 Professor Tait remarked that an excellent and inter 

 esting instance of the incapacity of metaphysicians to 

 understand even the most elementary mathematical 

 demonstrations, had been of late revived under the 

 auspices of Dr. J. H. Stirling. His name, with those of 

 Berkeley and Hegel, formed a sufficient warrant for 

 calling attention to the point. 



It is where Newton, seeking the fluxion of a product, 

 as ab, writes it in a form equivalent to 



^ [(a + i adt) (b + \bdt} -(a- \adt] (b - 



which gives, at once, the correct value 



ab + ba. 



Now Berkeley, Hegel, Stirling, and others have all in 

 turn censured this process as a mere trick (or in terms 



