of Edinburgh, Session 1868 - 69 . 
483 
mere axioms and definitions. We must by synthetic geometry, 
by actual seeing, learn the qualities of lines and angles before we 
can begin to use analysis. Then, given so many synthetic pro- 
positions, we can deduce others by algebra ; but only by a use of 
actual intuition, first , in translating the geometrical enunciation 
into algebraic formulae ; and, second , in translating the algebraic 
result (if that result is not merely quantitative) into its geometric 
meaning. The answer to a proposition in analytical geometry is 
simply a rule to guide us in actually constructing, by a new use of 
our eyes or imagination, the new lines which we must have to inter- 
pret the result. Analysis does not enable us to dispense with syn- 
thetic constructions, but simply serves to guide us in these construc- 
tions, and so to dispense more or less completely with the tact re- 
quired to find out the geometrical 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 bya 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 geo- 
metry — the discovery of a whole class of new guides to construction. 
Professor Tait remarked that an excellent and interesting 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 
-7- J ( a + \ adt) (b + \ bd£) — (a — 
CLL I 
\ ddt) {b — \ bdt) 
