92 Mr. Woodhouse on the Truth of Conclusions 
maticians, who, in questions of abstract science, profess never 
to rest contented with “ a rational faith and moral persuasion,” 
the principle of explanation just adduced must needs be unsa- 
tisfactory; for, whatever extension of meaning be allowed to 
the term analogy, still this is certain, that a proof by analogy 
is inferior to strict demonstration. What is it that determines 
the nature of this analogy ? Or how can its several coincidences, 
interruptions, and limitations be ascertained, except by sepa- 
rate and direct investigations of the properties of the circle 
and hyperbola ? If the analogy between the two curves depends 
on investigation, and is limited thereby, then all operations 
with imaginary expressions are perfectly nugatory; since we 
are not warranted to adopt a single conclusion obtained by 
their aid, except such conclusion be verified by a distinct and 
rigorous demonstration. 
The author of the principle of analogy allows that it is 
imperfect ; and I perceive no sure method of ascertaining the 
restrictions to which it is subject, except by the forms that 
result from actual investigation. 
To shew that the principle of analogy ought to be aban- 
doned, and a more natural and satisfactory one sought for, an 
argument maybe used, similar to the one employed against those 
who maintain operations by imaginary symbols to be perfectly 
unintelligible'; that, since arguments have been invented, which, 
if they do not satisfy, yet afford the mind a glimpse and indis- 
tinct perception of the reason why certain processes lead to 
truth, it may be presumed possible to convert such probable 
arguments into certain proofs, and to discipline a vague, peril- 
ous, and irregular analogy, into a strict, sure, and formal 
> demonstration. 
