iiS Mr. Woodhouse on the Truth of Conclusions 
The nature of the subject has obliged me to give this paper, 
in several of its parts, somewhat of a controversial cast : for 
having used the freedom of discussion in matters of pure sci- 
ence, an apology is unnecessary ; the memoir of the ingenious 
person whose opinion I have formally controverted, I can most 
sincerely commend for every thing, except the justness of the 
principle of explanation. 
To excuse the prolixity that may appear in the explanation 
of the operations, and in the proofs of their justness, I wish it to 
be considered, that it was necessary to examine the notions on 
which calculation ultimately rests ; to explain the meaning of 
imaginary symbols, by tracing their derivation ; to establish 
by separate and independent proofs, rules for the combination 
of impossible quantities, and not by inference from their simi- 
larity to rules for like combinations of real quantities; and 
carefully to distinguish between what is proved on evident 
principles, and what is only consequent from arbitrary as- 
sumptions. 
Mathematical science has been at times embarrassed with 
contradictions and paradoxes; yet they are not to be imputed 
to imaginary symbols, rather than to any other symbols in- 
vented for the purpose of rendering demonstration compendious 
and expeditious. It may, however, be justly remarked, that 
me unimportant. A useful consequence from this possibility of resolution, is said to 
be the integration of the form ^ -^-x j, P and Q_being rational functions of x; now, 
when -77 is expressed by a series of fractions, a form as * — — - presents 
itself to be integrated ; but the actual value of the integration cannot be assigned,, 
without knowing what A, B are.; and A, B, cannot be determined, except ( 3 i: are 
known. 
