go Mr . Woodhouse on the Truth of Conclusions 
it is said, all just reasoning is suspended, and the mind is 
bewildered by exhibitions that resemble the juggling tricks of 
mechanical dexterity. 
The arguments that seem to render all operations performed 
with impossible quantities unintelligible, may be included under 
the following statement. Algebra is a species of short-hand 
writing ; a language, or system of characters or signs, invented 
for the purpose of facilitating the comparison and combination 
of ideas. Now all demonstration by signs, must ultimately 
rest on observations made on individual objects; and all the 
varieties of the transformation and combination of signs, except 
what are arbitrary and conventional, must be regulated by pro- 
perties observed to belong to the things of which the signs are 
the representatives. Demonstration by signs is shewn to be 
true, by referring to the individual things the signs represent; 
and is shewn to be general, by remarking that the operation is 
the same, whatever is the thing signified, or, in other words, 
that the operation is independent of the things signified. Yet, 
against this statement, from the very concessions of the mathe- 
maticians that have opposed the use of impossible quantities, is 
to be derived a powerful argument, an argument sufficiently 
satisfactory to the mind, that operations with impossible quan- 
tities are really regulated by the rules of a logic equally just 
with the logic of possible quantities. It is conceded, and men- 
tioned as a paradox, that the conclusions obtained by the aid 
of imaginary quantities are most true and certain. Now, if 
operations with any characters or signs lead to just conclusions, 
such operations must be true by virtue of some principle or 
other; and the objections against imaginary quantities, ought to 
be diverted upon the unsatisfactory explanation given of their 
