obtained by Means of imaginary Quantities . 91 
nature and uses. It would indeed be a singular paradox, or 
a rare felicity, if truth, not always attained by meditation, should 
unexpectedly result from un-ideal operations conducted without 
principle, purpose, or regularity. 
The paradox, that a process in which no idea is introduced 
conducts to truth, and that operations by unintelligible charac- 
ters lead to certain and just conclusions, has been expressly 
treated in a paper presented to the Royal Society. The 
ingenious author, confining his enquiry concerning impos- 
sible quantities to their use in calculating the values of sines, 
cosines, &c. has attempted to shew, that operations with such 
quantities are true, on the principle of analogy. He is of 
opinion, that, “ The operations performed with imaginary cha- 
tacters, though destitute of meaning themselves, are yet notes 
of reference to others which are significant. They point out 
indirectly a method of demonstrating a certain property of 
the hyperbola, and then leave us to conclude from analogy, 
that the same property belongs also to the circle. All that 
we are assured of by the imaginary investigation is, that its 
conclusion may, with all the strictness of mathematical reason- 
ing, be proved of the hyperbola ; but if from thence we would 
transfer that conclusion to the circle, it must be in consequence 
of the principle just now mentioned. The investigation there- 
fore resolves itself ultimately into an argument from analogy ; 
and, after the strictest examination, will be found without any 
other claim to the evidence of demonstration/' By this explana- 
tion, the operations of imaginary quantities, before disorderly and 
confused, assume some appearance of purpose and regularity ; 
and the assent of the mind, if not compelled by certain proof, 
is at least solicited by probable arguments. But, to mathe- 
N 2 
