"My object has been to notice particularly several points in the principles of algebra and geometry, which have not obtained their due importance in our elementary works... The perusal of the opinions of an individual, offered simply as such, may excite many to become inquirers, who would otherwise have been workers of rules and followers of dogmas. ...It has been my endeavor to avoid entering into the purely metaphysical part of the difficulties of algebra. The student is, in my opinion, little the better for such discussions, though he may derive such conviction of the truth of results by deduction from particular cases, as no à priori reasoning can give to a beginner. In treating, therefore, on the negative sign, on impossible quantities, and on fractions of the form \frac{0}{0}, etc., I have followed the method adopted by several of the most esteemed continental writers, of referring the explanation to some particular problem, and showing how to gain the same from any other. Those who admit such expressions as -a, \sqrt{- a}, \frac{0}{0}, etc., have never produced any clearer method; while those who call them absurdities, and would reject them altogether, must, I think, be forced to admit the fact that in algebra the different species of contradictions in problems are attended with distinct absurdities, resulting from them as necessarily as different numerical results from different numerical data. ...[D]ifferent misconceptions... give rise to the various expressions above alluded to."