The difficulties connected with several peculiar symbols to which algebraic calculations sometimes lead, and especially to the expressions called _imaginary_, have been, I think, much exaggerated through purely metaphysical considerations, which have been forced upon them, in the place of regarding these abnormal results in their true point of view as simple analytical facts. Viewing them thus, we readily see that, since the spirit of mathematical analysis consists in considering magnitudes in reference to their relations only, and without any regard to their determinate value, analysts are obliged to admit indifferently every kind of expression which can be engendered by algebraic combinations. The interdiction of even one expression because of its apparent singularity would destroy the generality of their conceptions. The common embarrassment on this subject seems to me to proceed essentially from an unconscious confusion between the idea of _function_ and the idea of _value_, or, what comes to the same thing, between the _algebraic_ and the _arithmetical_ point of view. A thorough examination would show mathematical analysis to be much more clear in its nature than even mathematicians commonly suppose. NEGATIVE QUANTITIES. As to negative quantities, which have given rise to so many misplaced discussions, as irrational as useless, we must distinguish between their _abstract_ signification and their _concrete_ interpretation, which have been almost always confounded up to the present day. Under the first point of view, the theory of negative quantities can be established in a complete manner by a single algebraical consideration. The necessity of admitting such expressions is the same as for imaginary quantities, as above indicated; and their employment as an analytical artifice, to render the formulas more comprehensive, is a mechanism of calculation which cannot really give rise to any serious difficulty. We may therefore regard the abstract theory of negative quantities as leaving nothing essential to desire; it presents no obstacles but those inappropriately introduced by sophistical considerations.