value. But it is easy, nevertheless, to explain very clearly, and in a quite general manner, how far the question must be simplified by such an artifice. For this purpose, it is necessary to begin by distinguishing different orders of infinitely small quantities, a very precise idea of which may be obtained by considering them as being either the successive powers of the same primitive infinitely small quantity, or as being quantities which may be regarded as having finite ratios with these powers; so that, to take an example, the second, third, &c., differentials of any one variable are classed as infinitely small quantities of the second order, the third, &c., because it is easy to discover in them finite multiples of the second, third, &c., powers of a certain first differential. These preliminary ideas being established, the spirit of the infinitesimal analysis consists in constantly neglecting the infinitely small quantities in comparison with finite quantities, and generally the infinitely small quantities of any order whatever in comparison with all those of an inferior order. It is at once apparent how much such a liberty must facilitate the formation of equations between the differentials of quantities, since, in the place of these differentials, we can substitute such other elements as we may choose, and as will be more simple to consider, only taking care to conform to this single condition, that the new elements differ from the preceding ones only by quantities infinitely small in comparison with them. It is thus that it will be possible, in geometry, to treat curved lines as composed of an infinity of rectilinear elements, curved surfaces as formed of plane elements, and, in mechanics, variable motions as an infinite series of uniform motions, succeeding one another at infinitely small intervals of time.
Examples. Considering the importance of this admirable conception, I think that I ought here to complete the illustration of its fundamental character by the summary indication of some leading examples.
- Tangents. Let it be required to determine, for each point of a plane curve, the equation