of which is given, the direction of its tangent; a question whose general solution was the primitive object of the inventors of the transcendental analysis. We will consider the tangent as a secant joining two points infinitely near to each other; and then, designating by dy and dx the infinitely small differences of the co-ordinates of those two points, the elementary principles of geometry will immediately give the equation t = dy/dx for the trigonometrical tangent of the angle which is made with the axis of the abscissas by the desired tangent, this being the most simple way of fixing its position in a system of rectilinear co-ordinates. This equation, common to all curves, being established, the question is reduced to a simple analytical problem, which will consist in eliminating the infinitesimals dx and dy, which were introduced as auxiliaries, by determining in each particular case, by means of the equation of the proposed curve, the ratio of dy to dx, which will be constantly done by uniform and very simple methods.
- Rectification of an Arc. In the second place, suppose that we wish to know the length of the arc of any curve, considered as a function of the co-ordinates of its extremities. It would be impossible to establish directly the equation between this arc s and these co-ordinates, while it is easy to find the corresponding relation between the differentials of these different magnitudes. The most simple theorems of elementary geometry will in fact give at once, considering the infinitely small arc ds as a right line, the equations
ds2 = dy2 + dx2, or ds2 = dx2 + dy2 + dz2,
according as the curve is of single or double curvature. In either case, the question is now entirely within the domain of analysis, which, by the elimination of the differentials (which is the peculiar object of the calculus of indirect functions), will carry us back from this relation to that which exists between the finite quantities themselves under examination.