APPLIED TO THE FINDING OF CURVES.

It is now manifest that, without an added condition, this could not even retain the name of a "question": for it is clear that the shorter the line chosen and the closer it approaches a vertical position, the shorter the time of descent upon it will be. For this reason, one cannot seek in an absolute sense the line upon which a falling heavy body glides most quickly or in the shortest time; rather, the quantity of the abscissaThe horizontal coordinate (x-axis) in a coordinate system. to which the curve to be found corresponds must be defined at the same time. Thus, among all curves corresponding to the same abscissa taken on an axis given in position, one would seek that curve upon which a heavy body would glide down most swiftly.

Nor, indeed, was that condition sufficient in this Problem to make it determined: but it was necessary to add this further condition, that the curve to be found must pass through two given points. This Problem had to be constrained by these conditions so that it might become determined—namely, to determine, among all curved lines passing through two given points, that one upon which a descending body completes the arc corresponding to the given abscissa in the shortest time.

Meanwhile, it should be noted here that the condition of passing through two points is not absolutely necessary in the statement, but in this Problem, it is brought in by the solution itself. For in the solution of this Problem, one arrives immediately at a differential equationAn equation involving derivatives, describing how one quantity changes in relation to another. of the second degree, which, when integrated twice, receives two arbitrary constantsValues that can be any number, usually denoted as C, which appear during the process of integration.. To determine these, one needs two points through which the curve is led, or other similar properties. And this same condition, as if of its own accord, attaches itself to all Problems of this kind whose solution leads immediately to a differential equation of the second degree.

However, in Problems which are resolved by a differential equation of the fourth or higher order, not even two points are sufficient to determine the curve, but as many points are needed as the degrees the differentials obtain. On the contrary, if the solution leads immediately to an algebraic equation, then the Problem will be perfectly determined without a condition of this kind, provided the length of the abscissa is defined. But all these things will be perceived more clearly when we arrive at the solutions of the Problems below: and there we shall explain these observations more fully. For here at the beginning, it seemed best only to mention these things so that