Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

...which are to be evaluated from the arbitrary constants that the solution introduces. Thus, in the solution of the Problem where a curve is sought which, among all curves of the same length original: "ejusdem longitudinis", encloses the greatest area with the abscissa abscissa: the horizontal coordinate (x) on a graph, representing the distance along the base line, two new constants enter; because of this, to make the Problem determinate In mathematics, a problem is "determinate" when it has a specific, unique solution rather than an infinite number of possibilities., it must be proposed in such a way that among all curves of the same length—which not only correspond to the same abscissa, but also pass through two given points—the one that yields the maximum area for the given abscissa is sought. And in a similar way, it can happen that four points, and sometimes even more, must be assumed at will so that the Problem becomes determinate; the judgment of this matter must be sought from the very nature of the Problems. Just as, however, in the isoperimetric problem isoperimetric: from the Greek for "having the same perimeter"; a problem where you compare shapes that all have the same boundary length, all curves from which the solution must be determined are assumed to be of the same length; so, in place of this property, any other property can be proposed which must be common to all of them. Thus, curves endowed with a property of maximum or minimum have been sought among only those curves related to the same abscissa which, when all are rotated around that abscissa, generate equal surface areas; and in a similar way, any other properties can be proposed. Furthermore, not just one, but several such properties can be prescribed, which must be common to all the curves among which the one containing some maximum or minimum is to be defined. For example, if a curve endowed with some property of maximum or minimum were sought among all curves corresponding to the same abscissa which were all equal to each other in length as well as enclosing equal areas.

S C H O L I O N I I.

15. Because of this distinction between the absolute and relative Method of maxima and minima, our treatment will be divided into two parts. First, we shall present the method for determining the curve endowed with a maximum or minimum property among all possible curves corresponding to the same abscissa. Then, we shall proceed to those types of Problems in which the curve of maximum or minimum pro- The text continues on the next page with the word "property" (original: "proprietate").