Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

...must be set apart original: "segregandæ", into which the same prescribed property fits; and then finally, from these set-apart curves, the one which is sought must be defined.

COROLLARY II.

12. Although by such a condition the number of all curves related to the same abscissa The horizontal 'x' coordinate on a graph. is greatly restricted, it will nonetheless still remain infinite. Indeed, even if not just one, but several properties are prescribed which all the curves—from which the sought one is to be determined—must possess, the number of curves will still remain infinite.

COROLLARY III.

13. Therefore, the more properties are proposed that must be common to those curves from which the sought one is to be defined, the more the number of curves among which the choice of the sought one must be made will be restricted, even if it remains infinite.

SCHOLION I.

14. From this class, in which we establish the relative Method of Maxima and Minima, at the beginning of this century, that famous Isoperimetric Problem original: "Problema Isoperimetricum" was first brought forward by Jakob BERNOULLI. In this problem, a curve endowed with a property of the maximum or minimum was sought, not among all curves related to the same abscissa, but only among those which were of the same length; for this reason, those curves from which the sought one had to be extracted were called isoperimetric original: "isoperimetræ"; from the Greek for "having the same perimeter" or boundary length.. Thus, if among all curves corresponding to the same abscissa and equal in length, one seeks the one which encloses the greatest area with the abscissa and the ordinate original: "applicata"; the vertical 'y' coordinate on a graph., the circular line is found to satisfy the requirement. This was, in fact, already known and demonstrated by geometers long before this method was invented. But in this case again, from the very nature of the problems, new conditions are added, just as in those which pertain to the absolute Method of Maxima and Minima;