Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

COROLLARY II.

25. Since, however, the property of the maximum or minimum must be proposed in such a way that it refers to a given and determined abscissa abscissa: the horizontal distance along the x-axis, usually denoted as x; the Formula of the maximum or minimum must also be referred to that definite abscissa.

COROLLARY III.

26. Therefore, the Formula of the maximum or minimum will be a variable quantity depending on the length of whatever abscissa it corresponds to. And in any Problem, that curve will be sought for which, at the definite abscissa, that Formula of the maximum or minimum obtains the maximum or minimum value.

COROLLARY IV.

27. Nor indeed can the Formula of the maximum or minimum depend on the abscissa alone: for if this were the case, it would obtain the same value for all curves corresponding to the same abscissa, and therefore all would satisfy the condition equally.

COROLLARY V.

28. For this reason, the Formula of the maximum or minimum—besides the abscissa common to all curves coming under consideration—must depend specifically on each particular curve; so that there is only one curve for which it can take on a maximum or minimum value.

SCHOLIUM I.

Fig. 1.

29. In order that all these things may be understood more clearly, and that the state of the Questions to be treated in the following section may be better grasped; let us suppose that—either among all curves whatsoever, or only among the countless ones having a certain common property—which correspond to the same abscissa AZ, that one curve must be determined The sentence continues onto the next page.