Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

ON THE METHOD OF MAXIMUMS AND MINIMUMS

...can be reduced original: reduci possunt, yet it is helpful to join them with the theory of curved lines. For if we wished to draw the mind away from curved lines and fix it on absolute quantities absolute quantities: variables considered as pure numbers without a geometric or physical context alone, the questions themselves would first become quite abstruse and inelegant, and their use and significance would be less apparent. Secondly, the method of solving questions of this kind, if proposed only in abstract quantities, would be too abstruse and troublesome; whereas the same is wonderfully aided and rendered easy to understand through the inspection of figures and the linear representation of quantities. For this reason, although questions of this sort can be applied to both abstract and concrete quantities, we shall most conveniently translate and solve them in terms of curved lines. That is, whenever an equation between x and y is sought such that a certain proposed formula composed of x and y (if the value of y is substituted from that sought equation and a determined value is assigned to x) becomes a maximum or minimum: then we will always transfer the question to finding a curved line whose abscissa abscissa: the horizontal coordinate, or distance along the axis is x, and whose ordinate ordinate: the vertical coordinate; Euler uses the Latin term applicata is y, for which that formula W becomes a maximum or minimum, if the abscissa x is taken to be of a given magnitude.

These things having been noted, the nature of questions of this kind is seen clearly enough—unless perhaps the ambiguous phrase "of maximum and minimum at the same time" still creates doubt for anyone. Truly, there is no ambiguity here either; for although the method itself shows maximums and minimums equally, yet in any case it will be easy to discern whether the solution provides a maximum or a minimum. Often, however, it can happen that in a given question both a maximum and a minimum hold a place, and in these cases the solution will be twofold, one showing the maximum, the other the minimum. Mostly, however, one or the other—namely, either the maximum or the minimum—is usually impossible; this happens if the formula for the maximum or minimum can grow or decrease infinitely; for in these cases, either a maximum or a minimum will not be given. It can also happen in practice original: Usu venire that the proposed formula W can both grow and decrease infinitely, and in these cases no solution at all takes place...