ON THE METHOD OF MAXIMUMS AND MINIMUMS

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Fig. 1. ...for which the value of formula $W$ becomes a maximum or minimum. Let us suppose that the curve $a m z$ satisfies this Question, so that, if any other curve is referred to the same definite abscissa abscissa: the horizontal distance along the axis, usually denoted as x $AZ$, the value of formula $W$ either becomes smaller than it is for this curve, or larger—depending on whether $W$ ought to be a maximum or a minimum for the curve that solves it. In this very broad question, we have, first, an abscissa of a determined length $AZ$; next, a curve must be sought, either from among absolutely all curves related to this same abscissa, or only from among the countless curves that share one or more properties. This depends on whether the question is adapted to the absolute or relative method of maximums and minimums Euler is distinguishing between "absolute" problems (finding a curve with no constraints) and "relative" problems (finding a curve from a set that shares a specific property, like having the same total length).. Thirdly, we have that quantity $W$ whose value must be a maximum or minimum on the sought curve $a m z$; this quantity $W$ will therefore be the "formula of the maximum or minimum," just as it was defined.

Now, it appears immediately that this formula $W$ must be constructed in such a way that it can be applied to all curves that can be imagined. First, it must depend on the quantity of the definite abscissa $AZ$, so that the value of $W$ changes as the value of $AZ$ changes. Secondly, it must also depend in a specific way on the nature of whatever curve is being considered; for if it were not so, it would result in the same value for all curves, and there would be no problem to solve. Therefore, the quantity $W$ must include within itself not only the abscissa, but also quantities pertaining to the curve itself. Since every curve is determined by the relationship between the abscissa and the ordinate ordinate: the vertical distance or height of a point on the curve, usually denoted as y. Latin: applicata, the quantity $W$ must be composed of the abscissa and the ordinate, and the quantities that depend on them. That is, if the indefinite abscissa is called $x$ and the corresponding indefinite ordinate is called $y$, the quantity $W$ must be a function of the two variables $x$ and $y$. Because of this, if any specific curve is considered, and the relationship between $y$ and $x$ derived from its nature is substituted into the formula $W$, it will yield a definite value belonging to that specific curve and its definite abscissa. Since the formula $W$ takes on different values for different curves—even if the same abscissa is used for all of them—it is clear that among those countless curves