Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

COROLLARY II.

40. In problems of this kind, therefore, where such a maximum or minimum: the greatest or least value of a quantity is sought, it is not necessary to define the specific length of the abscissa: the horizontal coordinate along the x-axis to which the maximum or minimum corresponds; rather, if for any single abscissa the formula $\int Z dx$ is a maximum or minimum, then for any other abscissa it will possess that same property.

COROLLARY III.

41. Therefore, problems of this kind will be solved if the individual particles of the sought curve are determined in such a way that for them the value of the formula $\int Z dx$ becomes a maximum or minimum. For then, at the same time, the whole curve and any portion of it will likewise be endowed with the same property of being a maximum or minimum.

SCHOLION original: "SCHOLION", an explanatory note or commentary common in mathematical and philosophical texts.

42. This property—which is possessed by curves where formulas of the type $\int Z dx$ are a maximum or minimum (where $Z$ is an algebraic or determined function: a function whose value at a point is fixed by the local variables x, y, and their derivatives of $x, y, p, q,$ etc.)—is of the greatest importance; for upon it rests the entire method of solving problems of this kind. However, it seemed best to bring forward this Proposition primarily so that this property—which is unique only to these formulas $\int Z dx$ where $Z$ is either an algebraic or a determined function—should not be thought to be common to absolutely all formulas that can be proposed. For in the following Proposition, we will demonstrate that if integral: the accumulation or sum of values over an interval formulas are contained within $Z$, then that same property no longer holds; from which, at the same time, the nature of questions of this kind will be more clearly understood. Moreover, the proof of this present Proposition is derived from this foundation: that the value of the formula $\int Z dx$ (provided $Z$ is either an algebraic or determined function of $x, y, p, q, r,$ etc.) which fits any portion of the abscissa $MN$, depends only on the corresponding portion of the curve $mn$, and is not affected by the rest of the curve, whether the preceding part $am$ or the following part $nz$. This reasoning ceases if in $Z$ there are contained for-