Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

PROOF.

The value of the formula $\int Z dx$ for the abscissa abscissa: the horizontal coordinate along the x-axis AZ is the sum of all the values of that same formula which correspond to the individual portions of the abscissa AZ. If, therefore, the abscissa AZ is conceived as being divided into any number of parts, one of which is MN, and the value of the formula $\int Z dx$ is shown for each of these individual parts; the sum of all these values will yield the value of the formula $\int Z dx$ that belongs to the whole abscissa AZ, and which will be a maximum or a minimum. However, since $Z$ is assumed to be an algebraic function of $x, y, p, q,$ etc. As noted previously, p and q represent the first and second derivatives (the slope and the rate of change of the slope) of the curve., the value of the formula $\int Z dx$ corresponding to the portion of the abscissa MN will depend only on the nature of the corresponding portion of the curve $mn$. It will remain the same regardless of how the remaining parts $am$ and $nz$ are varied; for the values of each of the individual letters $x, y, p, q,$ etc., are determined by the portion of the curve $mn$ alone. If, therefore, the values of the formula $\int Z dx$ that fit the portions of the abscissa AM, MN, and NZ are set as $P, Q,$ and $R,$ these quantities $P, Q,$ and $R$ will not depend on one another. Wherefore, since their sum $P + Q + R$ is a maximum or a minimum, it is necessary that each individual part also be endowed with the property of the maximum or minimum. For this reason, if in the curve $amz$ the formula $\int Z dx$ has a maximum or minimum value, and the quantity $Z$ is an algebraic function of $x, y, p, q,$ etc., then for any portion of that curve, the same formula $\int Z dx$ will enjoy the property of being a maximum or minimum. Which was to be demonstrated. original: "Q. E. D." (Quod Erat Demonstrandum)

COROLLARY I.

39. Therefore, if a curve $amz$ has been found which, for a given abscissa AZ, has a maximum or minimum value of the formula $\int Z dx$, and $Z$ is an algebraic or determined function, then any portion of that same curve, with respect to its own corresponding abscissa, will also enjoy the same property of being a maximum or minimum.