Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

and so on; whatever formula is required to be a maximum or minimum in the sought curve, it will always be of this form $\int Z dx$, namely the integral of some finite quantity $Z$ multiplied by the differential differential: an infinitely small change, denoted here as dx $dx$. Moreover, $Z$ must be a quantity of such a kind that if an equation is established between $x$ and $y$, the integral $\int Z dx$ obtains a determined value. From this, $Z$ will be a function of the quantities $x$ and $y$, and of those depending on them, $p, q, r,$ etc. Euler uses $p, q, r$ to represent the first, second, and third derivatives of $y$ with respect to $x$., being either algebraic (that is, determined), or it will furthermore encompass within itself indeterminate integral formulas; this distinction must be carefully maintained. Thus, if the formula of the maximum or minimum $W$ were $\int y dx$, or $\int y dx \sqrt{(1 + p^2)}$; the quantity $Z$ will be algebraic. But if it were $W = \int y dx \int y dx$, then $Z$ would be $y \int y dx$; that is, the quantity $Z$ itself will be indeterminate, for its value cannot be shown unless the relationship between $x$ and $y$ is given. Indeed, it can even happen that the value of $Z$ itself cannot be expressed by an expanded formula of this kind, but must only be extracted through a differential equation, such as if $dZ = y dx + Z^2 dx$; from which equation the value of $Z$ cannot even be displayed in terms of $x$ and $y$. Hence, three classes of formulas $\int Z dx$ arise here, which must become maxima or minima in the sought curves:

The first of these encompasses those formulas in which $Z$ is an algebraic or determined function of $x, y, p, q, r,$ etc. To the second class, we refer those formulas in which the quantity $Z$ itself further involves integral formulas. In the third class are contained those formulas in which the value of $Z$ is determined by a differential equation whose integration is not known.

PROPOSITION II. THEOREM.

38. If $amz$ were a curve in which the value of the formula $\int Z dx$ is a maximum or minimum, and $Z$ is an algebraic or determined function of $x, y, p, q, r,$ etc., then any portion $mn$ of that same curve will enjoy the same prerogative original: "praerogativa", meaning it shares the same mathematical property, so that for that portion, when referred to its own abscissa abscissa: the horizontal coordinate (the x-axis segment) $MN$, the value of $\int Z dx$ is likewise a maximum or minimum.