Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

...formula W must be so composed that its value, when applied to a specific curve amz, depends on the position of each individual element of this curve situated between the boundaries a and z. However, this cannot happen unless the quantity W is an indefinite integral formula indefinite integral: here Euler means a functional, an expression where the total value depends on the entire path of the curve rather than just its endpoints, which generally does not allow for integration without an assumed equation between x and y. Which was to be demonstrated. original: "Q. E. D." (Quod Erat Demonstrandum)

COROLLARY I.

35. Therefore, unless the formula of the maximum or minimum W is an indefinite integral quantity, no curved line can even be determined in which the value of W itself is a maximum or minimum; and thus the question of finding a curve in which W would be a maximum or minimum will be null original: "nulla", meaning the problem is mathematically impossible or trivial if the formula is not an integral.

COROLLARY II.

36. Therefore, so that a curve can be assigned in which the value of W is a maximum or minimum compared to others, the formula W must have the form $\int Zdx$; and the quantity Z must be so composed that the differential differential: an infinitely small change in a quantity Zdx cannot be integrated unless a relationship is established between x and y.

SCHOLARLY NOTE

37. Since the formula of the maximum or minimum W must be the integral of an indefinite differential formula of the first degree—that is, one whose integral becomes a finite quantity—this differential formula can always be reduced to this form Zdx, by means of the letters p, q, r, etc. Euler uses these letters to represent the derivatives or slopes of the curve: p is the first derivative (dy/dx), q the second, and so on.. For this reason, in the following sections, the maximum or minimum formula will perpetually be indicated by us as $\int Zdx$. Moreover, Z will be a function not only of the quantities x and y, but will also contain the letters p, q, r, etc. Thus, if the area AazZ is to be a maximum or minimum, the formula W will become $\int ydx$; and if the surface of a solid of revolution solid of revolution: a 3D shape created by spinning a 2D curve around an axis generated by the rotation of the curve amz around the axis AZ is to be a maximum or minimum, it will be $W = \int ydx\sqrt{(1+pp)}$;