Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

ON THE METHOD OF MAXIMA AND MINIMA.

...indeterminate integral formulas. For the values of the quantities x, y, p, q, r, etc. Euler uses $p, q, r$ to denote the successive derivatives of $y$ with respect to $x$: $p = \frac{dy}{dx}$, $q = \frac{dp}{dx}$, etc. which are obtained for the arc of the curve mn, depend only on the position of the elements of this arc mn, and on some contiguous elements which do not constitute an arc of finite quantity; from which also a quantity composed in any way from these letters will be determined by the nature of the arc mn alone, unless there were present integral quantities, such as $\int y dx$, which would introduce the whole preceding area AamM, or $\int dx \sqrt{(1 + p^2)}$, which would involve the whole preceding arc from m.

Hence, therefore, it is more clearly understood what we wish to denote by a determined function: a function whose value at any specific point depends only on the coordinates and derivatives at that point of these variables x, y, p, q, r, etc.: Namely, a determined function is so constituted that, for any given place, it depends only on the present values of the letters x, y, p, q, etc., and does not include their preceding values within itself.

An indeterminate function: a function whose value depends on the entire path or "history" of the curve up to that point, however, is such that its value in any given place cannot be determined solely from the values which these letters x, y, p, q, etc. obtain in that place, but additionally requires for its determination all the values which those letters obtained in all the preceding places. Thus it is evident that all algebraic functions: functions involving only the standard operations of addition, subtraction, multiplication, division, and roots are at the same time determined; furthermore, all transcendental functions: functions such as logarithms or exponentials that "transcend" simple algebra which do not depend on the relation between x and y are determined, such as $\log \sqrt{x^2 + y^2}$, $e^{\frac{p}{y}}$, and the arc-sine of $\frac{p}{q}$; the values of which in any place can be assigned from the values of the letters which they obtain in that place alone.

When, however, indeterminate integral formulas are present in any function, which depend on the mutual relation between x and y that holds everywhere, then their value in a given place cannot be known from the values which these letters have in that place, but it is necessary additionally to know all the values in every preceding place—that is, the general relation between the coordinates: the x and y values defining a point's position x and y. We call such functions "indeterminate"; since they are original: "toto cœlo diversæ", literally "different by the whole sky", meaning they are poles apart or diametrically opposed entirely different from those we have called determined.