Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

APPLIED TO FINDING CURVED LINES.

...one must be that in which the value of formula W becomes a maximum or minimum; and for finding this curve for any given determined problem, the method we are presenting is designed for this purpose.

COROLLARY VI.

30. Therefore, the formula for the maximum or minimum, W, will be a certain function of two variables x and y: of which one, x, denotes the abscissa abscissa: the horizontal coordinate along the axis, and the other, y, denotes the ordinate ordinate: the vertical coordinate; Euler uses the Latin term applicata. Therefore, not only the variables x and y themselves can be present in W, but also all quantities depending on them, such as p, q, r, s, etc. In Euler’s notation, these letters typically represent the first, second, and higher-order derivatives (e.g., $p = \frac{dy}{dx}$), which describe the slope and curvature., the meanings of which we have provided above. Furthermore, any integral formulas arising from these can be present in W: indeed, they even ought to be, if the problem is to be determined, as we shall soon show.

COROLLARY VII.

31. Therefore, when such a formula W, or function of x and y, is proposed, if the problem belongs to the absolute method of maximums and minimums This refers to finding an optimal curve without any additional constraints or "side conditions" (like a fixed length)., a specific equation between x and y is sought, such that, if the value of y determined through x is substituted into W, and a definite value is assigned to x, a larger quantity for W results (or a smaller one) than if any other equation between x and y had been assumed.

COROLLARY VIII.

32. In this way, therefore, problems pertaining to the theory of curved lines can be reduced to pure analysis Euler is highlighting a major shift in mathematics: turning geometric shape problems into algebraic and calculus-based equations.. And conversely, if a problem of this kind is proposed in pure analysis, it can be referred to and resolved by the theory of curved lines.

SCHOLION II.

33. Although problems of this kind [belong] to pure analysis...