Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

APPLIED TO FINDING CURVES.

...will have. These distinctions, moreover, the calculus itself will perpetually demonstrate once the solution is found.

PROPOSITION I. THEOREM.

34. In order that a curve amz may be determined by a formula W of a maximum or minimum, which satisfies the condition above all other possible curves, the formula W must be an indefinite integral quantity indefinite integral quantity: a value representing the total accumulation of a function over an interval, rather than a single point, which cannot be integrated unless a relationship between x and y is assumed.

DEMONSTRATION.

For let us suppose that the formula W does not involve indefinite integrals; it would then be a function of the quantities x and y, and the values depending on them, p, q, r, s, etc. Euler uses these letters to represent the derivatives of the curve: p is the first derivative or slope, q the second, and so on., whether algebraic or of such a transcendental transcendental: a value, like a logarithm or circular arc, that cannot be produced by simple arithmetic or algebra alone nature that it could be expressed without assuming a relationship between x and y. This occurs if logarithms of these quantities, circular arcs, or other such defined transcendental quantities—which are to be considered equivalent to algebraic ones—are involved.

Now, if W is set as a function of x and y alone, it is manifest that the value of the formula W (which it obtains for a given curve amz related to a given abscissa abscissa: the horizontal coordinate along the axis, usually x AZ) depends only on the final ordinate ordinate: the vertical distance from the axis to the curve; Euler uses the Latin term applicata Zz; and for all curves having the same ordinate Zz at Z, the value of W would be the same. Thus, the character of the entire curve would not be determined by such a formula W, but only the position of its endpoint z. If, besides x and y, the quantity p is also present in W, then besides the length of the ordinate Zz, the position of the tangent to the curve at z—or the position of the final element at z—will be determined.

If, moreover, q is included, then the position of two contiguous elements of the curve at z will be determined, and so forth. From this it follows that if W were a determinate function of x, y, p, q, r, etc., then only an infinitely small portion of the curve around the extremity z would be determined by it; and for all curves ending at the same extremity, the same value of W would be produced. Therefore, so that the entire curve amz, insofar as it corresponds to the whole abscissa AZ, may be defined by the formula W, the for-