Methodus Inveniendi Lineas Curvas (Calculus of Variations, 1744)

the sought original: "Quæsitæ" curve must pass through; indeed, as many points as there are arbitrary quantities Variables that can take any value until fixed by specific conditions, similar to the constant '+ C' in modern integration. in the discovered equation will make that equation determined. In place of points, however, to perfectly determine the sought curve, one may also employ just as many tangents which touch the sought curve; and, if the contact must occur at a given point on the tangent, this condition will be equivalent to two points. Furthermore, any other conditions whatsoever can be substituted in place of points, provided they are of such a nature that the arbitrary quantities contained in the discovered equation may be determined through them. Nor is it necessary to bring the solution to a conclusion before this judgment is undertaken; rather, certain criteria will be provided below, by the help of which one can immediately distinguish—from that variable quantity which must be a maximum or minimum—which new constants will enter into the equation for the curve that were not contained in the original question. Moreover, these arbitrary constants arise from the degree of the differentials The 'order' of the derivatives in the equation; for example, a second-degree (second-order) differential equation requires two conditions to solve. to which the equation for the sought curve rises; for as many degrees as the differential equation for the sought curve produces, that many arbitrary quantities must be considered to potentially exist within it. Hence, just as many conditions will be needed to determine the curve. This same thing also occurs in the solution of all problems when a differential equation of either the first or a higher degree is found; so that, for our present purpose, no peculiar difficulty should be thought to reside here.

DEFINITION III.

10. The relative method of maxima and minima teaches how to determine the curve which enjoys the property of being a maximum or minimum, not from among absolutely all curves responding to the same abscissa The horizontal coordinate or x-axis., but only from among those which share a certain prescribed common property.

COROLLARY I.

11. To solve problems of this kind, therefore, first, from absolutely all curves responding to the same abscissa, those are