ON THE METHOD OF MAXIMA AND MINIMA

COROLLARY II.

3. However, since the same curve can be made similar to itself in infinite ways, the problem would be highly indeterminate, and thus impossible to solve, unless some restriction is applied. For whatever curve might be presented as possessing a property of a maximum or minimum, another curve—either similar or dissimilar to it—could always be shown that would contain that same property to a greater or lesser degree.

COROLLARY III.

4. Therefore, since an adequate knowledge of curves requires that they be related to some axis given by position, and to any portions of it which are called abscissas The horizontal coordinate (x-axis) in a coordinate system.: the first and most important restriction must be sought from the quantity of the abscissa.

COROLLARY IV.

5. Problems belonging to this method ought to be proposed in such a way that we seek curved lines related to an axis given by position, which, among all other curves corresponding to the same abscissa, are endowed with the property of a maximum or minimum.

SCHOLION A scholarly explanatory note or commentary on a mathematical text..

6. This method of maxima and minima, therefore, differs greatly from the one we have explained elsewhere. In that other method, for a given and determined curved line, we determined the location where a certain proposed variable quantity belonging to the curve becomes a maximum or minimum. Here, however, the curved line itself is sought, in which a certain proposed quantity becomes a maximum or minimum. This method began to be cultivated in the previous century, soon after the discovery of the Analysis of Infinites original: "Analysis infinitorum"; this refers to the early development of calculus., by the celebrated Bernoulli brothers, and from that time it has received the greatest advancements. Indeed, the first problem of this kind to be treated pertained to Mechanics, and in it was sought the curved line upon which a falling heavy body descends most quickly; to which the name brachystochrone curve, or Line of swiftest descent, was given. In this problem