ON THE METHOD OF MAXIMA AND MINIMA

let us remove perverse ideas regarding the determination of problems of this kind.

DEFINITION II.

7. The absolute method of maxima and minima teaches how to determine, among absolutely all curves related to the same abscissa The horizontal coordinate (x-axis) in a coordinate system., that one in which a certain proposed variable quantity obtains the maximum or minimum value.

COROLLARY.

8. Therefore, in problems pertaining to this method, the axis is given by position; and, among all curves which can be related to this axis and its determined portion, that one is determined in which a certain variable quantity becomes maximum or minimum.

SCHOLION A scholarly explanatory note or commentary..

9. We do not add here, in general, any other condition for the determination of the maximum or minimum besides the quantity of the abscissa. For problems are given which are perfectly determined in this way, as will appear more distinctly below. For even if problems of this kind occur, for the determination of which two or more points can additionally be prescribed through which the sought curve must pass, nevertheless this will finally be perceived from the solution of each problem itself. For if one arrives at an equation of this kind for the sought curve, in which new constant quantities have entered through integration The mathematical process of finding the integral; here referring to the constants of integration that arise when solving differential equations. which were not present in the question itself, then the solution must be deemed ambiguous and vague; because it encompasses within itself innumerable curved lines which can arise from the determination of those constant and arbitrary quantities. In these cases, therefore, it must be concluded that the problem, by its nature, is not entirely determined: but for its full determination, besides the quantity of the abscissa, as many new conditions must be added as are necessary to reduce those arbitrary constants to determined values. For conditions of this kind, the points through which the sought curves original: "quæsitæ" pass are most conveniently assumed.