The next remarkable problems on maxima and minima are said to have been investigated by Zenodorus, 1 and were preserved by Pappus and Theon of Alexandria. Of these we may mention the following:—(1) among regular polygons of equal perimeter that of the greatest number of sides contains the greatest area; (2) of polygons of the same perimeter and the same number of sides the regular polygon contains the greatest area; (3) the circle contains a greater area than any other curve or polygon of the same perimeter; (4) the sphere contains the greatest volume for a given superficial area.

In the progress of mathematics the terms maxima and minima have come to be used to imply, not the absolutely greatest and least values of a variable magnitude, but the value which it has at the moment it ceases to increase and begins to decrease, or vice versa. For example, if it be said that the height of the barometer is a maximum at any instant it means that up to that time the barometer was rising and then began to fall. In this way it is possible that there should be several maxima and minima in the course of one day, and that one of the minima should be greater than one of the maxima.

The theory of maxima and minima, in the differential calculus point of view, is very simple. Thus, if u be a given function of a variable x, the values of x for which u has a maximum or a minimum value are, in general, determined by the equation — 0. Again, if % be a function of two variables x and y, then the maximum or minimum values of u must satisfy the equations — = 0 and du λ . , 1 .

— = 0. There is, however, no real maximum or minimum dy s ° iuti ° n if (⅛) 2 is ≡ reater than S $ a short ac · count of this method, illustrated by examples, is given in vol, xiii. pp. 23, 24.

John Bernoulli’s problem (Acta Eruditorum, June 1696) of the “brachystochrone,” i.e., of the curve of quickest descent under the action of gravity, differed essentially 1 Montucla {Hist, de Math., tom. i. p. 113) erroneously attributed these theorems on isoperimetry to Pythagoras, but his statement is based on a misinterpretation of a passage in Diogenes Laertius, See Bretschneider, Die Geometrie vor Exιklides, pp. 89, 90.

from all problems on maxima and minima which had been previously solved. In this he introduced into mathematics the conception of a new and most important class of problems, James Bernoulli, also, in his solution of this problem of his brother, proposed a more general question, which may be stated as follows :—“Of all curves of the same length described on a given base to determine one such that the area of a second curve, each of whose ordinates is a given function of the corresponding ordinate, or arc, of the first, may bβ a maximum, or a mininum.” Such problems were styled “isoperimetrical,” and come under a class now called relative maxima and minima, in which the maximum or minimum curve is to be determined, not from all possible curves, but from among those which possess a given property. The investigations of the Bernoullis were extended and generalized by other eminent mathematicians, but more especially by Euler, and culminated in the invention of the calculus of variations, with an appropriate notation, by Lagrange.