MAXIMUM
, denotes the greatest state or quantity attainable in any given case, or the greatest value of a variable quantity. By which it stands opposed to Minimum, which is the least possible quantity in any case.
As in the algebraical expression , where a and b are constant or invariable quantities, and x a variable one. Now it is evident that the value of this remainder or difference, , will increase as the term bx, or x, decreases; and therefore that will be the greatest when this is the smallest; that is, is a maximum, when x is the least, or nothing at all.
Again, the expression or difference , evidently increases as the fraction b/x diminishes; and this diminisnes as x increases; therefore the given expression will be the greatest, or a maximum, when x is the greatest, or infinite.
Also, if along the diameter KZ (the 3d fig. below) of a circle, a perpendicular ordinate LM b[ecedil] conceived to move, from K towards Z; it is evident that, from K it increases continually till it arrive at the centre, in the position NO, where it is at the greatest state; and from thence it continually decreases again, as it moves along from N to Z, and quite vanishes at the point Z. So that the maximum state of the ordinate is NO, equal to the radius of the circle.
Methodus de Maximis et Minimis, a method of finding the greatest or least state or value of a variable quantity.
Some quantities continually increase, and so have no maximum but what is insinite; as the ordinates BC, DE of the parabola ACE: Some continually decrease, and so their least or minimum state is nothing; as the ordinates FG, HI, to the asymptotes of the hyperbola. Others increase to a certain magnitude, which is their maximum, and then decrease again; as the ordinates LM &c of the circle. And others again decrease to a certain magnitude TV, which is their minimum, and then increase again; as the ordinates of the curve SVY. While others admit of several maxima and minima; as the ordinates of the curve abcde, where at b and d they are maxima, and a, c, e, minima. And thus the maxima and minima of all other variable quantities may be conceived; expressing those quantities by the ordinates of some curves.
The first maxima and minima are found in the Elements of Euclid, or flow immediately from them: thus, it appears, by the 5th prop. of book 2, that the greatest rectangle that can be made of the two parts of a given line, any how divided, is when the line is divided equally in the middle; prob. 7, book 3, shews that the greatest line that can be drawn from a given point within a circle, is that which passes through the centre; and that the least line that can be so drawn, is the continuation of the same to the other side of the circle: prop. 8 ib. shews the same for lines drawn from a point without the circle: and thus other instances might be pointed out in the Elements.— Other writers on the Maxima and Minima, are, Apollonius, in the whole 5th book of his Conic Sections;| and in the Preface or Dedication to that book, he says others had then also treated the subject, though in a slighter manner.—Archimedes; as in prop. 9 of his Treatise on the Sphere and Cylinder, where he demonstrates that, of all spherical segments under equal superficies, the hemisphere is the greatest.—Serenus, in his 2d book, or that on the Conic Sections.— Pappus, in many parts of his Mathematical Collections; as in lib. 3, prop. 28 &c, lib. 6, prop. 31 &c, where he treats of some curious cases of variable geometrical quantities, shewing how some increase and decrease both ways to infinity; while others proceed only one way, by increase or decrease, to infinity, and the other way to a certain magnitude; and others again both ways to a certain magnitude, giving a maximum and minimum; also lib. 7, prop. 13, 14, 165, 166, &c. And all these are the geometrical Maxima and Minima of the Ancients; to which may be added some others of the same kind, viz. Viviani De Maximis & Minimis Geometrica Divinatio in quintum Conicorum Apollonii Pergæi, in fol. at Flor. 1659; also an ingenious little tract in Thomas Simpson's Geometry, on the Maxima and Minima of Geometrical Quantities.
Other writings on the Maxima and Minima are chiefly treated in a more general way by the modern analysis; and first among these perhaps may be placed that of Fermat. This, and other methods, are best referred to, and explained by the ordinates of curves. For when the ordinate of a curve increases to a certain magnitude, where it is greatest, and afterwards decreases again, it is evident that two ordinates on the contrary sides of the greatest ordinate may be equal to each other; and the ordinates decrease to a certain point, where they are at the least, and afterwards increase again; there may also be two equal ordinates, one on each side of the least ordinate. Hence then an equal ordinate corresponds to two different abscisses, or for every value of an ordinate there are two values of abscisses. Now as the difference between the two abscisses is conceived to become less and less, it is evident that the two equal ordinates, corresponding to them, approach nearer and nearer together; and when the differences of the abscisses are infinitely little, or nothing, then the equal ordinates unite in one, which is either the maximum or minimum. The method hence derived then, is this: Find two values of an ordinate, expressed in terms of the abscisses: put those two values equal to each other, cancelling the parts that are common to both, and dividing all the remaining terms by the difference between the abscisses, which will be a common factor in them: next, supposing the abscisses to become equal, that the equal ordinates may concur in the maximum or minimum, that difference will vanish, as well as all the terms of the equation that include it; and therefore, striking those terms out of the equation, the remaining terms will give the value of the absciss corresponding to the maximum or minimum.
For example, suppose it were required to find the greatest ordinate in a circle KMQ. Put the diameter KZ = a, the absciss KL = x, the ordinate LM = y; hence the other part of the diameter is , and consequently, by the nature of the circle being equal LM2, or . Again, put another absciss , where d is the difference LP, the ordinate PQ, being equal to LM or y; here then again , or : put now these two values of y2 equal to each other, so shall ; cancel the common terms ax and x2, then , or ; divide all by d, so shall , a general equation derived from the equality of the two ordinates. Now, bringing the two equal ordinates together, or making the two abscisses equal, their difference d vanishes, and the last equation becomes barely 2x = a, or x = (1/2)a, = KN, the value of the absciss KN when the ordinate NO is a maximum, viz, the greatest ordinate bisects the diameter. And the operation and conclusion it is evident will be the same, to divide a given line into two parts, so that their rectangle shall be the greatest possible.
For a second example, let it be required to divide the given line AB into two such parts, that the one line drawn into the square of the other may be the greatest possible. Putting the given line AB = a, and one part AC = x; then the other part CB will be , and therefore is the product of one part by the square of the other. Again, let one part be , then the other part is , and . Then, putting these two products equal to each other, cancelling the common terms , and dividing the remainder by d, there results ; hence, cancelling all the terms that contain d, there remains , or 3x = 2a, and, x = (2/3)a; that is, the given line must be divided into two parts in the ratio of 3 to 2. See Fermat's Opera Varia, pa. 63, and his Letters to F. Mersenne.
The next method was that of John Hudde, given by Schooten among the additions to Des Cartes's Geometry, near the end of the 1st vol. of his edition. This method is also drawn from the property of an equation that has two equal roots. He there demonstrates that, having ranged the terms of an equation, that has two roots equal, according to the order of the exponents of the unknown quantity, taking all the terms over to one side, and so making them equal to nothing on the other side; if then the terms in that order be multiplied by the terms of any arithmetical progression, the resulting equation will still have one of its roots equal to one of the two equal roots of the former equation. Now since, by what has been said of the foregoing method, when the ordinate of a curve, admitting of a maximum or minimum, is expressed in terms of the abscissa, that abscissa, or the value of x, will be two-fold, because there are two ordinates of the same value; that is, the equation has at least two unequal roots or values of x: but when the ordinate becomes a maximum or minimum, the two abscisses unite in one, and the two roots, or values of x, are equal; therefore, from the above said property, the terms of this equation for the maximum or minimum being multiplied by the terms of any arithmetical progression, the root of the resulting equa-| tion will be one of the said equal roots, or the value of the absciss x when the ordinate is a maximum.
Although the terms of any arithmetic progression may be used for this purpose, some are more convenient than others; and Mr. Hudde directs to make use of that progression which is formed by the exponents of x, viz, to multiply each term by the exponent of its power, and putting all the resulting products equal to nothing; which, it is evident, is exactly the same process as taking the fluxions of all the terms, and putting them equal to nothing; being the common process now used for the same purpose.
Thus, in the former of the two foregoing examples, where , or y2, is to be a maximum; mult. by 1 2 gives ; hence 2x = a, and x = (1/2)a, as before.
And in the 2d example, where , is to be a maximum; mult. by - - 2 3 gives - - - - - ; hence , or 3x = 2a, and x = (2/3)a, as before.
The next general method, and which is now usually practised, is that of Newton, or the method of Fluxions, which proceeds upon a principle different from that of the two former methods of Fermat and Hudde. These proceed upon the idea of the two equal ordinates of a curve uniting into one, at the place of the maximum and minimum; but Newton's upon the principle, that the fluxion or increment of an ordinate is nothing, at the point of the maximum or minimum; a circumstance which immediately follows from the nature of that doctrine: for, since a quantity ceafes to increase at the maximum, and to decrease at the minimum, at those points it neither increases nor decreases; and since the fluxion of a quantity is proportional to its increase or decrease, therefore the fluxion is nothing at the maximum or minimum. Hence this rule. Take the fluxion of the algebraical expression denoting the maximum or minimum, and put it equal to nothing; and that equation will determine the value of the unknown letter or quantity in question.
So in the first of the two foregoing examples, where it is required to determine x when is a maximum: the fluxion of this is ; divide by x., so shall , or a = 2x, and x = (1/2)a.
Also, in the 2d example, where must be a maximum: here the fluxion is ; hence , or 2a = 3x, and x = (2/3)a.
When a quantity becomes a maximum or minimum, and is expressed by two or more affirmative and negative terms, in which only one variable letter is contained; it is evident that the fluxion of the affirmative terms will be equal to the fluxion of the negative ones; since their difference is equal to nothing.
And when, in the expression for the fluxion of a maximum or minimum, there are two or more fluxionary letters, each contained in both affirmative and negative terms; the sum of the terms containing the fluxion of each letter, will be equal to nothing: For, in order that any expression be a maximum or minimum, which contains two or more variable quantities, it must produce a maximum or minimum, if but one of those quantities be supposed variable. So if denote a minimum; its fluxion is ; hence , and ; from the former of these y = (1/2)a, and from the latter x = (1/2)b. Or, in such a case, take the fluxion of the whole expression, supposing only one quantity variable; then take the sluxion again, supposing another quantity only variable: and so on, for all the several variable quantities; which will give the same number of equations for determining those quantities. So, in the above example, , the fluxion is , supposing only x variable; which gives y = (1/2)a: and the fluxion is , when y only is variable; which gives x = (1/2)b; the same as before.
Farther, when any quantity is a maximum or minimum, all the powers or roots of it will be so too; as will also the result be, when it is increased or decreased, or multiplied, or divided by a given or constant quantity; and the logarithm of the same will be also a maximum or minimum.
To find whether a proposed algebraic quantity admits of a maximum or minimum.—Every algebraic expression does not admit of a maximum or minimum, properly so called; for it may either increase continually to infinity, or decrease continually to nothing; in both which cases there is neither a proper maximum nor minimum; for the true maximum is that value to which an expression increases, and after which it decreases again; and the minimum is that value to which the expression decreases, and after that it increases again. Therefore when the expression admits of a maximum, its fluxion is positive before that point, and negative after it; but when it admits of a minimum, its fluxion is negative before, and positive after it. Hence, take the fluxion of the expression a little before the fluxion is equal to nothing, and a little after it; if the first fluxion be positive, and the last negative, the middle state is a maximum; but if the first fluxion be negative, and the last positive, the middle state is a minimum. See Maclaurin's Fluxions, book 1, chap. 9, and book 2, chap. 5, art. 859.
