Part 12

MAWKMAI (Burmese _Maukmè_), one of the largest states in the eastern division of the southern Shan States of Burma. It lies approximately between 19° 30´ and 20° 30´ N. and 97° 30´ and 98° 15´ E., and has an area of 2,787 sq. m. The central portion of the state consists of a wide plain well watered and under rice cultivation. The rest is chiefly hills in ranges running north and south. There is a good deal of teak in the state, but it has been ruinously worked. The sawbwa now works as contractor for government, which takes one-third of the net profits. Rice is the chief crop, but much tobacco of good quality is grown in the Langkö district on the Têng river. There is also a great deal of cattle-breeding. The population in 1901 was 29,454, over two-thirds of whom were Shans and the remainder Taungthu, Burmese, Yangsek and Red Karens. The capital, MAWKMAI, stands in a fine rice plain in 20° 9´ N. and 97° 25´ E. It had about 150 houses when it first submitted in 1887, but was burnt out by the Red Karens in the following year. It has since recovered. There are very fine orange groves a few miles south of the town at Kantu-awn, called Kadugate by the Burmese.

MAXENTIUS, MARCUS AURELIUS VALERIUS, Roman emperor from A.D. 306 to 312, was the son of Maximianus Herculius, and the son-in-law of Galerius. Owing to his vices and incapacity he was left out of account in the division of the empire which took place in 305. A variety of causes, however, had produced strong dissatisfaction at Rome with many of the arrangements established by Diocletian, and on the 28th of October 306, the public discontent found expression in the massacre of those magistrates who remained loyal to Flavius Valerius Severus and in the election of Maxentius to the imperial dignity. With the help of his father, Maxentius was enabled to put Severus to death and to repel the invasion of Galerius; his next steps were first to banish Maximianus, and then, after achieving a military success in Africa against the rebellious governor, L. Domitius Alexander, to declare war against Constantine as having brought about the death of his father Maximianus. His intention of carrying the war into Gaul was anticipated by Constantine, who marched into Italy. Maxentius was defeated at Saxa Rubra near Rome and drowned in the Tiber while attempting to make his way across the Milvian bridge into Rome. He was a man of brutal and worthless character; but although Gibbon's statement that he was "just, humane and even partial towards the afflicted Christians" may be exaggerated, it is probable that he never exhibited any special hostility towards them.

See De Broglie, _L'Église et l'empire Romain au quatrième siècle_ (1856-1866), and on the attitude of the Romans towards Christianity generally, app. 8 in vol. ii. of J. B. Bury's edition of Gibbon (Zosimus ii. 9-18; Zonaras xii. 33, xiii. 1; Aurelius Victor, _Epit._ 40; Eutropius, x. 2).

MAXIM, SIR HIRAM STEVENS (1840- ), Anglo-American engineer and inventor, was born at Sangerville, Maine, U.S.A., on the 5th of February 1840. After serving an apprenticeship with a coachbuilder, he entered the machine works of his uncle, Levi Stevens, at Fitchburg, Massachusetts, in 1864, and four years later he became a draughtsman in the Novelty Iron Works and Shipbuilding Company in New York City. About this period he produced several inventions connected with illumination by gas; and from 1877 he was one of the numerous inventors who were trying to solve the problem of making an efficient and durable incandescent electric lamp, in this connexion introducing the widely-used process of treating the carbon filaments by heating them in an atmosphere of hydrocarbon vapour. In 1880 he came to Europe, and soon began to devote himself to the construction of a machine-gun which should be automatically loaded and fired by the energy of the recoil (see MACHINE-GUN). In order to realize the full usefulness of the weapon, which was first exhibited in an underground range at Hatton Garden, London, in 1884, he felt the necessity of employing a smokeless powder, and accordingly he devised maximite, a mixture of trinitrocellulose, nitroglycerine and castor oil, which was patented in 1889. He also undertook to make a flying machine, and after numerous preliminary experiments constructed an apparatus which was tried at Bexley Heath, Kent, in 1894. (See FLIGHT.) Having been naturalized as a British subject, he was knighted in 1901. His younger brother, Hudson Maxim (b. 1853), took out numerous patents in connexion with explosives.

MAXIMA AND MINIMA, in mathematics. By the _maximum_ or _minimum_ value of an expression or quantity is meant primarily the "greatest" or "least" value that it can receive. In general, however, there are points at which its value ceases to increase and begins to decrease; its value at such a point is called a maximum. So there are points at which its value ceases to decrease and begins to increase; such a value is called a minimum. There may be several maxima or minima, and a minimum is not necessarily less than a maximum. For instance, the expression (x² + x + 2)/(x - 1) can take all values from -[oo] to -1 and from +7 to +[oo], but has, so long as x is real, no value between -1 and +7. Here -1 is a maximum value, and +7 is a minimum value of the expression, though it can be made greater or less than any assignable quantity.

The first general method of investigating maxima and minima seems to have been published in A.D. 1629 by Pierre Fermat. Particular cases had been discussed. Thus Euclid in book III. of the _Elements_ finds the greatest and least straight lines that can be drawn from a point to the circumference of a circle, and in book VI. (in a proposition generally omitted from editions of his works) finds the parallelogram of greatest area with a given perimeter. Apollonius investigated the greatest and least distances of a point from the perimeter of a conic section, and discovered them to be the normals, and that their feet were the intersections of the conic with a rectangular hyperbola. Some remarkable theorems on maximum areas are attributed to Zenodorus, and preserved by Pappus and Theon of Alexandria. The most noteworthy of them are the following:--

1. Of polygons of n sides with a given perimeter the regular polygon encloses the greatest area.

2. Of two regular polygons of the same perimeter, that with the greater number of sides encloses the greater area.

3. The circle encloses a greater area than any polygon of the same perimeter.

4. The sum of the areas of two isosceles triangles on given bases, the sum of whose perimeters is given, is greatest when the triangles are similar.

5. Of segments of a circle of given perimeter, the semicircle encloses the greatest area.

6. The sphere is the surface of given area which encloses the greatest volume.

Serenus of Antissa investigated the somewhat trifling problem of finding the triangle of greatest area whose sides are formed by the intersections with the base and curved surface of a right circular cone of a plane drawn through its vertex.

The next problem on maxima and minima of which there appears to be any record occurs in a letter from Regiomontanus to Roder (July 4, 1471), and is a particular numerical example of the problem of finding the point on a given straight line at which two given points subtend a maximum angle. N. Tartaglia in his _General trattato de numeri et mesuri_ (c. 1556) gives, without proof, a rule for dividing a number into two parts such that the continued product of the numbers and their difference is a maximum.

Fermat investigated maxima and minima by means of the principle that in the neighbourhood of a maximum or minimum the differences of the values of a function are insensible, a method virtually the same as that of the differential calculus, and of great use in dealing with geometrical maxima and minima. His method was developed by Huygens, Leibnitz, Newton and others, and in particular by John Hudde, who investigated maxima and minima of functions of more than one independent variable, and made some attempt to discriminate between maxima and minima, a question first definitely settled, so far as one variable is concerned, by Colin Maclaurin in his _Treatise on Fluxions_ (1742). The method of the differential calculus was perfected by Euler and Lagrange.

John Bernoulli's famous problem of the "brachistochrone," or curve of quickest descent from one point to another under the action of gravity, proposed in 1696, gave rise to a new kind of maximum and minimum problem in which we have to find a curve and not points on a given curve. From these problems arose the "Calculus of Variations." (See VARIATIONS, CALCULUS OF.)

The only general methods of attacking problems on maxima and minima are those of the differential calculus or, in geometrical problems, what is practically Fermat's method. Some problems may be solved by algebra; thus if y = f(x) ÷ [phi](x), where f(x) and [phi](x) are polynomials in x, the limits to the values of y[phi] may be found from the consideration that the equation y[phi](x) - f(x) = 0 must have real roots. This is a useful method in the case in which [phi](x) and f(x) are quadratics, but scarcely ever in any other case. The problem of finding the maximum product of n positive quantities whose sum is given may also be found, algebraically, thus. If a and b are any two real unequal quantities whatever {½(a + b)}² > ab, so that we can increase the product leaving the sum unaltered by replacing any two terms by half their sum, and so long as any two of the quantities are unequal we can increase the product. Now, the quantities being all positive, the product cannot be increased without limit and must somewhere attain a maximum, and no other form of the product than that in which they are all equal can be the maximum, so that the product is a maximum when they are all equal. Its minimum value is obviously zero. If the restriction that all the quantities shall be positive is removed, the product can be made equal to any quantity, positive or negative. So other theorems of algebra, which are stated as theorems on inequalities, may be regarded as algebraic solutions of problems on maxima and minima.

For purely geometrical questions the only general method available is practically that employed by Fermat. If a quantity depends on the position of some point P on a curve, and if its value is equal at two neighbouring points P and P´, then at some position between P and P´ it attains a maximum or minimum, and this position may be found by making P and P´ approach each other indefinitely. Take for instance the problem of Regiomontanus "to find a point on a given straight line which subtends a maximum angle at two given points A and B." Let P and P´ be two near points on the given straight line such that the angles APB and AP´B are equal. Then ABPP´ lie on a circle. By making P and P´ approach each other we see that for a maximum or minimum value of the angle APB, P is a point in which a circle drawn through AB touches the given straight line. There are two such points, and unless the given straight line is at right angles to AB the two angles obtained are not the same. It is easily seen that both angles are maxima, one for points on the given straight line on one side of its intersection with AB, the other for points on the other side. For further examples of this method together with most other geometrical problems on maxima and minima of any interest or importance the reader may consult such a book as J. W. Russell's _A Sequel lo Elementary Geometry_ (Oxford, 1907).

The method of the differential calculus is theoretically very simple. Let u be a function of several variables x1, x2, x3 ... x_n, supposed for the present independent; if u is a maximum or minimum for the set of values x1, x2, x3, ... x_n, and u becomes u + [delta]u, when x1, x2, x3 ... x_n receive small increments [delta]x1, [delta]x2, ... [delta]x_n; then [delta]u must have the same sign for all possible values of [delta]x1, [delta]2 ... [delta]x_n.

Now _ _ __ [delta]u | __ [delta]²u __ [delta]³u | [delta]u = \ --------- [delta]x1 + ½ | \ ---------- + 2 \ ------------------- [delta]x1 [delta]x2 ... | + ... /__ [delta]x1 |_ /__ [delta]x1² /__ [delta]x1 [delta]x2 _|

The sign of this expression in general is that of [Sigma]([delta]u/[delta]x1)[delta]x1, which cannot be one-signed when x1, x2, ... x_n can take all possible values, for a set of increments [delta]x1, [delta]x2 ... [delta]x_n, will give an opposite sign to the set -[delta]x1, -[delta]x2, ... -[delta]x_n. Hence [Sigma]([delta]u/[delta]x1)[delta]x1 must vanish for all sets of increments [delta]x1, ... [delta]x_n, and since these are independent, we must have [delta]u/[delta]x1 = 0, [delta]u/[delta]x2 = 0, ... [delta]u/[delta]x_n = 0. A value of u given by a set of solutions of these equations is called a "critical value" of u. The value of [delta]u now becomes _ _ | __ [delta]²u __ [delta]²u | ½ | \ --------- [delta]x1² + 2 \ ------------------- [delta]x1 [delta]x2 + ... |; |_ /__ [delta]x1² /__ [delta]x1 [delta]x2 _|

for u to be a maximum or minimum this must have always the same sign. For the case of a single variable x, corresponding to a value of x given by the equation du/dx = 0, u is a maximum or minimum as d²u/dx² is negative or positive. If d²u/dx² vanishes, then there is no maximum or minimum unless d²u/dx² vanishes, and there is a maximum or minimum according as d^4u/dx^4 is negative or positive. Generally, if the first differential coefficient which does not vanish is even, there is a maximum or minimum according as this is negative or positive. If it is odd, there is no maximum or minimum.

In the case of several variables, the quadratic

__ [delta]²u __ [delta]²u \ ---------- [delta]x1² + 2 \ ------------------- + ... /__ [delta]x1² /__ [delta]x1 [delta]x2

must be one-signed. The condition for this is that the series of discriminants

a11 , | a11 a12 | , | a11 a12 a13 | , ... | a21 a22 | | a21 a22 a23 | | a31 a32 a33 |

where a_pq denotes [delta]²u/[delta]a_p[delta]a_q should be all positive, if the quadratic is always positive, and alternately negative and positive, if the quadratic is always negative. If the first condition is satisfied the critical value is a minimum, if the second it is a maximum. For the case of two variables the conditions are

[delta]²u [delta]²u / [delta]² \² ---------- · ---------- > ( ------------------- ) [delta]x1² [delta]x2² \ [delta]x1 [delta]x2 /

for a maximum or minimum at all and [delta]²u/[delta]x1² and [delta]²u/[delta]x2² both negative for a maximum, and both positive for a minimum. It is important to notice that by the quadratic being one-signed is meant that it cannot be made to vanish except when [delta]x1, [delta]x2, ... [delta]x_n all vanish. If, in the case of two variables,

[delta]²u [delta]²u / [delta]²u \² ---------- · ---------- = ( ------------------- ) [delta]x1² [delta]x2² \ [delta]x1 [delta]x2 /

then the quadratic is one-signed unless it vanishes, but the value of u is not necessarily a maximum or minimum, and the terms of the third and possibly fourth order must be taken account of.

Take for instance the function u = x² - xy² + y². Here the values x = 0, y = 0 satisfy the equations [delta]u/[delta]x = 0, [delta]u/[delta]y = 0, so that zero is a critical value of u, but it is neither a maximum nor a minimum although the terms of the second order are ([delta]x)², and are never negative. Here [delta]u = [delta]x² - [delta]x[delta]y² + [delta]y², and by putting [delta]x = 0 or an infinitesimal of the same order as [delta]y², we can make the sign of [delta]u depend on that of [delta]y², and so be positive or negative as we please. On the other hand, if we take the function u = x² - xy² + y^4, x = 0, y = 0 make zero a critical value of u, and here [delta]u = [delta]x² - [delta]x[delta]y² + [delta]y^4, which is always positive, because we can write it as the sum of two squares, viz. ([delta]x - ½[delta]y²)² + ¾[delta]y^4; so that in this case zero is a minimum value of u.

A critical value usually gives a maximum or minimum in the case of a function of one variable, and often in the case of several independent variables, but all maxima and minima, particularly absolutely greatest and least values, are not necessarily critical values. If, for example, x is restricted to lie between the values a and b and [phi]´(x) = 0 has no roots in this interval, it follows that [phi]´(x) is one-signed as x increases from a to b, so that [phi](x) is increasing or diminishing all the time, and the greatest and least values of [phi](x) are [phi](a) and [phi](b), though neither of them is a critical value. Consider the following example: A person in a boat a miles from the nearest point of the beach wishes to reach as quickly as possible a point b miles from that point along the shore. The ratio of his rate of walking to his rate of rowing is cosec [alpha]. Where should he land?

Here let AB be the direction of the beach, A the nearest point to the boat O, and B the point he wishes to reach. Clearly he must land, if at all, between A and B. Suppose he lands at P. Let the angle AOP be [theta], so that OP = a sec[theta], and PB = b - a tan [theta]. If his rate of rowing is V miles an hour his time will be a sec [theta]/V + (b - a tan [theta]) sin [alpha]/V hours. Call this T. Then to the first power of [delta][theta], [delta]T = (a/V) sec²[theta] (sin [theta] - sin [alpha])[delta][theta], so that if AOB > [alpha], [delta]T and [delta][theta] have opposite signs from [theta] = 0 to [theta] = [alpha], and the same signs from [theta] = [alpha] to [theta] = AOB. So that when AOB is > [alpha], T decreases from [theta] = 0 to [theta] = [alpha], and then increases, so that he should land at a point distant a tan [alpha] from A, unless a tan [alpha] > b. When this is the case, [delta]T and [delta][theta] have opposite signs throughout the whole range of [theta], so that T decreases as [theta] increases, and he should row direct to B. In the first case the minimum value of T is also a critical value; in the second case it is not.

The greatest and least values of the bending moments of loaded rods are often at the extremities of the divisions of the rods and not at points given by critical values.

In the case of a function of several variables, X1, x2, ... x_n, not independent but connected by m functional relations u1 = 0, u2 = 0, ..., u_m = 0, we might proceed to eliminate m of the variables; but Lagrange's "Method of undetermined Multipliers" is more elegant and generally more useful.

We have [delta]u1 = 0, [delta]u2 = 0, ..., [delta]u_m = 0. Consider instead of [delta]u, what is the same thing, viz., [delta]u + [lambda]1[delta]u1 + [lambda]2[delta]u2 + ... + [lambda]_m[delta]u_m, where [lambda]1, [lambda]2, ... [lambda]_m, are arbitrary multipliers. The terms of the first order in this expression are

__ [delta]u __ [delta]u1 __ [delta]u_m \ --------- [delta]x1 + [lambda]1 \ --------- [delta]x1 + ... + [lambda]_m \ ---------- [delta]x1. /__ [delta]x1 /__ [delta]x1 /__ [delta]x1

We can choose [lambda]1, ... [lambda]_m, to make the coefficients of [delta]x1, [delta]x2, ... [delta]x_m, vanish, and the remaining [delta]x_(m+1) to [delta]x_n may be regarded as independent, so that, when u has a critical value, their coefficients must also vanish. So that we put

[delta]u [delta]u1 [delta]u_m ---------- + [lambda]1 ---------- + ... + [lambda]_m ---------- = 0 [delta]x_r [delta]x_r [delta]x_r

for all values of r. These equations with the equations u1 = 0, ..., u_m = 0 are exactly enough to determine [lambda]1, ..., [lambda]_m, x1 x2, ..., x_n, so that we find critical values of u, and examine the terms of the second order to decide whether we obtain a maximum or minimum.

To take a very simple illustration; consider the problem of determining the maximum and minimum radii vectors of the ellipsoid x²/a² + y²/b² + z²/c² = 1, where a² > b² > c². Here we require the maximum and minimum values of x² + y² + z² where x²/a² + y²/b² + z²/c² = 1.

We have

/ [lambda]\ / [lambda]\ / [lambda]\ [delta]u = 2x [delta]x ( 1 + -------- ) + 2y [delta]y ( 1 + -------- ) + 2z [delta]z ( 1 + -------- ) \ a² / \ b² / \ c² /

/ [lambda]\ / [lambda]\ / [lambda]\ + [delta]x² ( 1 + -------- ) + [delta]y² ( 1 + -------- ) + [delta]z² ( 1 + -------- ). \ a² / \ b² / \ c² /

To make the terms of the first order disappear, we have the three equations:--

x(1 + [lambda]/a²) = 0, y(1 + [lambda]/b²) = 0, z(1 + [lambda]/c²) = 0.

These have three sets of solutions consistent with the conditions x²/a² + y²/b² + z²/c² = 1, a² > b² > c², viz.:--

(1) y = 0, z = 0, [lambda] = -a²; (2) z = 0, x = 0, [lambda] = -b²;

(3) x = 0, y = 0, [lambda] = -c².

In the case of (1) [delta]u = [delta]y² (1 - a²/b²) + [delta]z² (1 - a²/c²), which is always negative, so that u = a² gives a maximum.

In the case of (3) [delta]u = [delta]x² (1 - c²/a²) + [delta]y² (1 - c²/b²), which is always positive, so that u = c² gives a minimum.

In the case of (2) [delta]u = [delta]x²(1 - b²/a²) - [delta]z²(b²/c² - 1), which can be made either positive or negative, or even zero if we move in the planes x²(1 - b²/a²) = z²(b²/c² - 1), which are well known to be the central planes of circular section. So that u = b², though a critical value, is neither a maximum nor minimum, and the central planes of circular section divide the ellipsoid into four portions in two of which a² > r² > b², and in the other two b² > r² > c². (A. E. J.)

MAXIMIANUS, a Latin elegiac poet who flourished during the 6th century A.D. He was an Etruscan by birth, and spent his youth at Rome, where he enjoyed a great reputation as an orator. At an advanced age he was sent on an important mission to the East, perhaps by Theodoric, if he is the Maximianus to whom that monarch addressed a letter preserved in Cassiodorus (_Variarum_, i. 21). The six elegies extant under his name, written in old age, in which he laments the loss of his youth, contain descriptions of various amours. They show the author's familiarity with the best writers of the Augustan age.