M A U M A X G43 international conference at Brussels in 1853, which pro duced the greatest benefit to navigation as well as indirectly to meteorology. One result was the establishment of the meteorological department of the English Board of Trade, now known as the Meteorological Office, which adopted Maury s model log-books. In 1853 he published his Letters on the Amazon and Atlantic Slopes of South America, and in 1855 he was promoted to the rank of commander. On the outbreak of the American civil war in 1861, Maury threw in his lot with the South, and, having lost nearly his all, retired to England, where he was presented with a handsome testimonial raised by public subscription. Afterwards he became imperial commissioner of emigration to Maximilian of Mexico, on whose death he took up his residence in Virginia, where he died on February 1, 1873. In 1848 Maury published a Treatise on Navigation, which was long used as a text-book in the United States navy. The work, however, by which he is best known is his Physical Geography of the Sea, the first edition of which was published in London in 1855, and in New York in 1856; it was translated into several European languages. The theories which it contains are now generally admitted to be quite erroneous. Maury s reputation rests on the eminent services he rendered to navigation and meteorology, he having been the first to show how the latter could be raised to the certainty of a science. He was essentially a practical man ; his great aim was to render navigation more secure and economical, and in this he was eminently successful. Other works published by Maury are the papers contributed by him to the Astronomical Observations of the United States Observatory, Letters concerning Lines for Steamers crossing the Atlantic (1854), Physical Geography (1864), and Manual of Geography (1871). In 1859 he began the publication of a series of nautical monographs. MAUSOLUS, or according to his coins Maussolus (Mavo-o-coXos), a king of Caria, whose reign probably began in 377 and terminated with his death in 353 B.C. The part he took in the revolt against Artaxerxes Memnon, his con quest of Lydia, Ionia, and several of the Greek islands along the coast, his co-operation with the Rhodians and their allies in the war against Athens, and the removal of his capital from Mylasa, the ancient seat of the Carian kings, to the city of Halicarnassus are the leading facts of his history. He is best known, however, from the tomb erected for him by his widow Artemisia with such cultured magnificence that the name of mausoleum has become the generic title of all similar monuments. One of the most curious of the inscriptions discovered at Mylasa details the punishment of certain conspirators who had attempted the life of Mausolus when he was attending a festival in a temple at Labranda. See HALICARNASSUS. MAXENTIUS, MARCUS AURELIUS VALERIUS, Roman emperor from 306 to 312, was the son of Maximianus Herculius, and the son-in-law of Galerius, but on account of his vices and incapacity 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 the public discontent on October 28, 306, found expression in the massacre of those magistrates who maintained their loyalty to Severus and in the election of Maxentius to the imperial dignity, an election in which the rest of Italy, as well as Africa, concurred. 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 Maximian, and then, after achieving a military success in Africa against one Alexander, to declare war against Constantine for the conduct towards the old emperor of which he in turn had been guilty at Marseilles. The contest resulted in the defeat of Maxentius at Saxa Rubra, and his death by drowning in the Tiber at the Milvian Bridge on October 28, 312. (See CONSTANTINE.) The general testimony to the worthlessness and brutality of his character is unambiguous and unanimous ; less apparent are the grounds for the particular statement of Gibbon that he was "just, humane, and even partial towards the afflicted Christians." MAXIMA AND MINIMA. The consideration of the greatest or the least value of a variable quantity, that is restricted by certain conditions, is a problem of which several simple cases were investigated by the early Greek geometers. Thus in Euclid iii. 7, 8 we find the determina tion of the greatest and least right lines that can be drawn from a point to the circumference of a circle. But the most characteristic problem of the kind in Euclid is that contained in vi. 27, 28, 29. Thus prop. 27, when reduced to its simplest form, is equivalent to the statement that if a right line be bisected the rectangle under the segments is greater than that under those made by any other point of division. Props. 29 and 30 are, when considered algebrai cally, reducible to the solution of the equations x(a -x) = l z and x(a + x) = & 2 , coupled with the determination of the maximum value of 5 for which the solution of the former is possible (see Matthiessen, Grundzuge der antiken und modernen Algebra, Leipsic, 1878). Apollonius extended the investigation of Euclid, bk. iii., to the problem of the greatest and least distances of a point from an ellipse, showing that it depended on drawing normals from the point to the curve ; and he reduced the latter problem to finding the points of intersection of the ellipse with a certain hyperbola. 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 con tains 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 sc, the values of x for which u has a maximum or a minimum value are, in general, de termined by the equation ^ = . Again, if u be a func tion of two variables x and y, then the maximum or minimum values ofw must satisfy the equations ~r = ^ and dy . . 0. There is, however, no real maximum or minimum solution if (myj than 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,, torn. i. p. 113) erroneously attributed these theorems on isoperimetry to Pythagoras, but his statement is based on a misinterpretation of a passage in Diogenes Laertms. See
Bretschneider, Die Geometric vor Euklides, pp. 89, 90.