644 M A X M A X 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 b 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 cul minated in the invention of the calculus of variations, with an appropriate notation, by Lagrange. M AXIMI A NUS, MARCUS AURELIUS VALERIUS, surnamed HERCULIUS, Roman emperor from 286 to 305, and again in a doubtful manner for some time prior to 308, was by birth a Pannonian peasant, but achieved great distinction in the course of long service in the army in almost every quarter of the empire, and, having been made Ctcsar by Diocletian in 285, received the title of Augustus in the following year (April 1, 286) with the honorary appel lation of Herculius. In 287 he suppressed the rising of the peasants (Bagauda?) in Gaul, but in 289, after a three years struggle, his colleague and he were compelled to acquiesce in the assumption by his lieutenant Carausius of the title of Augustus in Britain. After 292. Maximian left the care of the Rhine frontier to Constantius Chlorus, who had been designated Caesar in that year, but in 297 his arms achieved a rapid and decisive victory over the barbarians of Mauretania, and in November 303 he shared at Rome the triumph of Diocletian, the last pageant of the kind ever witnessed by that city. On May 1, 305, the day of Diocletian s abdication, he also, but without his colleague s sincerity, divested himself of the imperial dignity at Milan, which had been his capital, and retired to a villa in Lucania ; in the following year, however, he was induced by his son Maxentius to reassume the purple. In 307 he brought the emperor Severus a captive to Rome, and also compelled the retreat of Galerius, but in 308 he was himself driven by Maxentius from Italy into Illyricum, whence again he was compelled to seek refuge at Aries, the court of his son-in-law Constantino. Here a false report was received, or invented, of the death of Con- stantine, at that time absent on the Rhine, and Maximian at once grasped at the succession, but was soon driven to Marseilles, where, having been delivered up to his pursuers, he strangled himself in 310 (February). MAXIMIANUS, GALERIUS VALERIUS, usually referred to by his name GALERIUS, Roman emperor from 305 to 311, was born near Sardica in Dacia, and originally followed his father s occupation, that of a herdsman, whence his surname of Armentarius. He served with distinction as a soldier under Aurelian and Probus, and in 292 was designated Ca?sar along with Constantius, receiving in marriage Diocletian s daughter Valeria, and at the same time having assigned to him as his special charge the care of the Illyrian provinces. In 296, at the beginning of the Persian war, he was removed from the Danube to the Euphrates ; his first campaign ended in a crushing defeat on the same field as that which had proved fatal to Crassus, but in 297, advancing through the mountains of Armenia, and taking the enemy by surprise, he gained a victory over Narses by which his military reputation was more than restored. In 305, on the abdication of Diocletian and Maximian, he at once assumed the title of Augustus, along with Constantius his former colleague, and, having pro cured the promotion to the rank of Ca?sar of Severus, a faithful servant, and Daza (Maximinus), his nephew, he hoped on the death of Constantius to become sole master of the Roman world. This scheme, however, was defeated by the sudden elevation of Constantino at York on the death of his father, and by the action of Maximian and Maxentius in Italy. After an unsuccessful invasion of Italy in 307 he elevated his friend Licinius to the rank of Augustus, and, moderating his ambition, devoted the few remaining years of his life " to the enjoyment of pleasure and to the execution of some works of public utility." He died of the mot-bus pedicularis, it is said in May 311. It was at the instance of Galerius that the first of the celebrated edicts of persecution against the Christians was published, on February 24, 303, and this policy of repres sion was maintained by him until the appearance of the general edict of toleration, running in his own name and in those of Licinius and Constantino. MAXIMILIAN I. (1459-1519), holy Roman emperor, the son of the emperor Frederick III., was born on the 22d of March 1459. In 1477 he married Mary, daughter of Charles the Bold of Burgundy, thus securing for his family the possessions of the house of Burgundy ; and by the mar riage of his son Philip with the infanta Joanna in 1496 he prepared the wayfor the association of Spain with the empire under his grandson, Charles V. In 1486 Maximilian was chosen king of the Romans, and in 1493, after the death of his father, he succeeded to the imperial throne. During the reign of Frederick III. the system of private war created profound discontent, and there were urgent demands for the reform of imperial institutions. Maximilian was never thoroughly in sympathy with this movement, but at his first diet, in 1495, he declared a perpetual public peace ; and he did something for the restoration of order by the establish ment, in the same year, of the imperial chamber (Heichskam- mergericld}, and, in 1501, of the imperial Aulic council (Reichshofmtli). Another important change was the division of Germany into six, afterwards (in 1512) into ten, circles (Kreise), over each of which was placed a captain, with a force for the punishment of disturbers of the peace. Standing troops, called Landsknechte, were for the first time organized by Maximilian, who also improved the artillery then in use, and issued good police regula tions. He encouraged science, art, and literature, devoted much attention to the universities, especially those of Vienna and Ingolstadt, collected medieval poems, and caused copies to be made of ancient chronicles and other important manuscripts. Through the influence of his second wife, Blanca Sforza, daughter of Duke Galeazzo Sforza of Milan, he was induced to contend for supremacy in Milan and Naples ; but his resources were inadequate for war on equal terms with the kings of France, Charles VIII. and Louis XII. In 1499 he carried on an unsuc cessful war with the Swiss confederates, the result of which was that, by the peace of Basel, the confederates became practically independent of the empire. On the other hand, he was singularly fortunate in increasing the power of the house of Austria. By the death of his cousin, the arch duke Sigismund, he inherited Tyrol ; lie also received Gorz, Gradisca, the Pusterthal, and a part of Bavaria; and by the marriage of two of his grandchildren with children of the king of Hungary and Bohemia he took the first step towards the ultimate incorporation of these countries with the Austrian hereditary territories. He wrote several books, and planned the " Weiss-Kunig," a kind of poetical autobiography, completed by his private secretary, Treizsaurwein von Erentreiz. Maximilian had
some part also in the preparation of Theuerdank, an