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 determination 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 algebraically, reducible to the solution of the equations x(a -x) = b 2 and x(a + x') = δ 2 , coupled with the determination of the maximum value of b for which the solution of the former is possible (see Matthiessen, Grundzüge 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....

MAXIMIANUS, Marcus Aurelius Valerius, surnamed Πerculius, 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 Caesar by Diocletian in 285, received the title of Augustus in the following year (April 1, 286) with the honorary appellation of Herculius. In 287 he suppressed the rising of the peasants (Bagaudae) 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 ids 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 Arles, the court of his son-in-law Constantine. Here a false report was received, or invented, of the death of Constantine, at that time absent on the Rhine, and Maximian at once grasped at the succession, but was soon driven ...

MAXIMILIAN I. (1459-1519), holy Roman emperor, the son of the emperor Frederick HI., 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 marriage 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 establishment, in the same year, of the imperial chamber (Reichskammergericht), and, in 1501, of the imperial Aulic council (Reichshofrath). 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 regulations. He encouraged science, art, and literature, devoted much attention to the universities, especially those of Vienna and Ingolstadt, collected mediaeval 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 ...

MAXIMILIAN II. (1527-1576), holy Roman emperor, son and successor of Ferdinand I., was born at Vienna on the 1st of August 1527. He was of a mild and tolerant disposition, and in youth received a favourable impression of Protestantism from his tutor, Wolfgang Severus,—an impression which was not effaced by a residence of three years at the Spanish court. In 1562 he became king of Bohemia and king of the Romans, in 1563 king of Hungary, and in 1564 emperor. At the time of his accession to the imperial throne Hungary was at war with Turkey. The sultan Soliman II. was conciliated by the cession of all the territories he had conquered in Hungary, and by the promise of a yearly tribute of 300,000 florins. Soon afterwards Soliman renewed the war on behalf of the prince of Transylvania; but after his death in 1566 his successor, Selim, concluded with Maximilian an armistice of eight years. Maximilian’s brothers, Ferdinand and Charles, fought incessantly against Protestantism in their respective lands; but it was tolerated by Maximilian in Austria, Bohemia, and Hungary. His authority, however, was greatly limited by the influence of the Jesuits. He died on the 12th of October 1576. Of his eight children (six sons and two daughters) two—Rudolph II. and Matthias—became emperors.

See Koch, Quellen zur Geschichte des Kaisers Maximilian II., 1857-61; and Wertheimer, Zur Geschichte des Türkenkriegs Maximilians II., 1875....

MAXIMINUS, CaiusJulius Verus, Roman emperor from 235 to 238, was of barbarian parentage, his father being a Goth and his mother an Alan, and was born in a village on the confines of Thrace, where his immense stature and enormous feats of strength first drew the attention of the emperor Septimius Severus. From being a shepherd he became a soldier, and under Caracalla rose to the rank of centurion. He carefully absented himself from court during the reign of Elagabalus, but rose to the first military command under his successor Alexander Severus. On March 19, 235, the troops saluted him emperor, and shortly afterwards Alexander was put to death. The three years of his reign, which were spent wholly~in the camp, were marked by great cruelty and oppression; the widespread discontent thus produced culminated in a revolt in Africa and the assumption of the purple by Gordianus (q.v.'). Maximin, who was in Pannonia at the time, marched against Rome, and passing over the Julian Alps descended on Aquileia; while detained before that city he and his son were murdered in their tent by a body of praetorians. Their heads were cut off and despatched to Rome, where they were burnt on the Campus Martius by the exultant crowd (May 238)....

MAXIMUS, the name of four Roman emperors. In chronological order the first was Μ. Clodius Pupienus Maximus, who was associated with Balbinus in the imperial dignity by the senate for a short time in 238, before and after the death of the hated Maximin. Tho second was Magnus Clemens Maximus, a native of Spain, who shared the imperial dignity with Valentinian and Theodosius from 383 to 388. He had accompanied Theodosius on several expeditions, and from 368 held high military rank in Britain. The disaffection of the Roman troops towards the emperor Gratian found expression in 383 in the proclamation of Maximus as emperor, —whether with or without his complicity in the act is uncertain. Voluntarily or under compulsion Maximus forthwith attacked Gratian in Gaul, and drove him from Paris to Lyons, where the fugitive was murdered. Circumstances made it difficult for Theodosius at the time to avenge the death of his colleague by war, and an agreement was therefore come to by which Maximus was recognized as Augustus and sole emperor in Gaul, Spain, and Britain, while Valentinian was to remain unmolested in Italy and Illyricum, Theodosius retaining his sovereignty in the East. A prosperous reign of four years having tempted Maximus, in 387, to pass the Alps, Valentinian was speedily put to flight, while the invader established himself in Milan, and for the time became master of Italy. Theodosius now took vigorous measures: advancing a powerful army by land, he utterly defeated the western, troops at Siscia (Sciszek) in Pannonia, and, passing the Julian Alps with great rapidity, came upon Maximus, who had fled to Aquileia, seized him, and caused him to be beheaded (August 388). The third, Maximus Tyrannus, was made emperor in Spain by the Roman general Gerontius, who had ...