MAXIMA AND MINIMA. 207 MAXIMIANUS II. B- ■< AC and A •< 0, and for a minimum B' < AG and A > 0. When B= = AC or A = B = C = 0, further investigation is necessary. A few of the important propositions of ph c raa.xima and minima are; (1) Of all triangles formed with the same two given sides, that is the nia.ximum whose sides contain a right angle; (2) of all isoperimetric triangles (those of equal perimeters) on the. same base, the isosceles is the ma.imum ; (3) of all isoperimetric triangles, that which is equilateral is the ma.ximum; (4) of all triangles having the same base and area, tile isosceles has the mininmm perimeter; (5) if a line of given length be bent and its ends joined by a straight line, the area of the figure inclosed is a maximum when the curved line has the form of a semicircle; (0) of all isoperi- metric plane figures, the maximum is a circle ; (7) of all isoperimetric polygons of a given num- ber of sides, the maximum is regular. Traces of the doctrine of ma.xima and minima are to be found in the works of Apollonius on conic sections, and among the theorems of Zeno- dorus. The Hindus displayed great ingenuity in solving, by ordinary algebra, problems of maxima and minima ; but thorough investigation of the subject requires the aid of the calculus, and Kepler, the Bernoulli brothers, Newton, -Maclaurin, Euler, and Lagrange distinguished themselves in this department. See CALCULUS. MAXIM GUN. See M.chi.ne Gun. MAXTMIA'NUS. A Latin elegiac poet of the sixth century a.d. The author, whose name may be merely borrowed from a character in the fourth of the six elegies, deals in a dramatic and realistic way with variations on the theme 'love and old age.' His diction and metre are far above the average of the period. The elegies were pojiular during the Jliddle Ages and were frequently imitated. They were translated in part in several early English manuscripts and entire by H. V[alUer?], (1089), who. like sev- eral early editors, ascribed them to Cornelius Callus; and were edited by Petschenig (1890), and by Richard Webster with commentary (Princeton, 1900). MAXIMIANUS I. A Roman Emperor. See DURLETIAN. MAXIMIANUS II. A name sometimes ap- plied to aierius Maximianus Galerius (q.v.), a Roman emperor. MAX'IMIL'IAN I. (14.591519). Holy Ro- man Emperor from 1493 to 1.519. He was the son and successor of Frederick III., and was born at Wiener-Xeustadt, near Vienna, JIarch 22, 1459. His fir.st wife, whom he married in 1477, was Mary of Burgundy, daughter and sole heiress of Charles the Bold (q.v.), Duke of Burgundy and sovereign of the Xetherlands. ^Maximilian became at once involved in war with Louis XI. of France, who laid claim to Burgundy and other parts of Mary's inheritance. Max- imilian won the battle of Guinegate in 1479, but was finally compelled in 14S2 to conclude the Treaty of Arras with Loiiis, who retained Bur- gimdy, Artois, and Franche-Comti'. The bulk of the Xetherlands, the 7nost opulent realm in Eu- rope, remained with the Hapsburgs. In I486 Maximilian was elected King of the Romans. In 1488 the Flemings rose against Maximilian, who was for a time held a jirisoner at Bruges. In 1490 he nuide a successful incursion into Hungary and soon after became master of the Austrian crownlands. He again took up arms against France, whose King, Charles VIII., had married Anne of Brittany, in order to acquire that great duchy, after a matrimonial alliance had been concluded between that princess and Maximilian. B}' the Treaty of Senlis, in 149;i, .Maximilian recovered Artois and Franche-Corate. In the same year he succeeded his father on the im- perial throne of Germany. Soon afterwards, Mary of Burgundy having died in 1482, he mar- ried Bianca, a daughter of the late Duke of Milan. Galeazzo Sforza, and thus was involved subsequently in the Italian wars. He joined the League of Cambrai against Venice in 1508 and the Holy League against France in 1513, and after Francis I.'s victory at Mclegnano (1515) was forced to cede Milan to the French. Xor was Maximilian more successful against the Swiss, who in 1499 completely separated themselves from the German Empire. By the marriage of Philip, the son of Maximilian, with the Infanta Joan, daughter of Ferdinand and Isabella, the House of Hapsburg, in 1516, ascended the throne of Spain in the person of Oiarles I. (ila.ximil- ian's successor in the Empire as Charles V.). The marriage of two of Maximilian's gi-andchil- dren with the son and daughter of Ladislas, King iif Hungary and Bohemia, ultimately brought both these kingdoms to the Austrian monarchy, ilaximilian died at Wels, in L'pper Austria, •January 12, 1519. As an administrator Max- imilian sought to strengthen the organization of the Empire. The evil of private warfare was partially remedied by the declaration of a per- petual peace for the Empire at Worms in 1495, and for the better maintenance of the peace a division of the Empire was made into circles. Reforms in the administration were introduced by the establishment of the Imperial Chamber and the Aulic Council (qq.v.). Maximilian was an ardent humanist and a zealous patron of art and learning. He wrote various works on the art of war, hunting, gardening, etc., and sketched the Wctsskunifi, an autobiographic poem. His frank and generous nature, his love for outdoor sports, and his many accomplishments gained him the title of 'Last of the Knights.' Consult: Heyck, Kaiser Maximilian I. (Bielefeld, 1898). MAXIMILIAN II. (1527-76). Holy Ro- man Emperor from 1564 to 1576. He was the eldest son of the Emperor Ferdinand I., and was born in Vienna, July 31, 1527. He was educated in Spain with his cousin, Philip II. In 1548 he married his cousin Maria, daughter of Cliarles V. Xotwithstanding the surround- ings in which he had been educated, his attitude toward the Reformation was so libera! that he was even considered by those about him to be at heart a heretic. The Protestants of Germany expected much from hia accession, even hoping that they might have a Protestant Emperor of the House of Hapsburg. These hopes re- mained unfulfilled. Lack of decison, dynastic policy, his personal ambitions, all united to keep Maximilian in his allegiance to the Church. He
