MAX. 206 MAXIMA AND MINIMA. choly Kun" (1S09, Hanil)urc; Gallery) and "The Anatomist" (1S09) — both strikingly somhre and pathetic, and illiistrative of the painter's mor- [lid tendencies. Patient sutTering is depicted in '"The Blind Lamp Seller in the Catacombs" (1871) ; in "Xydia," the blind Thessalian flower girl of Buhver's Last Dai/s of Pompeii (1874). One of his most touching pictures is "The Last Token" ( 1S74, Jletropolitan Museum, New York). The tragic element is best represented by "The Lion's Bride" (187.5). probal)Iy his best- known work. The "Handkerchief of Saint Ver- onica" I 1874) created a sensation as a pictorial phenomenon, the Saviour's eyes appearing by turns to open and close. His spiritualistic ten- dencies are revealed in such paintings as "Spirit Greeting" (1879). Consult: K]pmt. dnhrirl Max Mild seine Werke (Vienna, 1SS7). and Meissner in Die h'loisl tinserer Zeit (ilunich, 1899). MAXENTIUS, maks-en'shi-us. Roman Em- peror, .'-^ec ( (PNST.^NTI.XE . MAXENTIUS, Ciiicu.s of. A circus on the Appian Way near Rome dedicated to Divus Romulus, son of Maxcntius, who died in A.i). .'509. It was .'?oO yai'ds by 8(i and it accommodated 18,000 spectators. The barriers and spina were .set oblir|Ucly to equalize the distance to be covered by contestants starting in different positions. MAX'EY, Samuel Bell (1825-9.5). An American soldier, born in Tompkinsville. Ky. He chevalier of the Legion of Honor, and was created a knight by the English Crown in 1901. MAXIM, HUDSO.N- (18,53—). An American invcnlor and engineer, born at Orneville, Me. He was educated in the local schools and at the Wesleyan Seminary, Kent's Hill, Me., and de- voted himself to the study of chemistry, engineer- ing, and natural science", at the age "of twenty- two formulating a hypothesis of the compound nature of atoms. From 1883 to 1888 he was engaged in the printing and publishing business at Pittsfiehl, Mass., devising a process for print- ing daily papers in colors. In 1888 he became interested in ordnance and explosives, and was among the first to make smokeless powder in the United States. Extending this business he developed the Maxim-Schupphaus smokeless pow- der, which was used by the United States Gov- ernment; in 1901 he sold to that same Government the secret of the high explosive, maximite (q,v. ). Later inventions include various processes con- nected with the electric furnace, a detonating fuze for high-explosive projectiles, automobile torpedoes, and "stal)illite," a smokeless powder invented by Iiini ami di'vcloped by him in connec- tion with the K. I. (hi I'niil lie Xcniours Company. MAXIMA AND MINIMA (Lat., neut. pi. of maximus, greatest, and minimus, least). In mathematics, the greatest and the least values of variable ipiantities or magnitudes. Strictly, a graduated at the United States Jlilitarv Academy maximum is not necessarily the gi'eatest of all
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in 184ti. and served throughout the Mexican War. In 1850 he began to practice law, but at the outbreak of the Civil War raised a Confederate regiment, liecoming its colonel: soon he was promoted brigadier-general. He served under General Bragg, took part in the attack on Buell's retreating army, and was at the first siege of Port Hu<lsnn. In 1803, when in command of the Indian Ti'rritory, he organized an army of 8000, defeated (ien. Frederick Steele, and captured his train of 227 wagons. For these services he was promoted major-general. In 1874 he was elected I'nited States Senator, and in 1881 reelected. MAX'IM, Sir Hiram Stevens (1840—). An engineer and inventor, born in Sangersville, Maine, where he received his early education. After being apprenticed to a coach-builder, he worked in a machine shop at Fitcbburg, JIass., later becoming foreman of an instrviment fac- tory. After this be worked at the Novelty Iron Works and Shipbuilding Company in New York. Mciinwhile he bad patented various improve- ments in steam-engines and had put on the mar- ket an automatic g:is machine. In 1878 he invented an improved incandescent lamp. In this field he made other important inventions, some of which wero exhibited at the Paris Ex- position of 1881. His most celebrateil invention was the Maxim gun (see Maciiixe Gfx), in which invention he is said to have developed an iilea of his boyhood. More than one- hundred international patents relating to petroleum and olher motors, and so on, were taken out by liini. He bought an estate near Bexley. England, where he erected his laboratories and experimenl;>l sta- tion, in which he carried on some elaborate experiments in aeronautics (q.v,). He beenmc a naturalized citizen of Great Britain because of the alleged unfair treatment of his inventions by the United States Government. He was made a the possible values of a variable; it is a value that is greater than the values immediateli/ pre- ceding and following it in series. Similarly, a minimum, strictly defined, is a value that is less than the values immediately preceding and fol- lowing it. Hence a function may have several maxima and minima, equal or unequal among themselves. Thus, in the accompanying figure, a,, Oo, U3, are maxinuim values of the ordinates of f [X), and i,, /)„, 63, are minimum values. The tangent of the angle which a line tangent at any point to the curve makes with the X-axis is zero at a maximum or minimum value of the ordin- ate. This means that the difTerential coellicient dy dx =: (see CALCin,us), and hence the abscis- sas corresponding to the maxima and minima are the roots of -r- = 0, dx Y A function of two ndependent variables, f(x, 1/), has a maximum value when ftx, y) > fix -f h, 1/ + A-), for all small values of h and k, positive or negative; and a minimum value when fix, y) < f{x -- h. y+ k) . The condi- tions for maxima and minima in the case of a function u of two variables are -^r- = 0, and 8-/ the = 0. If A further conditions S" = for -^ Sx -■ '"■'• , and C = SxSi/ 5--1 a maximum arc
