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302 ALGEBRA ALGERIA gave the solution of equations of the third de- gree, by an operation which is still known among all mathematicians as the formula of Cardan ; those of the fourth degree were solved by his pupil Ludovico Ferrari, and published in the Ars Magna, in which also he makes the distinc- tion between positive, negative, and irrational solutions. At the same time Stifelius in Germany invented the signs +, , and -y, which did so much to simplify the formulas ; he published his Arithmetica Integra in Nuremberg in 1544. In 1552 Robert Recorde published in England "The Whetstone of Witte," in which for the first time the sign of equality (=) is intro- duced. From that time not much progress was made till Vieta in France perfected the alge- braic operations and transformations of for- mulas, and even advanced so far as the general solution of equations of all degrees. He first applied algebra to geometry, and he also found the remarkable expression which solved nu- merically the problem of the quadrature of the circle. His works were written about the year 1600, but only published long after his death, by Schooten. Among the eminent mathema- ticians of that time we must also mention Ge- rard in Flanders, who was the first to indicate the important use of the negative roots of equations in geometrical constructions, while in England Harriot introduced the signs > and <, and Oughtred first wrote the decimal frac- tions simply by the decimal point, as we do now, without writing the denominator always, as was customary till his time. The 17th cen- tury was the most brilliant of all centuries in mathematical discoveries, producing the im- mortal Descartes, Fermat, Wallis, Galileo, Huyghens, Kepler, Newton, Leibnitz, Ber- noulli, and many others not less illustrious; and that century closed with the important discovery of the logarithms and of the dif- ferential calculus. The 18th century enriched the vast domain transmitted, and men like Laplace, La Grange, D'Alembert, Maupertuis, Maclaurin, "Waring, Lambert, Cutler, Stirling, De Moivre, and above all Euler, developed and perfected all the branches of the science. The operations of algebra are founded on a mutual agreement concerning signs and sym- bols. The first letters of the alphabet, a, &, c, fec., are used to represent known quantities, whether of space, time, or number, and the last, z, y, a*, &c., are used for the unknown quantities. They are connected by the signs + , , x , and -:-, meaning respectively addition, subtraction, multiplication, and division. The powers of quantities are expressed by superior numbers, as a" for a x a, a 6 for a x a x a x a x a ; the roots by the sign ^/, or J and ^/, &c. The small space in which a long operation can be indicated by these signs may be illustrated by the following algebraic expression : 17ax(c (Q' + c'x i/(g*& a ) a Xc 4/3aft V(Jxc) (a &)x(c d) + acd 8a* which is an ordinary expression involving so many operations that to describe them clearly would occupy a whole page. ALGECIRAS, a seaport and town of Spain, in Andalusia, province of Cadiz, on the W. side of Gibraltar bay, opposite and 6 m. W. of Gib- raltar; pop. 18,000. Charles III. rebuilt it in 1760, as a point of annoyance against Gibraltar. It is constructed of stone, and presents a fair appearance compared with many of the smaller Spanish towns. The port is guarded by a bat- tery called the Fuerte de Santiago. The town is supplied with water conveyed by an aque- duct over the Miel. The principal trade is the export of coal from the neighboring mountains, charcoal, and leather. Two considerable sea fights took place off Algeciras in July, 1801, between the English and French squadrons, and in the second the English were victorious. ALGERBA, the third star in the constellation of Leo. It is a noted star among astronomers, being used as a test for telescopes, which prove it to be double. One of its constituents is orange, the other green. M.I.I K. 1 1 (i ratio. Jr., an American author, born at Revere, near Boston, Mass., Jan. 13, 1834. He graduated at Harvard college in 1852, and was afterward engaged partly in teach- ing and partly in writing, being for a time editorially connected with two Boston news- papers. He then spent a year in travel in Europe, corresponding with American papers. Upon his return he resumed teaching and writing. In 1866 he took up his residence in New York, where he became deeply inter- ested in the condition of the street boys. This has given form to most of his later writ- ings. Prominent among these are the "Rag- ged Dick " series and the " Tattered Tom " series. With the exception of " Helen Ford, a Novel," and various magazine papers, Mr. Algcr's writings belong mainly to the class of books for the young. ALGER, William Rounseyllle, an American clergyman and author, cousin of the preceding, born at Freetown, Mass., in 1823. He gradu- ated at Harvard college in 1847, studied for the ministry, and became pastor of a Unitarian church in Roxbury. In 1 855 he removed to Bos- ton, where he succeeded Theodore Parker as minister of the " Liberal Christians " who wor- ship in Music Hall, Boston, where he still contin- ues to preach (1873). His works comprise " The Poetry of the Orient, or Metrical Specimens of the Thought, Sentiment, and Fancy of the East" (1856); "A Critical History of the Doctrine of a Future Life" (1861); "The Genius of Solitude, or the Loneliness of Hu- man Life" (1867); and " Friendships of Wo- men " (1870). ALGERIA, a division of N. Africa, formerly the Turkish pashalic of Algiers, but since 1831 included in the foreign dominions of France, bounded N. by the Mediterranean, E. by Tunis, W. by Morocco, S. by the Great Sahara. It is, in the main, situated between lat. 32 and 37 N., and Ion. 2 W. and 9 E. The bounda-