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LEFT TANCRED 243 TANGENT nople, whence Tancred crossed the Bos- porus disguised as a common soldier, that so he might escape from swearing allegiance to the emperor. Alexios fol- lowed the Crusaders, and Tancred re- luctantly, yielding to Bohemond's coun- sel, took the required oath. At Dory- laion (July 4, 1097) his bravery saved the camp of priests and women; his ban- ner was the first to float from the towers of Tarsus, though Baldwin's jealousy dislodged it thence. In the siege of An- tioch he slew, say chroniclers, 700 infi- dels; with Robert of Normandy he first set foot in the Holy City July 15, 1099. Appointed by Godfrey de Bouillon prince of Galilee, he founded churches in Naz- areth, in Tiberias, and on Mount Tabor, and helped at Ascalon to guard the new Christian kingdom against the Fatimite caliph. His efforts on Godfrey's death (1100) to secure the crown of Jerusalem to Bohemond only roused Baldwin's jealousy again; but his own principality he held successfully against both Turks and Greeks, even Edessa owning his su- premacy. He was busy with plans for bringing the Syrian chieftains under his sway, when he died in Antioch of a wound received in battle (1112). TANDAH, a town in the district of Fyzabad, Oude, British India; 3 miles from the Ghogra river; 100 miles S. E. of Lucknow. It is the seat of the largest weaving colony in the province, and manufactures both coarse cloth and fine muslin. TANEY, ROGER BROOKE, an Ameri- can statesman; born in Calvert co., Md., March 17, 1777; was graduated at Dick- inson College in 1795; admitted to the bar in 1799, and elected to the house of delegates in the autumn of the same year. During the war with Great Brit- ain he led the wing of the Federal party that upheld the policy of the govern- ment. In 1816 he was sent to the State senate; in 1827 became attorney-general of Maryland and in December, 1831, suc- ceeded John M. Berrien as attorney-gen- eral of the United States. He was ap- pointed Secretary of the Treasury under President Jackson on Sept. 24, 1833, but was forced to resign the next year, owing to his action with regard to the removal of the treasury deposits. On Dec. 26, 1835, however, he was nominated Chief- Justice of the United States and con- firmed by the United States Senate on March 15, 1836. While in this office he rendered decisions on many important cases, notably those of Dred Scott, and Sherman M. Booth, both bearing on the Fugitive Slave Law. He died in "Wash- ington, D. C, Oct. 12, 1864. TANGANYIKA (-ye'ka), a lake of Central Africa, to the S. of Lake Albert Nyanza. It extends from about lat. 3" 25' to 8° 40' S., and from Ion. 29° 20' to 32° 20' E. It is 420 miles long, has an average breadth of about 30 miles, and is 2,700 feet above the level of the sea. The basin in which it lies is inclosed by an almost continuous series of hills and mountains. It is fed by numerous rivers and streamlets, and discharges by the river Lukuga into the Kongo. There are several London Missionary Society stations on Tanganyika, and on the E. shore is the Arab town of Ujiji. A car- riage road, 210 miles, runs to Nyassa. Tanganyika was discovered by Speke and Burton in 1858. Since the Peace Treaty of 1919 it belongs to the British post of the "Department of Tanganyika," shared with Belgium. The lake was in the war zone in 1916-1918. TANGENT, in geometry, a straight line which meets or touches a circle or curve in one point, and which, being pro- duced, will not cut it. In Euclid (III. 16, Cor.) it is proved that any line drawn at right angles to the diameter of a circle at its extremity is a tangent to the circle. In trigonometry, the tangent of an arc or angle is a straight line, touching the circle of which the arc is a part at one extremity of the arc, and meeting the diameter passing through the other ex- tremity; or it is that portion of a tan- gent drawn at the first extremity of an arc, and limited by a secant drawn through the second extremity. The tan- gent is always drawn through the ini- tial extremity of the arc, and is reck- oned positive upward, and consequently, negative downward. The tangent of an arc or angle is also the tangent of its supplement. The arc and its tangent have always a certain relation to each other, and when the one is given in parts of the radius, the other can always be computed by means of an infinite series. Tables of tangents for every arc from 0° to 99°, as well as of sines, cosines, etc., are computed and formed into tables for trigonometrical purposes. Two curves are tangent to each other at a common point, when they have a' common rectilinear tangent at this point. A tan- gent plane to a curved surface is the limit of all secant planes to the surface through the point. The point is called the point of contact. Two surfaces are tangent to each other when they have, at least, one point in common; through which, if any number of planes be passed, the sections cut out by each plane will be tangent to each other at the point. This point is called the point of contact. Another definition is this: Two surfaces