TANGENT TANNAHILL. 



521 



cred's tower. During the scenes of horror which 

 attended the capture of Jerusalem (July 19, 1099), 

 he conducted himself with humanity, and saved the 

 lives of thousands of the enemy, at the peril of his 

 own. For this he was accused of being an enemy 

 to the priests and to religion ! The sultan of 

 Egypt was now advancing to attempt tha recovery 

 of Jerusalem, but was totally defeated by Tancred", 

 with the loss of his camp, before Ascalon (August 

 12). Tancred captured Tiberias, besieged Jaffa, 

 and, after the death of Godfrey, endeavoured to 

 effect the election of Bohemond as king of Jerusalem ; 

 but the unworthy Baldwin obtained the throne, and 

 Tancred, while engaged in the field against the emir 

 of Damascus, was summoned to appear before the 

 new king, on a charge of rebellion. But, secure 

 in the attachment of his vassals, Tancred, now 

 prince of Galilee, despised the base arts of Baldwin, 

 and hastened to Antioch, whose prince, Bohemond, 

 had been captured by the Turks. The city was 

 equally threatened by the Turks and the false 

 Greeks ; but Tancred alternately made head against 

 both, restored his friend to liberty, and, with the 

 utmost disinterestedness, gave him back his terri- 

 tories. When Bohemond returned to Europe to 

 obtain recruits, Tancred was left to protect An- 

 tioch, which was menaced at once from Aleppo and 

 by the Greek armies. He was even obliged to en- 

 counter the attacks of Baldwin, count of Edessa, 

 and Josselin de Courtenay. Bohemond died at 

 Salerno, and his soldiers either dispersed or entered 

 the service of the Greek emperor : still Tancred 

 succeeded in forcing the Turkish sultan to retreat 

 over the Euphrates. This was his last exploit. 

 He died soon after, in 1112, in his thirty-fifth year. 

 Tancred was the flower and pattern of chivalry. 



Tasso has immortalized him An account of his 



life may be found in Raoul de Caen's Gestes de 

 Tancrede, and in Delabarre's Histoire de Tancrede 

 (Paris, 1822). 



TANGENT, in general ; every straight line 

 which has one single point in common with, and 

 lies entirely outside of, a curve (at least of every 

 such curve as can be cut by a straight line in two 

 points only). This is the geometrical tangent. In 

 trigonometry, the name is applied particularly to 

 that part of the tangent to the circle which stands 

 perpendicular at the end of one of the radii, includ- 

 ing a particular arc, and is cut by the prolonged 

 radius passing through the other end of the arc (the 

 second). Trigonometrical tangents, used with the 

 sine and cosine, &c., for the solution of triangles 

 (see Trigonometry}, have been calculated according 

 to their relative value (i. e. with reference to a 

 radius of a certain magnitude) for every arc ; and 

 these relative values, or their logarithms, are gene- 

 rally to be found in the trigonometrical tables, with 

 the sines and cosines of the same arcs. How this 

 calculation of trigonometrical tangents, in reference 

 to sines, cosines and radii, is performed, may be 

 easily understood by a mere comparison of the two 

 similar triangles which originate when we draw 

 these lines and the corresponding arc. The differ- 

 ential calculus gives a very simple method for cal- 

 culating the tangents by means of the subtangents, 

 under the name of the direct method of the tangents. 

 To this direct method the higher analysis adds an 

 inverted method, called the inverse method of tan- 

 gents. 



Tangential Force. In order to have a clear idea 

 how the planets are made to revolve in consequence 

 of the attraction which the sun, situated in one 



focus of their elliptical orbits, exercises upon them, 

 we may imagine that they originally received an 

 impulse urging them forward in a straight line. 

 With this impulse the attraction of the sun (cen- 

 tripetal force ; see Central Forces') being united, 

 the planet was thus made to describe the diagonal 

 of a parallelogram, whose sides represent the direc- 

 tions of these forces. As there is nothing to 

 diminish the impulse which we have supposed ori- 

 ginally given to the planet, it would continue its 

 path in the direction of the diagonal ; but the cen- 

 tripetal force, operating continually upon the direc- 

 tion which the planet has obtained, makes it change 

 its direction incessantly. In this way originates 

 (as a diagram, drawn according to what we have 

 said, clearly shows) a motion around the centre of 

 forces. (See Circular Motion, and Central Forces.) 

 The planet has at each point of its path a certain 

 tendency (the consequence of its previous motion ; 

 hence, properly speaking, the effect of its inertness) 

 to continue its last received diagonal direction, and 

 thus to recede from the centre of forces. To this 

 tendency, the centripetal force, directed towards 

 this point, is opposed. The centripetal force may 

 again be divided into two forces, the first of which 

 (the normal force) operates perpendicularly to the 

 orbit, and only contributes to retain the planet in 

 the same, in order to prevent the curved motion 

 from degenerating into a straight one : the latter, 

 however, coincides with the direction of the orbit 

 itself, and, therefore, only affects the velocity. 

 This latter force is the tangential force, so called 

 because the element of the curve coincides with the 

 tangent. The doctrine of central forces is so im- 

 portant, because our imagination, unaided by theory, 

 is almost incapable of conceiving a body which turns 

 around another, exercising an attraction upon it, 

 yet without ever coming in contact with the at- 

 tracting body. But what has been said shows that 

 a correct proportion of the centripetal force to the 

 original impulse, renders the contact of the body 

 with the sun impossible. Generally, the endeavour 

 of the planet to recede from the centre of forces, 

 is called the centrifugal force ; but can we, proper- 

 ly, call that a force which is evidently the effect of 

 inertness ? The original impulse may be compared 

 to the first impulse which sets the pendulum in 

 motion ; after which, if we omit other influences, 

 it would continue its oscillations for eternity, from 

 the mere influence of gravity. 



TANGIER, OB TANJAH (anciently Tingis) ; 

 a town of Morocco, situated at the west entrance 

 of the straits of Gibraltar, thirty-eight miles south- 

 west of Gibraltar; Ion. 5 50 7 W.; lat. 35 48' N. 

 The population is about 7000. Tangier was pos- 

 sessed by the British from 1662 to 1784. It after- 

 wards became a distinguished station for piracy; 

 but the disuse of this practice in Morocco has di- 

 minished the importance of the town. It now sub- 

 sists chiefly by supplying the British garrison of 

 Gibraltar with cattle and vegetables. The bay of 

 Tangier is not safe when the wind is in the west, 

 having been encumbered by the ruins of the mole 

 and fortification ; the cables are liable to be torn, 

 and the ships to be driven on shore. Tangier, 

 viewed from the sea, presents a pretty regular as- 

 pect; but within it exhibits the most disgusting 

 wretchedness. It is the residence of the European 

 and American consuls. 



TANNAHILL, ROBERT, a very popular writer 

 of Scottish songs, was born in Paisley, on the 3d 

 of June, 1774, of parents, who, though in humble 



