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TBIESTE 507 TBIGONOKETBY Theresa. Since 1816 Trieste has borne the title of the "Most Loyal of Towns." Charles Lever and Sir Richard Burton were consuls there. The city suffered heavily in the World War from bombard- ment. In 1918 it was united to Italy. TRIGGS, OSCAR LOVELL, an Amer- ican author; born in Greenwood, 111., Oct. 2, 1865; was graduated at the Uni- versity of Minneasota in 1889, and then studied in the Universities of Oxford and Berlin. In 1892 he became connec- ted with the English Department of the University of Chicago He was the au- thor of "Browning and Whitman: a Study in Democracy" (1893) ; "The Changing Order" (1901) ; and many re- view articles. He also edited Lydgate's "Assembly of Gods" (1895); and "Se- lections from Prose and Poetry of Wal- ter Whitman" (1898). TRIGONOMETRY, o r i g i n a 11 y the branch of geometry which had to do with the measurement of plain triangles. This gradually resolved itself into the in- vestigation of the relations between the angles of the triangle, for the simple reason that all triangles having the same set of angles are similar, so that if, in addition, one side is given the other two at once follow. It is easy to show from the Sixth Book of Euclid that, if we fix the values of the angles of a triangle, the ratio of the sides containing any one of these angles is the same whatever be the size of the triangle. This ratio is a definite function of the angles; and it is with the properties of such ratios that trigonometry has now to deal. The fundamental ratios are obtained from a Tight-angled triangle, of which one angle is the angle under consideration. It will suffice to show what these ratios are and how they have received their names. Let POM be the angle considered, PM being dravm perpendicular to OM. With center O describe the two circles PA and MQ. The appropriate measure of the angle at O is the ratio of the subtended arc to the radius — i. e., either AP/OP or MQ/OM (see Circle). This measure we shall adopt throughout, and shall represent it by the symbol &, If QN is dravm per- pendicular to OM, then the ratio of any pair of sides of the triangle OQN is equal to the ratio of the corresponding sides of triangle 0PM. All the possible ratios which can be formed are the so-called trigonometrical or circular functions of the angle e. Thus the ratio PM/OP or QN/OQ is the sine of 0. It is evidently half the chord of the angle 2/0; and its value is numerically less than 6, because PM being less than the chord PA is less than the arc PA. Again, the ratio PM/ OM is the tangent of 0. MP is, in fact, the geometrical tangent drawn from the one extremity of the arc MQ till it meets the radius through the other extremity. For a similar reason the ratio OP/OM or OQ/ON is called the secant of the angle 0. In the same way the ratios OM/OP, OM/PM, OP/PM are respectively the sine, tangent, and secant of the angle 0PM, which is the complement of the angle POM. Hence these ratios, regarded as functions of 0, are called the cosine, cotangent, and cosecant of 0. For any given angle there are, then, six trigono- metrical functions. It is obvious that these functions are mutually dependent. Indeed, if any one is given the other five can be at once calculated. For instance, the well-known relation OM^-f MP"=OP* gives at once by dividing by OP* (sin. e)='+ (cos. 0)^1, or, as it is usually written, sin." + COS." 0=1. Then, again, the cosecant is the reciprocal of the sine, and the secant of the cosine. The tangent is the ratio of the sine to the cosine; and the cotangent is the recipro- cal of the tangent. The sine and cosine are never greater than unity, and the se- cant and cosecant are never less than uni- ty. The tangent is less or greater than unity according as the angle POM is less or greater than half a right angle. Suppose OP to rotate counter-clock- wise. Then as the angle AOP increases from zero to a right angle the sine evi- dently grows from zero to unity; while at the same time the cosine diminishes from unity to zero. Continuing the in- crease so that AOP becomes an obtuse angle, we find that the sine begins to di- minish, and that the cosine begins to increase numerically but toward the left