866 TRIGONOMETRY TRILLIUM and the tangent of 89 3' is 60-3058. The re- ciprocals of the sine, cosine, and tangent (that are called the cosecant, secant, and cotangent of the angle at C. If the cosine be subtracted from 1, the remainder is called the "versed sine;" and if the sine be subtracted from 1, the remainder is called the "coversed sine." In practice these names are always abbreviated. Instead of " sine of 30 " it is always written sin 30, and, putting C for the angle, the abbreviations are as follows : sin C, cos C, tan C, cosec C, sec C, cotan C, covers 0, and vers 0. These terms all indicate numbers depending on the value of the angle, and are called the " trigonometrical functions." The value of these functions has been calculated for all possible angles which our most delicate instruments enable us to measure, and these values are recorded in tables, so that, any angle being given, the functions can be found, or any function being given, the angle can be found, by simply look- ing in the tables. The numbers employed in trigonometry, especially where great accuracy is required, often contain so many digits that the labor of calculation would be intolerable were it not for the use of logarithms. The tables generally used in practice contain, not the actual values of the functions, but the loga- rithms of those values. Tables of the actual values are also published, and they can be easily found, if wanted, from their logarithmic values by means of a table of the logarithms of numbers. A single example of the use made of these functions will show how measure- ments can be made which without them would be inconvenient or impossible. Suppose a per- son at B, fig. 2, on the bank of a river, on FIG. 2. the opposite side of which is a lofty hill, whose highest peak H he can see with his telescope. He wishes to know the perpendicular height of the peak (H X) above the plain B. Sup- posing him to be provided with the proper instruments for measuring angles, he takes a sight at the peak H and finds that the angle of elevation X B H is 28 41'. Subtracting this from 180, he finds the angle H B = 151 19'. Next he measures back from the river say 1,000 ft. to 0, and then takes another sight at the peak and finds that the angle H X is 18 4'. The rest is matter of calculation and looking in the tables. The angles are quickly and easily measured, and the only physical labor of any consequence is the carrying his instruments from B to C and measuring the distance of 1,000 ft. between them. Any other distance than '1,000 ft. would have answered the purpose ; but, for reasons which it is not necessary to enter into, it will save trouble and insure accuracy to have the distance BC as near as a rough guess will give to B H. Geom- etry tells us that if from the angle H B X = 28 41' we subtract the angle H C B = 18 4', we shall get the angle C H B, between the two lines of sight. We thus find C H B = 10 37'. The text books on trigonometry show that in every triangle the sines of any two angles are to each other as the sides opposite the angles. Looking in a table of natural sines (that is', of the actual values, and not the logarithms), we find the sine of 10 37' is the decimal fraction 0-18424, and the sine of 18 4' is 0-31012. The side opposite the angle CHB we have mea- sured, and hence we have the proportion, or "sum in the rule of three:" as 0-18424 is to 0-31012, so is 1,000 to B H, the side opposite the angle HOB. Making the calculations, which are much more easily made by means of logarithms, we get 1683-28 ft. as the distance from B to H. We now apply the same process to the triangle B H X. The angle B H X is a right angle, and its sine is 1. The sine of 28 41' is 0-47997; hence, as 1 is to 0-47997, so is 1683-28 to HX, the height which we wished to find ; making the calculations, we find it to be 807*92 ft., or, taking the nearest foot, we say the peak is 808 ft. high. We have only made use of the sines ; but all the other func- tions may come into play, according to the nature of the problem. The great mathema- ticians of modern times have shown how trig- onometry can be treated as a branch of pure algebra, and all its formulas developed without any reference to triangles. They have also shown how in this abstract form it can be ap- plied to geometry, and a perfectly intelligible explanation given to what are called imaginary or impossible quantities. Treated in this man- ner, it constitutes the connecting link between the mathematical sciences of the present and those higher but as yet undeveloped branches of the mathematics of the future that have been referred to in the article GEOMETRY, and the foundations of which have been laid in the " Quaternions " of Hamilton, the Ausdehnungs- lehre of Grassmann, and the " Linear Associa- tive Algebra" of Peirce. Among the multi- tude of works on the science, the following are of special excellence: A. De Morgan, "Trigonometry and Double Algebra" (Lon- don, 1849); J Todhunter, "Plane Trigonom- etry" (4th ed., London, 1869) and " Spherical Trigonometry" (3d ed., 1871); L. Mack, 60- niometrie und Trigonometrie (Stuttgart, 1860); and C. Briot and A. Bouquet, Lemons nouvelles de trigonometric (4th ed., Paris, 1862). (For the application of trigonometry to surveying, see COAST SURVEY, and SURVEYING.) TRILLIUM (Lat. trilix, triple, the parts being in threes), a genus of North American plants,
