Trigonometry (Gr. -τρίγωνον, a triangle, and μετρείν, to measure), the branch of mathematics which treats of the measurement of triangles. The practical object in nearly all applications of the science is to measure indirectly some height or some distance the direct measurement of which would be inconvenient or impossible. The labors of the civil engineer and the astronomer consist in great part in a constant application of the principles of trigonometry, and the best treatises on the subject, like that of Prof. Peirce, include also treatises on surveying, navigation, and spherical astronomy. Trigonometry is divided into plane and spherical, the former treating of plane triangles, the latter of spherical triangles. In surveying and ordinary engineering operations plane trigonometry is mostly employed; in the higher problems of navigation, in engineering operations conducted on a grand scale, as in the coast survey, and in astronomy, spherical trigonometry is indispensable. But the general principles are the same in both branches. As spherical trigonometry consists essentially in an extension of the principles of plane trigonometry, we shall confine our attention to the latter.

In every plane triangle there are six elements to be considered, three sides and three angles. The angles depend upon the proportions of the sides, and conversely the proportions of the sides depend upon the angles. If we know the three angles, we can find the ratio which any one side bears to each of the others, but we cannot find the length of any one of them; hence it is necessary for the complete determination of all the elements of a triangle, that we should know the length of at least one side. In calculating the unknown elements of a triangle certain ratios are employed, called " trigonometrical functions," which depend upon the angles. One quantity is said to be a function of another when its value depends upon the value of the other. The ordinary method of measuring angles is explained under Angle. - There are two methods of explaining the trigonometrical functions. The one, which may be called the ancient method, is presented in nearly all the text books in use before the middle of the present century; the other or modern method is followed in the best text books of recent date, and is fast superseding the former. In the old system the trigonometrical functions are lines, in the new system they are abstract numbers expressing the ratios of lines.

A brief explanation of the modern system will enable the general reader to form an idea of the nature and objects of the science. Draw two lines, C A, C B, fig. 1, forming an angle at C. At any point in either line, say at P in the line C B, erect a perpendicular to C B, intersecting C A in D. It matters not where in the line C B the point P is; so long as the angle at C remains unchanged, the proportions of the lines CD, CP, and PD will remain the same. In the figure the angle at C is intended to be an angle of 30°; and with this angle, if C D is an inch, P D will be half an inch, and if C D is ten miles, PD will be five miles; in other words, with an angle of 30°, PD is always half of C D. The number ½ is called the " sine " of 30°, orPD/CD = ½ = sine of 30°. If the angle C be altered, the ratio will change, and hence the sine is said to be a function of the angle. But the sine does not vary directly as the angle. When the angle is a right angle or 90°, the lines C D and P D fall together and become one line, and their ratio is 1, or the sine of 90 = 1; and although the angle is three times 30°, the sine is only twice the sine of 30°. The ratio of C P to C D, or CP/CD, is called the "cosine" of the angle at C. The cosine of 30° is the decimal fraction 0.866 very nearly.

The ratio of the sine to the cosine, or of the line PD to CP, is called the "tangent" of the angle at C. The tangent of 30° is -J- divided by 866/1000,or in decimals correct to three places, 0.577. The sine and cosine are never greater than 1, and hence in all cases except where the line C D coincides with one of the other lines, the sine and cosine are fractions. The tangent may have any value. Thus the sine of 89° 3' is 0.99986, and the cosine is 0.01658; both are fractions less than 1, but the former contains the latter more then 60 times, and the tangent of 89° 3' is 60-8058. The reciprocals of the sine, cosine, and tangent (that are called respectively the cosecant, secant, and cotangent of the angle at 0. 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 C. 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 looking 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 logarithms 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 measurements can be made which without them would be inconvenient or impossible. Suppose a person at B, fig. 2, on the bank of a river, on the opposite sido of which is a lofty hill, whoso highest peak II he can see with his telescope. He wishes to know the perpendicular height of the peak (II X) above the plain C B. Supposing him to be provided with the proper instruments for measuring angles, he takes a sight at the peak II and finds that the angle of elevation X B II is 28° 41'. Subtracting this from 180°, he finds the angle II B C = 151° 19'. Next he measures back from the river say 1,000 ft. to C, and then takes another sight at the peak and finds that the angle HC 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 B C as near as a rough guess will give to BH. Geometry tells us that if from the angle II B X = 28° 41' we subtract the angle II C B = 18° 4', we shall get the angle CHB, 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 C H B we have measured, 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 II, the side opposite the angle HC B. Making the calculations, which are much more easily made by means of logarithms, we get 1683.28 ft, as the distance from B to II. We now apply the same process to the triangle.B H X. The angle BHX 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 H X, the height which we wished to find; making the calculations, Ave find it to be 807.92 ft., or, taking the nearest foot, Ave say the peak is 808 ft. high.

We have only made use of the sines; but all the other functions may come into play, according to the nature of the problem. - The great mathematicians of modern times have shown how trigonometry 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 applied to geometry, and a perfectly intelligible explanation given to what are called imaginary or impossible quantities. Treated in this manner, 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 Ausdehni(ngs~ lehre of Grassmann, and the "Linear Associative Algebra" of Peirce. - Among the multitude of works on the science, the following are of special excellence: A. De Morgan, "Trigonometry and Double Algebra" (London, 1849); J Todhunter, "Plane Trigonometry" (4th ed., London, 1869) and " Spherical Trigonometry" (3d ed., 1871); L. Mack, Go-niometrieund Trigonometric (Stuttgart, 1860); and C. Briot and A. Bouquet, Lecons nouvelles de trigonometric (4th ed., Paris, 1862). (For the application of trigonometry to surveying, see Coast Survey, and Surveying). Fig. 1.  Fig. 2.