Page:The American Cyclopædia (1879) Volume XV.djvu/895

 TRIGG of local self-government; and by tho constitution of Dec. 21, 1867, they wore made a constituent part of tho Littoral province. TRIGG, a S. W. county of Kentucky, border- ing on Tennessee, bounded V. by the Tennes- see river and drained by the Cumberland river ; area, 530 sq. m. ; pop. in 1870, 18,686, of whom 3,806 were colored. Tho surface is hilly and tho soil fertile in parts. Horses, cattle, mules, and swine are exported in great numbers. Iron, bituminous coal, and limestone are found. The chief productions in 1870 were 99,871 bushels of wheat, 589,820 of Indian corn, 16,- 114 of oats, 14,805 of Irish and 18,882 of sweet potatoes, 3,614,363 Ibs. of tobacco, 18,442 of wool, 83,808 of butter, and 584 tons of hay. There wore 2,678 horses, 1,908 mules and asses, 2,440 milch cows, 8,311 other cattle, 9,439 sheep, and 21,288 swine ; 4 flour mills, 1 manu- factory of pig iron, 4 tanneries, 4 currying es- tablishments, and 8 saw mills. Capital, Cadiz. TRICOiVOMETRY (Gr. rp/ywiw, a triangle, and peTpetv, to measure), tho branch of mathematics which treats of the measurement of triangles. Tho practical object in nearly all applications of tho science is to measure indirectly some height or some distance tho direct measure- ment of which would bo inconvenient or im- possible. Tho labors of the civil engineer and tho astronomer consist in groat part in a con- stant application of tho principles of trigonom- etry, an<l the. heat treatises on the subject, liko that of Prof. Poirce, include also treatises on surveying, navigation, and spherical as- tronomy. Trigonometry is divided into plane and spherical, tho former treating of piano triangles, tho latter of spherical triangles. In surveying and ordinary engineering operations Elano trigonometry is mostly employed ; in tho igher problems of navigation, in engineering operations conducted on a grand scale, as in the coast survey, and in astronomy, spherical trigonometry is indispensable. But tho gen- eral principles arc tho same in both branches. As spherical trigonometry consists essentially in an extension of tho principles of plane trig- onometry, we shall confine our attention to tho latter. In every plane triangle there are six elements to be considered, three sides and three angles. Tho angles depend upon the proportions of tho sides, and conversely the proportions of tho sides depend upon tho angles. If wo know the three angles, we can find (,ho ratio which any ono^ido bears to each of the others, but we cannot find tho length of any one of them ; hence it is necessary for tho complete determination of all tho elements of a triangle, that wo should know the length of at least one side. In calculating the un- known elements of a triangle certain ratios are employed, called "trigonometrical func- tions," which depend upon the angles. One quantity is said to bo a function of another when its value depends upon tho value of the other. Tho ordinary method of measuring angles is explained under ANGLE. There are TRIGONOMETRY 660 two methods of explaining tho trigonometrical functions. The one, which may I,,, culled tho ancient method, is presented in nearlv all tho text books in use before the middle of tho present cenlui-y ; the other or modern method is followed in tho bost text books of ivonit <late, and is fast superseding the former. In the old system tho trigonometrical functions aro linos, in tho now system Ihey are abstract numbers expressing the ratios of lines. A brief explanation of tho modern system will enable the general reader to form an idea of t| m nature and objects of tho science. Draw two lines, A, B, fig. 1, forming an an C. At any point in either line, say at, 1' in the, line C B, erect a perpendicular to B, inter- secting A in D. It matters not where in t ho line C B the point P is ; so long as tho angle at C remains unchanged, the proportions of tho lines CD, OP, and PD will remain tho same. In the figure tho angle at is intended to bo an angle of 80; and with this angle, if C D is an inch, P D will be half an inch, and if D is ton miles, PI) will be five miles; in other words, with an angle of 80, PD is always half of CD. Tho number is called tho "sine" of 80, or =i = sine of 80. If tho angle CD I'll C bo altered, tho ratio (!|) will change, and hence tho sine is said to bo a function of the angle. But the sine does not, vary directly as the angle. When tho angle is a right angle or <)<)", the lines C I) and I'D fall together and be- como one lino, and their ratio is 1, or tho sine of 90 = 1; and although the angle in three times 80, the sine is only twice tho sine of 80. Tho ratio of P to D, or, is called the " cosine " of tho angle at C. The cosine of 80 is the decimal fraction 0-866 very near- ly. The ratio of the sine to the cosine, or of the line PD to OP, is called the " tangent" of the angle at 0. The tangent of 80 is di- vided by T WV> or * n d ec ' ma l 9 correct to three places, 0-577. The sino and cosine are never greater than 1, and hence in all cases except where tho lino CD coincides with one of the other lines, the sino and cosine are fractions. Tho tangent may have any value. Thus Mm sine of 89 8' is 0-990KC,, and the cosine is 0-01668; both aro fractions less than 1, but the former contains the latter more then 60 times,