2

- for 3; (3) from (1) on writing a for a; and (4) from (3), as (2) from (1). Dividing (4) by (1), and both terms of the resulting righthand quotient by cos a cos B, there results: tan (a-3)=(tan a-tan ):(+tan a tan 3), which with the relation, tan (a + 3) = (tan a + tan 3) (1-tan a tan ẞ), constitutes the Addition Theorem for the Tangent. The analogous relations for the remaining functions are omitted as being but little used.

Prostha pheretic Formule-From the addition theorems, which concern functions of angle sums and differences, may be easily deduced equally important formulæ concerning function sums and differences. Replacing a +3 by u and a-ẞ by v in (1), (2), (3), (4), adding (3) and (4), then (1) and (2), and then taking the corresponding differences, one obtains the four formula:

(5)

(6)

(7)

sin u + sin v = 2 sin (u+v) cos (u —v), cos u + cos v = 2 COS (u +1') cos (u —v), sin usin v=2 cos (u+v) sin (u —v), (8) cos u cos v2 sin (u+v) sin (u-v). These relations, which have been named prostha pheretic, express sums and differences in terms of products, and so render them suitable for logarithmic computation.

Some Important Deductions from Foregoing Formula.-Setting a=3 in (2) and (3), there

result

(9) (10)

COS 2 α = cos2a - sin'a,

sin 2 a 2 sin a cos a. combined

The former
I=sin'a +cos'a yields

(11)

(12)

with the relation

I +COS 2α = 2 cos'α, ICOS 2α = 2 sin2a.

Again, putting a =ẞ in addition theorem of tangent, the result is tan 2a = 2 tan a:(1-tan2a). Division of (12) by (11) yields (1-cos 2α): (1+cos 2 a) =tan'a, whence follow

secant relation is cosec'a = I + cot'a. Hosts of other more or less useful and interesting kindred formulæ, readily deducible, may be found in the current text-books and in pocket manuals for engineers. This paragraph will be closed with a deduction of the above-presented Law of Tangents. By the Law of Sines, a:b=sina: sin ß; whence, by "composition and division,» (a + b) : (a −b) = (sin a +sin ẞ): (sin a-sin 3); expanding the right-hand member by (4) and (3), and applying the definition of tangent, the relation sought is found to be 1 · +b): (a - b) =tan (a): (tan (a-3).

Solution of Triangles.-Etymologically TRIGO NOMETRY is triangle measurement, and, though the science wonderfully exceeds the verbal significance of its name, yet measurement of triangles is a very important, and, at the same time, the most generally familiar, one of its manifold applications. A triangle is determined by three independent data, of which the simplest are: two sides and an angle; two angles and a side; three sides. The three angles are not independent, any pair of them determine the third angle. Let a, b, c denote the lengths of the sides, and a, B, 7 the corresponding (opposite) angles, of any triangle. The data being those mentioned, three cases arise: (i) given a pair of opposite parts, and one other, as a, a, and b, or a, a, and 3, to find the remaining parts; (ii) given three adjacent parts, as a, r, and c, or a, c, and ẞ, to find the rest; (iii) given three alternate parts, a, b, and c, to find the angles. In case (i), it is sufficient to employ the Law of Sines; in (ii), the Law of Tangents; in (iii), the Law of Cosines. In (i), if the "one other» part be a side, as b, the Sine Law yields the sine of B, the opposite angle. But, as sin sin (-), the problem presents an ambiguity, which, in every actual example, is readily resoluble by easy considerations explained in every text-book of trigonometry. The Cosine Law is equivalent to the equation: COS a

=

= (b2+c2 — a2): 2bc. The numerator not being a product, the formula is not adapted to logarithmic use. From it, however, is readily derived an adequate formula that is so adapted. It is

a=18° 27′ 23′′, a = 36° 54′ 46′′.

In like manner one may find -50° 32′ 32"; and 7=92° 32' 44". As a check one finds a+B+r=180° 0' 2", an excess regarded in practice as very slight and in most work negligible. To secure more accurate results, which is seldom necessary, it suffices to employ logarithms of more than five decimal places.

Inverse Trigonometric Functions.-The symbols sin-1n, cos-1n, tan-1n, etc., denote respectively an angle whose sine is n, whose cosine is n, etc. They are variously read inverse sine, cosine, etc., of n, or anti-sine, etc., of n, or, again, the arc or angle whose sine, cosine, etc., is n. They are called inverse trigonometric or circular functions, being related to the direct (so-called) trigonometric or circular functions much as are the integral and the derivative of the Calculus (which see), or the logarithm and the exponential of Algebra (which see). Like analogies abound. It should be noted that sin-1, cos-, etc., do not signify reciprocal of

sin, cos, etc. Moreover, unless the contrary is expressed, it is generally understood that sin-in, etc., shall signify the smallest one of the infinitely many angles whose sine, etc., is n. Thus sin-1 will ordinarily mean 30°, though, taken in full generality, it would signify 30°2ия, ог 150° ±2n, n being any integer. The direct functions are one-valued

(c) COS α=I

+

I.2 1.2.3.4 I 2... 8

It may be proved algebraically, and is proved by means of Maclaurin's Expansion (see ĈALCULUS), that the series (s) and (c) respectively represent or define sin a and cos a for every finite value of the angle a reckoned in terms of the radian. The precise meaning is that, if Sn denote the sum of the first n terms of (s), then the limit of Sn as n increases endlessly is sin a. Similarly for (c). The algebraic proof is too long for insertion here, and that by the Calculus rests on presuppositions not appropriate in this article. As a compromise it is edifying and interesting to assume the validity of equations (s) and (c) and then after the manner of natural science to test them, regarded as hypotheses, by their implications, or consequences. Rigorous proof is not thus obtainable, but certainty can be thus more and more nearly approximated. Any consequence of (s) or (c) or both that is known to be untrue would alone suffice to invalidate one or both assumptions absolutely. while any number of consequences known to be true merely tend to support but do not suffice ing consequences may be noted. to prove the hypotheses. Some such supportIf a=0, series (s) and (c) become respectively o and 1, as should be the case, since sin oo and cos o = 1.

If a be replaced by -a, each term of (s) is reversed in sign, while (c) is unchanged; and this, too, should be so, for, as before seen, the sin is an odd, and the cos an even, function of the angle. Again, it is proved in algebra (see ALGEBRA, also SERIES) that, e being the Napieri.

an base,

or

Replacing i by −i,

The product of the last two equations yields I=cos'a + sin'a, another result known to be true. Once more,

also,

çia. ¿i3 = çi(a+?) =cos (a +ß) +i sin(a +ß) ;

eia. ei? =(cos a + i sin a) (cos ẞ+i sin ẞ) =cos a cos 3-sin a sin ß

whence

+i(sin a cos +cos a sin 3);

cos (a +ẞ) =cos a cos 3-sin a sin 3,

and

sin (a+ß) = sin a cos ẞ+cos a sin ß, two equations of the known addition theorem for sine and cosine. It is indeed a fact that the whole body of trigonometric relations deduced or deducible from the original definitions, Fig. 1, of the functions, are obtainable analytically from (s) and (c) regarded as definitions, and, like the latter, would then be free from geometric reference. Each of the other functions, direct or inverse, is representable in the form of a series analogous to (s) and (c). Such series may be found in books of trigonometry and of the calculus. De Moivre's Theorem.-Frome ia =cos a+isin a follows

sin c = cos (90°-a) cos (90° — C), sin b =cos (90° - a) cos (90° — B), sin (90°-B) = cos b cos (90° — C), sin (90° —C) =cos c cos (90° — B);

eia =cos a +i sin a, we get ei=i; squaring, we obtain e1 = -1; also ei-i, and e2xi1· -=0. The last is especially noteworthy as involving the most notable set of five numbers in mathematics: o, I, i, e, π. Further developments would lead into the doctrine of Circle Partition (Kreistheilung), which belongs to the Theory of Functions of the Complex Variable, to which the reader is referred.

SPHERICAL TRIGONOMETRY.

A spherical triangle is the figure bounded by three arcs (of great circles) on the surface of a sphere. Spherical trigonometry has for its principal problem that of determining the numerical values of the three remaining parts of a spherical triangle when three parts are given. This note is confined to triangles whose sides ore each less than a semi-circumference and whose angles are each less than, or 180°. Any spherical angle is measured by the corresponding diedral angle, and the latter by the plane angle of intersecting lines (in the faces of the diedral angle) drawn perpendicular to its edge. For convenience the triangle will be supposed to be on a sphere of unit radius; the

Arranging the parts in some such cyclical scheme as in Fig. 11, it will be seen that any part being taken as middle part, there are two adjacent parts, and two others that may be and are called opposite. By inspecting the first half of the preceding table it appears that the sine of any middle part is equal to the product of the cosines of its opposite parts, and, from. the second half, that the sine of a middle part is the product of the tangents of its adjacent parts. Such are Napier's rules for circular parts, the more readily remembered by virtu❤

These functions are most instructively introduced through the integral calculus. For their geometric interpretation the reader is referred to such works as W. B. Smith's Infinitesimal Analysis, Vol. I., and Greenhill's 'Differential and Integral Calculus.'

Pseudo-spherical Trigonometry.-A given sphere has constant positive curvature (see CALCULUS). If the radius be infinite, the curvature is zero. The plane is a sphere of constant zero curvature. Suppose the curvature to be constant and negative. The surface is then called pseudo-sphere. This, too, has its trigonometry. Its formulæ are obtainable from those of spherical trigonometry by replacing the circular functions by the corresponding hyperbolic functions. Bibliography.-Chauvenet, Treatise on Plane and Spherical Trigonometry); Loney, Plane Trigonometry); Todhunter, Plane Trigonometry' and 'Spherical Trigonometry.' CASSIUS J. KEYSER, Professor of Mathematics, Columbia University. Trigonometry, History of the Elements of. Among the ancients trigonometry was simply an adjunct to astronomy, and it so remained until comparatively recent times. A slight trace of its application to mensuration is found in the famous papyrus of Ahmes (see ALGEBRA, HISTORY OF THE ELEMENTS OF), where a quotient called seqt is mentioned. In the case of the pyramids the seqt seems to have been the cosine of the angle of slope of the edge, or in some cases the tangent of the angle of slope of the face. Among the Greeks frequent reference to trigonometry is found among the writings of the astronomers. Hypsicles (c. 190 B.C.) used the Babylonian division of the circumference into 360 degrees, and from this time the sexagesimal fraction became common in astronomy. Hipparchus (c. 150 B.C.) was the first to compute a table of chords, the ancients generally using the chord instead of the half-chord or sine. Hero of Alexandria (see HERO OF ALEXANDRIA) gave rules which are the equivalent of certain modern formulas, and in par

2π

ticular computed the values of cot for all

n

values of n from 3 to 12 inclusive. Menelaus of Alexandria (c. 100 A.D.) carried the study of Spherics to a considerable prominence, his celebrated Regula sex quantitatum relating to the transversal of the sides of a spherical triangle, and he wrote six books on the calculation of chords. It is, however, to Claude Ptolemy (q.v.), c. 125 A.D., that is due the introduction of a formal spherical trigonometry into astronomy. The Almagest made the sexagesimal fraction more widely known, and Ptolemy calculated the chords of arcs to a half degree.

The Hindu astronomers used the half chord instead of the chord which the Greeks usually (but not always) employed. They thus used the sine, and they added the versed sine and the cosine, computing tables for these ratios. They also knew the relation, sin2 x + cos2 r=1.

The Arabs made the greatest advance in trigonometry of any peoples before the Renaissance. Al Battani, or Albategnius as the Latin writers called him, c. 900 A.D., brought into greater prominence the use of the sine, and computed a table of values of sin x/cos x and

its reciprocal, thus practically using the tangent and cotangent. The present names for the various functions are mostly modern. The name sinus seems first to have been used by Gherardo of Cremona, c. 1150, although often attributed to Plato of Tivoli (also c. 1150) in his translation of Al Battani. Among the western Arabs Jabir ibn Aflah, often known as Geber, was prominent, his trigonometry covering both the plane and the spherical parts.

In Christian Europe the science is first seriously considered in the work of Regiomontanus (q.v.), the famous pupil of Peuerbach (q.v.). The latter had done some excellent work in trigonometry, but he died before he could write his projected treatise, and Regiomontanus carried out his plans. The result was a work which influenced subsequent text-books much as Euclid's 'Elements' influenced plane geometry. The principal formulas of plane and spherical trigonometry are set forth, and the elementary science became crystallized. Subsequent advances have been chiefly in the nomenclature, the symbolism, and the computation of tables. particularly of logarithmic tables. Among the most prominent computers of the values of the functions and of logarithms should be mentioned Rhæticus (1514-76), Pitiscus (1561-1613), Bürgi (1552-1632), Napier (1550-1617), Briggs (1560-1630), and Vlacq, whose tables appeared in 1628.

Bibliography.-Braunmühl, A. von, 'Geschichte der Trigonometrie (Leipsic, 1900, 1904); Cantor, M., Geschichte der Mathematik) (Leipsic, 2d ed. 1894); Gow, J., History of Greek Mathematics' (Cambridge, 1884).

DAVID EUGENE SMITH, Professor of Mathematics, Teachers College. Columbia University, New York.

Trikala, tré'kä-lä, or Trikkala, Greece, the capital of the nome of Trikala in northwestern Thessaly, situated 38 miles west of Larissa, with which it has railroad connection by way of the Gulf of Volos. The chief industries are tanning, dyeing, and the manufacture of cotton and woolen goods. The ancient town, Trikka, had a famous temple of Esculapius. Pop. about 24,000.

Trikoupis, Charilaos, Greek statesman, son of Spiridion Trikoupis (q.v.): b. Nauplia 23 July 1832; d. Cannes, France, II April 1896. He inherited his father's literary bent of mind and later began the study of jurisprudence in Athens, afterward studying and completing his course at Paris. His mind then turned toward diplomacy and in 1852 he entered the service of his country as attaché to the legation in London, and in 1863 was promoted chargé d'affaires. In 1865 he was in charge of the negotiations with England which brought about the cession of the Ionian Islands. He was elected to the Boulé the latter part of the same year, and joining the Radical party became successively minister of foreign affairs in 1866, premier in 1875, and again in 1877 minister of foreign affairs in the coalition ministry under Canaris. He then again became premier in 1878-80, and served also the terms 1882-5, 1886-90, 1891-3, and 1893-5, but at the next election he was defeated and even lost his seat in the Boulé. This was probably due to the low state of the finances of the gov ernment, which he had in vain attempted to raise