History.

An account of Greek trigonometry is given under Ptolemy (q.v.).

The Indians, who were much more apt calculators than the Greeks, availed themselves of the Greek geometry which came from Alexandria, and made it the basis of trigonometrical calculations. The principal improvement which they introduced consists in the formation of tables of half-chords or sines instead of chords. Like the Greeks, they divided the circumference of the circle into 360 degrees or 21,600 minutes, and they found the length in minutes of the arc which can be straightened out into the radius to be 3438'. The value of the ratio of the circumference of the circle to the diameter used to make this determination is 62832 :20000, or π = 3∙1416, which value was given by the astronomer Âryabhata (476-550; see Sanskrit, vol. xxi. p. 294) in a work called Ârya-bhatîya, written in verse, which was republished in Sanskrit by Dr Kern at Leyden in 1874. The relations between the sines and cosines of the same and of complementary arcs were known, and the formula sin ½ a = √1719(3438 - cos a) was applied to the determination of the sine of a half angle when the sine and cosine of the whole angle were known. In the Sûrya-Siddhânta, an astronomical treatise which has been translated by Ebenezer Bourgess in vol. vi. of the Journal of the American Oriental Society (New Haven, 1860), the sines of angles at an interval of 3° 45' up to 90° are given; these were probably obtained from the sines of 60° and 45° by continual application of the dimidi-ary formula given above and by the use of the complementary angle. The values sin 15° = 890', sin 7° 30' = 449', sin 3° 45' = 225', were thus obtained. Now the angle 3° 45' is itself 225'; thus the arc and the sine of 1 / 96 th of the circumference were found to be the same, and consequently special importance was attached to this arc, which was called the right sine. From the tables of sines of angles at intervals of 3° 45' the law expressed by the equation sin (n +1.225') - sin (n . 225') = sin (n . 225') - sin (n - 1.225')

-sin[(n . 225')/(225)]

was discovered empirically, and used for the purpose of recalculation. Bhâskara (fl . 1150) used the method, to which we have now returned, of expressing sines and cosines as fractions of the radius; he obtained the more correct values sin 3° 45'= 100/1529, cos 3° 45' = 466/467, and showed how to form a table, according to degrees, from the values sin 1° = 10/573, cos 1° = 6568/6569, which are much more accurate than Ptolemy’s values. The Indians did not apply their trigonometrical knowledge to the solution of triangles; for astronomical purposes they solved right-angled plane and spherical triangles by geometry.

The Arabs were acquainted with Ptolemy’s Almagest, and they probably learned from the Indians the use of the sine. The celebrated astronomer of Batnae, Abú 'Abdallah Mohammed b. Jābir al-Battání (Bategnius), who died in 929/930 A.D., and whose Tables were translated in the 12th century by Plato of Tivoli into Latin, under the title De scientia stellarum, employed the sine regularly, and was fully conscious of the advantage of the sine over the chord; indeed, he remarks that the continual doubling is saved by the use of the former. He was the first to calculate sin ϕ from the equation sin ϕ ∕cos ϕ = k, and he also made a table of the lengths of shadows of a vertical object of height 12 for altitudes 1°, 2°, . .. of the sun; this is a sort of cotangent table. He was acquainted, not only with the triangle formulae in the Almagest, but also with the formula cos a = cos b cos c + sin b sin c cos A for a spherical triangle ABC. Abú 'l-Wafá of Baghdad (b. 940) was the first to introduce the tangent as an independent function: his “umbra” is the half of the tangent of the double arc, and the secant he defines as the “diameter umbrae.” He employed the umbra to find the angle from a table and not merely as an abbreviation for sin/cos; this improvement was, however, afterwards forgotten, and the tangent was re-invented in the 15th century. Ibn Yúnos of Cairo, who died in 1008, showed even more skill than Al-Battání in the solution of problems in spherical trigonometry and gave improved approximate formulae for the calculation of sines. Among the West Arabs, Abú Mohammed Jābir b. Aflah, known as Geber b. Aflah, who lived at Seville in the 11th century, wrote an astronomy in nine books, which was translated into Latin in the 12th century by Gerard of Cremona and was published in 1534. The first book contains a trigonometry which is a considerable improvement on that in the Almagest. He gave proofs of the formulae for right-angled spherical triangles, depending on a rule of four quantities, instead of Ptolemy’s rule of six quantities. The formulae cos B = cos b sin A, cos c = cot A cot B, in a triangle of which C is a right angle had escaped the notice of Ptolemy and were given for the first time by Geber. Strangely enough, he made no progress in plane trigonometry. Arrachel, a Spanish Arab who lived in the 12th century, wrote a work of which we have an analysis by Purbach, in which, like the Indians, he made the sine and the arc for the value 3° 45' coincide.

Purbach (1423-1461), professor of mathematics at Vienna, wrote a work entitled Tractatus super propositiones Ptole-maei de sinubus et chordis (Nuremberg, 1541). This treatise consists of a development of Arrachel’s method of interpolation for the calculation of tables of sines, and was published by Regiomontanus at the end of one of his works. Johannes Müller (1436-1476), known as Regiomontanus (q.v. ), was a pupil of Purbach and taught astronomy at Padua; he wrote an exposition of the Almagest and a more important work, De triangulis planis et sphericis cum tabulis sinuum, which was published in 1533, a later edition appearing in 1561. He re-invented the tangent and calculated a table of tangents for each degree, but did not make any practical applications of this table, and did not use formulae involving the tangent. His work was the first complete European treatise on trigonometry, and contains a number of interesting problems; but his methods were in some respects behind those of the Arabs. Copernicus (1473-1543) gave the first simple demonstration of the fundamental formula of spherical trigonometry; the Trigonometria Copernici was published by Rheticus in 1542. George Joachim (1514-1576), known as Rheticus (q.v.) wrote Opus Palatinum de triangulis (see Tables, p. 9 above), which contains tables of sines, tangents, and secants of arcs at intervals of 10" from 0° to 90°. His method of calculation depends upon the formulae which give sin nα and cos nα in terms of the sines and cosines of (n - 1 )α and (n - 2)α; thus these formulae may be regarded as due to him. Rheticus found the formulae for the sines of the half and third of an angle in terms of the sine of the whole angle. In 1599 there appeared an important work by Pitiscus (1561-1613), entitled Trigonometries seu de dimensione triangulorum ; this contained several important theorems on the trigonometrical functions of two angles, some of which had been given before by Finck, Landsberg, and Adriaan van Roomen. François Viète or Vieta ( q.v. ) (1540-1603) employed the equation (2 cos½ φ ) 3 - 3(2cos½φ) = 2 cos φ to solve the cubic x 3 — 3a 2 x = a 2 b (a > ½ b) ; he obtained, however, only one root of the cubic. In 1593 Van Roomen proposed, as a problem for all mathematicians, to solve the equation

45 y — 3795 y 3 + 95634 y 5 —... + 945 y 41 — 45 y 43 + y 45 = C.

Viète gave y = 2 sin 1 / 45 φ , where (7 = 2 sin φ , as a solution, and also twenty-two of the other solutions, but he failed to obtain the negative roots. In his work Ad angulares sectiones Viète gave formulae for the chords of multiples of a given arc in terms of the chord of the simple arc.

A new stage in the development of the science was commenced after Napier’s invention of logarithms in 1614. Napier also simplified the solution of spherical triangles by his well-known analogies and by his rules for the solution of right-angled triangles. The first tables of logarithmic sines and tangents were constructed by Edmund Gunter (1581-1626), professor of astronomy at Gresham College, London; he was also the first to employ the expressions cosine, cotangent, and cosecant for the sine, tangent, and secant of the complement of an arc. A treatise by Albert Girard (1590-1634), published at The Hague in 1626, contains the theorems which give areas of spherical triangles and polygons, and applications of the properties of the supplementary triangles to the reduction of the number of different cases in the solution of spherical triangles. He used the notation sin, tan, sec for the sine, tangent, and secant of an arc. In the second half of the 17th century the theory of infinite series was developed by Wallis, Gregory, Mercator, and afterwards by Newton and Leibnitz. In the Analysis per aequationes numero terminorum infinitas, which was written before 1669, Newton gave the series for the arc in powers of its sine; from this he obtained the series for the sine and cosine in powers of the arc; but these series were given in such a form that the law of the formation of the coefficients was hidden. James Gregory discovered in 1670 the series for the arc in powers of the tangent and for the tangent and secant in powers of the arc. The first of these series was also discovered independently by Leibnitz in 1673, and published without proof in the Acta eruditorum for 1682. The series for the sine in powers of the arc he published in 1693; this he obtained by differentiation of a series with undetermined coefficients.

In the 18th century the science began to take a more analytical form; evidence of this is given in the works of Kresa in 1720 and Mayer in 1727. Oppel’s Analysis triangulorum (1746) was the first complete work on analytical trigonometry. None of these mathematicians used the notation sin, cos, tan, which is the more surprising in the case of Oppel, since Euler had in 1744 employed it in a memoir in the Acta eruditorum. John Bernoulli was the first to obtain real results by the use of the symbol

√-1; he published in 1712 the general formula for tan nφ in terms of tanφ, which he obtained by means of transformation of the arc into imaginary logarithms. The greatest advance was, however, made by Euler, who brought the science in all essential respects into the state in which it is at present. He introduced the present notation into general use, whereas until his time the trigonometrical functions had been, except by Girard, indicated by special letters, and had been regarded as certain straight lines the absolute lengths of which depended on the radius of the circle in which they were drawn. Euler’s great improvement consisted in his regarding the sine, cosine, &c., as functions of the angle only, thereby giving to equations connecting these functions a purely analytical interpretation, instead of a geometrical one as heretofore. The exponential values of the sine and cosine, De Moivre’s theorem, and a great number of other analytical properties of the trigonometrical functions are due to Euler, most of whose writings are to be found in the Memoirs of the St Petersburg Academy.

The preceding sketch has been mainly drawn from the following sources: — Cantor, Gesch. d. Math.; Hankel, Gesch. d. Math.·, Marie, Hist. des sc. math. ; Suter, Gesch. d. Math. ; Klügel, Math. Wörterbuch. Plane Trigonometry.

Imagine a straight line terminated at a fixed point O, and initially coincident with a fixed straight line OA, to revolve round O, and finally to take up any position OB. We shall suppose that, when this revolving straight line is turning in one direction, say that opposite to that in which the hands of a clock turn, it is describing a positive angle, and when it is turning in the other direction it is describing a negative angle. Before finally taking up the position OB the straight line may have passed any number of times through the position OB, making any number of complete revolutions round O in either direction. Each time that the straight line makes a complete revolution round O we consider it to have described four right angles, taken with the positive or negative sign according to the direction in which it has revolved; thus, when it stops in the position OB, it may have revolved through any one of an infinite number of positive or negative angles any two of which differ from one another by a positive or negative multiple of four right angles, and all of which have the same bounding lines OA and OB. If OB' is the final position of the revolving line, the smallest positive angle which can have been described is that described by the revolving line making more than one-half and less than the whole of a complete revolution, so that in this case we have a positive angle greater than two and less than four right angles. We have thus shown how we may conceive an angle not restricted to less than two right angles, but of any positive or negative magnitude, to be generated.

Two systems of numerical measurement of angular magnitudes are in ordinary use. For practical measurements the sexagesimal system is the one employed: the ninetieth part of a right angle is taken as the unit and is called a degree; the degree is divided into sixty equal parts called minutes; and the minute into sixty equal parts called seconds; angles smaller than a second are usually measured as decimals of a second, the “thirds,” “fourths,” &c., not being in ordinary use. In the common notation an angle, for example, of 120 degrees, 17 minutes, and 14∙36 seconds is written 120° 17' 14"∙36. The decimal system measurement of angles has never come into ordinary use. In analytical trigonometry the circular measure of an angle is employed. In this system the unit angle is the angle subtended at the centre of a circle by an arc equal in length to the radius. The constancy of this angle follows from the geometrical propositions—(1) the circumferences of different circles vary as their radii; (2) in the same circle angles at the centre are proportional to the arcs which subtend them. It thus follows that the unit mentioned above is an angle independent of the particular circle used in defining it. The constant ratio of the circumference of a circle to its diameter is a quantity incommensurable with unity, usually denoted by π. We shall indicate later on (p. 571 sq. ) some of the methods which have been employed to approximate to the value of this quantity. Its value to 20 places is 3 ∙14159265358979323846; its reciprocal to the same number of places is ∙31830988618379067153. In circular measure every angle is measured by the ratio which it bears to the unit angle. Two light angles are measured by the quantity π, and, since the same angle is 180°, we see that the number of degrees in an angle of circular measure θ is obtained from the formula 180 × θ∕π. The value of the unit of circular measure has been found to 41 places of decimals by Glaisher (Proc. London Math. Sοc., vol. iv.); the value of 1 / π from which the unit can be easily calculated, is given to 140 places of decimals in Grunert's Archiv, vol. i., 1841. To 10 decimal places the value of the unit angle is 57° 17'44"∙8062470964. The unit of circular measure is too large to be convenient for practical purposes, but its use introduces a simplification into the series in analytical trigonometry, owing to the fact that the sine of an angle and the angle itself in this measure, when the magnitude of the angle is indefinitely diminished, are ultimately in a ratio of equality.

If a point moves from a position A to another position B on a straight line, it has described a length AB of the straight line. It is convenient to have a simple mode of indicating in which direction on the straight line the length AB has been described; this may be done by supposing that a point moving in one specified direction is describing a positive length, and when moving in the opposite direction a negative length. Thus, if a point moving from A to B ∣ is moving in the positive direction, we consider the length AB as: positive; and, since a point moving from B to A is moving in the negative direction, we consider the length BA as negative. Hence any portion of an infinite straight line is considered to be positive or negative according to the direction in which we suppose this portion to be described by a moving point; which direction is the positive one is, of course, a matter of convention.

If perpendiculars AL, BM be drawn from two points A, B on any straight line, not necessarily in the same plane with AB, the length LM, taken with the positive or negative sign according to the convention as stated above, is called the projection of AB on the given straight line; the projection of BA being ML has the opposite sign to the projection of AB. If two points A, B be joined by a number of lines in any manner, the algebraical sum of the projections of all these lines is LM,— that is, the same as the projection of AB. Hence the sum of the projections of all the sides of any closed polygon, not necessarily plane, on any straight line, is zero. This principle of projections we shall apply below to obtain some of the most important propositions in trigonometry.

Let us now return to the conception of the generation of an angle as in fig. 1. Draw BOB' at right angles to and equal to AA'.

We shall suppose that the direction from A' to A is the’ positive one for the straight line AO A', and that from B' to B for BOB'. Suppose OP of fixed length, equal to OA, and let PM, PN be drawn perpendicular to A'A, B'B respectively; then OM and ON, taken with their proper signs, are the projections of OP on A'A and B'B. The ratio of the projection of OP on B'B...