# Ptolemy

### angle sine positive straight line angles negative cosine sines direction

PTOLEMY (q.v.).

Indian. 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 = 3.1416, which value was given by the astronomer Aryabhata (476-550 ; see SANSKRIT, vol. xxi. p. 294) in a work called lryabltatiya, 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 a = 1/1719(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 Sarya-Siddhelnta, 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 dimidiary 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 Ath 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 - . 225') ,(7i.2251 - sin was discovered empirically, and used for the purpose of recalculation. BhSskara (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° = 656816569, 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, Arabian. and they probably learned from the Indians the use of the sine. The celebrated astronomer of Batnze, Abu 'Abdallah Mohammed b. Jabir al-Battani (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 I See also vol. ii. of the Asiatic Researches (Calcutta).

by the use of the former. He was the first to calculate silly!) from the equation sin4/coss6 =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 e cos A for a spherical triangle ABC. Abu '1-Wafa 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 Yilnos of Cairo, who died in 1008, showed even more skill than Al-Battani in the solution of problems in spherical trigonometry and gave improved approximate formulae for the calculation of sines. Among the West Arabs, Abfi Mohammed Jabir b. Allah, 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 formulzz 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.

Modern. Purbach (1423-1461), professor of mathematics at Vienna, wrote a work entitled Tractatus super propositiones Ptolemxi de sinuous 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 Muller (1436-1476), known as REGIOMONTANTIS (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 split-wit-is 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 formulm which give sin na and cos na in terms of the sines and cosines of (n - 1)a and (n - 2)a ; thus these formulae may be regarded as due to him. Rheticus found the formulm 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 Trigon,ometrim seu de dirnensione 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. Frangois Viete or VIETA (q.v.) (1540-1603) employed the equation (2 cos22-4))3 - 3(2 cos 4 0) = 2 cos s5 to solve the cubic x3 - 3a2x = a2b(a>,1,-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 45y - 3795y3+ 95634e - + 945y41 - 452/43+ y43= a Viete gave y= 2 sin 41,16, where C= 2 sin 95, 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 Viete 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 mquationes 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 ,r--1-; he published in 1712 the general formula for tan nqS in terms of tan (1,, 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, Gesell,. d. Math. ; &Rigel, Math. Worterbuch.

Plane Trigonometry.

Concep- Imagine a straight line terminated at a fixed point 0, and initially tion of coincident with a fixed straight line OA, to revolve round 0, and angles of finally to take up any position OB.

any mag- We shall suppose that, when this renitnde. volving 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 revo lutions round 0 in either direction. Fig. 1.

- Each time that the straight line makes a complete revolution round 0 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.

formula 180 x BIT. The value of the unit of circular measure has been found to 41 places of decimals by Glaisher (Proc. London Math,.

Soc., vol. iv.); the value of !, from which the unit can be easily calculated, is given to 140 places of decimals in Gruncrt's Archly, vol. 1, 1841. To 10 decimal places the value of the unit angle is 57° 17'440-8062470964. 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 drawn perpendicular to A'A, B'B respectively ; then 0.3! and ON, Nowr taken with their proper signs, are the projections of OP on A'A and B'B. The ratio of • Fig. 2. the projection of OP on B'B to the absolute length of OP is dependent only on the magnitude of the angle POA, and is called the sine of that angle ; the ratio of the projection of OP on A'A to the length OP is called the cosine of the angle POA. The ratio of the sine of an angle to its cosine is called the tangent of the angle, and that of the cosine to the sine the cotangent of the angle ; the reciprocal of the cosine is called the secant, and that of the sine the cosecant of the angle. These functions of an angle of magnitude a are denoted by sin a, cos a, tan a, cot a, sec a, cosec a respectively. If any straight line BS be drawn parallel to OP, the projection of ES on either of the straight lines A'A, B'B can be easily seen to bear to RS the same ratios which the corresponding projections of OP bear to OP: thus, if a be the angle which BS makes with A'A, the projections of RS on A'A, B'B are RS cos a and RS sin a respectively, where RS denotes the absolute length RS. It must be observed that the line S12 is to be considered as parallel not to OP but to OP", and therefore makes an angle r + a with A'A; this is consistent with the fact that the projections of SR are of opposite sign to those of RS. By observing the signs of the projections of OP for the positions P, F, P', P" of P we see that the sine and cosine of the angle POA are both positive ; the sine of the angle P'0A is positive and its cosine is negative ; both the sine and the cosine of the angle FOA are negative ; and the sine of the angle P'"0.4 is negative and its cosine positive. If a be the numerical value of the smallest angle of which OP and OA are boundaries, we see that, since these straight lines also bound all the angles 2nr+ a, where as is any positive or negative integer, the sines and cosines of all these angles are the same as the sine and cosine of a. Hence the sine of any angle 2nr + a is positive if a is between 0 and r and negative if a is between n- and 2r, and the cosine of the same angle is positive if a is between 0 and tr or Orr and 2n- and negative if a is between isr and pr.

In fig. 2 if the angle POA is a, the angle P"OA is – a, P'OA is n- – a, •'0A is 1-+ a, POB is a. By observing the signs of the projections we see that Also sin(r+a)=sin(r – tr – a) = sin(tr – a)= cos a, cos(3r + a)=cos(r – tr – a)= – cos( r – a) = – sin a.

From these equations we have tan( – a) – tan a, tai(a- – a) = – tan a, tan(r + = – tan a, tan(rr – a) = cot a, tan(r + a) = – cot a, with corresponding equations for the cotangent.

The only angles for which the projection of OP on B'B is the same as for the given angle POA (=a) are the two sets of angles bounded by OP, OA and OP', 0.A; these angles are 2nr +a and 2nr + – a, and are all included in the formula rar + ( – 1)Pa, where r is any integer ; this therefore is the formula for all angles having the same sine as a. The only angles which have the same cosine as a are those hounded by OA, OP and OA, OP", and these are all included in the formula 2nr±a. Similarly it can be shown be done by supposing that a point moving in one specified direction Sign of is describing a positive length, and when moving in the opposite portions direction a negative length. Thus, if a point moving from A to B of an in-is moving in the positive direction, we consider the length AB as finite positive ; and, since a point moving from B to A is moving in the straight negative direction, we consider the length BA as negative. Hence line. 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 oue is, of course, a matter of convention.

If perpendiculars AL, BM be drawn from two points A, B on Projecany straight line, not necessarily in the same plane with AB, the tions of length LM, taken with the positive or negative sign according to straight the convention as stated above, is called the projection of AB on lines on the given straight line ; the projection of BA being ML has the each opposite sign to the projection of AB. If two points A, B be joined other. by a number of lines in any manner, the algebraical sum of the projections of all these lines is L31 - that is, the same as the projection of AB. Hence the sum of the projections of all the sides f 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 Definiangle as in fig. 1. DrawB0/3' at right angles to and equal to AA'. tion of We shall suppose that the direction from A' trigonoAlli■ to A is the- positive one for the straight line metrical