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. Georg Purbach (1423-1461), professor of mathematics at Vienna, wrote a work entitled Tractatus super propositiones Ptolemaes 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 Muller (1436-1476), known as Regiomontanus, 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 reinvented 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, wrote Opus palatinum de triangulis (see TABLES, MATHEMATICAL), which contains tables of sines, tangents and secants of arcs at intervals of 10" from o° to 90°. His method of calculation depends upon the formulae 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 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 Bartholomew Pitiscus (1561-1613), entitled Trigonometriae 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 (or Lansberghe de Meuleblecke) and Adriaan van Roomen. François Viète or Vieta (1540-1603) employed the equation (2 cos 1)-3(2 cos (6)=2 costo solve the cubic x-30x=ab(a> 1b); 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-3795y+95634y* — Vieta gave y = 2 sin, where C-2 sin 6, 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 Vieta 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 John 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 (15901634), 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 John 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. Friedrich Wilhelm v. 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 surprin the case of Oppel, since Leonhard Euler had in 1744 employed it in a memoir in the Acta eruditorum. Jean Bernoulli was the first to obtain real results by the use of the symbol V-1; he published in 1712 the general formula for tan no in terms of tan, 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. Plane Trigonometry. sup Conception of any B FIG. 1. initially coincident with a fixed straight line OA, to revolve round 0, 1. Imagine a straight line terminated at a fixed point 0, and and finally to take up any position OB. We shall pose that, when this revolv- of Angles ng straight line is turning Magnitude. 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 Before finally taking up the angle. 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 differ from one another by a positive or negative multiple of four an infinite number of positive or negative angles any two of which right angles, and all of which have the same bounding lines 04 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 be less than two right angles, but of any positive or negative magnitude, to be generated. Numerical Angular 2. 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 Measure. degree is divided into sixty equal parts called minutes; ment of 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 Magnitudes. 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 or radian 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 radian 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 number incommensurable with unity, usually denoted by. We shall indicate later on some of the methods which have been employed to approximate to the value of this number. Its value to 20 places is 3.14159265358979323846: its reciprocal to the same number of places is 0-31830988618379067153. In circular measure every angle is measured by the ratio which it bears to the unit angle. Two right angles are measured by the number, and, since the same angle is 180°, we see that the number of degrees in an angle of circular measure is obtained from the formula 180X0/. The value of the radian has been found to 41 places of decimals by Glaisher (Proc. London Math. Soc. vol. iv.); the value of 1/, from which the unit can easily be calculated, is given to 140 places of decimals in Grunerts Archiv (1841), vol. i. 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 | From these equations we have tan(-a)=-tan a, tan(x-a)= angle is indefinitely diminished, are ultimately in a ratio of equality.-tan a, tan (+a)=-tan a, tan(-a)=cot a. tan (}r+a)= 3. If a point moves from a position A to another position B on a -cot a, with corresponding equations for the cotangent. straight line, it has described a length AB of the straight line. It The only angles for which the projection of OP on B'B is the is convenient to have a simple mode of indicating in same as for the given angle POA(a) are the two sets of angles Sign of which direction on the straight line the length AB has bounded by OP, OA and OP', OA; these angles are 2n+a and Portions of been described; this may be done by supposing that a 2n+(-a), and are all included in the formula +(-1)'a, ao lafialte Straight point moving in one specified direction is describing where is any integer; this therefore is the formula for all angles Line. a positive length, and when moving in the opposite having the same sine as a. The only angles which have the same direction a negative length. Thus, if a point moving cosine as a are those bounded by OA, OP and OA, OP", and these from A to B is moving in the positive direction, we consider the are all included in the formula 2na. Similarly it can be shown length AB as positive; and, since a point moving from B to A is that n+a includes all the angles which have the same tangent 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 Projections according to the convention as stated above, is called of Straight the projection of AB on the given straight line; the Lines on projection of BA being ML has the opposite sign to the cach other. 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, taken in order, 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. IN B M A FIG. 2. as a. metrical From the Pythagorean theorem, the sum of the squares of the projections of any straight line upon two straight lines at right angles to one another is equal to the square on the Relations projected line, we get sin'a+cos?a=1, and from this between by the help of the definitions of the other functions we Trigono deduce the relations + tan'a = sec2a, 1 + cot'a = cosec1a. We have now six relations between the six Functions. functions; these enable us to express any five of these functions in terms of the sixth. The following table shows the values of the trigonometrical functions of the angles o,, *, }, 2T, and the signs of the functions of angles between these values; I denotes numerical increase and D numerical decrease:-- Angle. Sine I -1 -D -D -D +1 -D I 4. 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'. Definition We shall suppose that the direction from A to A is the ef Trigono- positive one for the straight line 404', and that from metrical B' to B for BOB'. Suppose OP of fixed length, equal Functions, 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 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 colangent of the angle; the reciprocal of the cosine is called the secant, and that of 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 RS be drawn parallel to OP. the projection of RS on either of the straight lines A'A, B'B can These are obtained as follows. (1). The sine and be easily seen to bear to RS the same ratios which the correspond- cosine of this angle are equal to one another, since sin ing projections of OP bear to OP; thus, if a be the angle which cos (1-1); and since the sum of the squares RS makes with A'A, the projections of RS on A'A, B'B are RS cos a of the sine and cosine is unity each is 1/√2. (2) # and T. metrical and RS sin a respectively, where RS denotes the absolute length Consider an equilateral triangle; the projection of one RS. It must be observed that the line SR is to be considered as side on another is obviously half a side; hence the cosine parallel not to OP but to OP, and therefore makes an angle +a of an angle of the triangle is or cos =4, and from with A'A; this is consistent with the fact that the projections of SR this the sine is found. (3) x/10, #/5, 2/5, 3/10. In the triangle are of opposite sign to those of RS. By observing the signs of the constructed in Euc. iv. 10 each angle at the base is , and the projections of OP for the positions P, P', P", P" of P we see that the vertical angle is . If a be a side and b the base, we have by sine and cosine of the angle POA are both positive; the sine of the the construction a(a-b) = b2; hence 2b=a (5-1); the sine angle P'OA is positive and its cosine is negative; both the sine and of /10 is b/2a or (5-1), and cos is a/2b=(√5+1). the cosine of the angle POA are negative; and the sine of the angle (4) . . Consider a right-angled triangle, having an angle . POA is negative and its cosine positive. If a be the numerical Bisect this angle, then the opposite side is cut by the bisector value of the smallest angle of which OP and OA are boundaries, we in the ratio of √3 to 2; hence the length of the smaller segment see that, since these straight lines also bound all the angles 2n+e, is to that of the whole in the ratio of √3 to 3+2, therefore where n is any positive or negative integer, the sines and cosines tan =√3/(√3+2)} tan # or tan=2√3. and from this of all these angles are the same as the sine and cosine of a. Hence we can obtain sin and cost. the sine of any angle 2nr+a is positive if a is between o and and negative if a is between and 2, and the cosine of the same angle is positive if a is between 0 and 1 or 1 and 2 and negative if a is between 1 and r. In fig. 2 the angle POA is a, the angle POA is -a, P'OA is -a, POA is +a, POB is {π-a. By observing the signs of the projections we see that sin(-a) sin a, sin(x-a)=sin a, sin (z+a)=sin a, Also sin(+a) = sin(x − {x− a) = CUS a, Cos(+a) = cos(x-x-α) = cos(} = − a) = —sin a, 45° cosine on a straight line making an angle + with OD, and is therefore equal to OF sin B; hence or OF cos (A+B)=OE cos A+EF cos (x+4) =OF (cos A cos B-sin A sin B), cos (A+B)=cos A cos B-sin A sin B. The angles A, B are absolutely unrestricted in magnitude, and thus this formula is perfectly general. We may change the sign of B, thus cos (A-B) =cos A cos (B)-sin A sin (-B). cos (AB)=cos A cos B+sin A sin B. or If we projected the sides of the triangle OEF on a straight line making an angle with OA we should obtain the formulae sin (AB)=sin A cos Bcos A sin B, which are really contained in the cosine formula, since we may put by the formulae -B for B. The formulae tan Atan B = E cot A cot BI cot Bcot A the above formulae. The equations (C+D) cos (C-D) cos (C-D), (C+D), (C+D) cos (C-D). (C+D) sin (C-D). tan (AB) = I tan A tan B cot (A ±B) = are immediately deducible from sin C+sin D=2 sin sin C-sin D-2 sin cos D+cos C=2 cos cos D-cos C=2 sin may be obtained directly by the method of projections. Take two equal straight lines OC, OD, making angles C, D, with OA, and draw OE perpendicular to CD. The angle which OE makes with OA is (C+D) and that which DC makes is (+C+D); the angle COE is (C-D). The sum of the projections of OD and DE on OA is equal to that of OE, and the sum of the projections of OC and CE is equal to that of OE; hence the sum of the projections of OC and OD is twice that of OE, or cos C+cos D = 2 cos (C+D) cos (C-D). The difference of the A projections of OD and OC an OA is equal to twice that of ED, hence we have the formula cos D-cos C =2 sin (C+D) sin (C-D). The other two formulae will be obtained by projecting on a straight line inclined at an angle + to OA. FIG. 4. As another example of the use of projections, we will find the sum of the series cos a +cos (a+B) +cos (a+28) + ... + cos(a+1-18). Suppose an unclosed polygon each angle which is -ẞ to be inscribed in a circle, and let A, A1, A2, Aa, A be n+1 consecutive angular points; let D be the diameter of the circle; and suppose a Arithmetical straight line drawn making an angle a with AA, then sin (A+A2+A1)=sin A1 cos A, cos A、 cos (A+A2+4)=cos A, cos A2 cos A2 We can by induction extend these formulae to the case of n angles. Assume sin (A1+A2+ . . . + A1) = S1 − S3+S¿ − ... cos (A+A+...+An) So - S2+S1~.. where S, denotes the sum of the products of the sines of r of the angles and the cosines of the remaining n-r angles; then we have sin (A1+A2+...+An+An+1)= cos A2+1(S1-S1+S1-...) +sin An(So-S2+S1~ ...). The right-hand side of this equation may be written -(S2 cos An+1+S2 sin An+ 1) + .... tan } A =(-1)" where is the integral part of A/2x, q the integral part of A/2x+}, and the integral part of A/T. Sin 4, cos 4 are given in terms of sin A by the formulae 2 sin A=(-1)♪'(1+sin A)}+(−1)(1 − sin A)♪, 2 cos A = (-1)♪′(1+sin A)} − (− 1)'' (1 − sin A)', where p' is the integral part of A/2+ and q' the integral part of A/2π-. 6. In any plane triangle ABC we will denote the lengths of the sides BC, CA, AB by a, b, c respectively, and the angles BAC, ABC, ACB by A, B, C respectively. The fact that the projec tions of b and c on a straight line perpendicular to the Properties of Triangles. side a are equal to one another is expressed by the equation b sin C=c sin B; this equation and the one obtained by projecting c and a on a straight line perpendicular to a may be written a/sin A=b/sin B=c/sin C. The equation a=b cos C+c cos B expresses the fact that the side a is equal to the sum of the projections of the sides b and c on itself; thus we obtain the equations s(s-a) ‚ sin A = ¿¿{s(s—a) (s — b) (s —c);'. where s denotes (a+b+c), can be deduced by means of the dimidiary formula. From any general relation between the sides and angles of a triangle other relations may be deduced by various methods of transformation, of which we give two examples. a. In any general relation between the sines and cosines of the angles A, B, C of a triangle we may substitute pA+qB+rC, TA +pB+qC, qA +rB+pC for A, B, C respectively, where P. q. are any quantities such that p+q++ is a positive or negative multiple of 6, provided that we change the signs of all the sines. Suppose p+g+r+1-6n, then the sum of the three angles 2n-(pA +gB+rC), 2n − (rA +pB+qC), 2nx − (qA +rB+pC) is = ; and, since the given relation follows from the condition A+B+C . we may substitute for A, B, C respectively any angles of which the sum is ; thus the transformation is admissible. - B. It may easily be shown that the sides and angles of the triangle formed by joining the feet of the perpendiculars from the angular points A, B, C on the opposite sides of the triangle ABC are respectively a cos A, b cos B, c cos C. -2A, 2B, -2C; we may therefore substitute these expressions for a, b, c, A, B, C respectively in any general formula. By drawing the perpendiculars of this second triangle and joining their feet as before, we obtain a triangle of which the sides are-a cos A cos 2A, b cos B cos 2B, - cos C cos 2C and the angles are 44, 4B-x. 4C-x; we may therefore substitute these expressions for the sides and angles of the original triangle; for example, we obtain thus the formula a cos? A cos 2A -b cos? B cos 2B-c2 cos C cost 2C cos 44 2bc cos B cos C cos 2B cos 2C This transformation obviously admits of further exten sion. = and Sub +(-1)2 Formulae cos 2A=cos2 A − sin2A =2 cos2 A-1=1-2 sin2 A, for Multiple 2 tan A sin 2A-2 sin A cos A, tan 24 = 1-tan' A' sin 343 sin A-4 sin3 A, cos 3A = 4 cos3 A −3 cos A, n (n − 1) (n − 2) sin nA = n cos"- A sin A cos"- A sin' A+... 3! ‚n (n − 1) . . . (n −2r) (27 +1)! Solution of Triangles. Z tan A=10+ log (s-b)+} log (sc)-} log - log (s-s) cos"-tr1 A sin*+ A, and two corresponding formulae for the other angles. (1) The three sides of a triangle ABC being given, the angles can be determined by the formula (2) The two sides a, b and the included angle C being given, the | OBC. angles A, B can be determined from the formulae A+B=-C. L tan (A-B)=log (a−b)—log (a+b) + L cot }C, and the side c is then obtained from the formula log clog a+L sin C-L sin A. (3) The two sides a, b and the angle A being given, the value of sin B may be found by means of the formula L sin B-L sin A+log blog a; this gives two supplementary values of the angle B, if b sin A <a. If b sin A > a there is no solution, and if b sin A= a there is one solution. In the case b sin A < a, both values of B give solutions provided b> a, but the acute value only of B is admissible if b <a. The other side c can be then determined as in case (2). (4) If two angles A, B and a side a are given, the angle C is determined from the formula C=-A-B and the side b from the formula log b = log a+L sin B-L sin A. Triangles The area of a triangle is half the product of Areas of a side into the perpendicular from the opposite and Quadri-angle on that side; thus we obtain the expressions be sin A, s(s-a) (s-b)(s-c) for the area of a laterals. triangle. A large collection of formulae for the area of a triangle are given in the Annals of Mathematics for 1885 by M. Baker. Let a, b, c, d denote the lengths of the sides AB, BC, CD, DA respectively of any plane quadrilateral and A+C=2a; we may obtain an expression for the area S of the quadrilateral in terms of the sides and the angle a. We have 2S ad sin A +bc sin(2a-A) and }(a2+d2-b2-c2) = ad cos A-bc cos (2a-A); hence 45+(a+ď2 — b2 — c2)2 = a2d+b2c-2abcd cos 2a. If 25 = a+b+c+d, the value of S may be written in the form S=1s(s-a)(s-b)(s—c)(s—d)-abcd cosa. Let R denote the radius of the circumscribed circle, r of the inscribed, and 71, 72, 7 of the escribed circles of a triangle Radil of Cir ABC; the values of these radii are given by the followcumscribed, ing formulae:R=abc/4S-a/2 sin A, Inscribed C, and Escribed =S/s (s-a)tan {A =4R sin A sin B sin T=S/(-a)=s tan 14 =4R sin 14 cos B cos C. Spherical Trigonometry. Circles of a Triangle. 7. We shall throughout assume such elementary propositions in spherical geometry as are required for the purpose of the investigation of formulae given below. A spherical triangle is the portion of the surface of a sphere bounded by three arcs of great circles of the sphere. If BC, CA, AB denote these arcs, the circular measure of the Definition angles subtended by these arcs respectively at the of Spherical centre of the sphere are the sides a, b, c of the spherical Triangle. triangle ABC; and, if the portions of planes passing through these arcs and the centre of the sphere be drawn, the angles between the portions of planes intersecting at A, B, C respectively are the angles A, B, C of the spherical triangle. It is not necessary to consider triangles in which a side is greater than r, since we may replace such a side by the remaining arc of the great circle to which it belongs. Since two great circles intersect Associated each other in two points, there are eight triangles of Triangles. which the sides are arcs of the same three great circles. If we consider one of these triangles ABC as the fundamental one, then one of the others is equal in all respects to ABC, and the remaining six have each one side equal to, or common with, a side of the triangle ABC, the opposite angle equal to the corresponding angle of ABC, and the other sides and angles supplementary to the corresponding sides and angles of ABC. These triangles may be called the associated triangles of the fundamental one ABC. It follows that from any general formula containing the sides and angles of a spherical triangle we may obtain other formulae by replacing two sides and the two angles opposite to them by their supplements, the remaining side and the remaining angle being unaltered, for such formulae are obtained by applying the given formulae to the associated triangles. If A'. B', C' are those poles of the arcs BC, CA, AB respectively which lie upon the same sides of them as the opposite angles A, B, C, then the triangle A'B'C' is called the polar triangle of the triangle ABC. The sides of the polar triangle are -A. -B, C, and the angles -a, T-b, x-c. Hence from any general formula connecting the sides and angles of a spherical triangle we may obtain another formula by changing each side into the supplement of the opposite angle and each angle into the supplement of the opposite side. FIG. 5 on 8. Let O be the centre of the sphere which is the spherical triangle ABC. Draw AL perpendicular to OC and AM perpendicular to the plane Then the projection of OA on OB is the sum of the projections of OL, LM, MA on the same straight line. Since AM has no projection on any straight line in the plane OBC, this gives angles. Funda mental Equations OA cos c=OL cos a+LM sin a. cos a=cos b cos c+sin b sin c cos A cos c=cos a cos b+sin a sin b cos C (1) These formulae (1) may be regarded as the fundamental equations In the figures we have AM=AL sin C=r sin b sin C, where r = we shall find below a symmetrical form for k. If we eliminate cos b between the first two formulae of (1) we have cos a sinc=sin b sin c cos A+sin c cos c_sin a cos B; cot a sin b=cot A sin C+cos b cos C follow at once from (a), (8), (7). cos ccot A cot B cos Acos A sin B cos B=cos b sin A These are the formulae which are used for the solution of right-angled triangles. Napier gave mnemonical rules for remembering them. The following proposition follows easily from the theorem in equation (3): If AD, BE, CF are three arcs drawn through A, B, C to meet the opposite sides in D, E, F respectively, and if these arcs pass through a point, the segments of the sides satisfy the relation sin BD sin CE sin AF-sin CD sin AE sin BF; and conversely if this relation is satisfied the arcs pass through a point. From this theorem it follows that the three perpendiculars from the angles on the opposite sides, the three bisectors of the angles, and the three arcs from the angles to the middle points of the opposite sides, each pass through a point. 9. If D be the point of intersection of the three Formulæ bisectors of the angles A, B, C, and if DE be drawn for Sine perpendicular to BC. it may be shown that BE and Cosine {(a + c - b) CE=(a+b-c), and the angles BDE, ADC are supplementary. We have Angles. sin ADB sin b sin ADC. therefore sin? 14 sin A sin CD sin A sin BD sin CD sin CDE sin BDE. But sin BD sin BDE=sin BE and by multiplication sin (a+c-b) sin (a+b−c) { 13. The formulae we have given are sufficient to determine three parts of a triangle when the other three parts are given; moreover (7) such formulae may always be chosen as are adapted Solution of to logarithmic calculation. The solutions will be unique Triangles. except in the two cases (1) where two sides and the angle opposite one of them are the given parts, and (2) where two angles and the side opposite one of them are given. sin A = 2{sin (a+b+c) sin (b+c−a) sin (c+a−b) sin §(a+b−c)14 Of Half cos= { a sin= cos (A+C-B) cos (A+B-C{} -cos (B+C-A) cos }(A+B+C {{ a -cos (B+C-A) cos (A+B+C { } tan-= cos (A+C-B) cos (A+B-CS we have. kk' = 1 (9) (10) Suppose a, b. A are the given parts. We determine B from the formula sin B-sin b sin A/sin a; this gives two Ambiguous supplementary values of B, one acute and the other Cases. obtuse. Then C and c are determined from the sin (a-b) sin (A+B) tan C=sin(a+b) cot }(A — B), tan }c= sin (4-B) tan (a-b). Now tan C, tan le, must both be positive; hence A-B and a-b must have the same sign. We shall distinguish three cases. First, (11) suppose sin b<sin a; then we have sin B<sin A. Hence A lies between the two values of B, and therefore only one of these values Ifk'-(1-cos A-cos B-cos C-2 cos A cos Bcos C/sin A sin B sin C, is admissible, the acute or the obtuse value according as a is greater (12) or less than b; there is therefore in this case always one solution. 10. Let E be the middle point of AB; draw ED at right angles to Secondly, if sin b>sin a, there is no solution when sin b sin A>sina; AB to meet AC in D: then DE but if sin b sin A<sin a there are two values of B, both greater bisects the angle ADB. or both less than A. If a is acute, a-b, and therefore A-B, is Delambre's Let CF bisect the angle negative; hence there are two solutions if A is acute and none if A Formulae. DCB and draw FG per-fa is obtuse there is no solution unless A is obtuse, and in that is obtuse. These two solutions fall together if sin b sin A = sin a. pendicular to BC, then Also sin CG sin CF sin CFG and sin EB sin BF sin EFB; sin(a+b)sinC=cos (A-B)sinc (15) (16) The four formulae (13), (14), (15) (16) were first given by Delambre in the Connaissance des Temps for 1808. Formulae equivalent to these were given by Mollweide in Zach's Monatliche Correspondenz for November 1808. They were also given by Gauss (Theoria motus, 1809), and are usually called after him. 11. From the same figure we have Napler's tan FG-tan FCG sin CG=tan FBG sin BG; Analogies. therefore cot Csin (a - b)tan}(A — B)sin}(a+b), I case there are two, which coincide as before if sin b sin Asin a. Hence in this case there are two solutions if sin b sin Asin a and the two parts A, a are both acute or both obtuse, these being coincident in case sin b sin Asin a; and there is no solution if one of the two A, a, is acute and the other obtuse, or if sin b sin A>sin a, Thirdly, if sin b=sin a then BA or A. If a is acute, a-b is zero or negative, hence A-B is zero or negative; thus there is no solution unless A is acute, and then there is one. Similarly, if a is obtuse, A must be so too in order that there may be a solution. If a=b, there is no solution unless A, and then there are an infinite number of solutions, since the values of C and c become indeterminate. The other case of ambiguity may be discussed in a similar manner, or the different cases may be deduced from the above by the use of the polar triangle transformation. The method of classification according to the three cases sin busin a was given by Professor Lloyd Tanner (Messenger of Math., vol. xiv.). 14. If is the angular radius of the small circle inscribed in the triangle ABC, we have at once tan tan 4 sin (sa), where 25=a+b+c; from this we can derive the formulae tan r=n cosec s = N sec A sec B sec |C= sin a sin B sin C sec 14 (21) where n, N denote the expressions Radil of {sin s sin (s-a) sin (s—b) sin (s-c)}}, associated triangles; thus, applying the above formulae to the (19) (20) The formulae (17), (18), (19), (20) are called Napier's " Analogies "; they were given in the Mirif. logar. canonis descriptio. The following relations follow from the formulae just given:- 12. If we use the values of sin la, sin lb, sin c, cos la, cos b. cos c, given by (9), (10) and the analogous formulae obtained by interchanging the letters we obtain by multiplication Schmeisser's sin la cos 4b sin C=sin c cos (B+C-A) cos la cos b sin C-cosiccos (A+B-C) Formulae. sin la sin lb sin C = cos }ccos } (A+B+C) These formulae were given by Schmiesser in Crelle's Journ., vol. x. tan r tan tan ra tan ran2, sin2 scot 7 tan, tan tan r1, 15. If E=A+B+C-, it may be shown that E or for Formulae Spherical hence 13 E) COS C = -cos (a+b) cos je + cs (a + b); |