sin. A+sin. (A+2B)=2 cos. Bx sin. (A+B), or sin. (A+2B)=2 cos. Bx sin. (A+B)—sin. A. By means of these theorems, a table of the sines, and consequently also of the cosines, of arcs of any number of degrees and minutes, from 0 to 90, sin. A the table of tangents cos. A' may be constructed. Then, because tan. A= is computed by dividing the sine of any arc by the cosine of the same arc. When the tangents have been found in this manner as far as 45°, the tangents for the other half of the quadrant may be found more easily by another rule. For the tangent of an arc above 45° being the co-tangent of an arc as much under 45°; and the radius being a mean proportional between the tangent and co-tangent of any arc (1. Cor. def. 9), it follows, if the difference between any arc and 45° be called D, that tan. (45°—D): 1 :: 1: tan. (45°+D), so that tan. (45°+D)=; 1 tan. (450-D)' Lastly, the secants are calculated from (Cor. 2. def. 9.) where it is shewn that the radius is a mean proportional between the cosine and the secant of any arc, so that if A be any arc, sec. A=; 1 cos. A The versed sines are found by subtracting the cosines from the radius. 5. The preceding Theorem is one of four, which, when arithmetically expressed, are frequently used in the application of trigonometry to the solution of problems. 1mo, If in the last Theorem, the arc AC=A, the arc BC=B, and the radius EC I, then AD=A+B, and AB-A-B; and by what has just been demonstrated, 1 cos. B: sin. A: sin. (A+B)+ sin. (A−B), sin. Ax cos. B= sin. (A+B)+ (A-B). 2do, Because BF, IK, DH are parallel, the straight lines BD and FH are cut proportionally, and therefore FH, the difference of the straight lines FE and HE, is bisected in K; and therefore, as was shewn in the last Theorem, KE is half the sum of FE and HE, that is, of the cosines of the arcs AB and AD. But because of the similar triangles EGC, EKI, EC :EI::GE: EK; now, GE is the cosine of AC, therefore, R: cos. BC or 1: cos. B:: cos. A: cos. AC: cos. AD+ cos. AB, cos. Ax cos. B= cos. 3tio, Again, the triangles IDM, CEG are equiangular, for the angles KIM, EID are equal, being each of them right angles, and therefore, taking away the angle EIM, the angle DIM is equal to the angle EIK, that is, to the angle ECG; and the angles DMI, CGE are also equal, being both right angles, and therefore the triangles IDM, CGE have the sides about their equal angles proportionals, and consequently, EC: CG :: DI : IM; now, IM is half the difference of the cosines FE and EH, therefore, R: sin. AC:: sin. BC: cos. AB- cos. AD, or 1: sin. A :: sin. B: cos. (A-B)- cos. (A+B); and also, sin. AX sin. B=cos. (A-B)- cos. (A+B). 4to, Lastly, in the same triangles ECG, DIM, EC: EG:: ID: DM; now, DM is half the difference of the sines DH and BE, therefore, R: cos. AC: sin. BC: sin. AD— sin. AB, or 1: cos. A :: sin. B : sin. (A+B)—{ sin. (A+B); and therefore, cos. Ax sin. B= sin. (A+B)— sin. (A—B). 6. If therefore A and B be any two arcs whatsoever, the radius being supposed 1; I. sin. Axcos B= sin. (A+B)+ sin. (A−B). sin. Axcos. B+cos. Ax sin. B=sin. (A+B). 7. Again, since by the first of the above theorems, sin. Ax cos. B={sin. (A+B)+} sin. (A—B), if A+B=S, and A-B=D, S-D S-D A+B sin. A-B 2 -=sin. S+ D. But as S and D may be any arcs whatever, to preserve the former notation, they may be called A and B, which also express any arcs whatever: thus, 2 -X cos. = sin. A+ sin. B, or sin. A+sin. B. In all these Theorems, the arc B is supposed less than A. 8. Theorems of the same kind with respect to the tangents of arcs may be deduced from the preceding. Because the tangent of any arc is equal to the sine of the arc divided by its cosine, sin. (A+B) tan. (A+B)= But it has just been shewn, that cos. (A+B) sin. (A+B)=sin. Ax cos. B+cos. Ax sin. B, and that = cos. (A+B)=cos. Ax cos. B-sin. A× sin. B; therefore, tan. (A+B) sin. Ax cos. B+cos. Ax sin. B cos. Ax cos. B-sin. AX sin. B' minator of this fraction by cos. Ax cos. B, tan. (A+B): and dividing both the numerator and deno -tan. A+tan. B 1 tan. Axtan. B 9. If the Theorem demonstrated in Prop. 3, be expressed in the same manner with those above, it gives sin. A+sin. B tan. (A+B) sin. A sin. B tan. Also by Cor. 1, to the 3d, cos. A+cos. B cot. = (A-B) (A+B) cos. A-cos. B tan. (A-B) And by Cor. 2, to the same proposition, sin. Asin. B tan. (A+B) cos. A+cos. B cos. A+cos. B = R = tan. } (A+B). or since R is here supposedi, 10. In all the preceding Theorems, R, the radius, is supposed =1, because in this way the propositions are most concisely expressed, and are also most readily applied to trigonometrical circulation. But if it be required to enunciate any of them geometrically, the multiplier R, which has disappeared, by being made = 1, must be restored, and it will always be evident from inspection in what terms this multiplier is wanting. Thus, Theor. 1, 2 sin. A x cos. B=sin. (A+B)+sin. (A-B), is a true proposition, taken arithmetically; but taken geometrically, is absurd, unless we supply the radius as a multiplier of the terms on the right hand of the sine of equality. It then becomes 2 sin. Ax cos. B=R (sin. (A+B)+sin. (A−B)); or twice the rectangle under the sine of A, and the cosine of B equal to the rectangle under the radius, and the sum of the sines of A+B and A-B. In general, the number of linear multipliers, that is, of lines whose numerical values are multiplied together, must be the same in every term, otherwise we will compare unlike magnitudes with one another. The propositions in this section are useful in many of the higher branches of the Mathematics, and are the foundation of what is called the Arithmetic of Sines. ELEMENTS OF SPHERICAL TRIGONOMETRY. PROP. I. If a sphere be cut by a plane through the centre, the section is a circle, having the same centre with the sphere, and equal to the circle by the revolution of which the sphere was described. FOR all the straight lines drawn from the centre to the superficies of the sphere are equal to the radius of the generating semicircle, (Def. 7. 3. Sup.). Therefore the common section of the spherical superficies, and of a plane passing through its centre, is a line, lying in one plane, and having all its points equally distant from the centre of the sphere; therefore it is the circumference of a circle (Def. 11. 1.), having for its centre the centre of the sphere, and for its radius the radius of the sphere, that is, of the semicircle by which the sphere has been described. It is equal, therefore, to the circle of which that semicircle was a part. DEFINITIONS. 1. ANY circle, which is a section of a sphere by a plane through its centre, is called a great circle of the sphere. COR. All great circles of a sphere are equal; and any two of them bisect one another. They are all equal, having all the same radii, as has just been shewn; and any two of them bisect one another, for as they have the same centre, their common section is a diameter of both, and therefore bisects both. 2. The pole of a great circle of a sphere is a point in the superficies of the sphere, from which all strai rht lines drawn to the circumference of the circle are equal. 3. A spherical angle is an angle on the superficies of a sphere, contained by the arcs of two great circles which intersect one another; and is the same with the inclination of the planes of these great circles. 4. A spherical triangle is a figure, upon the superficies of a sphere, comprehended by three arcs of three great circles, each of which is less than a semicircle. PROP. II. The arc of a great circle, between the pole and the circumference of another great circle, is a quadrant. Let ABC be a great circle, and D its pole; if DC, an arc of a great circle, pass through D, and meet ABC in C, the arc DC is a quadrant. D Let the circle, of which CD is an arc, meet ABC again in A, and let AC be the common section of the planes of these great circles, which will pass through E, the centre of the sphere: Join DA, DC. Because AD=DC, (Def. 2.), and equal straight lines, in the same circle, cut off equal arcs (28. 3.), the arc AD = the arc DC; but ADC is a semicircle, therefore the arcs AD, DC are each of them quadrants. B COR. 1. If DE be drawn, the angle AED is a right angle; and DE being therefore at right angles to every line it meets with in the plane of the circle ABC, is at right angles to that plane (4. 2. Sup.). Therefore the straight line drawn from the pole of any great circle to the centre of the sphere is at right angles to the plane of that circle; and, conversely, a straight line drawn from the centre of the sphere perpendicular to the plane of any greater circle, meets the superficies of the sphere in the pole of that circle. COR. 2. The circle ABC has two poles, one on each side of its plane, which are the extremities of a diameter of the sphere perpendicular to the plane ABC; and no other points but these two can be poles of the circle ABC. PROP. III. If the pole of a great circle be the same with the intersection of other two great circles: the arc of the first mentioned circle intercepted between the other two, is the measure of the spherical angle which the same two circles make with one another. Let the great circles BA, CA on the superficies of a sphere, of which the centre is D, intersect one another in A, and let BC be an arc of another great circle, of which the pole is A; BC is the measure of the spherical angle BAC. Join AD, DB, DC; since A is the pole of BC, AB, AC are quadrants (2.), and the angles ADB, ADC are right angles: therefore (4. def. 2. Sup.), the angle CDB is the inclination of the planes of B |