7. Again, since by the first of the above theorems, sin AXcos B sin(A+B)+ sin (A-B),if A+B=S,and A-B=D, then (Lem. 2.) A = S+D S-D wherefore sin ; S+D X cos S-D 2 sin S+ sin D. But as S and D may be any arches whatpreserve the former notation, they may be called A and B, which also express any arches whatever, thus, ever, to A+B A B 2 cos X cos In the same manner, from Theor. 2. is derived, cos B+cos A. From the 3d, 2 2 2 2 In all these Theorems, the arch B is supposed less than A. 8. Theorems of the same kind with respect to the tangents of arches may be deduced from the preceding. Because the tangent of any arch is equal to the sine of the arch divided by its cosine, tan (A+B) sin (A+B) But it has just shewn, that cos (A+B) sin (A+B)=sin AXcos B+cos AX sin B, and that cos (A+B)= cos AXcos B-sin Axsin B; therefore tan (A+B) and dividing both the numerator and denominator of this fraction by cos AX cos B, tan (A+B) tan A+tan B 1-tan AXtan B In like manner, tan (A-B) tan A--tan B 1+tan Axtan B 1 9. If the theorem demonstrated in Prop. 3. be expressed in the same manner with those above, it gives cos A 1, 10. In all the preceding theorems, R, the radius is supposed because in this way the propositions are most concisely expressed, and are also most readily applied to trigonometrical calculation. 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 AXcos 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 AXcos 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. Fo OR 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. Q. E. D. DEFINITIONS. 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. II. The pole of a great circle of a sphere is a point in the superficies of the sphere, from which all straight lines drawn to the circumference of the circle are equal. III. A spherical angle is an angle on the superficies of a sphere, contained by the arches of two great circles which intersect one another; and is the same with the inclination of the planes of these great circles. IV. A spherical triangle is a figure, upon the superficies of a sphere, comprehended by three arches of three great circles, each of which is less than a semicircle. PROP. II. The arch 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 arch of a great circle, pass through D, and meet ABC in C, the arch DC is a quadrant. Let the circle, of which CD is an arch, 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 arches, (28. 3.) the arch AD=the arch DC; but ADC is a semicircle, therefore the arches AD, DC are each of them quadrants. Q. E. D. Ak E C 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 great 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 arch 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 arch 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 the circles AB, AC, and is (def. 3.) equal to the spherical angle BAC ; but the arch BC measures the angle BDC, therefore it also measures the spherical angle BAC. *Q. E. D. B COR. If two arches of great circles, AB and AC, which intersect one another in A, be each of them quadrants, A will be the pole of the great circle which passes through B and C, the extremities of those arches. For since the arches AB and AC are quadrants, the angles ADB, ADC are right angles, and AD is therefore perpendicular to the plane BDC, that is to the plane of the great circle which passes through B and C. The point A is therefore (Cor. 1. 2.) the pole of the great circle which passes through B and C. PROP. VI. If the planes of two great circles of a sphere be at right angles to one another, the circumference of each of the circles passes through the poles of the other; and if the circumference of one great circle pass through the poles of another, the planes of these circles are at right angles. Let ACBD, AEBF be two great circles, the planes of which are at right angles to one another, the poles of the circle AEBF are in the circumference ACBD, and the poles of the circle ACBD in the circumference AEBF. From G the centre of the sphere, draw GC in the plane ACBD perpendicular to AB. Then, because GC in the plane ACBD, at *When in any reference no mention is made of a Book, or of the Plane Trigonome. try, the Spherical Trigonometry is meant. |