PROPOSITION X.

If A and B be any two arcs, it is proposed to shew, geometrically, that

F

D

H

K/L

Let C E be the arc A, and CD the arc B; draw the sines DL and E K, and produce E K to meet the circle again in I. Draw D M F parallel to the diameter GHC; draw the radii E H, D H, and 1 H; and join E F, FI, and ID. Now (Theo. 60. Geo.) EI is bisected in K, therefore M I, the sum of M K and K I, is the sum of the sines KE and LD; and ME is the difference of the same sines. The arc DCI, or the sum of the arcs CD and C E, is the measure of the angle at the centre D HI; therefore the half of D I, or half the sum C D and CE, is the measure of the angle D F I, at the circumference and in the same manner may half the difference of CD and CE be shewn to be the measure of the angle D F E.

But as F M is perpendicular to E I, therefore MI: ME: IF M: tan EFM; or as sin A+ sin B: sin A sin B

tan

tan

A+ B

A B

: tan

2

2

Cor. 1. As H K is the cosine of E C, and H L is the cosine of DC, therefore FM is the sum, and K L or MD the difference of the cosines of C D and C E.

But FM: MD:: tan FIM: tan DIM, or :: cot M FI: tan A+ B 2

EF D. That is, cos A+ cos B: cos B

cos A : cot

A-B

2

: tan

Cor. 2. FM:MI:

+ sin B: rad: tan

rad: tan IF M; or cos A+ cos B: sin A A+ B

2

Cor 3. FM: ME: : rad: tan MFE; or cos A + cos B: sin A

Cor 4. IM: MD:: rad: tan MID; cos sin A+ sin B: cos B

The properties demonstrated in this proposition have been otherwise deduced in the scholium to Prop. 2; they are there stated in the form of equations among the formulæ marked (4.)

PROPOSITION XI.

The secant of any arc is equal to the sum of its tangent, and the tangent of half its complement.

B

D

Let A B be any arc, A D its tangent, and C D its secant; produce A D till DE is equal to D C, and join CE. Then the angle E is the complement of A CE; and as D C and D E are equal, the angle E is also the complement of half the angle D. Hence A C E is equal to half A D C, or to half the complement of A C D, and consequently A E is the tangent of half the comple

A

E

ment of ACD; therefore D A the tangent of A B, and AE the tangent of half its complement, are together equal to D C, the secant of the same arc.

Remark. By this proposition, when the natural tangents have been computed, the natural secants may readily be obtained.

EXAMPLES FOR EXERCISE IN THE THEORY OF TRIGONOMETRY.

A

1. SHEW geometrically that rad. rad + cos A = 2 cos2. ; that

3. Demonstrate geometrically that rad. sect 2 A = tan A. tan 2 A + rad2.

4. Shew that in any plane triangle the base is to the sum of the other two sides as the sine of half the vertical angle is to the cosine of half the difference of the angles at the base.

5. Shew that the base of a plane triangle is to the difference of the other two sides as the cosine of half the vertical angle is to the sine of half the difference of the angles at the base.

6. The difference of two sides of a triangle is to the difference of the segments of the third side, made by a perpendicular from the opposite angle, as the sine of half the vertical angle is to the cosine of half the difference of the angles at the base; required the proof?

7. a, b, c, being the sides of a plane triangle, and A the angle opposite the side a, and S half the sum of a, b, and c; shew that

9. Demonstrate geometrically that rad. sin 2 A = tan A. suvers 2 A. 10. As the greater of two sides of a triangle is to the less, so radius is to the tangent of a certain angle. Shew that radius is to the tangent of the difference between this angle and half a right angle, as the tangent of half the sum of the angles at the base is to the tangent of half their difference.

ELEMENTARY PRINCIPLES

OF

SPHERICAL TRIGONOMETRY.

1. A SPHERE is a solid bounded by a curve surface, every point of which is at the same distance from a point within, called the centre. Cor. 1. A sphere may be conceived to be generated by the revolution of a semicircle, about its diameter,

Cor. 2. The section of a sphere, by a plane passing through the centre, is a circle; for every straight line drawn in the plane, from the centre to the surface of the sphere, is equal to the radius.

Cor. 3. All circles of a sphere whose planes pass through the centre are equal; and any two of them bisect each other.

2. The circles of a sphere whose planes pass through the centre are called great circles; and those whose planes do not pass through the centre are called less circles.

3. The pole of a circle is a point on the surface of the sphere equally distant from every point in the circumference of the circle. 4. A spherical angle is an angle on the surface of a sphere contained by the arcs of two great circles which intersect each other; and it is the same with the inclination of the planes of the circles; or with the plane angle formed by the tangents of the arcs at the point of intersection.

5. A spherical triangle is a figure formed upon the surface of a sphere by the intersection of three planes which meet in the centre of the sphere.

PROPOSITION I.

Every section of a sphere by a plane is a circle.

If the plane pass through the centre, then as every point in the surface of the sphere is equidistant from its centre the section is a plane figure, every point of whose periphery is equidistant from a certain point within it, and the figure is therefore a circle.

But if the plane do not pass through the centre, let CDEF be the section, and C, D any two

B

points in its periphery. Let A B be a perpendicular from the centre A on the plane of the section, and join A C, A D. Then the angles A B C, A B D are right angles, A C is equal to A D, and the perpendicular A B common to both the right angled triangles A B C, A B D; therefore the bases BC and BD are equal. Hence all lines drawn from B in the plane C D E F, to the periphery of that plane are equal; and the figure is consequently a circle.

PROPOSITION II.

The arc of a great circle intercepted between the pole and the circumference of another great circle is a quadrant.

E

Let D be the pole of the circle A B C, and A D C a great circle passing through D, then as ADC and ABC are semicircles, A C the common sections of their planes is a diameter, and consequently E the centre of the sphere is in the line A C; join A D, D C, and D E. Then as A D and D C are equal, and the two sides A E, E D are respectively equal to C E and E D, each of them indeed being the radius of the sphere, therefore the angles A E D and DE C are equal, and they are consequently right angles; whence A D and DC are quadrants.

B

Cor. Every circle has two poles, one on each side of its plane, and they are the extremities of a diameter perpendicular to that plane,

PROPOSITION III.

If two great circles, as B A, CA, intersect each other in A, on the surface of a sphere whose centre is D, and if B C be an arc of the great circle, whose pole is A, B C is the measure of the spherical angle BA C. For join A D, BD, and C D, then as A is the pole of B C, AB, and AC are quadrants, and therefore the angles AD B and A D C are right angles,

Hence the angle B D C is the inclination of the planes of A B and A C, and it is therefore equal to the spherical angle B A C. But B D C is measured by the arc B C, therefore the spherical angle BAC is also measured by the arc B C.

B

A

Cor. 1. As AD is perpendicular to D B and D C, it is perpendicular to the plane B D C, therefore the planes A D B and A D C which pass through A D are also perpendicular to the plane BDC; and consequently the spherical angles A B C and A C B are right angles. Hence the great circles which pass through the pole of another great circle, cut it at right angles.

Cor. 2. Great circles, whose planes are at right angles to the plane of one and the same great circle, meet in the poles of that circle.

Cor. 3. If the planes of two great circles be at right angles to each other, 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.

PROPOSITION IV.

In isosceles spherical triangles the angles which are opposite the equal sides are equal; and if two angles of a spherical triangle are equal, the sides which are opposite those angles are equal.

E

Let A B C be an isosceles spherical triangle, A B and A C being the equal sides, and let D be the centre of the sphere. Let BE and CE be tangents to A B and A C, and B F and CF two tangents drawn from B and C in the plane DB C, intersecting each other in F, and join F E. Then as A B and A C are equal, their tangents BE and CE are equal; and they have therefore the common secant D E; and (Geo. Theo. 67. Cor. 2.) B F is equal to F C, and F E common to the triangles BFE and CFE, consequently the plane angle E B F is equal to the plane angle E CF, or the spherical angles A B C and A C B are equal.

B F

C

Again, the same construction being made, suppose the spherical angles A B C and A C B, or the plane angles EBF and ECF to be equal; then as DB is at right angles to B E and B F, it is at right angles to the plane B E F; and therefore the plane BDC which passes through D B is at right angles to the plane B E F. For a like reason, the plane B D C is at right angles to the plane E C F, and therefore FE the common section of the planes E B F and E F C is at right angles to the plane D B C. Hence E F B and E F C are right angles, and consequently as the angles EBF and ECF are equal, and the side B F equal to the side F C, the triangles BEF and CEF are identical; whence B E is equal to C E, and as D B and D C are also equal, and the angles D B E and DCE being right angles are equal, the angle EDB is equal to the angle E D C, therefore A C is equal to A B.

PROPOSITION V.

Any two sides of a spherical triangle are together greater than the third side.

Let A B C be a spherical triangle, and let D be the centre of the sphere, join A D, B D, and C D.. Then the solid angle at D is contained by three plane angles, A D B, A D C, and B D C, any two of which are together greater than the third. (Geo. Theo. 99.)

D

B

A

But these angles are measured respectively by the arcs A B, A C, and B C, therefore any two sides of the spherical triangle A B C are together greater than the third side.