For AF is the secant of the angle A, to the radius AD; and AC is to AB, as AF is to AD, that is AC is to AB as the secant of A, is to the radius.

Theorem.

The sides of a plane triangle are to one another as the sines of the opposite angles.

B

From any angle of a triangle, ABC, draw BD perpendicular to AC. Then, by the preceding theorem AB is to BD as the radius is to the sine of the angle A, and BD is to BC as the sine of the angle C, is to the radius. Therefore (as is proved by a geometrical proposition) inversely, AB is to BC, as the sine of the angle C, to the radius.

EXAMPLES OF THEOREMS IN SPHERICAL TRIGO

NOMETRY.

Theorem.

If a sphere be cut, by a plane, through the centre, the section is a circle.

The truth of this proposition is evident from the definition which geometry gives of a sphere. A sphere is a solid figure, described by the revolution of a semi-circle about a diameter.

Definitions.

1. Any circle, which is a section of a sphere, by a plane passing through its centre, is called a great circle of the sphere.

Corollary. All great circles of a sphere are equal, and the centre of the sphere is their common centre, and any two of them bisect each other.

2. The pole of a great circle of the sphere is a point in the superficies of the sphere, from which all straight lines drawn to the circumference of the circle are equal.

3. A spherical angle is that which, on the superficies of a sphere, is contained by two arches of great circles, and is the same with the inclination of the planes of those great circles.

4. A spherical triangle is a figure, upon the superficies of a sphere, comprehended by three arches of three great circles, each of which arches is less than a semi-circle.

Theorem.

The arch of a great circle between the pole and the circumference of another circle, is a quadrant.

D

E

A

B

D

Let ABC be a great circle, and D its pole. Let the great circle ADC pass through D, and let AEC be the common section of the planes of the two circles, which will pass through E the centre of the circle ADC. Join DA, and DC. Because the chord DA is equal to the chord DC (by the second definition) the arch DA is equal to the arch DC. Now ADC is a semicircle; therefore the arches AD and DC are quadrants.

Corollary 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. Therefore the right 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.

Corollary 2. The circle has two poles D, D', one on each side of its plane: which poles are the extremities of a diameter of the sphere perpendicular to the plane ABC.

Theorem.

Two great circles, whose planes are perpendicular to each other, pass through each others poles.

Let A CBD, AE BF, be two great circles, the planes of which are at right angles to each other. From G the centre of the sphere, draw GC in the plane AD BC, perpendicular to AB; then GC is also perpendicular to the plane AE BF (by a geometrical theorem); therefore C is the pole of the circle AE BF; and if CG be produced to D, D is the other pole of the circle AE BF.

In the same manner, by drawing GE in the plane AEBF, perpendicular to AB, and producing it to F, it is shown that E and F are the poles of the circle ADBC.

Corollary 1. If two great circles pass through each others poles, their planes are perpendicular to each other.

Corollary 2. If, of two great circles, the first pass through the poles of the second, the

second also passes through the poles of the first.

QUESTIONS.

How is proved the theorem," If in a right-angled triangle, the hypothenuse be made the radius, the sides become the sines of the opposite angles, and if one of the sides be made the radius, the other side becomes the tangent of the opposite angle, and the hypothenuse becomes its secant?" How is proved the theorem, "The sides of a plane triangle are to one another, as the sines of the opposite angles?" What are the definitions of spherical trigonometry? How is demonstrated the theorem, "The arch of a great circle, between the pole and the circumference of another circle, is a quadrant ?" How is demonstrated the theorem, "Two great circles, whose planes are perpendicular to each other, pass through each others poles ?"

CHAP. VII.

CONIC SECTIONS.

CONIC Sections, as the name implies, are such curve lines as are produced by the mutual intersection of a plane and the surface of a solid cone.

The nature and properties of these figures, were the subject of an extensive branch of ancient geometry, particularly among the Greeks, to whose quick and enquiring minds this speculation was, in a peculiar manner, adapted. In modern times, the conic geometry is intimately connected with every part of the higher mathematics, and natural philosophy.

The discovery of the curves, called conic sections, is attributed to the philosophers of the school of Plato; and, by some authors, it is