PROP. IV.

The angle formed by two arcs of great circles, is equal to the angle contained by the tangents drawn to these arcs at their point of intersection, and is measured by the arc described from their point of intersection or pole, intercepted by the arcs containing the angle.

Let ZPN, ZQN, arcs of great circles, intersect in Z.

Draw ZT, ZT', tangents to the arcs at the point Z.

With Z as pole, describe the arc PQ.

Take O the centre of the sphere, and join OP, OQ.

Then, the spherical angle PZQ is equal to the angle TZT', and is measured by the arc PQ.

For the tangent ZT drawn in the plane ZPN, is perpendicular to radius OZ.

N

T

And the tangent ZT' drawn in the plane ZQN, is perpendicular to radius OZ.

Hence, the angle TZT' is equal to the angle contained by these two planes, that is, to the spherical angle PZQ. (Geom. of Planes).

Again, since the arcs ZP, ZQ, are each of them equal to a quadrant ;

.. Each of the angles ZOP, ZOQ, is a right angle,

.... The angle QOP is the angle contained by the planes ZPN, ZQN, and is =TZT.

.. The arc PQ, which measures the angle POQ, measures the angle between the planes, that is, the spherical angle PZQ.

Cor. 1. The angle under two great circles is measured by the distance between their poles. For the axis of the great circles drawn through their poles being perpendicular to the planes of the circles, the angles under these axes will be equal to the angle between the circles; but the angle under the axes is obviously measured by the arc which joins their extremities, that is, by the distance between their poles.

Cor. 2. The angle under two great circles is measured by the arc of a common secondary intercepted between them.

For, since the secondary passes through the poles of both, taking away from the equal quadrants of the secondary between each circle and its pole, the common arc intercepted between one circle and the pole of the other, the remainders are the intercept of the common secondary between the two circles, and the distance between their poles, and these are therefore equal. But the latter is, by the last Cor., the measure of the angle.

Cor. 3. Vertical spherical angles, such as QPW, QPS, are equal, for each of them is the angle formed by the planes QPS, WPR.

Also, when two arcs cut each other, the two adjacent angles QPW, QPR, when taken together, are always equal to two right angles.

W

R S

PROP. V

If from the angular points of a spherical triangle considered as poles, three arcs be described forming another triangle, then, reciprocally, the angular points of this last triangle will be the poles of the sides opposite to them in the first.

Let ABC be a spherical triangle.

From the points A, B, C, considered as poles, describe the arcs B'C', A'C', A'B', forming the spherical triangle A'B'C'.

Then, A' will be the pole of BC, B' of AC, and C' of AB.

For, since B is the pole of A'C', the distance from B to A' is a quadrant.

And, since C is the pole of A'B', the distance from C to A' is a quadrant.

Thus, it appears that the point A' is distant by a quadrant from the points B and C.

.. A' is the pole of the arc BC.

Similarly, it may be shown that B' is the pole of AC, and C' the pole of AB.

PROP. VI.

The same things being given as in the last proposition, each angle in either of the triangles will be measured by the supplement of the side opposite to it in the other triangle.

Produce the sides of the first triangle to D, E, F, G, H, K.

Then, since A is the pole of B'C', the angle A is measured by the arc EK.

For the same reason, the angles B and C are measured by the arcs DH and FG respectively. Because B' is the pole of FK, the arc B'K is a quadrant.

Because C' is the pole of DE, the arc C'E is a quadrant.

D

A

F

C'

H

K

C

B

B'

G

But the arcs EK, DH, FG, are the measures of the angles A, B, C, respectively,.. 180° - B'C', 180°

---

A'C', 180° A'B', or the supplements of B'C',

A'C', and A'B', are the measures of these angles.

Again, since A' is the pole of HG, the angle A' is measured by GH.

For the same reason, the angles B', C', are measured by the arcs FK and

DE respectively.

Because B is the pole of A'C', the arc BH is a quadrant

Because C is the pole of A B', the arc CG is a quadrant.

And GH, FK, DE, are the measures of the angles A', B', C', respectively. These triangles ABC, A'B'C', are, from their properties, usually called Polar triangles, or Supplemental triangles.

PROP. VII.

In any spherical triangle any one side is less than the sum of the two others.

Let ABC be a spherical triangle, O the centre of the sphere. Draw the radii OA, OB, OC.

Then the three plane angles AOB, AOC, BOC,. form a solid angle at the point O, and these three angles are measured by the arcs AB, AC, BC.

But each of the plane angles which form the solid angle, is less than the sum of the two others. Hence each of the arcs AB, AC, BC, which measures these angles, is less than the sum of the two others.

A

C

PROP. VIII.

The sum of the three sides of a spherical triangle is less than the circumference of a great circle.

CONIC SECTIONS.

THERE are three curves, whose properties are extensively applied in Mathematical investigations, which, being the sections of a cone made by a plane in different positions, are called the Conic Sections (see page 437). These are,

1. THE PARABOLA.

2. THE ELLIPSE.

3. THE HYPERBOLA.

Before entering upon the discussion of their properties, it may be useful to enumerate the more useful theorems of proportion which have been proved in the treatises on Algebra and Geometry, or which are immediately deducible from those already established. For convenience in reference, they may be arranged in the following

PARABOLA.

DEFINITIONS.

1. A PARABOLA is a plane curve, such, that if from any point in the curve two straight lines be drawn; one to a given fixed point, the other perpendicular to a straight line given in position: these two straight lines will always be equal to one another.

2. The given fixed point is called the focus of the parabola.

3. The straight line given in position, is called the directrix of the parabola.

the directrix, and cutting the curve, is called a diameter; and the point in which it cuts the curve is called the vertex of the diameter.

5. The diameter which passes through the focus is called the axis, and the point in which it cuts the curve is called the principal vertex.

Thus: draw N1 P1 W1, N2 P2 W2, N3 P3' Ws, KASX, through the points P1, P2, P3, S, perpendicular to the directrix; each of these lines is a diameter; P1, P2, P3, A, are the vertices of these diameters; ASX is the axis of the parabola, A the principal vertex.

6. A straight line which meets the curve in any point, but which, when produced both ways, does not cut it, is called a tangent to the curve at that point.

N

Νι

Wi

N2

P

-W2

KA

Na

S

X

W3

P3

n

7. A straight line drawn from any point in the curve, parallel to the tangent at the vertex of any diameter, and terminated both ways by the curve, is called an ordinate to that diameter.

8. The ordinate which passes through the focus, is called the parameter of that diameter.