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

PROP. VIII.

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

Let ABC be any spherical triangle.

Produce the sides AB, AC, to meet in D.

Then, since two great circles always bisect each other (Prop. 1, cor.) the arcs ABD, ACD, are semicircles.

Now, in the triangle BCD,

BCBD + DC, by Prop. vII.; .: AB+ AC+ BC ≤ AB + BD + AC + DC

ABD + ACD

< circumference of great circle.

A

B

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).

1. THE PARABOLA.

2. THE ELLIPSE.

3. THE HYPERBOLA.

These are,

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

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, NË P2 W2, N3 P3' W3, 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.

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.

9. The part of a diameter intercepted between its vertex and the point in which it is intersected by one of its own ordinates, is called the abscissa of the diameter.

10. The part of a diameter intercepted between one of its own ordinates and its intersection with a tangent, at the extremity of the ordinate, is called the sub-tangent of the diameter.

Thus: let TPt be a tangent at P, the vertex of the diameter PW.

From any point Q in the curve draw Qq parallel to Tt and cutting PW in v. Through S draw RSr parallel to Tt.

Let QZ, a tangent at Q, cut WP, produced in Z.

Then Qq is an ordinate to the diameter PW; Rr is the parameter of PW.

Po is the abscissa of PW, corresponding to

the point Q.

vZ is the sub-tangent of PW, corresponding to the point Q.

11. A straight line drawn from any point in the curve, perpendicular to the axis, and terminated both ways by the curve, is called an ordinate to the axis.

12. The ordinate to the axis which passes through the focus is called the principal parameter, or latus rectum of the parabola.

13. The part of the axis intercepted between its vertex and the point in which it is intersected by one of its own ordinates, is called the abscissa of the

axis.

14. The part of the axis intercepted between one of its own ordinates, and its intersection with a tangent at the extremity of the ordinate, is called the subtangent of the axis.

MT is the sub-tangent of the axis corresponding to the point P.

It will be proved in Prop. 3, that the tangent at the principal vertex is perpendicular to the axis; hence, the four last definitions are in reality included in the four which immediately precede them.

Cor. It is manifest from def. 1, that the parts of the curve on each side of the axis are similar and equal, and that every ordinate Pp is bisected by the axis.

15. If a tangent be drawn at any point, and a straight line be drawn from the point of contact perpendicular to it, and terminated by the curve, that straight line is called a normal.

16. The part of the axis intercepted between the intersections of the normal and the ordinate, is called the sub-normal.

P.

Thus: let TP be a tangent at any point

From P draw PG perpendicular to the tangent, and PM perpendicular to the axis.

Then PG is the normal corresponding to the point P; MG is the sub-normal corresponding to the point P.

AS M

PROP. I.

The distance of the focus from any point in the curve, is equal to the sum of the abscissa of the axis corresponding to that point, and the distance from the focus to the vertex.

That is,

The latus rectum is equal to four times the distance from the focus to the vertex. That is,