Then, by Props. vii. and vIII.,

DO: DH2 :: LM: EF,

and AN AG2 :: IK : BC.

But, since AN, AG3, are equal to DO3, DH2; therefore, IK: BC:: LM: EF But BC is equal to EF, by hypothesis; therefore, IK is also equal to LM.

In the same manner, it is shown that any other sections, at equal distance from the vertex, are equal to each other.

Since, then, every section in the cone, is equal to the corresponding section in the pyramids, and the heights are equal, the solids ABC, DEF, composed of those sections, must be equal also. Q. E. D.

PROP. XIII.

Every pyramid of a triangular base, is the third part of a prism of the same base and altitude.

Let ABCDEF be a prism, and BDEF a pyramid, upon the same triangular base DEF; then will the pyramid BDEF be a third part of the prism ABCDEF.

For, in the planes of the three sides of the prism, draw the diagonals BF, BD, CD. Then the two planes BDF, BCD, divide the whole prism into the three pyramids BDEF, DABC, DBCF; which are proved to be all equal to one another as follows:

B

E

Since the opposite ends of the prism are equal to each other, the pyramid whose base is ABC and vertex D, is equal to the pyramid whose base is DEF and vertex B (Prop. xII.), being pyramids of equal base and altitude.

But the latter pyramid, whose base is DEF and vertex B, is the same solid as the pyramid whose base is BEF and vertex D, and this is equal to the third pyramid, whose base is BCF and vertex D, being pyramids of the same altitude and equal bases BEF, BCF.

Consequently, all the three pyramids which compose the prism, are equal to each other, and each pyramid is the third part of the prism, or the prism is triple of the pyramid. Q. E. D.

Corol. 1. Every pyramid, whatever its figure may be, is the third part of a prism of the same base and altitude; since the base of the prism, whatever be its figure, may be divided into triangles, and the whole solid into triangular prisms and pyramids.

Cor. 2. Any right cone is the third part of a cylinder, or of a prism, of equal base and altitude; since it has been proved that a cylinder is equal to a prism, and a cone equal to two pyramids, of equal base and altitude.

SCHOLIUM. Whatever has been demonstrated of the proportionality of prisms, or cylinders, holds equally true of pyramids or cones, the former being always triple the latter; viz. that similar pyramids or cones, are as the cubes of their like linear sides, or diameters, or altitudes, &c.

SPHERICAL GEOMETRY.

DEFINITIONS.

1. A SPHERE is a solid terminated by a curve surface, and is such that all the points of the surface are equally distant from an interior point, which is called the centre of the sphere.

We may conceive a sphere to be generated by the revolution of a semicircle APB about its diameter AB; for the surface described by the motion of the curve ABP will have all its points equally distant from the centre O

2. The radius of a sphere is a straight line drawn from the centre to any point on the surface.

The diameter or axis of a sphere is a straight line drawn through the centre, and terminated both ways by the surface.

Of

P

B

It appears from Def. 1, that all the radii of the same sphere are equal, and that all the diameters are equal, and each double of the radius.

3. It will be demonstrated, (Prop. 1.), that every section of a sphere, made by a plane, is a circle; this being assumed,

A great circle of a sphere is the section made by a plane passing through the centre of the sphere.

A small circle of a sphere is the section made by a plane which does not pass through the centre of the sphere.

4. The pole of a circle of a sphere is a point on the surface of the sphere equally distant from all the points in the circumference of that circle.

It will be seen, (Prop. 11.), that all circles, whether great or small, have two poles.

5. A spherical triangle is the portion of the surface of a sphere included by the arcs of three great circles.

6. These arcs are called the sides of the triangle, and each is supposed to be less than half of the circumference.

7. The angles of a spherical triangle are the angles contained between the planes in which the sides lie.

8. A plane is said to be a tangent to a sphere, when it contains only one point in common with the surface of the sphere.

PROP. I.

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

Let AZBX be a sphere whose centre is O.
Let XPZ be a section made by the plane XZ.
From O draw OC perpendicular to the plane XZ.
In XPZ take any points P1, P2, P3, . . . . . .
Join CP1; CP2; CP3; ...... also, OP1; OP2;

X

R

Hence, the angles OCP1, OCP2, OCP3, ...... are right angles,

Hence, XPZ is a circle whose centre is C, and every other section of a sphere made by a plane may, in like manner, be proved to be a circle.

Cor. 1. If the plane pass through the centre of the sphere, then OC = o, and the radius of the circle will be equal to the radius of the sphere.

Cor. 2. Hence, all great circles are equal to one another, since the radius of each is equal to the radius of the sphere.

Cor. 3. Hence, also, two great circles always bisect each other, for their common intersection passing through the centre is a diameter.

Cor. 4. The centre of a small circle and that of the sphere, are in a straight line, which is perpendicular to the plane of the small circle.

Cor. 5. We can always draw one, and only one, great circle through any two points on the surface of a sphere, for the two given points and the centre of the sphere give three points, which determine the position of a plane.

If, however, the two given points are the extremities of a diameter, then these two points and the centre of the sphere are in the same straight line, and an infinite number of great circles may be drawn through the two points.

Distances on the surface of a sphere are measured by the arcs of great circles. The reason for this is, that the shortest line which can be drawn upon the surface of a sphere, between any two points, is the arc of a great circle joining them. (See Int. Calculus.)

PROP. II.

If a diameter be drawn perpendicular to the plane of a great circle, the extremities of the diameter will be the poles of that circle, and of all the small circles whose planes are parallel to it.

Let APB be a great circle of the sphere whose centre is O.

Draw ZN a diameter perpendicular to the plane of circle APB.

Then, Z and N, the extremities of this diameter, are the poles of the great circle APB, and all the small circles, such as apb, whose planes are parallel to that of APB.

Pi

Pa

P

B

A

10

PA

Pa

Through each of these points respectively, and the points Z and N, describe great circles, ZP,N, ZP,N.

Join OP1, OP 2, ....

Then, since ZO is perpendicular to the plane of APB, it is perpendicular to all the straight lines OP 1, OP 2, drawn through its foot in that plane. Hence, all the angles ZOP1, ZOP 2, ... ... are right angles, and .. the arcs ZP 1, ZP 2, ... ... are quadrants.

Thus, it appears that the points Z and N are equally distant from all the points in the circumference of APB, and are .. the poles of that great circle. Again, since ZO is perpendicular to the plane APB, it is also perpendicular to the plane apb, which is parallel to the former.

Hence, the oblique lines Zp1, Zp2,..... drawn to P1, P2, in the circumference of apb, will be equal to each other. (Geometry of Planes.) equal, the arcs Zpi, Zp2,...

.. The chords Zp1, Zp2,..... being

which they subtend, will also be equal.

.. The point Z is the pole of the circle apb; and, for the same reason, the point N is also a pole.

Cor. 1. Every arc P1Z drawn from a point in the circumference of a great circle to its pole, is a quadrant, and this arc P、Z makes a right angle with the arc AP,B. For, the straight line ZO being perpendicular to the plane APB, every plane which passes through this straight line will be perpendicular to the plane APB (Geometry of Planes); hence, the angle between these planes is a right angle, or, by (Def. 7), the angle of the arcs AP1 and ZP1 is a right angle.

Cor. 2. In order to find the pole of a given arc AP1 of a great circle, take PZ equal to a quadrant, and perpendicular to AP1, the point Z will be a pole of the arc AP1; or, from the points A and P1 draw two arcs AZ and P1Z perpendicular to AP1, the point Z in which they meet is a pole of AP1.

Cor. 3. Reciprocally, if the distance of the point Z from each of the points A and P1 is equal to a quadrant, then the point Z is the pole of AP1, and each of the angles ZAP1, ZP,A, is a right angle.

For, let O be the centre of the sphere, draw the radii OA, OP¿, OZ;

Then, since the angles AOZ, P,OZ, are right angles, the straight line OZ is perpendicular to the straight lines OA, OP1, and is. perpendicular to their

plane; hence, by Prop., the point Z is the pole of AP1, and .. the angle ZAP1, ZPA, are right angles.

Cor. 4. Great circles, such as ZA, ZP1, whose planes are at right angles to the plane of another great circle, as APB, are called its secondaries; and it appears from the foregoing corollaries, that,

1. The planes of all secondaries pass through the axis, and their circumferences through the poles of their primary; and that the poles of any great circle may always be determined by the intersection of any two of its secondaries. 2. The arcs of all secondaries intercepted between the primary and its poles, are 90°.

3. A secondary bisects all circles parallel to its primary.

Cor. 5. Let the radius of the sphere = R, radius of small circle parallel to it Distance of two circles, or Oo = d.

= r.

Cor. 6. Two secondaries intercept similar arcs of circles parallel to their primary, and these arcs are to each other as the cosines of the arcs of the secondaries between the parallels and the primary.

For the arcs of the parallels subtend at their respective centres, angles equal to the inclinations of the planes of the secondaries, and these arcs will therefore be similar.

be the radii of two small parallels, the rest of notation as

Every plane perpendicular to a radius at its extremity, is a tangent to the