THEOREM VI.

The lateral area of a cylinder is equal to the product of the perimeter of a right section of the cylinder by the length of an element of the surface.

Corollary I. The lateral area of a cylinder of revolution is equal to the product of the circumference of the base and the altitude.

S=2πrh.

Corollary II. The lateral areas of similar cylinders of revolution are to each other as the squares of the altitudes, or as the squares of the radii of the bases.

THEOREM VII.

The volume of a circular cylinder is equal to the product of its base and its altitude.

V = r2h.

Corollary. The volumes of similar cylinders of revolution are to each other as the cubes of the altitudes, or as the cubes of the radii of the bases.

THE CONE.

THEOREM VIII.

Every section of a cone made by a plane passing through the vertex is a triangle.

THEOREM IX.

Every section of a circular cone made by a plane parallel to its base is a circle, the centre of which is the intersection of the plane with the axis.

THEOREM X.

A right circular cone may be generated by the revolution of a right triangle about one of its sides as an axis.

THEOREM XI.

A plane passing through a tangent to the base of a circular cone and the element drawn through the point of contact is tangent to the cone.

Corollary. If a plane is tangent to a circular cone, its intersection with the plane of the base is tangent to the base.

THEOREM XII.

If a pyramid, the base of which is a regular polygon, be inscribed in or circumscribed about a given circular cone, the volume of the pyramid will approach the volume of the cone as its limit, and the lateral area of the pyramid will approach the area of the convex surface of the cone as its limit, as the number of sides of the base is indefinitely increased.

THEOREM XIII.

The lateral area of a cone of revolution is equal to the product of the circumference of the base and half the slant height.

S=πrl.

Corollary. The lateral areas of similar cones of revolution are to each other as the squares of the slant heights, or as the squares of the altitudes, or as the squares of the radii of the bases.

THEOREM XIV.

The volume of a circular cone is equal to one third the product of its base and its altitude.

Corollary. Similar cones of revolution are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases.

BOOK IX.

THE SPHERE.

THEOREM I.

Every section of a sphere made by a plane is a circle. Corollary I. All great circles of the same sphere are equal. Corollary II. Every great circle divides the sphere into two equal parts.

Corollary III. Any two great circles on the same sphere bisect each other.

Corollary IV. An arc of a great circle may be drawn through any two given points on the surface of a sphere, and, unless the points are the opposite extremities of a diameter, only one such arc can be drawn.

Corollary V. Through any three points on the surface of a sphere one and only one circle can be drawn.

THEOREM II.

All the points in the circumference of a circle on a sphere are equally distant from either of its poles.

Corollary I. All the arcs of great circles drawn from a pole of a circle to points in its circumference are equal.

Corollary II. The polar distance of a great circle is a quadrant.

Corollary III. If a point on the surface of a sphere is at a quadrant's distance from each of two given points of the surface which are not opposite extremities of a diameter, it is the pole of the great circle passing through them.

THEOREM III.

A sphere may be generated by the revolution of a semicircle about its diameter.

THEOREM IV.

Through four points not lying in a plane one sphere, and only one, can be drawn.

THEOREM V.

A plane tangent to a sphere is perpendicular to the radius drawn to the point of contact.

Conversely, a plane perpendicular to a radius of a sphere at its extremity is tangent to the sphere.

THEOREM VI.

The intersection of two spheres is a circle the plane of which is perpendicular to the straight line joining the centres of the spheres, and the centre of which is in that line.

THEOREM VII.

The angle formed by two arcs of great circles is equal to the angle between the planes of the circles, and is measured by the arc of a great circle described from its vertex as a pole and included between its sides (produced if necessary).

Corollary. All arcs of great circles drawn through the pole of a given great circle are perpendicular to its circumference.

SPHERICAL TRIANGLES AND POLYGONS.

THEOREM VIII.

If the first of two spherical triangles is the polar triangle of the second, then, reciprocally, the second is the polar triangle of the first.

THEOREM IX.

In two polar triangles, each angle of one is measured by the supplement of the side lying opposite to it in the other.

THEOREM X.

Two triangles on the same sphere are either equal or symmetrical when two sides and the included angle of one are respectively equal to two sides and the included angle of the other.

THEOREM XI.

Two triangles on the same sphere are either equal or symmetrical when a side and the two adjacent angles of one are respectively equal to a side and the two adjacent angles of the other.

THEOREM XII.

Two triangles on the same sphere are either equal or symmetrical when the three sides of one are respectively equal to the three sides of the other.

THEOREM XIII.

If two triangles on the same sphere are mutually equiangular, they are mutually equilateral, and are either equal or symmetrical.

THEOREM XIV.

Any side of a spherical triangle is less than the sum of the other two.

THEOREM XV.

The sum of the sides of a convex spherical polygon is less than the circumference of a great circle.

THEOREM XVI.

The sum of the angles of a spherical triangle is greater than two, and less than six, right angles.

THEOREM XVII.

Two symmetrical spherical triangles are equivalent.