(Th. iv), and since ABCDE is a regular polygon, it follows that the sides ab, bc, cd, de, and ea, are all equal to each other.

Hence, the trapezoids EAae, ABba, &c., which form the convex surface of the frustum are equal. But the perpendicular distance between the parallel sides of these trapezoids is equal to Ef, the slant height of the frustum.

Now, the area of either of the trapezoids, as AEea, is equal to half the product of Ffx (EA+ea) (Bk. IV. Th. x): hence, the area of all of them, that is, the convex surface of the frustum, is equal to half the sum of the perimeters of the upper and lower bases, multiplied by the slant height.

The convex surface of a cone is equal to half the product circumference of the base multiplied by the slant heigh.

In the circle which forms the base of the cone, inscribe a regular polygon, and join the vertices with the vertex S, of the cone. We shall then have a right pyramid inscribed in the cone.

The convex surface of this pyramid will be equal to half the product

the

Of the Cone.

of the perimeter of the base by the slant height (Th. vi).

Let us now suppose the number of sides of the polygon to be indefinitely increased: the polygon will then coincide with the base of the cone, the pyramid will become the cone, and the line Sf, which measures the slant height of the pyramid, will then measure the slant height of the cone.

S

Hence, the convex surface of the cone is equal to half the product of the slant height by the circumference of the base.

THEOREM IX.

The convex surface of the frustum of a cone is equal to half the sum of the circumferences of its two bases multiplied by the slant height.

For, if we suppose the frustum of a right pyramid to be inscribed in the frustum of a cone, its convex surface will be equal to half the product of its slant height by the perimeters of its two bases. But if we increase the number of sides of the

polygon indefinitely, the frustum of the pyramid will become the frustum of the cone: hence, the area of the frustum of the cone is equal to half the sum of the circumferences of its twe bases multiplied by the slant height.

Of the Sphere.

23. Two cylinders are similar, when the diameters of their bases are proportional to their altitudes.

24. Two cones are also similar, when the diameters of heir bases are proportional to their altitudes.

25. A sphere is a solid terminated by a curved surface, all the points of which are equally distant from a certain point within called the centre.

Of the Sphere.

29. All diameters of a sphere are equal to each other; and each is double a radius.

30. The axis of a sphere is any line about which it re volves; and the points at which the axis meets the surfa re called the poles.

Of Planes.

THEOREM XI.

If two angles, not situated in the same plane, have their sides parallel and lying in the same direction, the angles will be equal.

Let the angles ACE and BDF have the sides AC parallel to BD,

E

and CE to DF: then will the angle ACE be equal to the angle BDF.

For, make AC equal to BD, and CE equal to DF, and join AB, CD, and EF; also, draw AE, BF.

Now since AC is equal and parallel to BD, the figure AD will be a parallelogram (Bk. I. Th. xxv); therefore, AB is equal and parallel to CD.

B

D

Again, since CE is equal and parallel to DF, CF will be a parallelogram, and EF will be equal and parallel to CD. Then, since AB and EF are both parallel to CD, they will be parallel to each other (Th. x); and since they are each equal to CD, they will be equal to each other. Hence, the figure BAEF is a parallelogram (Bk. I. Th. xxv), and consequently, AE is equal to BF. Hence, the two triangles ACE and BDF have the three sides of the one equal to the three sides of the other, each to each, and therefore the angle ACE is equal to the angle BDF (Bk. I. Th. viii).

THEOREM XII.

If two planes are parallel, a straight line which is perpendicular to the on will also be perpendicular to the other.