each solid angle formed by three plane angles of 108°. In the formula used in (148), let a = b = c = 108°, and it becomes

Hence if a right angled triangle be constructed in which the square of one side is one fifth of the square of the hypotenuse, the angle opposite to the other side will be the supplement of the angle A.

BOOK IV.

On the Cylinder, the Cone, and the Sphere.

(162) DEF. A cylinder is a solid produced by the revolution of a rectangle ABCD, (fig. (Art. 172)) which is conceived to turn on the immovable side A B as an axis.

(163) By this motion the sides BC and AD move in planes which are perpendicular to A B, and their extremities C and D describe circles in those planes with the points B and A as centres, and the lines BC and AD as radii. The side D C evidently describes a surface concave towards the fixed central line A B. (164) What has been just stated of the sides BC and A D is equally true of any other line perpendicular to A B, which also moves in a plane perpendicular to AB, and its extremity describes a circle on that plane, and which circle is also on the surface of the cylinder. Hence it follows, that every section of the cylinder perpendicular to the line AB is a circle equal to the circular ends A and B.

(165) DEF.-The line AB is called the axis of the cylinder, and the circular ends are called its bases.

(166) COR. 1.-Every section of a cylinder by a plane through the axis is a rectangle equal to twice the generating rectangle ABCD.

(167) COR. 2.-If from any point C in the circumference of the base of a cylinder a straight line be drawn perpendicular to the plane of the base, that line must be in the cylindrical surface. For it coincides with the side of the generating rectangle when the extremity of that line is at the point C.

(168) COR. 3.-The right line CD is the intersection of the cylindrical surface with every plane through C parallel to the axis A B or perpendicular to the base. Hence it appears, that the intersection of a cylindrical surface, and a plane which is parallel to its axis, is a right line parallel to the axis; or, as the plane will meet the cylindrical surface twice, it intersects it in two right lines parallel to the axis, and therefore to each other.

The intersections of this plane with the bases of the cylinder are parallel to each other and perpendicular to the axis, and therefore also perpendicular to the intersections with the cylindrical surface. Hence the entire intersection of the plane with the cylinder is a rectangle, whose sides are parallel and perpendicular to the axis.

(169) COR. 4.-It is evident from (167) that a
right line drawn through any point in the cylindrical
surface parallel to its axis is wholly in the surface.
(170) DEF.-Such a line is called a side of the cylinder.

PROPOSITION I.

(171) If a plane ABCD be drawn through the axis of a cylinder, and another at right angles FCDG to this and passing through the side C D, the plane FCDG will be entirely outside the cylindrical surface except in the line CD in which it meets it.

For let any other plane AI H B be drawn through the axis of the cylinder intersecting the plane FCDG. In the right angled triangle À DI the hypotenuse AI is greater than the side AD. Since AI is greater than the radius of the circular base, the point I must be outside the cylinder, and the same may be proved of H, and every point in the line H1, and, in general, for every line in the plane FCDG parallel to CD. Hence every part of the plane except the line CD lies outside the cylinder. (172) DEF. Such a plane is called a tangent plane to the cylindrical surface.

B

H

(173) COR. 1.-Hence all tangent planes are parallel to the axis, and their lines of contact are sides of the cylinder.

(174) COR. 2.-Tangent planes which pass through the extremities of the same diameter of the base are parallel, and vice versá.

(175) If the base of a cylinder be divided at three or more points, and sides be drawn through these points, and produced to the opposite base, these sides will divide the opposite base similarly with the first; and if planes be drawn through every

pair of adjacent sides, these planes will form a prism whose sides are those of the cylinder passing through the points of division, and whose bases are formed by the chords of the arcs into which the circular bases of the cylinder are divided. Such a prism is said to be inscribed in the cylinder.

(176) If several planes touch the same cylinder intersecting each other, and also the planes of the bases of the cylinder produced, their intersections with the planes of the bases will be tangents to the bases themselves, and the planes may be so disposed, that these tangents shall form polygons circumscribing the bases. Hence a prism will be formed of which the tangent planes are lateral faces, and the polygons are bases.

Such a prism is said to circumscribe the cylinder.

(177) It is evident, that both the volume and surface of a cylinder are greater than those of any inscribed prism, and are less than those of any circumscribed prism.

(178) This observation is applicable to the surfaces, whether the bases be considered as parts of them or not.

(179) It is also evident, that as the number of sides of the inscribed or circumscribed prism is increased, the difference between its volume or surface and that of the cylinder is diminished, and that the sides may be so increased in number as to render this difference less than any given magnitude.

PROPOSITION II.

(180) If a cylinder and right prism have equal bases and altitudes, they will have equal volumes.

Let V be the volume of the cylinder, and V' that of the prism. If they be not equal, V must either be greater or less than V'. First. Let V be greater than V'. Let a prism be inscribed in the cylinder V, such that the difference between its volume and that of the cylinder shall be less than the difference between V and V', and let the volume of this prism be P. It follows that P is greater than V'; but since these prisms have equal altitudes, the base of P must be greater than the base of V', and therefore greater than the base of V. But the base of P is a polygon inscribed in the base of V, and therefore cannot be greater than the base of V; hence the volume V cannot be greater than V'.

Secondly. Let V be less than V'. Let a prism be circumscribed round the cylinder V, such that the difference between it and the cylinder shall be less than the difference between V and V', and let the volume of this prism be P. Then V' is greater than P, and therefore the base of V' is greater than the base of P, and hence the base of the cylinder is greater than that of its circumscribed prism, which cannot be. Hence V cannot be less than V', and since it can neither be greater than V', or less, they must be equal.

(181) COR. 1.-Hence the volume of a cylinder is expressed numerically by the product of its base and altitude.

(182) COR. 2.-Let a be the altitude of a cylinder, and r the radius of its base, and let be the number which expresses the approximate ratio of the circumference of a circle to its diameter. Then the area of the base is r2, and the volume of the cylinder is z r2 a.

(183) COR. 3.-Since the areas of circles are as the squares of their radii, or diameters, it follows that the volumes of cylinders are as the products of their altitudes and the squares of their diameters, or in a ratio compounded of their altitudes and the squares of their diameters.

(184) COR. 4.-The volumes of cylinders with equal bases are as their altitudes, and those with equal altitudes are as their bases.

(185) DEF. Similar cylinders are those whose altitudes are proportional to their diameters.

(186) COR. 1.-The volumes of similar cylinders are as the cubes of their altitudes or diameters.

(187) COR. 2.-In equal cylinders the bases and altitudes are reciprocally proportional, and vice versa.

(188) COR. 3.-In equal cylinders the altitudes are inversely as the squares of the diameters, and vice versa.

PROPOSITION III.

(189) If a cylinder and right prism have equal altitudes and isoperimetrical bases, they will have equal convex surfaces.

(By the convex surfaces is meant those parts of the surfaces which are included between the bases.)

Let S be the surface of the cylinder, and S' that of the prism. If these be not equal,

1o. Let S be greater than S', and let a prism be inscribed in the cylinder, such that the difference between its surface and that of the cylinder shall be less than the difference between S