the properties of cylindrical surfaces, this part of geometry is necessary to the right understanding and practice of architecture, painting, and those arts of design in which the effects of lights and shadows are to be investigated or represented.

(400.) The position and form of lines in space are expressed, in the higher geometry, by determining the projection of these lines on planes placed at right angles to each other. Two such projections being given, the line in question will be perfectly known.

From every point of the line whose form and position are to be determined let perpendiculars be supposed to be drawn to a horizontal plane, such as the floor of a room; these perpendiculars will form a cylindrical surface, of which the line in question is the generatrix or base. The points of the horizontal plane where the perpendiculars meet it will form the horizontal projection of the line, and will be the other base of the cylinder. If perpendiculars be in like manner drawn from the line to a vertical plane, such as one of the walls of a room, they will form another cylindrical surface, of which the line is also the base or generatrix ; and another projection of it, forming the other base of the cylinder, will be formed on the vertical plane.

If these two projections, one on the horizontal and the other on the vertical plane, were given, the line of which they are the projections would be found by constructing two cylindrical surfaces, having these two projections as their bases, and perpendicular respectively to the two planes on which the projections are given. The line formed by the intersection of these two cylindrical surfaces would be the line sought.

(401.) If a cylinder be laid with its side upon a plane, the points at which it will meet the plane will lie in a straight line, forming the side or one of the positions of the directrix of the cylinder. All other points of the plane will lie outside the cylinder. The plane is in this case a tangent plane to the cylindrical surface. If the cylinder be rolled upon the plane, each line of

contact which it assumes with the plane will be parallel to all former lines of contact. In fact, the line of contact of the cylinder with the plane will move parallel to itself, and will be parallel to the axis of the cylinder, which likewise moves parallel to itself.

If the cylinder be a right circular cylinder, its axis will, during such motion, move in a plane parallel to that on which the cylinder rolls, and at a distance above it equal to the radius of the cylinder.

(402.) The form of a plane is imparted to soft substances by virtue of this property of the right circular cylinder. In agriculture, when the surface of a tilled field is required to be made plane by breaking or pressing down the rough mould which the plough or harrow has left upon it, and in gardening, when the rough surface of loose gravel forming a walk or road is required to be rendered even and plane, a heavy cylindrical roller of iron or stone is passed over it, which, forcing itself into contact by its weight with the surface on which it rolls, reduces that surface to the plane form, without which continued contact with it would be impossible. Since, however, a cylindrical roller passing in one direction only will not produce a level surface, in the formation of a plane where great precision is required the roller should be passed over frequently and in various directions.

fig. 165.

(403.) If two right circular cylinders be placed with their axes parallel one to the other, and so that the distance A A' (fig. 165.) between the axes shall be equal to the sum of their radii; then the surfaces of these cylinders will touch each other, and their line of contact will be a straight line parallel to their axes, being, in fact, a side of the cylinder.

If two cylindrical surfaces thus placed be intersected by a plane at right angles to their axes, their section by that plane will be two circles equal to

the bases of the cylinders, which will touch each other externally, as represented in fig. 165.

(404.) If one of the cylinders, A, thus placed be rolled upon the other, their line of contact will move parallel to itself, being always a common side of the two cylinders; and the axis A of one cylinder will move parallel to itself round the axis of the other, describing the surface of a right circular cylinder, whose radius A A ́ is equal to the sum of the radii of the two given cylinders, and whose axis is the axis A' of the fixed cylinder.

(405.) If the surfaces of two cylinders thus placed in contact and pressed together be so rough that one cannot move without moving the other with it, and that both be capable of revolving upon their axes, then any motion of revolution which is given to one cylinder will be imparted to the other, the surfaces of the two cylinders moving at the same rate.

(406.) It is on this principle that wheel work in machinery acts. The moving power, whatever it may be, gives motion to one wheel or cylinder, the edge of which, pressing on another, imparts motion to it, and that again acts on another, and so on. As the actual velocity of the edges of the wheels in contact will be the same, the velocities of revolution are varied by varying the magnitudes of the wheels. If the diameter of the wheel A (fig. 165.) be half the diameter of the wheel A', then it will require two revolutions of the former to produce one of the latter, and the velocity of revolution of the former wheel will be double that of the latter. In fact, the velocities of revolution of each pair of contiguous wheels will be in the inverse proportion of their diameters.

(407.) If the surfaces of the cylinders thus in contact were perfectly smooth, the revolution of one upon its axis would not impart motion to the other, but the surface of the one would slide on that of the other. In proportion to the roughness of the surfaces, friction will

be produced between them; and the resistance attending this friction will cause the surface of the second cylinder to be pushed round, and that the one cylinder, instead of sliding, shall roll upon the other. If so great a resistance, however, be opposed to the motion of the second cylinder as to exceed that produced by the friction of the surfaces, then, notwithstanding the friction, the surface of the one will still slide upon the surface of the other without imparting motion to it. In this case the resistance due to friction is increased either by coating the surfaces of the cylinders with leather, or some other rough material; or if they be wood, by cutting them with their grains in contrary directions. But where the resistance is considerable, or where the inaccuracies of motion produced by the occasional and accidental slipping of one surface on another must be avoided, as in the case of watchwork, then the surfaces are formed into teeth, of equal and uniform magnitude and form, which insert themselves between one another, and render any inequality of motion impossible, unless by the fracture of a tooth.

(408.) Whatever be the surface in contact with which a right circular cylinder is rolled, its axis will move in a parallel surface; and the same will be true of whatever may be supported by such an axis. A wheel carriage moving along a road is therefore carried in lines parallel to the road; since the wheels are right circular cylinders in contact with the road. Hence it is that all inequalities of the road produce corresponding inequalities of motion in every part of the carriage.

(409.) If a cylinder be in contact with any surface on which it is prevented from sliding by the resistance attending its friction with it, and if at the same time the surface on which it is placed be fixed and incapable of moving under it; then any motion of revolution which may be imparted to the cylinder must, at the same time, give to the cylinder a progressive motion

along that surface. For as the surface of the cylinder is prevented from rubbing or slipping on the surface on which it rests, it cannot turn round except by rolling on that surface; and it cannot roll on that surface without advancing along it with a progressive motion.

Thus, if any force be applied to the spokes of the wheels of a carriage, so as to compel the wheel to turn round, and if by the pressure of the wheel upon the road it is prevented from slipping as it revolves; then the carriage must roll onwards by the revolution of the wheels, in the same manner as if it were drawn forwards in the common way by horses or any other tractive power. (410.) It is on this principle that the steam engine is applied to produce the progressive motion of carriages upon railways. The wheels of the engine are fixed upon their axle, so as to turn with it, and not upon it, as in common carriages. On the axle of these wheels is formed a crank or arm like the handle of a winch or windlass. The piston-rod of the steam engine lays hold of this arm, and as the piston is driven backwards and forwards in the cylinder causes the arm to revolve. As the arm revolves the axle on which it revolves also revolves, and with this axle the wheels fixed upon it are made to revolve. Now, these wheels resting upon the rails, with the incumbent weight of the engine upon them, must, as they revolve, either slip on the rails or roll forward, causing the engine to roll with them; and as the resistance produced by their pressure upon the rails is so great as to prevent their slipping, the engine is compelled to roll forwards, and to draw after it the train of carriages or waggons.

(411.) In the modern printing presses the properties of a cylinder moving in contact with a plane is brought into frequent operation. In letterpress printing the stereotype plates, having upon their faces in relief the letters to be printed on the paper, are bent so as to correspond to the form of a large cylinder or roller to which they are attached. Another cylinder or roller is