cogs, or teeth, which being acted upon by any applied force, cause the wheel to revolve; and the axle being similarly furnished with teeth, or cogs, is termed a pinion. The wheel and pinion, therefore, bear a similar relationship to each other as the wheel and axle, and their power must be calculated in the same manner. Suppose AB, fig. 29, to be a shaft on which the handle A C, of twelve inches radius, and the pinion D, of one inch radius, are fixed; and the teeth of the wheel E, of twelve inches radius, acting in those of the pinion D, and upon the shaft of E is fixed the pinion F, of one inch radius, communicating with the wheel G, of twelve inches radius, upon the shaft of which the pulley H, of one inch radius, is fastened; we shall then have the handle AC representing the radius of a wheel, and the pinion D in the situation of the axle; so that there will be a gain of twelve to one: and the wheel E, bearing the same proportion to the pinion F, will also gain in a similar ratio, and G being to H, as E to F, the gain will again be augmented to the same extent; so a force equal to one at C will operate as twelve at D; and twelve at D will operate as a hundred and forty-four at F; and at H as seventeen hundred and twenty-eight. Thus one pound at C will raise seventeen hundred and twenty-eight pounds at H, and the handle C will have to pass through seventeen hundred and twenty-eight times the distance through which the weight I will move. By this form and disposition of wheels and pinions, an accession of power is obtained; but if velocity be required at the expense of power, this train should be inverted. For, if we suppose the pulley H to be turned by a force so as to cause the weight I to pass through one foot, the periphery of the wheel G will have passed through twelve feet, and the periphery of the pinion will have gone through the same distance; but the wheel E being twelve times the diameter of F, it will have passed through twelve times that distance, or a hundred and forty-four feet; and the pinion D, in like manner, will cause C to pass through twelve times that amount of space, or seventeen hundred and twenty-eight feet; whilst the force required at H to cause this motion, must be seventeen hundred and twentyeight times the resistance at C.

As the circumferences of wheels are proportionate to the circumferences of the pinions they have to act upon, or be acted upon by, so must the number of teeth in the one be to those in the other, otherwise the size of the teeth would not be similar; thus, a wheel that is twelve inches diameter, and a pinion one inch, the circumferences of circles being in proportion to their diameters, the wheel should have

C

twelve times as many teeth as the pinion, therefore, in practice, the number of teeth may be taken as data to estimate the power or velocity. Suppose a pinion has five teeth, and a wheel sixty, their power will be as twelve to one, as five will go twelve times in sixty; that is, the pinion will have to turn twelve times to move the wheel once; and if turned by a handle A C, whose radius is equal to the wheel, the power gained will be in the same ratio; and if the pinion is driven by the wheel, the velocity obtained will, in like manner, increase; consequently the velocities, or powers of any combination of wheels, may be estimated by their diameters, circumferences, or number of teeth.

Although this mode of communicating motion is used to a very great extent in applying wheel-work to machinery, yet, in peculiar cases, straps, chains, and cordage, of various descriptions, are beneficially introduced to transfer the action of wheels.

Combinations of the wedge are not very common: but its properties are introduced under several modifications, and afford methods of obtaining power of considerable pressure in short distances. For instance, that common and well-known part of mechanical construction, called the camb, or eccentric, is a wedge applied by one of its faces to a cylinder, which, by being turned by means of a lever, is capable of producing a powerful action. Fig. 30 represents a cylinder A, with a wedge B wound round it, but which, in this position, is denominated a camb, or eccentric piece; by the motion of the lever C to the situation C1, the cylinder A, with the camb B, is brought into the position B 1; thereby raising the obstacle D to D1. The power gained in this operation may be ascertained thus: as the length of the lever C from the center of A exceeds the radius of A, so will the force applied at C be increased, at the point E, where it may be supposed to act, against the wedge or camb B; and the effort to raise D may be known by considering the proportion of E F to EH, which is the portion of the circumference that must be considered as its base. Thus, if we call the length of the lever C three, and the radius of A one, if the force acting at C be one, its power at E will be three; and should the height EF be one third of the base of the camb B, this power will be again raised by three; thus, 1 at C will counterbalance 9 at D. This movement is extremely common in order to obtain power, or a regular direct motion. It is quicker than a screw, and capable of considerable

accuracy.

Fig. 31 is another modification of the wedge, placed on

the internal face of the circle E, acting with its face F, and causing by its movement the obstacle I to approach nearer to the centre G; this is called the snail movement, and might with propriety be termed a concentric.

Another method of placing a wedge, so as to apply its effects to a revolving motion, is represented in a side and top view at fig. 32, where the wedge AB is placed upon a circular plate CD, turning upon the axis E, and consequently creating motion in the obstacle upon which it acts to the amount of the line G A.

Another movement of considerable accuracy is obtained by the turning of a cone, the principle of whose action is referable to the wedge. Fig: 33 represents a cone fixed upon its axis ik. If an obstacle be presented at a, and the cone be caused to pass forward in the direction ki, the surface a c will operate as a wedge at abc, raising the obstacle to c; but if during that direct motion the cone is likewise caused to revolve on its axis, the obstacle, instead of passing over a c, will pass over the spiral line aeg d, to the point d; by this means the operation of a wedge, whose line of inclination is equal to the spiral line aegd, and whose height is equal to bc, is brought into action; and if the number of revolutions of the cone be increased during its direct motion, it is plain that the effect of a wedge of infinite elongation may be produced.

The screw is introduced both singly and in a state of combination in many parts of machinery. The combined action of two screws, which avoid the necessity of using a screw of greater fineness, in which the threads would be weakened, is represented at fig. 34, where they are applied to a press. Suppose AA to be a screw fitted in a female screw in the rail BC; and D, a screw that works in the inside of A, having its lower end joined to the upper board of the press H, so that it shall not turn round: now if the screw A A, and the screw D, contain exactly the same number of threads in the inch, by turning A A one revolution, it will proceed downwards exactly the same amount that the screw D will, by the same action, proceed upwards, and the board H will not be moved. But we will suppose that the screw A A contains four threads in the inch, and the screw D six, then, by one revolution, A A will move downwards one quarter of an inch, and D will at the same time, and by the same action, be raised onesixth of an inch, therefore the board H will move downwards the difference between one quarter and one-sixth, or onetwelfth part of an inch, by every single revolution: which

effect is similar to that which would be produced by using a screw of twelve threads to the inch.

For further elucidation we shall refer the action of each screw to that of a wedge from which the screw has been shown to be derived Fig. 35 represents two wedges, abh and ecd, each of which may be supposed to represent one lap of a screw of the respective fineness which their heights bh and ec denote. If the wedge a bh be caused to pass to the situation a' a h', and is supposed to operate upon the level surface ef, the line a e will be compressed to the line hic, by that movement; but if, whilst this action takes place, the wedge ecd be moved to the position e' c' e, and the effect takes place upon its upper surface e d, the line a e will only be reduced to the line g' e, equal to hd, and will consequently only be compressed to the amount g' a, which is in effect equal to what a wedge of the fineness of abg would have produced, whose height or line gb is just equal to the difference between e c and h b, as was the case with the screws. As a gain of power is attainable by two wedges of unequal fineness, performing equal numbers of revolutions, so is the same effect attainable by the unequal revolutions of two screws or wedges of equal fineness.

screws

or

MILL GEERING.

the

UNDER this head we purpose to treat of the best formation of the teeth of wheels, of the connection of shafts, termed couplings, of the disengaging and reengaging of the moving parts, and of the equalization of motion; and to them we shall annex some further observations upon general construction of Machinery. To avoid unnecessary repetition, we shall, previously to entering upon the formation of the teeth of wheels, give a general definition of the terms most commonly in use.

Cog-wheel is the general name of any wheel which has a number of teeth or cogs placed round its circumference.

Pinion is a small cog-wheel that has not in general more than twelve teeth; though, when two-toothed wheels act upon one another, the smallest is not unfrequently distinguished by this term; as is also the trundle, lantern, or wallower, when talking of the action of two wheels.

Trundle, lantern, or wallower, is sometimes used in lieu of a pinion. It is represented at fig. 36.

When the teeth of a wheel are made of the same material, and formed of one piece with the body of the wheel, they are called teeth; when of wood, or some other material, and affixed to the outer rim of the wheel, cogs; in a pinion they are called leaves; in a trundle staves.

When speaking of the action of wheel-work in general, the wheel which acts as a mover is called the leader, and the one upon which it acts the follower.

If a wheel and pinion are to be so constructed that the one shall give, and the other receive, impulse, so that the pinion shall perform four revolutions in the time that the wheel is performing one, they must be represented by twocircles, which are in proportion to each other as four is to one. When these two circles are so placed that their outer rims shall touch each other, a line drawn from the centre of the one to the centre of the other is termed the line of centres; and the radii of the two circles the proportional radii. These circles are sometimes called proportional circles, but by mill-wrights in general pitch lines.

The teeth which are to communicate motion must be formed upon these two circles. The distance from the centres of two circles to the extremities of their respective teeth, is called the real radii; and, in practice, the distance between the centres of two contiguous teeth, that is, the distance from the centres of two teeth measured upon their pitch line, is called the pitch of the wheel. The straight part of a tooth which receives the impulse is called the flank, and the curved part that imparts the impulse, the face.

Two wheels acting upon one another in the same plane, having their axes parallel to each other, are called spur geer; when their axes are at right, or other angles, bevelled geer.

TO DESCRIBE THE CYCLOID AND EPICYCLOID.

FIG. 37. If the circle 1, having a point a marked on its circumference, moves along the straight line A C, and at the same time revolves on its axis, the curved line which the point a describes is called the cycloid. The point a in circle 1 is at its starting place, at B it has reached its greatest height, and at C its lowest depth; and the curved line A B C described by that point, is the cycloid.

Fig. 38. If the circle 1 rolls on another circle, as on the circumference of circle 2, the point a describes, in a similar manner to the preceding, the curve a g h de, and the circles 3, 4, 5, 6, exhibit the point a in the several positions of a1, a2, a', a; ca the portion of circle 3 being equal to ca, c2 a