thing but its winding course. This is the external screw. The internal screw is formed by applying the parallelogram BEFC to a hollow cylinder, equal to the former, and making a groove where the thread falls to fit the protuberant thread of the external screw. This internal screw is often called a nut, and the other the screw. When the two screws are thus adapted to each other, the external or the internal screw, as the case requires, may be moved by means of a lever about their common axis, as in Figs. 54, 55. The force being applied to the lever at right angles to it, in a plane parallel to the base of the cylinder.

The screw and nut thus applied to each other, resemble two inclined planes, such as BHG and HBE, one of which is laid upon, and slides down the other; and as the planes wind round the cylinder a rotatory motion ensues. the machine is worked, the weight is laid upon the nut, and thus causes its inclined plane to press upon that of the screw in the direction of gravity. The consequence would be, that the nut and weight with it would begin to slide down the thread of the screw and descend, but this is prevented by confining the nut so that it cannot have a rotatory motion, but only one of ascent or descent. The screw is then turned round by means of a lever passing through its head, and thus its inclined thread sliding under that of the nut, forces the nut and the weight upon it to ascend, just as by pushing the inclined plane EBH in the direction EB, the plane GBH would be made to ascend. One turn of the screw raises the weight through an altitude equal to the distance between two threads. Sometimes, however, the nut is firmly fixed so as to admit of no motion whatever, (as in Fig. 54); and then the thread of the screw, in sliding under that of the nut, forces the screw to descend and press violently against any obstacle which may be opposed to it. In some cases the weight is not applied to the nut, but to the screw; but as the two inclined planes are perfectly equal and similar, it will require the same force to support a weight on one as on the other, and for this reason one investigation will serve for both.

As before observed, the screw is worked by applying a power Pat the end of a lever; and the moment of P to turn the screw round

= P x length of the lever,

and therefore P is equivalent to a force

Px length of the lever
rad. cylinder

acting immediately at the thread of the screw in a horizontal direction parallel to that in which Pacts. Now the inclined plane on which Wrests, by means of the nut, is only BH wrapped round the cylinder; its inclination to the horizon or base of the cylinder is therefore HBE.

But the radii of circles are proportional to their circum

As the distance between two successive threads can be made very small, and the circumference described by the power as large as we please, the advantage of this machine is very great; and it is remarkable, that it does not depend upon the thickness of the screw.

VI. On the Wedge.

216. A wedge is the solid figure defined by Euclid (Book XII. Def. 4) as a triangular prism. Its two ends are equal and similar triangles, and its three sides rectangular parallelograms, (see fig. 56). It is principally used in splitting timber, and separating bodies which are very strongly united, and in raising very heavy weights through a small altitude, for the purpose of introducing a lever, or some other more convenient machine. AB is called its edge, CDEF its head, CABD and FABE its faces.

When used, its edge is introduced into a small cleft prepared to receive it, and then by violent blows with a hammer on its head its body is driven between the substances, which are thus separated by an interval equal to the breadth of the head. After this, a larger wedge may be introduced, if necessary, and treated as before, until the requisite degree of separation is effected.

As the wedge is driven in. by violent blows, if its sides were perfectly smooth it would start back by the pressure of the obstacles upon them in the interval between the strokes; and thus we should fail in effecting and maintaining the requisite degree of separation, and the machine would be rendered useless. In practice, however, the friction in this machine is always so great as to prevent any recoil, and forms, in fact, the principal resistance to be overcome in driving the wedge. The mode of working this machine will at once present itself to the reader as being totally different in principle from that of all the other machines we have described. These are made to work by the constant and steady exertion of a power, uniformly pressing upon that point of the machine at which it is applied, and gradually producing motion in the weight; but in this machine motion is accumulated in a hammer, by suffering it to descend from an altitude, and is suddenly by an impulse transferred to the wedge. In this case it must evidently be a useless labour to attempt to calculate the ratio of P to W, when they act by pressures, as in the other mechanical powers, and are in equilibrium. It is true, when we know this ratio, a slight increase* of P will gradually produce a motion in W, and thus separate the obstacles; but this mode of working the machine is so widely different from that actually practised, that it would be a waste of time and labour to attempt an explication on Statical principles. A slight stroke with a hammer is found to be far more effective than several tons of pressure. The only theoretical property of the wedge which agrees with practice is that its advantage is increased by diminishing its angle DBE.

All cutting instruments, such as knives, swords, hatchets, chisels, planes used by carpenters, nails, pins, needles, &c. are modifications of the wedge. Of these, knives, planes, pins and needles, are usually worked by pressure, but swords, hatchets, chisels, nails, &c. are worked by percussion.

GENERAL PROPERTY OF MACHINES.

217. If the nature of a machine be such, that when the power and weight balance each other in one position of the machine they will balance in every position of it, a very remarkable property appertains to it, deducible from the principle of virtual velocities, which we may state as follows:

The power is to the weight as the space moved through by the weight when the machine is put in motion is to the space moved through by the power in the same time; the spaces being measured respectively in the directions in which the power and weight act.

Let the whole space (measured thus) through which the power P moves be divided into a very large number of spaces 81, 82..., and let 8'1, 82... be the corresponding spaces described by the weight W; then

8 = 81 + 82 +......

8 = 81+82+......

But because P and W are always in a position of equilibrium ; 81, 8'1, are their virtual velocities for the first position;

.. PS1 + W81 = 0,

Similarly Ps2 + W82 = 0, for the 2nd position

This equation expresses the property enunciated. The negative sign points to the fact, that the direction of the action of one of the two forces P, W is opposed to the direction in which the point moves on which it acts.

Mechanical powers possessing this property are ;

(1) The straight lever supporting weights.

(2) All the pulleys in which the strings are parallel. (3) The Wheel and Axle.

(4) The Screw.

(5) The Inclined Plane, only when the Power hangs by a string passing over the top of the plane.

WHITE'S PULLEY.

218. In the common systems of pulleys each pulley has its own independent centre of motion; and consequently as they all move with different velocities and with different degrees of pressure, some of them will be liable to greater