horizontally over the space AC; then the Weight has passed vertically over the space BC. Hence the space passed over by the Weight estimated in the direction of the Weight is to the space passed over by the Power estimated in the direction of the Power as the Power is to the Weight in the state of equilibrium. Thus we have here a fresh illustration of the important principle of Art. 208, and an indication, as in Art. 248, of the way in which it is to be understood.

252. The Wedge is a hard solid body bounded by five plane figures, two of which are triangles and the others are four-sided figures. The four sided-figures are often rectangles, and then the triangles are in parallel planes.

B

253. The Wedge may be employed to separate bodies. We may suppose the Wedge urged forwards by a force A P acting on one of the four-sided faces, and urged backwards by two resistances Q and R arising from the bodies which the Wedge is employed to separate, and acting on the other four-sided faces. These forces will be supposed all to act in one plane which is perpendicular to the edge of the Wedge; and we shall assume that the Wedge is smooth, so that the force on each face is at right angles to the face. Let the triangle ABC represent a section of the Wedge made by a plane perpendicular to its edge; and suppose the Wedge kept in equilibrium by the forces P, Q, R at right angles to AB, BC, CA respectively: then by reasoning which we do not give here it is shewn that P, Q, R are in the same proportion to each other as AB, BC, CA respectively. If AC and BC are equal the Wedge is called an Isosceles Wedge; in this case Q and R must be equal, and each of them be in the same proportion to P that AC is to AB.

254. There is very little value or interest in the preceding Article, because the circumstances there supposed

scarcely ever occur in practice. A nail is sometimes given as an example of the Wedge; but when the nail is at rest the resistances on its sides are balanced by friction, and not by a Power at the head. The nail is indeed driven into its place by blows on the head; but the discussion of the motion produced by such blows in conjunction with the resistances and the friction is too difficult for a work like the present.

255. The Screw. Everybody is familiar with the use of a Screw for fastening pieces of wood together, and this will

W

supply great help towards understanding the action of a Screw as a Mechanical Power. The Screw consists of a right circular cylinder AB, with a uniform projecting thread abcd... traced round its surface, making always the same angle with straight lines parallel to the axis of the cylinder. This cylinder fits into a block C pierced with an equal cylindrical aperture, on the inner surface of which is cut a groove, the exact counterpart of the projecting thread abcd...Thus when the block is fixed and the cylinder is introduced into it, the only manner in which the cylinder can move is backwards or forwards by turning round its axis.

256. In practice the forms of the threads of screws may vary, as we see exemplified in the annexed diagrams.

f

M

N

G

257. We may obtain in the following way some notion of the most essential characteristic of the Screw, namely its making at every point the same angle with the straight lines parallel to the axis of the cylinder. Let ABNM be any rectangle. Take any point Cin BN, and make CD, DE, EF,... all equal to BC. Join CA, and through D, E, F,...draw straight lines parallel to CA, meeting AM at the points c, d, e,...respectively. Then if we conceive ABNM to be formed into the convex surface of a right cylinder, the straight lines AC, cD, dE, eF,...will compose a connected curve which takes the shape of a thread of a Screw, supposing the thread to be excessively fine. In this diagram BC represents what is called the distance between two consecutive threads of the Screw.

e

d

E

D

с

C

B

258. Suppose the axis of the Screw to be vertical, and let a Weight W be placed on the Screw. Then the Weight, by its tendency to descend, would cause the Screw to turn round in the block unless this motion were prevented by some Power. We will suppose this Power P to act at the end of a horizontal arm perpendicular to the lever, and horizontally; the arm is firmly attached to the cylinder, as is shewn in the diagram of Art. 255, in which the arm is represented as attached to the cylinder at A. The distance between the axis of the cylinder and the point of application of the Power we shall call the Power-arm.

It is found that when there is equilibrium P is to W in the same proportion as the distance between two consecutive threads of the Screw is to the circumference of the circle having the Power-arm for radius. The reasoning on which this depends is not simple enough to find a place here, so that this may be taken as an experimental fact. It is interesting however to observe its agreement with the principle of Art. 208, as illustrated in Arts. 248 and 251. For suppose the Power to be a little greater than is necessary for equilibrium, then the Weight will be moved; by turning the Screw once round the Weight will be raised through a space equal to the distance between two consecutive threads. Also the whole space passed over by the end of the Power-arm, estimated in the direction of the Power, consists of a multitude of small spaces which are together equivalent to the circumference of the circle having the Power-arm for radius.

259. The most common use of a screw is not to support a Weight, but to exert a Pressure. Thus suppose a fixed horizontal board above the body denoted by W in the diagram of Art. 255; then by turning the Screw the body will be compressed between the head of the Screw and the fixed board. A bookbinder's press is an example of this mode of using the Screw. The proportion between P and W will be that stated in Art. 258, where W now denotes the whole force exerted parallel to the axis of the Screw by the body which is compressed; a force arising partly from the weight of the body, but mainly from the resistance which it offers to compression. The Screw-pile is another exemplification of the same thing; it is used for foundations which are to be laid under water. The thread of a Screw is cut out in the lower part of a wooden or metal pile; and by means of a capstan the pile is gradually screwed down to the depth which it is required to take: this process is found to succeed where it would be practically impossible to drive a pile down by blows.

260. In practice there will be much friction in the use of a screw; in the familiar case to which we allude at the beginning of Art. 255 this friction is in fact the Weight which the Power has to overcome.

XVII. COMPOUND MACHINES.

261. We have already spoken of the mechanical advantage of a machine, and have defined it to be the proportion of the Weight to the Power when the machine is in equilibrium: see Art. 207. Now we might theoretically obtain any amount of mechanical advantage by the use of any of the Mechanical Powers. For example, in the Wheel and Axle the advantage is the proportion of the radius of the Wheel to the radius of the Axle, and this proportion can be made theoretically as great as we please; but practically if the radius of the Axle is very small the machine is not strong enough for use, and if the radius of the Wheel is very great the machine becomes of an inconvenient size. Hence it is found advisable to employ various compound machines, by which great mechanical advantage may be obtained, combined with due strength and convenient size. We will now consider a few of these compound machines.

262. Combination of Levers. Let AB, BC, CD be three Levers, having fulcrums at K, L, M respectively.

Suppose all the Levers to be horizontal, and let the middle Lever have each end in contact with an end of one of the other Levers. Suppose the system in equilibrium with a Power P acting downwards at A, and a weight W acting downwards at D. It is easy to see that equilibrium can be secured by a proper adjustment of P and W; for P tends to raise the right-hand end of the Lever which has K for its fulcrum; thus the left-hand end of the Lever which has L for its fulcrum is pressed upwards, and therefore the right-hand end of the same Lever is pressed downwards: then the left-hand end of the Lever having M for its fulcrum is pressed downwards, and therefore the right-hand end of the same Lever is pressed