cant of the angle formed between one of the ropes and the perpendicular is to the radius, fo is the fum of the weights E and F to the weight G.

Hence it follows, that the general deduction concerning pulleys and weights is only true when the ropes are parallel.

The pulley or tackle is of fuch general utility, that it is needlefs to point out any particular instance.

CHAP. IV.

OF THE INCLINED PLANE, AND OF THE WEDGE.

W THE inclined plane has in its effects a near analogy to the lever. Let A B be an horizontal plane on which the weight E is placed, and let E D represent the force exerted by the weight. AB may also be conceived to act as the arm of a lever, whose fulcrum is A. Let this lever revolve on its fulcrum from в to C, then the weight E will be found at e, and will act on the plane AC with an oblique force ed, Refolve ed into eb per

equal and parallel to ED. pendicular, and bd, parallel to AC, and the force e b will be deftroyed by the reaction of the plane. With the other force bd, the weight will proceed with x an accelerated motion towards A. Whence it may be obferved, that the inclined plane, acting against e in the manner of a lever, deftroys that force which is exerted in the direction of the tangent of its line of motion, and that the acting force in this inftrument is that which in treating of the lever

was

was rejected (56, v), as having no effect. The y force with which any weight on an inclined plane tends downwards in the direction of the plane, is to the weight itself as bd to de; or as er to A e, which is the ratio of the height of the plane to its length, because the triangles bed and Fea are fimilar. But er is to Ae as the fine of the angle the inclined plane makes with the horizon is to radius. See fig. 26. Therefore, as the faid fine z is to radius, fo is the force tending downwards in the direction of the plane to the weight. And because radius is a conftant quantity, the forces by which the fame weight tends downwards in the directions of various planes will be as the fines of their inclinations.

This inftrument is not much used in its fimple form.

A

If it be required to fhew what force in the direction ep parallel to A B (fig. 27) will fuftain the weight e in equilibrio-Set off em equal to bd, which will reprefent its force or tendency in the direction of the plane; and equal, but on the contrary fide, fet off en, which will reprefent the force that, applied in the oppofite direction, will sustain the weight in equilibrio. Draw np perpendicular to AC and ep parallel to A B, interfecting n p in P ep will be the force required; for it is compofed of en and np, and np being perpendicular to the direction of the tendency of e avails nothing. Join pd and this last found force is to the whole weight of e, B as pe to ed, or as e F to FA, which is the ratio of

VOL. I.

F

the

the perpendicular height of the plane to its horizontal base, for the triangles ped and e FA are c fimilar*. And fince action and reaction are equal, and in contrary directions (22, R), it is evident that the fame force ep, which fuftains e on the fixed inclined plane C A B, applied in the contrary direction would, if the plane be fuppofed moveable in the direction of its bafe A B, and the body, e fixed by the application of an obstacle qr, sustain the effort with which the faid body tends to impel the plane from e towards p.

D The wedge is compofed of two inclined planes joined together at their common base, in the direction of which the power is impreffed.

E Let ABC (fig. 28) reprefent a wedge, whose vertex A is inserted between the two bodies D and E, which, being fixed in pofition, refift in a certain degree any force which tends to feparate them. This refiftance usually is, like the weight in the inclined plane, perpendicular to the base AF, and the power, or force, employed to overcome it, is

The fimilarity of thefe triangles not being obviously deducible is proved thus:

Prolong pe and d m till they meet in s; and the right angled triangle mse will be equal and fimilar to the triangle n pe, se being equal to e p..

The triangles sed, ped have the two fides s e, e p, equal, the fide ed common to both, and the included angles sed, ped are both right angles. Consequently the triangles are equal and alike in all refpects. Euclid 4.

1

But it is eafily fhewn, that the triangle sed is fimilar to

the triangle e A, and fo likewife must ped.

1

impreffed

impreffed as was just mentioned, in the direction of the faid bafe. Therefore, by the property of the inclined plane, the force required to keep one half CFA of the wedge in equilibrio with the preffure of the body D, is to that preffure as CF to FA. But as the preffure on the other half of the wedge acts with equal effect, a double force will be required to preferve the equilibrium, that is, a force as C B to F A. Or, in general terms; in any F wedge, as the line c B, joining the two equal fides AB and A C, is to the distance between the vertex A, and the middle point F of c B, fo is the force impreffed to the refiftance in D and E.

This inftrument is commonly used in cleaving G wood, and was formerly applied in engines for ftamping watch-plates. The force impreffed is commonly a blow, which is found to be much more effectual than a weight or preffure. This difference is ufually accounted for, by fuppofing that the tremulous motion produced by the stroke, confiderably diminishes the very great friction at the fides. But there is no doubt, that it is chiefly referable to the principles that obtain when refifting bodies are penetrated (37, M).

All cutting inftruments may be referred to the H wedge. A chizel, or an axe, is a fimple wedge. A faw is a number of chizels fixed in a line. A knife may be confidered as a wedge when employed in splitting, but if attention be paid to the edge, it is found to be a fine faw, as is evident from the much greater effect all knives produce by a drawing

F 2

a drawing ftroke, than what would have followed

from a direct action of the edge.

CHA P. V.

OF THE SCREW, AND OF MECHANICAL ENGINES IN GENERAL.

THE Screw is compofed of two parts, one of which is called the screw, and confists of a spiral protuberance, called the thread, which is wound or wrapt round a cylinder; and the other, called the nut, is perforated to the dimenfions of the cylinder, and in the internal cavity is cut a spiral groove adapted to receive the thread.

K Let A DGE (fig. 29) reprefent a cylinder, and ABC any flexible substance of a thickness altogether inconfiderable or evanefcent. Suppofe A B C to be a triangle, having a right angle at A, and one of the legs A B containing the right angle to be applied to the cylinder in a line parallel to its axis. Imagine now the cylinder to turn on its axis fo that the triangle ABC may be rolled or wrapped close on its furface. The lines BC, and all others, as IK, LR parallel to it, will then be contiguous to, or coincident with, the peripheries of circles, whofe planes are all at right angles to the axis, and consequently parallel to each other. But the line AC will become a curve AQL M'a NO P, &c. L which is called an helix. This curve will always, or in every part, proceed from one towards the

other