infpecting fig. the 7th, where it is evident, that if the weight be raised one foot, the ropes muft be fhortened of a foot each, viz. a foot from B to A, and another foot from D to C; hence whilst the length A B paffes over the pulley BD, twice that length must pass over the pulley F;

fo that the

pulley F, if equal in diameter to BD, must make two revolutions, whilft the pulley BD makes one revolution. It is alfo evident, that if the pulley F were of double the circumference, or, which is the fame thing, of double the diameter of BD, then each of the pulleys would make one revolution in the fame time. Now returning to the construction of fig. 9, it will be eafily comprehended, that as the three grooves of the upper pulley, as also the three of the lower pulley, belong to one folid body, they must revolve in the fame time; therefore, their diameters, or their circumferences, must be made in the proportion of the quantity of rope, which muft pafs over them in the fame time. Thus whilft one foot length of the rope paffes over the first groove a, two feet of rope muft pafs over the fecond groove b, three feet of rope muft pafs over the third groove c, and fo forth. Therefore, the diameter of the second groove b, must be twice the diameter of the first groove a; the diameter of the third groove c, muft be three times that of the first a; the diameter of the fourth groove d, must be four times that of a; &c. or, in other words,

the

the diameters of the grooves a, b, c, d, e, f, muft be in arithmetic progreffion; the difference of the terms being equal to the diameter of the firft or fmalleft groove.

It is evident, that in this construction, in order to raife the weight, fix ropes must be shortened, and of course the power muft move through fix times the space that the weight moves through, confequently the equilibrium takes place when the power is equal to the fixth part of the weight W.

The principal advantage which is attributed to this conftruction, is the reduction of friction; for in this, there are only two axes and four furfaces which rub against the blocks; whereas in the conftruction of fig. 8, where the pulleys are all feparate, there are fix axes and 12 furfaces which rub against the blocks. But, in my opinion, this advantage is more than compensated by the imperfections which are peculiar to this construction; for, in the first place, if the grooves are not made exactly in arithmetic progreffion, or if they become otherwife by, the accumulation of dirt, &c. then the rope muft partly flide over them, which will occafion a confiderable degree of friction; and fecondly, even when the grooves are of the proper dimensions, if the rope happens to ftretch more in one place than in another, which is generally the cafe, then the above-mentioned fliding and friction will alfo take place.

THE

THE INCLINED PLANE.

A plane fuperficies inclined to the horizon, is another mechanical power; its ufe being to raise weights from one level to another, by the application of much less force than would be neceffary to raise them perpendicularly. Thus in fig. 10, Plate VI. AB reprefents a plane inclined to the horizontal plane AC; where if the weight D be rolled upwards from A to B, the force neceffary for the purpose will be found to be much less than that which would be required to raise it directly and perpendicularly from C to B.

In this cafe the effect which is produced, confifts in the raifing of the weight from the level of AC to the level of B; but to effect this, the power must have moved from A to B; (for the power acts in that direction, whilft the weight or gravity of the body acts in the direction of the perpendicular CB;) therefore the velocity of the weight in this engine, being to the velocity of the power, as the perpendicular height BC of the plane is to its length AB, the equilibrium takes place when the weight is to the power, as the length of the plane is to its perpendicular height.

This property may be clearly fhewn by the following experiment :-Let AB, fig. 11, Plate VI. be a plane moveable upon the horizontal plane AC; viz. so as to admit of its being placed at any re...quired

quired angle of inclination, which is eafily accomplifhed by means of a hinge at A, and a prop between the two planes. The upper part of the plane must be furnished with a pulley B, over which a ftring may eafily run. Let the cylindrical weight D be made to turn upon slender pins in the frame F, in which the hook e is faftened with a ftring e BH, which paffing over the pulley B, holds the weight E fufpended at its other extremity.-The pulley should be fituated so that the rope e B, be parallel to the plane.

may

This plane may be fixed at any angle of inclination, and it will always be found, that if the weight of the body E be to the weight of the body D, together with that of its frame F, as the perpendicular height CB of the plane is to its length AB, the power E will juft fupport the cylinder D, with its frame F upon the plane, and the leaft touch of a finger will cause the cylinder D to afcend or defcend; the counterpoife or power E moving. at the fame time the contrary way.

It is evident, that the fmaller the angle of inclination is, the lefs force is required to draw up the weight D; and of course when the angle of inclination vanishes or becomes nothing, the leaft force will be fufficient to move the body; that is, when the plane AB becomes parallel to the horizon, or upon an horizontal plane, the heaviest body might be moved with the leaft power, were it not for the friction

friction, which is occafioned by the irregularity of the contiguous furfaces, &c. (1.)

THE

(1.) The above-mentioned explanation of the property of the inclined plane, applies only to one direction of the power; namely when the power acts in a direction parallel to the plane; but the general theory will be found in the following propofition:

When a body or weight W is fuftained upon a plane, which is inclined to the horizon; viz. when the power Pis juft fufficient to balance the weight upon that plane; then the power is to the weight, as the fine of the plane's inclination is to the fine which the direction of the power makes with a line perpendicular to the plane.

Let AB fig. 12, Plate VI. be the plane inclined to the horizon AC, and let a weight at O be supported partly by the plane, and partly by a power which acts in the direction OV. Through O draw EOC perpendicular to AB, and at C, where EC meets the horizontal plane, erect CV përpendicular to the horizon, to meet the direction of the power as at V.

Now the body W, fituated at O, is balanced, or kept at reft, by three powers, which (fee prop. IV. chap. VIII!) have the fame proportion to each other as have the right lines parallel to their refpective directions, and terminated by their mutual concurfe; namely, by the power which is as OV; by the gravitating power, which is as VC; and by the reaction of the plane, which is as OC; hence the power is to the weight, viz. P: W::OV: VC; or (fince the fides of plane trianglès are as the fines of their oppoffte angles) P: W:: fin. OCV, or BAC: (for those angles are equal fince the right-angled triangles BOC, and BAC

VOL. I.

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have