Of all the various directions of drawing bodies up an inclined plane, or sustaining them on it, the most favourable is where it is parallel to the plane BC, and passing through the centre of the weight; a direction which is easily given to it, by fixing a pulley at D, so that a cord passing over it, and fixed to the weight, may act or draw parallel to the plane. In every other position, it would require a greater power to support the body on the plane, or to draw it up. For if one end of the line be fixed at w, and the other end inclined down towards B, below the direction wD, the body would be drawn down against the plane, and the power must be increased in proportion to the greater difficulty of the traction. And, on the other hand, if the line were carried above the direction of the plane, the power must be also increased; but here only in proportion as it endeavours to lift the body off the plane.

If the length BC of the plane be equal to any number of times its perp. height AB, as suppose 3 times; then a power r of 1 pound hanging freely, will balance a weight w of 3 pounds, laid on the plane: and a power of 2 pounds, will balance a weight w of 6 pounds; and so on, always 3 times as much. But then if they be set a-moving, the perp. descent of the power r, will be equal to 3 times as much as the perp. ascent of the weight w. For, though the weight w ascends up the direction of the oblique plane, Bc, just as fast as the power r descends perpendicularly, yet the weight rises only the perp. height AB, while it ascends up the whole length of the plane вc, which is 3 times as much; that is, for every foot of the perp. rise, of the weight, it ascends 3 feet up in the direction of the plane, and the power r descends as much, or 3 feet.

PROPOSITION XXXVI.

197. When a Wedge is in Equilibrio; the Power acting against the Back, is to the Force acting Perpendicularly against either Side, as the Breadth of the Back AB, is to the Length of the Side Aç or Bс.

which are per. to them.

And therefore the thinner a wedge is, the greater is its effect in splitting any body, or in overcoming any resistance against the sides of the wedge.

SCHOLIUM.

199. But it must be observed, that the resistance, or the forces above-mentioned, respect one side of the wedge only. For if those against both sides be taken in, then, in the foregoing proportions, we must take only half the back AD, or else we must take double the line AC or DC.

1

1

In the wedge, the friction against the sides is very great, at least equal to the force to be overcome, because the wedge retains any position to which it is driven; and therefore the resistance is double by the friction. But then the wedge has a great advantage over all the other powers, arising from the force of percussion or blow with which the back is struck, which is a force incomparably greater than any dead weight or pressure, such as is employed in other machines. And accordingly we find it produces effects vastly superior to those of any other power; such as the splitting and raising the largest and hardest rocks, the raising and lifting the largest ship, by driving a wedge below it, which a man can do by the blow of a mallet: and thus it appears that the small blow of a hammer, on the back of a wedge, is incomparably greater than any mere pressure, and will overcome it,

OF

OF THE SCREW.

200. THE SCREW is one of six mechanical powers, chiefly used in pressing or squeezing bodies close, though sometimes also in raising weights.

The screw is a spiral thread or groove cut round a cylinder, and every where making the same angle with the length of it. So that if the surface of the cylinder, with this spiral thread on it, were unfolded and stretched into a plane, the spiral thread would form a straight inclined plane, whose length would be to its height, as the circumference of the cylinder, is to the distance between two threads of the screw : as is evident by considering that, in making one round, the spiral rises along the cylinder the distance between the two threads.

PROPOSITION XXXVII.

201. The Force of a Power applied to turn a Screw round, is to the Force with which it presses upward or downward, setting aside the Fraction, as the Distance between two Threads, is to the Circumference where the Power is applied.

THE screw being an inclined plane, or half wedge, whose height is the distance between two threads, and its base the circumference of the screw; and the force in the horizontal direction, being to that in the vertical one, as the lines perpendicular to them, namely, as the height of the plane, or distance of the two threads, is to the base of the plane, or circumference of the screw; therefore the power is to the pressure, as the distance of two threads is to that circumference. But, by means of a handle or lever, the gain in power is increased in the proportion of the radius of the screw to the radius of the power, or length of the handle, or as their circumferences. Therefore, finally, the power is to the pressure, as the distance of the threads, is to the circumference described by the power.

202. Corol. When the screw is put in motion; then the power is to the weight which would keep it in equilibrio, as the velocity of the latter is to that of the former; and hence their two momenta are equal, which are produced by multiplying each weight or power by its own velocity. So that this is a general property in all the mechanical powers, namely, that the momentum of a power is equal to that of the weight which would balance it in equilibrio; or that each of them is reciprocally proportional to its velocity.

SCHOLIUM.

203. Hence we can easily compute the force of any machine turned by a screw. Let the annexed figure represent a press driven by a screw, whose threads are each a quarter of an inch asunder; and let the screw be turned by a handle of 4 feet long, from A to B; then, if the natural force of a man, by which he can lift, pull,

or draw, be 150 pounds; and it be required to determine with what force the screw will press on the boad at D, when the man turns the handle at A and B, with his whole force. Then the diameter AB of the power being 4 feet, or 48 inches, its circumference is 48 × 3-1416 or 150 nearly; and the distance of the threads being of an inch; therefore the power is to the pressure as 1 to 6031: but the power is equal to 150lb; theref. as 1: 6031:: 150: 90480; and consequently the pressure at D is equal to a weight of 90480 pounds, independent of friction.

f

204. Again, if the endless screw AB be turned by a handle ac of 20 inches, the threads of the screw being distant half an inch each; and the screw turns a toothed wheel E, whose pinion L turns another wheel F, and the pinion of this another wheel G, to the pinion or barrel of which is hung a weight w; it is required to determine what weight the man will be able to raise, working at the handle c; supposing the diameters of the wheels to be 18 inches, and those of the pinions and barrel 2 inches; the teeth and pinions being all of a size.

Here

Here 20×3.1416×2=125·664, is the circumference of the power.

And 125-664 to, or 251-328 to 1, is the force of the screw alone.

Also, 18 to 2, or 9 to 1, being the proportion of the wheels to the pinions; and as there are three of them, therefore 93 to 13, or 729 to 1, is the power gained by the wheels.

Consequently 251-328×729, to 1, or 183218 to 1 nearly, is the ratio of the power to the weight, arising from the advantage both of the screw and the wheels.

But the power is 150lb; therefore 150 × 183218, or 27482716 pounds, is the weight the man can sustain, which is equal to 12269 tons weight.

But the power has to overcome, not only the weight, but also the friction of the screw, which is very great, in some cases equal to the weight itself, since it is sometimes sufficient to sustain the weight, when the power is taken off.

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ON THE CENTRE OF GRAVITY,

:

205. THE CENTRE of GRAVITY of a body, is a certain point within it, on which the body being freely suspended, it will rest in any position; and it will always descend to the lowest place to which it can get, in other positions.

PROPOSITION XXXVIII.

206. If a Perpendicular to the Horizon, from the centre of Gravity of any body, fall within the Base of the Body, it will rest in that Position; but if the Perpendicular fall without the Base, the Body will not rest in that Position, but will tumble down.

E; contrary to the nature of that centre, which only rests when in the lowest place. For the same reason, the body will not fall towards D. And therefore it will stand in that

position.

VOL. II.

23

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