force of gravity acts on the body constantly, so as to draw it to the earth, by a uniformly accelerated motion; and the result is, that the ball will move in a curve, and this curve may be easily shown to be that of the parabola. The resistance of the air being taken into account together with these circumstances, constitute the basis of the science of gunnery.

We shall give a simple example, to show the application of the former part of this subject. One force will cause the body A to move 20 miles in a day, and another, acting at right angles, will cause it to move 18 miles a day; draw these lines 20 and 18 from the line of lines on the sector, as the sides AB, AC, of a parallelogram, and complete it: draw the diagonal, then measure it, and it will be found to be 269, the resulting motion; and the angle being measured, will give the direction.-There are other methods of doing this by calculation, but this is simple, and is sufficient to show the principle.

MECHANICAL POWERS.

1. A MACHINE is any instrument employed to regulate motion, so as to save either time or force. No instrument can be employed by man so as to save both time and force; for it is a maxim in mechanics, that whatever we gain in the one of these two, must be at the expense of the other.

2. The simple machines, or those of which all others are constructed, are usually reckoned six: the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw. To these the funicular machine is sometimes

added.

3. The weight signifies the body to be moved, or the resistance to be overcome; and the power is the force em ployed to overcome that resistance, or move that body They are frequently represented by the first letters of their names, W and P.

THE LEVER.

4. A LEVER is an inflexible bar, either straight or bent, supposed capable of turning round a fixed point, called the fulcrum

According to the relative positions of the weight, power and fulcrum, on the lever, it is said to be of three kinds. viz. when the fulcrum is somewhere betwixt the weight and power, it is of the first kind; when the weight is between the power and the fulcrum, it is of the second kind; and when the power is between the weight and the fulcrum, it is of the third kind: thus,

8. In the first and second kinds there is an advantage of power, but a proportionate loss of velocity; and in the third kind, there is an advantage in velocity, but a loss of power.

=

9. When the weight x its distance from the fulcrum the power its distance from the fulcrum, then the lever will be at rest, or in equilibrio; but if one of these products be greater than the other, the lever will turn round the fulcrum in the direction of that side whose product is the greater.

10. In all the three kinds of levers, any of these quantities, the weight or its distance from the fulcrum, or, the power or its distance from the fulcrum, may be found from the rest, such, that when applied to the lever, it will remain at rest, or the weight and power will balance each other.

15. In the first kind of lever, the pressure upon the fulcrum = the sum of weight and power; in the second and third = their difference.

16. If there be several weights on both sides of the fulcrum, they may be reckoned powers on the one side of the fulcrum, and weights on the other. Then, if the sum of the product of all the weights × their distances from the

fulcrum be to the sum of the products of all the powers X their distances from the fulcrum, the lever will be at rest, if not, it will turn round the fulcrum in the direction of that side whose products are greatest.

17. In these calculations, the weight of the lever is not taken into account; but if it is, it is just reckoned like any other weight or power acting at the centre of gravity.

18. When two, three, or more levers act upon each other in succession, then the entire mechanical advantage which they give, is found by taking the product of their separate advantages.

19. It is to be observed, in general, before applying these observations to practice, that if the lever be bent, the distances from the fulcrum must be taken, as perpendiculars drawn from the lines of direction of the weight and power to the fulcrum.

Ex. In a lever of the first kind, the weight is 16, its distance from the fulcrum 12, and the power is 8; therefore, by No. 13 of this chapter,

16 x 12

8

-= 24, the distance of power from the fulcrum. In a lever of the second kind, a power of 3 acts at a distance of 12; what weight can be balanced at a distance of 4 from the fulcrum? Here, by No. 12,

In a lever of the third kind, the weight is 60, and its distance 12, and the power acts at a distance of 9 from the fulcrum; therefore, by No. 11,

If there be a lever of the first kind, having three weights, 7, 8, and 9, at the respective distances of 6, 15, and 29, from the fulcrum on one side, and a power of 17 at the distance of 9 on the other side of the fulcrum; then a power is to be applied at the distance of 12 from the fulcrum, on the last mentioned side: what must that power be to keep the lever in balance?

Here (6 × 7) + (15 x 8) + (29 × 9) = 423 = the effect of the three weights on the one side of the fulcrum; and 17 x 9 153 = the effect of the power on the other side. Now, it is clear that the effect of the weight is

far greater than the effect of the power; and the difference 423 - 153 = 270 requires to be balanced by a power applied at the distance of 12, which will evidently be found by dividing 270 by 12, which gives 22.5, the weight required.

20. The Roman steel-yard is a lever of the first kind, so contrived that only one movable weight is employed.

The common weighing balance is also a lever of the first kind. The requisites of a good balance are: that the points of suspension of the scales and the centre of motion, or fulcrum of the beam, be all in one straight line-that the arms of the beam be equal to each other in every respect that they be as long as possible-that the centre of gravity of the beam be a very little below the centre of motion—that the beam be balanced when the scales are empty, &c. But we may ascertain the true weight of any body even by a false balance, thus: weigh the body first in one scale, then in the other, and multiply their weights together; then the square root of this product will be the true weight.

THE WHEEL AND AXLE.

21. THE wheel and axle is a kind of lever, so contrived as to have a continued motion about its fulcrum, or centre of motion, where the power acts at the circumference of the wheel, whose radius may be reckoned one arm of the lever, the length of the other arm being the radius of the axle, on which the weight acts. If the power acts at the end of a handspike fixed in the rim of the wheel, then this increases the leverage of the power, by the length of the handspike.

The wheel and axle consists of a wheel having a cylindric axis passing through its centre. The power is applied to the cir- B cumference of the wheel, and the weight to the circumference of the axle.

In the wheel and axle, an equilibrium takes place when the power multiplied by PO the radius of the wheel, is equal to the weight multiplied by the radius of the axle; or, P: W:: CA: CB.

For the wheel and axle being nothing else but a lever so contrived as to have a continued motion about its ful

crum C, the arms of which may be represented by AC and BC, therefore, by the property of the lever, P: W:: CA : СВ.

If the power does not act at right angles to CB, but obliquely, draw CD perpendicular to the direction of the power, then, by the property of the lever, P: W:: CA: CD.

22. It will be easily seen, that if two wheels fastened together and turning round the same centre, be so adjusted, that while they turn round they will coil on their circumferences strings, to which weights are suspended; one of those wheels being larger than the other, the larger wheel will coil up a greater length of the string than the smaller one will do in the same time, and this will depend either on the radii or circumferences of the two wheels. The velocity of the weight will be in proportion to the length of string coiled in a given time; therefore, the velocity of the weight will be greater as the wheel is larger. Now, as in the lever we saw that a small weight required a great velocity to balance a large weight with a small velocity, we may infer, that the rules given for levers will also apply to the wheel and axle; since the velocity of any body on a lever depends upon its distance from the fulcrum.

Ex.-A weight of 13 lbs. is to be raised at a velocity of 14 feet per second, by a power whose velocity is 20 feet per second; how great must that power be?

If the velocity of the weight, be to that of the power, as 14 to 20, and the radius of the axle on which the weight is coiled be 7; then,

10, radius of wheel on which the power acts.

If a weight of 36 lbs, is to be raised by an axle 3 inches diameter; what must be the power applied at the end of a handspike 4 inches long, fixed in the rim of the wheel connected with the axle, the wheel being 6 inches diameter ?

Here the handspike will increase the distance of the power from the fulcrum, and will add to the diameter of the wheel twice its own length; therefore, 8+ 6 = 14;hence, 14 : 3 :: 36: 777, the power required to keep the weight in equilibrio.