A 2 3 B b F.1. annexed diagram. Let ABC, Fig. 1. be a right-angled triangle: divide AB and BC into the same number of equal parts, as here 5 for instance, through each of those divisions draw horizontal, perpendicular, and diagonal lines, as in the figure which is thus divided into 25 small triangles, all similar to ABC, and similar and equal to one another. Suppose a body to fall, by the constantly-increasing action of gravity, from A towards B, in 5 successive small but equal portions of time; then the horizontal lines la, C2b, 3c, &c. will represent the velocity of the body, at the end of the 1st, 2d, 3d, &c. portions of time, in which it continues to fall. In the 1st instant falling from A to 1, the velocity will be represented by the base la of the triangle Ala, which will represent the space passed over by the falling body in the 1st instant. In the next instant the body will fall to 2, its velocity will be shewn by 26, which is double 1a, and the space passed over by the body will be represented by the figure 1a, b2. Now this figure contains 3 triangles, similar and equal to the first triangle Ala: the body at the end of the 2d instant has therefore acquired a new velocity double the original, which added to it, produces a treble velocity. In the 3d instant the body falls to 3; the velocity will then be 3c, and the space passed over in that instant is represented by the figure 2b c3. But this space contains 5 triangles equal A1a; being greater than the space of the 2d instant by twice the 1st space Ala; consequently if to the space passed over by a falling body, in any given period of time, be added twice the space passed over in the 1st equal period, the space to be gone through in the next succeeding period will be obtained. The spaces passed through by falling bodies will therefore increase in the proportion of the numbers 1, 3, 5, 7, 9, 11, 13, 15, &c. as was before stated. The same diagram will give an idea of retarded motion, by proceeding in a contrary direction, ascending upwards from B to A. A body projected perpendicularly upwards from B, with a velocity represented by the line BC, will, by the counteraction of gravity, when it arrives at the end of the 1st instant at 4, have its velocity diminished to 4d: when it ascends to 2, it will have only a velocity equal to 2b, and on arriving at A, its projectile force being completely balanced and exhausted by gravity, the body will proceed no higher, will there stop, and, being acted on by gravity alone, it will begin to fall down with accelerated velocity to B, where it began to ascend. The reader will be aware, that experiments on the motion of falling bodies ought to be made above and not below the surface of the earth: because attraction being inherent in every particle of matter, a body sunk below the surface must be attracted upwards, by the particles above it, in diminution of the attraction of those below it, and its specific gravity is consequently lessened: so that were an aperture formed through the centre of the earth from surface to surface, and a pistol bullet dropped into it, instead of passing through the globe, it would, from the velocity acquired in falling from the surface, pass a little beyond the centre, then return back for a still shorter space, and so after a few vibrations, like the pendulum of a clock, remain stationary in the heart of the earth. Experiments on falling bodies from the top of a lofty tower will be satisfactory but those made in a deep well or pit must be inaccurate. It was before shewn that a body acted upon by two uniform forces, in different directions, will move in a direction between the two, and in a straight line. If, however, one of the forces be regularly diminished, while the other continues unabated, the moving body will gradually draw nearer and nearer to the direction of the uniform force, and consequently describe a curve line. Thus, when a ball is thrown from a cannon (not perpendicularly upwards) it receives such an impulse from the powder as would carry it on in a straight line: but the resistance produced by the air gradually diminishes this original projecting force, so that it becomes less and less able to overcome the power of gravity, by which the ball is drawn down towards the centre of the earth. The consequence of this is, that the ball which proceeded from the mouth of the cannon in a course differing very little from a straight line, describes a curve bending more and more downwards, and at last falls to the ground. This curve was once supposed to be in certain cases a parabola, or that formed by cutting a cone by a plane parallel to one of the inclined sides: but experience has shown that, from the resistance of the air, and other causes, the curve described by cannon-ball is very different from this section of the cone. On a knowledge of the curves described by projected bodies is founded the art of gunnery or artillery. The force with which a body moves, or which it would exert upon another body opposed to it, is always in proportion to its velocity multiplied by its weight, that is, by its quantity of matter. This force is called the momentum of the body, and upon the application of this property depends the whole art of constructing machines. ATTRACTION AND REPULSION.-By attraction we mean the tendency of bodies, in certain circumstances, to draw near to each other, agreeably to the import of the term. Attraction is divided into various kinds; but as their causes are all equally unknown, it is uncertain whether they may not be all modifications of the same principle. These are the attraction of cohesion, of gravitation, of electricity, of magnetism, and chemical attraction. The attraction of cohesion takes place between bodies, only when they are at very small distances from each other; and by this it is that bodies preserve their form, and are prevented from falling in pieces. It would however appear that, in some cases, bodies when very near together, repel each other, so that although they appear to be in contact, they are not truly so, but will require considerable force to bring them to touch one another. two pieces of lead be scraped very clean with a knife, and strongly squeezed together, they will adhere so firmly that they can scarcely be separated. The same effect will take place with plates of glass or marble wetted with water. The different degrees of cohesive attraction may be the cause of the different degrees of tenacity or hardness observable in bodies. If The attraction of gravity or gravitation, is that property by which all masses of matter tend towards each other, and which they exert at all distances. It is by this property that a stone falls from a height to the ground: and it is by this that the heavenly bodies are retained in their several places, by their action on one another. It is a law of nature ascertained by the immortal Newton, that every particle of matter gravitates, or has a tendency to approach towards every other particle. This law is the grand leading principle of the Newtonian system of natural philosophy. The planets and comets all gravitate towards the sun, and towards each other, as does the sun towards them; each body in proportion to the quantity of matter it contains. All bodies on this earth gravitate towards a point at, or very near its centre, consequently they fall every where perpendicularly to its surface; hence the direction of a falling body on one side of the globe must be directly opposite to that of one falling at a point diametrically opposite to it. As this force of gravitation or of attraction is always in proportion to the quantity of matter in bodies, it is this which constitutes their respective weights. If two bodies containing equal quantities of matter were placed at any given distance asunder, in free open space, where nothing could interrupt their motion, they would be mutually and equally attracted, and at last meet in a point in the middle of their original distance. If, however, one of the bodies were greater than the other, as double its weight, the point of meeting would be as much nearer to the greater body as this exceeded the smaller body in weight. In all places equally distant from the centre of the earth the force of gravity is equal; but this earth not being a perfect sphere, as was explained when treating of Geography, its surface is not in all parta equally distant from its centre. The diameter at the equator is about 34 miles longer than that at the poles, consequently gravity is weaker under the equator than at the poles: hence it is, that the pendulum of a clock, (which is moved by its gravitation towards the centre,) adapted to the latitude of London, in 514 degrees, will require to be lengthened if carried towards the north pole, and to be shortened at the equator, in order to keep time as it did in London. All bodies possess some measure of gravity or weight, and, if at liberty, would equally tend to the centre of the earth: but smoke, vapour, &c. mount up in the air. This, however, is no proof that these substances have no real weight, but only that they are lighter than the air in which they float, which falls to the ground, and thereby forces the smoke, vapour, &c. to rise until they come to a region where the air is of the same gravity with themselves. When any substance is placed in one scale of a balance, so heavy as to make the other scale containing a weight to rise up, we do not suppose this weight and its scale to be absolutely light, but only that their gravity is overcome by that of the substance in the scale at the other end of the beam. The force of gravity regularly increases as bodies approach the surface of the earth: but below the surface this force regularly diminishes; so that it is greatest on the surface itself. In proportion as any body penetrates below the surface of the earth, more and more matter is collected above it, the attraction of which must act in a direction opposite to that towards the centre; so that a body at the centre of the earth being equally attracted in every direction all around, may be said to possess no real gravity. That great bodies of the earth possess attraction was clearly proved in 1774, from a set of experiments purposely instituted under direction of the late Astronomer Royal, the Rev. Dr. Maskelyne, 2 and other members of the Royal Society of London. The experiments were made on Shehallien, a remarkable conical mountain, detached from all others, near Loch Tay, in Scotland, rising to an equal height with Snowdon, in Wales, viz. 3550 feet above the sea. A pendulum suspended by ingenious mechanism, on opposite points of the mountain, was found to be constantly drawn towards it, out of the true perpendicular line. It was already said, that all motion produced by one power must be in a right line: when, therefore, we see bodies moving in curved lines, we may be sure that they are acted upon by more powers than one: for if all these forces cease to act save one, the body will again move in a right line in the direction of this single force. A stone in a sling whirled about by the hand acquires a force by which, if at liberty, it would be carried forward in a straight line to a considerable distance. Being retained, however, by the string, the stone is compelled to describe a circular motion until the string be let go, when it will proceed in a straight line, at right angles, to the direction of the string at the moment of its discharge, or in the tangent of the circle of its re volution. The stone in whirling round is acted upon by two powers, the one of the force impressed upon it by the whirling, by which it would fly off in the tangent, and so escape from the centre of the circle: hence this is called centrifugal force. On the other hand being held back by the string, the stone continues to revolve at equal distances round the centre of motion: hence this is called centripetal force. it If we lift a bar of iron, a block of wood, or any other solid body, in such a way that all its parts, in any position, will exactly balance one another, the point by which it is supported is called the centre of gravity. Suppose a solid block of timber, stone, &c. of a regular form, be set upon end on a level floor, it will have no tendency to fall to either side, and a line from the centre of gravity perpendicular to the surface of the earth, will pass exactly through the middle of the end on the floor. Let now the block be inclined to one side, the perpendicular from its centre will gradually approach that side to which the block leans, until just coincide with the angle of the end and the lower side. At this point the block will have no inclination, either to return to its former erect position, or to fall over on its side: but if by leaning it still more, the perpendicular from its centre fall on the outside of the end next the floor, then the greater part of the weight being on the outside of the base, the equal balance will be destroyed, and the block will fall over on its side. The common centre of gravity of two bodies is that about which they would just balance each other. Thus, if two balls of lead, A and B, be fixed to the ends of an iron rod, and that A weigh pounds, while B weighs 4 pounds, the point in the rod between them, on which the balls will balance each other, will be exactly as much nearer to the greater ball than to the less, as this last is lighter than the first. For, in order to make a balance or equilibrium between two bodies, the centre of gravity or point of support must be so situated that the distance of the one ball multiplied by its weight, shall be equal to the distance of the other ball multiplied by its weight. The weight of A is 2 pounds, and that of B is 4 pounds; the length of the rod connecting them is 12 inches: the distance of A from the centre of sup port, must therefore be double that of B, consequently if the rod be divided into 3 equal parts, 2 of them or 8 inches will be the distance C from A to the centre, and one of them or 4 inches, that between B and the centre, For 2 multiplied by 4 are just equal to 4 multiplied by 2. On this principle is formed the steel-yard, or balance, with arms of unequal length, as will be afterwards explained. When the perpendicular from the centre of gravity of our body falls within the base of our feet, we stand firm: if it fall without that base, we fall down on that side and it is surprising to consider the various postures and methods to which a man has recourse, as it were instinctively, to retain or to recover his proper position. Thus, we bend our body forward when we rise from a chair, or in going up a stair, to assist our motion: and for the same reason a man leans forward when he carries a load on his back, and bends backward when he carries it before him, and to the right or left side, according as he carries it on the left or the right. In all these cases, the object is to place the body and the load in such a position that the perpendicular from the centre of gravity of both, may fall within the base of the feet. Hence appears the great error of persons rising hastily up, in a carriage or a boat, when likely to be overturned; for by so doing, the centre of gravity is thrown still farther from the base, and the machine must inevitably be overset. Whereas, had they instantly fallen down in the bottom, the centre of gravity would have been thrown more within the base, and the accident, probably prevented. In regular bodies of an uniform substance, as in dry timber, stone, metals, &c. the centre of gravity will in general coincide with the centre of the figure, that is in the middle of the length, breadth and thickness. But in bodies of irregular shape the centre of gravity is found in this way. Suspend such a body by any point, with one side perpendicular to the horizon, and from the same point hang a plumb line; on, the body draw a line where the string passes: do the same at any other point of suspension, and where these two lines meet, will point out the centre of gravity: for this being in each line it must be at the point where they meet. .* MECHANICAL POWERS. The mechanical powers are engines for enabling us to move and raise heavy bodies, and to overcome resistances, which, without their aid, we could neither raise nor move. In all machines three things are to be considered, the weight to be raised or moved, the power by which it is moved, and the instrument or engine by which the motion is effected. The mechanic powers are commonly counted to be six in number, viz. the Lever, the Pulley, the Axis and Wheel, the Inclined Plane, the Wedge, and the Screw: But these may all be reduced to two; for the pulley and the wheel are only assemblages of levers, and the wedge and screw are inclined planes. The Lever is the simplest of all machines, only a straight bar of wood, iron, or other substance, called a crow or a handspike, supported on a prop called a fulcrum. Levers are distinguished into three sorts, according to the different positions of the prop or fulcrum, and the power employed to move the lever: 1. When the prop is placed between the power and the weight; prop is at one end of the lever, the power at the other, and the weight between them; and 3. When the prop is at one end, the weight at the her, and the power between them. Of the first sort of lever is the 2. When the |