This term, in the common acceptation, implies no more than the nature of what is called the mechanical powers, together with the combination of these powers in the construction of machines. But as the general properties of matter and central forces are necessary in order to a thorough knowledge of mechanics, we have joined all these subjects together under the general name of Mechanics.

Of Matter, and its Properties.

By the word matter is here meant every thing that has length, breadth, and thickness, and resists the touch.

The inherent properties of matter are solidity, inactivity, mobility, and divisibility.

The solidity of matter arises from its having length, breadth, thickness; and hence it is that all bodies are comprehended under some shape or other, and that every particular body hinders all others from occupying the same part of space which it possesses. Thus, if a piece of wood or metal be squeezed ever so hard between two plates, they cannot be brought into contact. And even water or air has this property; for if a small quantity of it be fixed between any other bodies, they cannot be brought to touch one another.

A second property of matter is inactivity, or passiveness; by which it always endeavours to continue in the state that it is in, whether of rest or motion. And therefore, if one body contains twice or thrice as much matter as another body does, it will have twice or thrice as much inactivity; that is, it will require twice or thrice as much force to give it an equal degree of motion, or to stop it after it hath been put into such a motion.

That matter can never put itself into a motion is allowed by all men. For a stone, lying on the plain surface of the earth, never removes itself from that place, nor does any one imagine it ever can. But most people are apt to believe, that all matter has a propensity to fall from a state of motion into a state of rest; because they see, that if a stone Note or a cannon-ball be put into ever so violent a motion, it soon stops; not considering that this stoppage is caused, 1. By the gravity or weight of the body, which sinks it to the ground in spite of the impulse; and, 2. By the resistance of the air through which it moves, and by which its velocity is retarded every moment till it falls.

A bowl moves but a short way upon a bowling-green; because the roughnesses and unevennesses of the grassy surface soon creates friction enough to stop it. But if the green were perfectly level, and smooth, and the bowl were perfectly hard, round, and smooth, it would go a great way further; as it would have nothing but the air to resist it: if then the air were taken away, the bowl would go on without any friction, and consequently without any diminution of the velocity it had at setting out: and therefore, if the green were extended quite around the earth, the bowl would go on round and round the earth, for ever.

If the bowl were carried several miles above the earth, and there projected in a horizontal direction, with such a velocity as would make it move more than a semidiameter of the earth, in the time it would take to fall to the earth by gravity: in that case, and if there were no resisting medium in the way, the bowl would not fall to the earth at all; but would continue to circulate round it, keeping always in the same tract, and returning to the same point from which it was projected, with the same velocity as at first. In this manner the moon moves round the earth, although she be as inactive and dead as any stone upon it.

The third property of matter is mobility; for we find that all matter is capable of being moved, if a sufficient degree of force be applied to overcome its inactivity or resistance.

The fourth property of matter is divisibility, of which there can be no end. For since matters can never be annihilated by cutting or breaking, we can never imagine it to be cut into such small particles, but that if one of them be laid on a table, the uppermost side of it will be further from the table than the undermost side.

Plate CV. fig. 1. That matter is infinitely divisible in a mathematical sense, is easy to be demonstrated. For, let AB be the length of a particle to be divided; and let it be touched at opposite ends by the parallel lines CD and EF, which suppose to be infinitely extended beyond D and F. Set off the equal divisions BG, GH, HI, &c. on the line EF, towards the right hand from B; and take a point, as at R, anywhere towards the left hand from A, in the line CD: Then, from this point draw the right lines RG, RH, RI, &c. each of which will cut off a part from the particle AB. But after any finite number of such lines are drawn, there will still remain a part, as AP, at the top of the particle, which can never be cut off: because the lines DR and EF being parallel, no line can ever be drawn from the point R to any point of the line EF that will coincide with the line RD. Therefore the particle AB contains more than any finite number of parts.

A fifth property of matter is attraction, which seems rather to be infused than inherent. Of this there are four kinds, viz. cohesion, gravitation, magnetism, and electricity.

Vol. II., No 70.

The attraction of cohesion is that by which the small parts of matter are made to stick and cohere together. Of this we have several instances, some of which follow.

1. If a small glass tube, open at both ends, be dipped in water, the water will rise up in the tube to a considerable height above its level in the basin; which must be owing to the attraction of a ring of particles of the glass all around in the tube, immediately above those to which the water at any instant rises. And when it has risen so high, that the weight of the column balances the attraction of the tube, it rises no higher. This can be no ways owing to the pressure of the air upon the water in the basin; for, as the tube is open at top, it is full of air above the water, which will press as much upon the water in the tube as the neighbouring air does upon any column of an equal diameter in the basin. Besides, if the same experiment be made in an exhausted receiver of the air-pump, there will be found no difference.

2. A piece of loaf-sugar will draw up a fluid, and a sponge will suck in water; and on the same principle sap ascends in trees.

3. If two drops of quicksilver be placed near each other, they will run together, and become one large drop.

4. If two pieces of lead be scraped clean, and pressed together with a twist, they will attract each other so strongly, as to require a force much greater than their own weight to separate them. And this cannot be owing to the pressure of the air, for the same thing will hold in an exhausted receiver.

5. If two polished plates of marble or brass be put together, with a little oil between them to fill up the pores in their surfaces, and prevent the lodgement of any air; they will cohere so strongly, even if suspended in an exhausted receiver, that the weight of the lower plate will not be able to separate it from the upper one. In putting these plates together, the one should be rubbed upon the other, as a joiner does two pieces of wood when he glues them.

6. If two pieces of cork, equal in weight, be put near each other in a basin of water, they will move equally fast toward each other, with an accelerated motion, until they meet; and then, if either of them be moved, it will draw the other after it. If two corks of unequal weights be placed near each other, they will approach with accelerated velocities inversely proportionate to their weights: that is, the lighter cork will move as much faster than the heavier, as the heavier exceeds the lighter in weight. This shews that the attraction of each cork is in direct proportion to its weight or quantity of matter.

This kind of attraction reaches but to a very small distance; for, if two drops of quicksilver be rolled in dust, they will not run together, because the particles of dust keeps them out of the sphere of each other's attraction.

Where the sphere of attraction ends, a repulsive force begins; thus, water repels most bodies till they are wet; and hence it is, that a small needle, if dry, swims upon water; and flies walk upon it without wetting their feet.

The repelling force of the particles of a fluid is but small; and therefore, if a fluid be divided, it easily unites again. But if glass, or any other hard substance, be broken... broke into small parts, they cannot be made to stick together again without being first wetted; the repulsion being too great to admit of a re-union.

The repelling force between water and oil is so great, that we find it almost impossible to mix them so as not to separate again. If a ball of light wood be dipped in oil, and then put into water, the water will recede so as to form a channel of some depth all around the ball.

The repulsive force of the particles of air is so great, that they can never be brought so near together by condensation as to make them stick or cohere. Hence it is, that when the weight of the incumbent atmosphere is taken off from any small quantity of air, that quantity will diffuse itself so as to occupy (in comparison) an infinitely greater portion of space than it did before.

Attraction of gravitation is that power by which distant bodies tend towards one another. Of this we have daily instances in the falling of bodies to the earth. By this power in the earth it is, that bodies, on whatever side, fall in lines perpendicular to its surface; and consequently, on opposite sides, they fall in opposite directions; all towards the centre, where the force of gravity is as it were accumulated; and by this power it is, that bodies on the earth's surface are kept to it on all sides, so that they cannot fall from it. And as it acts upon all bodies in proportion to their respective quantities of matter, without any regard to their bulks or figures, it accordingly constitutes their weight. Hence,

If two bodies which contain equal quantities of matter, were placed at ever so great a distance from one another, and then left at liberty in free space; if there were no other bodies in the universe to affect them, they would fall equally swift towards one another by the power of gravity, with velocities accelerated as they approached each other; and would meet in a point which was half way between them at first. Or, if two bodies containing unequal quantities of matter, were placed at any distance, and left in the same manner at liberty, they would fall towards one another with velocities which would be in an inverse proportion to their respective quantities of matter; and moving faster and faster in their mutual approach, would at last meet in a point as much nearer to the place from which the heavier body began to fall, than to the place from which the lighter body began to fall, as the quantity of matter in the former exceeded that in the latter.

All bodies that we know of have gravity or weight. For, that there is no such thing as positive levity, even in smoke, vapours, and fumes, is demonstrable by experiments on the air-pump: which shews, that although the smoke of a candle ascends to the top of a tall receiver, when full of air; yet upon the air's being exhausted out of the receiver, the smoke falls down to the bottom of it. So, if a piece of wood be immersed in a jar of water, the wood will rise to the top of the water, because it has a less degree of weight than its bulk of water has; but if the jar be emptied of water, the wood falls to the bottom.

As every particle of matter has its proper gravity, the effect of the whole must be in proportion to the number of the attracting particles; that is, as the quantity of matter in the whole body. This is demonstrable by experiments on pendulums; for if they are of equal lengths, whatever their weights be, they vibrate in equal times. Now it is plain, that if one be double or triple the weight of another, it must require a double or triple power of gravity to make it move with the same celerity; just as it would require a double or triple force to project a bullet of twenty or thirty pound weight with the same degree of swiftness that a bullet of ten pounds would require. Hence, it is evident, that the power or force of gravity is always proportional to the quantity of matter in bodies, whatever their bulks or figures are.

Gravity also, like all other virtues or emanations which proceed or issue from a centre, decreases as the distance multiplied by itself increases: that is, a body at twice the distance of another attracts with only a fourth part of the force; at thrice the distance, with a ninth part; at four times the distance, with a sixteenth part; and so on. This too is confirmed by comparing the distance which the moon falls in a minute from a right line touching her orbit, with the distance through which heavy bodies near the earth fall in that time; and also by comparing the forces which retain Jupiter's moons in their orbits, with their respective distances from Jupiter. These forces will be explained afterwards.

The velocity which bodies near the earth acquire in descending freely by the force of gravity, is proportional to the times of their descent. For, as the power of gravity does not confine in a single impulse, but is always operating in a constant and uniform manner, it must produce equal effects in equal times; and consequently in a double or triple time, a double or triple effect; and so, by acting uniformly on the body, must accelerate its motion proportionally to the time of its descent.

To be a little more particular on this subject, let us suppose that a body begins to move with a celerity constantly and gradually increasing, in such a manner, as would carry it through a mile in a minute; at the end of this space it will have acquired such a degree of celerity, as is sufficient to carry it two miles the next minute, tho' it should then receive no new impulse from the cause by which its motion had been accelerated; but if the same accelerating cause continues, it will carry the body a mile farther; on which account, it will have run through four miles at the end of two minutes; and then it will have acquired such a degree of celerity, as is sufficient to carry it through a double space in as much more time, or eight miles in two minutes, even though the accelerating force should act upon it no more. But this force still continuing to operate in an uniform manner, will again, in an equal time, produce an equal effect; and so, by carrying it a mile further, cause it to move through five miles the third minute; for, the celerity already acquired, and the celerity still acquiring, will each have its complete effect. Hence we learn, that if the body should move one mile the first minute, it would move three the second, five the third, seven the fourth, nine the fifth, and so on in proportion.

And thus it appears, that the spaces described in successive equal parts of time, by an uniformly accelerated motion, are always as the odd numbers 1, 3, 5, 7, 9, &c. and consequently, the whole spaces are as the squares of the times, or of the last acquired velocities. For, the continued addition of the odd numbers yields the squares of all numbers from unity upwards. Thus, 1 is the first odd number, and the square of 1 is 1; 3 is the second odd number, and this added to 1 makes 4, the square of 2; 5 is the third odd number, which added to 4 makes 9, the square of 3; and so on for ever. Since, therefore, the times and velocities proceed evenly and constantly as 1, 2, 3, 4, &c., but the spaces described in each equal time are as 1, 3, 5, 7, &c., it is evident that the space described,

In 1 minute will be - - 1 = square of 1 In 2 minutes - - 1+3 = 4 = square of 2 In 3 minutes - - 1+3+5 = 9 = square of 3 In 4 minutes 1+3+5+7 = 16 = square of 4 &c.

As heavy bodies are uniformly accelerated by the power of gravity in their descent, it is plain that they must be uniformly retarded by the same power in their ascent. Therefore, the velocity which a body acquires by falling, is sufficient to carry it up again to the same height from whence it fell; allowance being made for the resistance of the air, or other medium in which the body is moved. Thus, the body D (fig. 2.) in rolling down the inclined plane AB, will acquire such a velocity by the time it arrives at B, as will carry it up to the inclined plane BC, almost to C; and would carry it quite up to C, if the body and plane were perfectly smooth, and the air gave no resistance.—So, if a pendulum were put into motion in a space quite void of air and all other resistance, and had no friction on the point of suspension, it would move for ever; for the velocity it had acquired in falling through the descending part of the arc, would be still sufficient to carry it equally high in the ascending part thereof.

The centre of gravity is that point of a body in which the whole force of its gravity or weight is united. Therefore, whatever supports that point bears the weight of the whole body; and whilst it is supported, the body cannot fall, because all its parts are in a perfect equilibrium about that point.

An imaginary line drawn from the centre of gravity of any body towards the centre of the earth, is called the line of direction. In this line all heavy bodies descend, if not obstructed.

Since the whole weight of a body is united in its centre of gravity, as that centre ascends or descends we must look upon the whole body to do so too. But as it is contrary to the nature of heavy bodies to ascend of their own accord, or not to descend when they are permitted; we may be sure that, unless the centre of gravity be supported, the whole body will tumble or fall. Hence it is, that bodies stand upon their bases when the line of direction falls within the base; for in this case the body cannot be made to fall without first raising the centre of gravity higher than it was before. Thus, the inclining body ABCD, (fig 3.) whose centre of gravity is E, stands firmly on its base CDIK, because the line of direction EF falls within the base. But if a weight, as ABGH, be laid upon the top of the body, the centre of gravity of the whole body and weight together is raised up to I; and then, as the line of direction ID falls without the base at D, the centre of gravity I is not supported; and the whole body and weight tumble down together.

The broader the base is, and the nearer the line of direction is to the middle or centre of it, the more firmly does the body stand. On the contrary, the narrower the base, and the nearer the line of direction is to the side of it, the more easily may the body be overthrown: a less change of position being sufficient to remove the line of direction out of the base in the latter case than in the former. And hence it is, that a sphere is so easily rolled upon a horizontal plane; and that it is so difficult, if not impossible, to make things which are sharp pointed to stand upright on the point.—From what hath been said, it plainly appears, that if the plane be inclined on which the heavy body is placed, the body will slide down upon the plane whilst the line of direction falls within the base; but it will tumble or roll down when that line falls without the base. Thus, the body A (fig. 4.) will only slide down the inclined plane CD, whilst the body B rolls down upon it.

When the line of direction falls within the base of our feet, we stand; and most firmly, when it is in the middle: but when it is out of that base, we immediately fall. And it is not only pleasing, but even surprising, to reflect upon the various and unthought of methods and postures which we use, to retain this position, or to recover it when it is lost. For this purpose, we bend our body forward when we rise from a chair, or when we go upstairs: and for this purpose a man leans forward when he carries a burden on his back, and backward when he carries it on his breast, and to the right or left side as he carries it on the opposite side.

The quantity of matter in all bodies is in exact proportion to their weights, bulk for bulk. Therefore, heavy bodies are as much more dense or compact than light bodies of the same bulk, as they exceed them in weight.

All bodies are full of pores, or spaces void of matter: and in gold, which is the heaviest of all known bodies, there is perhaps a greater quantity of space than of matter. For the particles of heat and magnetism find an easy passage through the pores of gold; and even water itself has been forced through them. Besides, if we consider how easily the rays of light pass through so solid a body as glass, in all manner of directions, we shall find reason to believe that bodies are exceedingly porous.

All bodies are some way or other affected by heat; and all metallic bodies are expanded in length, breadth, and thickness thereby.—The proportion of the expansion of several metals, according to the best experiments, is nearly thus. Iron and glass as 3, steel 4, copper 4 and one eighth, brass 5, tin 6, lead 6 and one eighth. An iron rod 3 feet long is about one seventh part of an inch longer in summer than in winter.

The expansion of metals by heat, is demonstrated by the following machine, called a pyrometer AA (fig. 5.) is a flat piece of mahogany, in which are fixed four brass fluids B,C,D,L; and two pins, one at F, and the other at H. On the pin F turns the crooked index EI, and upon the pin H the straight index GK; against which a piece of watch-spring R bears gently, and so presses it towards towards the beginning of the scale MN, over which the point of that index moves. This scale is divided into inches and tenth parts of an inch: the first inch is marked 1000; the second 2000, and so on. A bar of metal O is laid into notches in the top of the studs C and D; one end of the bar bearing against the adjusting screw P, and the other end against the crooked index EI, at a 20th part of its length from its centre of motion F.—Now it is plain, that however much the bar O lengthens, it will move that part of the index EI against which it bears just as far: but the crooked end of the same index, near H, being 20 times as far from the centre of motion F as the point is against which the bar bears, it will move 20 times as far as the bar lengthens. And as this crooked end bears against the index GK at only a 20th part of its whole length GS from its centre of motion H, the point S will move through 20 times the space that the point of bearing near H does. Hence, as 20 multiplied by 20 produces 400, it is evident, that if the bar lengthens but a 400th part of an inch, the point S will move a whole inch on the scale; and as every inch is divided into 10 equal parts, if the bar lengthens but the 10th part of the 400th part of an inch, which is only the 4000th part of an inch, the point S will move the tenth part of an inch, which is very perceptible.

To find how much a bar lengthens by heat, first lay it cold into the notches of the studs, and turn the adjusting screw P until the spring R brings the point S of the index GK to the beginning of the divisions of the scale at M: then, without altering the screw any farther, take off the bar, and rub it with a dry woollen cloth till it feels warm: and then, laying it on where it was, observe how far it pushes the point S upon the scale by means of the crooked index EI; and the point S will shew exactly how much the bar has lengthened by the heat of rubbing. As the bar cools, the spring R bearing against the index KG, will cause its point S to move gradually back towards M in the scale: and when the bar is quite cold, the index will rest at M, where it was before the bar was made warm by rubbing. The indexes have small rollers under them at I and K; which, by turning round on the smooth wood as the indexes move, make their motions the easier, by taking off a great part of the friction, which would otherwise be on the pins F and H, and of the points of the indexes themselves on the wood.

Besides the universal properties above mentioned, there are bodies which have properties peculiar to themselves: such as the loadstone, in which the most remarkable are these, 1. It attracts iron and steel only. 2. It constantly turns one of its sides to the north and another to the south, when suspended by a thread that does not twist. 3. It communicates all its properties to a piece of steel when rubbed upon it, without losing any itself.

According to Dr Helsham's experiments, the attraction of the loadstone decreases as the square of the distance increases. Thus, if a loadstone be suspended at one end of a balance, and counterpoised by weights at the other end, and a flat piece of iron be placed beneath it, at the distance of four tenths of an inch, the stone will immediately descend and adhere to the iron. But if the stone be again removed to the same distance, and as many grains be put into the scale at the other end as will exactly counterbalance the attraction, then, if the iron be brought twice as near the stone as before, that is, only two tenths parts of an inch from it, there must be four times as many grains put into the scale as before, in order to be a just counterbalance to the attractive force, or to hinder the stone from descending and adhering to the iron. So if four grains will do in the former case, there must be sixteen in the latter. But from some later experiments, made with the greatest accuracy, it is found that the force of magnetism decreases in a ratio between the reciprocal of the square and the reciprocal of the cube of the distance; approaching to the one or the other, as the magnitudes of the attracting bodies are varied.

Several bodies, particularly amber, glass, jet, sealing-wax, agate, and almost all precious stones, have a peculiar property of attracting and repelling light bodies when heated by rubbing. This is called electrical attraction; for the properties of which, see Electricity.

Of Central Forces.

We have already mentioned it as a necessary consequence arising from the deadness or inactivity of matter, that all bodies endeavour to continue in the state they are in, whether of rest or motion. If the body A (fig. 6.) were placed in any part of free space, where nothing either draws or impels it any way, it would for ever remain in that part of space, because it could have no tendency of itself to remove any way from thence. If it receives a single impulse any way, as suppose from A towards B, it will go on in that direction; for, of itself it could never swerve from a right line, nor stop its course.—When it has gone through the space AB, and met with no resistance, its velocity will be the same at B as it was at A; and this velocity, in as much more time, will carry it through as much more space, from B to C; and so on for ever. Therefore, when we see a body in motion; we conclude that some other substance must have given it that motion; and when we see a body fall from motion to rest, we conclude that some other body or cause stopt it.

As all motion is naturally rectilineal, it appears, that a bullet projected by the hand, or shot from a cannon, would for ever continue to move in the same direction it received at first, if no other power diverted its course. Therefore, when we see a body move in a curve of any kind whatever, we conclude it must be acted upon by two powers at least; one putting it in motion, and another drawing it off from the rectilineal course it would otherwise have continued to move in: and whenever that power, which bent the motion of the body from a straight line into a curve, ceases to act, the body will again move on in a straight line, touching that point of the curve in which it was when the action of that power ceased. For example, a pebble moved round in a sling ever so long a time, will fly off the moment it is set at liberty by slipping one end of the sling cord; and will go on in a line touching the circle it described before; which line would actually be a straight one, if the earth's attraction did not affect the pebble, and bring it down to the ground. This shews, that the natural tendency of the pebble, when put into motion, is to continue moving in a straight line, although by the force that moves the sling it be made to revolve in a circle.

The change of motion produced is in proportion to the force impressed; for the effects of natural causes are always proportionate to the force or power of those causes.

By these laws it is easy to prove that a body will describe the diagonal of a square or parallelogram, by two forces conjoined, in the same time that it would describe either of the sides by one force singly. Thus, suppose the body A (fig. 7.) to represent a ship at sea; and that it is drove by the wind, in the right line AB, with such a force as would carry it uniformly from A to B in a minute: then, suppose a stream or current of water running in the direction AD, with such a force as would carry the ship through an equal space from A to D in a minute. By these two forces, acting together at right angles to each other, the ship will describe the line AEC in a minute: which line (because the forces are equal and perpendicular to each other,) will be the diagonal of an exact square. To confirm this law by an experiment, let there be a wooden square ABCD (fig. 8.) so contrived, as to have the part BEFC made to draw out or push into the square at pleasure. To this part let the pulley H be joined, so as to turn freely on an axis, which will be at H when the piece is pushed in, and at b when it is drawn out. To this part let the ends of a straight wire k be fixed, so as to move along with it, under the pulley: and let the ball G be made to slide easily on the wire. A thread m is fixed to this ball, and goes over the pulley to I; by this thread the ball may be drawn up on the wire, parallel to the side AD, when the part BEFC is pushed as far as it will go into the square. But, if this part be drawn out, it will carry the ball along with it, parallel to the bottom of the square DC. By this means, the ball G may either be drawn perpendicularly upward by pulling the thread m, or moved horizontally along by pulling out the part BEFC, in equal times, and through equal spaces; each power acting equably and separately upon it. But if, when the ball is at G, the upper end of the thread be tied to the pin I, in the corner A of the fixed square, and the moveable part BEFG be drawn out, the ball will then be acted upon by both the powers together; for it will be drawn up by the thread towards the top of the square, and at the same time carried with its wire k towards the right hand BC, moving all the while in the diagonal line L; and will be found at g when the sliding part is drawn out as far as it was before, which then will have caused the thread to draw up the ball to the top of the inside of the square, just as high as it was before, when drawn up singly by the thread without moving the sliding part.

If the acting forces are equal, but at oblique angles to each other, so will the sides of the parallelogram be: and the diagonal run through by the moving body will be longer or shorter, according as the obliquity is greater or smaller. Thus, if two equal forces act conjointly upon the body A, (fig. 9.) one having a tendency to move it through the space AB in the same time that the other has a tendency to move it through an equal space AD; it will describe the diagonal AGC in the same time that either of the single forces would have caused it to describe either of the sides. If one of the forces be greater than the other, then one side of the parallelogram will be so much longer than the other. For if one force singly would carry the body through the space AE, in the same time that the other would have carried the space AD, the joint action of both will carry it in the same time through the space AHF, which is the diagonal of the oblique parallelogram ADEF.

If both forces act upon the body in such a manner, as to move it uniformly, the diagonal described will be a straight line; but if one of the forces acts in such a manner as to make the body move faster and faster as it goes forward, then the line described will be a curve. And this is the case of all bodies which are projected in rectilineal directions, and at the same time acted upon by the power of gravity, which has a constant tendency to accelerate their motions in the direction wherein it acts.

Laws of the Planetary motions.

From the uniform projectile motion of bodies in straight lines, and the universal power of gravity or attraction, arises the curvilinear motion of all the heavenly bodies. If the body A (fig. 10.) be projected along the straight line AFH in open space, where it meets with no resistance, and is not drawn aside by any power, it will go on forever with the same velocity, and in the same direction. But if, at the same moment the projectile force is given it at A, the body S begins to attract it with a force duly adjusted*, and perpendicular to its motion at A, it will then be drawn from the straight line AFH, and forced to revolve about S in the circle ATW; in the same manner, and by the same law, that a pebble is moved round in a sling. And if, when the body is in any part of its orbit (as suppose at K) a smaller body as L, within the sphere of attraction of the body K, be projected in the right line LM, with a force duly adjusted, and perpendicular to the line of attraction LK; then, the small body L will revolve about the large body K in the orbit NO, and accompany it in its whole course round the yet larger body S. But then, the body K will no longer move in the circle ATW; for that circle will now be described by the common centre of gravity between K and L. Nay, even the great body S will not keep in the centre; for it will be the common centre of gravity between all the three bodies S, K, and L, that will remain immoveable there. So, if we suppose S and K connected by a wire P that has no weight, and K and L connected by a wire q that has no weight, the common centre of gravity of all these three bodies will be a point in the wire P near S; which point being supported, the bodies will be all in equilibrium as they move round it. Though indeed, strictly speaking, the common centre of gravity of all the three bodies will not be in the wire.

* To make the projectile force a just balance to the gravitating power, so as to keep the planet moving in a circle, it must give such a velocity as the planet would acquire by gravity when it had fallen through half the semidiameter of that circle. but when these bodies are all in a right line. Here, S may represent the sun, K the earth, and L the moon.

In order to form an idea of the curves described by two bodies revolving about their common centre of gravity, whilst they themselves with a third body are in motion round the common centre of gravity of all the three; let us first suppose E (Plate CVI. fig. 1.) to be the sun, and e the earth going round him with any moon; and their moving forces regulated as above. In this case, whilst the earth goes round the sun in the dotted circle RTUWX, &c., the sun will go round the circle ABD, whose centre C is the common centre of gravity between the sun and earth: the right line BD representing the mutual attraction between them, by which they are as firmly connected as if they were fixed at the two ends of an iron bar strong enough to hold them. So, when the earth is at e, the sun will be at E; when the earth is at T, the sun will be at F; and when the earth is at g, the sun will be at G, &c.

Now, let us take in the moon q (at the top of the figure,) and suppose the earth to have no progressive motion about the sun; in which case, whilst the moon revolves about the earth in her orbit HBCD, the earth will revolve in the circle S 13, whose centre R is the common centre of gravity of the earth and moon; they being connected by the mutual attraction between them in the same manner as the earth and sun are.

But the truth is, that whilst the moon revolves about the earth, the earth is in motion about the sun; and now, the moon will cause the earth to describe an irregular curve, and not a true circle, round the sun; it being the common centre of gravity of the earth and moon that will then describe the same circle which the earth would have moved in, if it had not been attended by a moon. For, supposing the moon to describe a quarter of her progressive orbit about the earth in the time that the earth moves from e to f, it is plain that when the earth comes to f, the moon will be found at r; in which time, their common centre of gravity will have described the dotted arc R 1 T, the earth the curve R 5 f, and the moon the curve q 14 r.

In the time that the moon describes another quarter of her orbit, the centre of gravity of the earth and moon will describe the dotted arc T 2 U, the earth the curve f 6 g, and the moon the curve r 15 s, and so on.—And thus, whilst the moon goes once round the earth in her progressive orbit, their common centre of gravity describes the regular portion of a circle R 1 T 2 U 3 V 4 W, the earth the irregular curve R 5 f 6 g 7 h 8 i, and the moon the yet more irregular curve q 14 r 15 s 16 t 17 u; and then, the same kind of tracks over again.

The centre of gravity of the earth and moon is 6000 miles from the earth's centre towards the moon; therefore the circle S 13 which the earth describes round that centre of gravity (in every course of the moon round her orbit) is 12000 miles in diameter. Consequently, the earth is 12000 miles nearer the sun at the time of full moon than at the time of new. [See the earth at f and at h.]

To avoid confusion in so small a figure, we have supposed the moon to go only twice and a half round the earth, in the time that the earth goes once round the sun: it being impossible to take in all the revolutions which she makes in a year; and to give a true figure of her path, unless we should make the semidiameter of the earth's orbit at least 84 inches; and then, the proportional semidiameter of the moon's orbit would be only a quarter of an inch.

If the moon made any complete number of revolutions about the earth in the time that the earth makes one revolution about the sun, the paths of the sun and moon would return into themselves at the end of every year; and so be the same over again: but they return not into themselves in less than 19 years nearly; in which time, the earth makes nearly 19 revolutions about the sun, and the moon 235 about the earth.

If the planet A (Plate CV. fig. 10.) be attracted towards the sun, with such a force as would make it fall from A to B, in the time that the projectile impulse would have carried it from A to F, it will describe the arc AG by the combined action of these forces, in the same time that the former would have caused it to fall from A to B, or the latter have carried it from A to F. But, if the projectile force had been twice as great, that is, such as would have carried the planet from A to H, in the same time that now, by the supposition, it carries it only from A to F; the sun's attraction must then have been four times as strong as formerly, to have kept the planet in the circle ATW; that is, it must have been such as would have caused the planet to fall from A to E, which is four times the distance of A from B, in the time that the projectile force singly would have carried it from A to H, which is only twice the distance of A from F. Thus, a double projectile force will balance a quadruple power of gravity in the same circle; as appears plain by the figures, and shall soon be confirmed by an experiment.

Plate CVI. fig. 2.—The whirling-table is a machine contrived for shewing experiments of this nature. AA is a strong frame of wood, B a winch or handle fixed on the axis C of the wheel D, round which is the catgut string F, which also goes round the small wheels G and K, crossing between them and the great wheel D. On the upper end of the axis of the wheel G, above the frame, is fixed the round board d, to which the bearer MSX may be fastened occasionally, and removed when it is not wanted. On the axis of the wheel H is fixed the bearer NTZ: and it is easy to see, that when the winch B is turned, the wheels and bearers are put into a whirling motion.

Each bearer has two wires, W, X, and Y, Z, fixed and screwed tight into them at the ends by nuts on the outside. And when these nuts are unscrewed, the wires may be drawn out in order to change the balls U and V, which slide upon the wires by means of braids loops fixed into the balls, which keep the balls up from touching the wood below them. A strong silk line goes through each ball, and is fixed to it at any length from the centre of the bearer to its end, as occasion requires, by a nut-screw at the top of the ball; the shank of the screw going into the centre of the ball and pressing the line against the under side of the hole that it goes through.—The line goes from the ball, and under a small pulley fixed in the middle of the bearer; then up through a socket in the round plate. plate (see S and T) in the middle of each bearer; then through a slit in the middle of the square top (O and P) of each tower, and, going over a small pulley on the top, comes down again the same way, and is at last fastened to the upper end of the socket fixed in the middle of the above-mentioned round plate. These plates S and T have each four round holes near their edges for letting them slide up and down upon the wires which make the corner of each tower. The balls and plates being thus connected each by its particular line, it is plain, that if the balls be drawn outward, or towards the ends M and N of their respective bearers, the round plates S and T will be drawn up to the top of their respective towers O and P.

There are several brass weights, some of two ounces, some of three, and some of four, to be occasionally put within the towers O and P, upon the round plates S and T: each weight having a round hole in the middle of it, for going upon the sockets or axes of the plates, and is slit from the edge to the hole, for allowing it to be slipped over the foresaid line which comes from each ball to its respective plate.

The experiments to be made by this machine are,

1. Take away the bearer MX, and take the ivory ball a, to which the line or silk cord b is fastened at one end; and having made a loop on the other end of the cord, put the loop over a pin fixed in the centre of the board d. Then, turning the winch B to give the board a whirling motion, you will see that the ball does not immediately begin to move with the board; but, on account of its inactivity, it endeavours to continue in the state of rest which it was in before.—Continue turning, until the board communicates an equal degree of motion with its own to the ball; and then turning on; you will perceive that the ball will remain upon one part of the board, keeping the same velocity with it, and having no relative motion upon it, as is the case with every thing that lies loose upon the plane surface of the earth, which having the motion of the earth communicated to it, never endeavours to remove from that place. But stop the board suddenly by hand, and the ball will go on, and continue to revolve upon the board, until the friction thereof stops its motion: which shews, that matter being once put into motion, would continue to move for ever, if it met with no resistance. In like manner, if a person stands upright in a boat before it begins to move, he can stand firm; but the moment the boat sets off, he is in danger of falling towards that place which the boat departs from: because, as matter, he has no natural propensity to move. But when he acquires the motion of the boat, let it be ever so swift, if it be smooth and uniform, he will stand as upright and firm as if he was on the plain shore; and if the boat strike against any obstacle, he will fall towards that obstacle; on account of the propensity he has, as matter, to keep the motion which the boat has put him into.

2. Take away this ball, and put a longer cord to it, which may be put down through the hollow axis of the bearer MX, and wheel G, and fix a weight to the end of the cord below the machine; which weight, if left at liberty, will draw the ball from the edge of the whirling-board to its centre.

Draw off the ball a little from the centre; and turn the winch; then the ball will go round and round with the board, and will gradually fly off farther and farther from the centre, and raise up the weight below the machine; which shews that all bodies revolving in circles have a tendency to fly off from these circles, and must have some power acting upon them from the centre of motion, to keep them from flying off. Stop the machine, and the ball will continue to revolve for some time upon the board; but as the friction gradually stops its motion, the weight acting upon it will bring it nearer and nearer to the centre in every revolution, until it brings it quite thither. This shews, that if the planets met with any resistance in going round the sun, its attractive power would bring them nearer and nearer to it in every revolution, until they fell into it.

3. Take hold of the cord below the machine with one hand, and with the other throw the ball upon the round board as it were at right angles to the cord, by which means it will go round and round upon the board. Then, observing with what velocity it moves, pull the cord below the machine, which will bring the ball nearer to the centre of the board, and you will see that the nearer the ball is drawn to the centre, the faster it will revolve; as those planets which are nearest the sun revolve faster than those which are more remote; and not only go round sooner, because they describe smaller circles, but even move faster in every part of their respective circles.

Take away this ball, and apply the bearer MX, whose centre of motion is in its middle at w, directly over the centre of the whirling-board d. Then put two balls (V and U) of equal weights upon their bearing wires; and having fixed them at equal distances from their respective centres of motion w and x upon their silk cords, by the screw-nuts, put equal weights in the towers O and P. Lastly, put the catgut strings E and F upon the grooves G and H of the small wheels; which being of equal diameters, will give equal velocities to the bearers above, when the winch B is turned; and the balls U and V will fly off towards M and N, and will raise the weights in the towers at the same instant. This shews, that when bodies of equal quantities of matter revolve in equal circles with equal velocities, their centrifugal forces are equal.

4. Take away these equal balls, and, instead of them, put a ball of six ounces into the bearer MX, at a sixth part of the distance wz from the centre, and put a ball of one ounce into the opposite bearer, at the whole distance xy, which is equal to wz from the centre of the bearer; and fix the ball at these distances on their cords, by the screw-nuts at top; then the ball U, which is six times as heavy as the ball V, will be at only a sixth part of the distance from its centre of motion; and consequently will revolve in a circle of only a sixth part of the circumference of the circle in which V revolves. Now, let any equal weights be put into the towers, and the machine be turned by the winch; which (as the catgut string is on equal wheels below) will cause the balls to revolve in equal times; but V will move six times as fast as U, because it revolves in a circle of six times its radius; and both the weights in the towers will rise at once. This shews, that the centrifugal forces of revolving bodies, (or their their tendencies to fly off from the circles they describe) are in direct proportion to their quantities of matter multiplied into their respective velocities, or into their distances from the centres of their respective circles. For, supposing U, which weighs 6 ounces, to be two inches from its centre of motion w, the weight multiplied by the distance is 12; and supposing V, which weighs only one ounce, to be 12 inches distant from its centre of motion x; the weight 1 ounce multiplied by the distance 12 inches is 12. And as they revolve in equal times, their velocities are as their distances from the centre, namely, as 1 to 6.

If these two balls be fixed at equal distances from their respective centres of motion, they will move with equal velocities; and if the tower O has 6 times as much weight put into it as the tower P has, the balls will raise their weight at the same moment. This shews, that the ball U, being six times as heavy as the ball V, has six times as much centrifugal force, in describing an equal circle with an equal velocity.

6. If bodies of equal weights revolve in equal circles with unequal velocities, their centrifugal forces are as the squares of the velocities. To prove this law by an experiment, let two-balls U and V of equal weights be fixed on their cords at equal distances from their respective centres of motion w and x; and then let the catgut string E be put round the wheel K (whose circumference is only one half of the circumference of the wheel H or G) and over the pulley s to keep it tight; and let four times as much weight be put in the tower P as in the tower O. Then turn the winch B, and the ball V will revolve twice as fast as the ball U in a circle of the same diameter, because they are equidistant from the centres of the circles in which they revolve; and the weights in the towers will both rise at the same instant; which shews, that a double velocity in the same circle will exactly balance a quadruple power of attraction in the centre of the circle. For the weights in the towers may be considered as the attractive forces in the centres, acting upon the revolving balls; which, moving in equal circles, is the same thing as if they both moved in one and the same circle.

7. If bodies of equal weights revolve in unequal circles, in such a manner that the squares of the times of their going round are as the cubes of their distances from the centres of the circles they describe; their centrifugal forces are inversely as the squares of their distances from those centres. For, the catgut string remaining as in the last experiment, let the distance of the ball V from the centre x be made equal to two of the cross divisions on its bearer, and the distance of the ball U from the centre w be three and a sixth part; the balls themselves being of equal weights, and V making two revolutions by turning the winch in the time that U makes one: so that if we suppose the ball V to revolve in one moment, the ball U will revolve in two moments, the squares of which are one and four: for the square of 1 is only 1, and the square of 2 is 4; wherefore the square of the period or revolution of the ball V is contained 4 times in the square of the ball U. But the distance of V is 2, the cube of which is 8; and the distance of U is 3½, the cube of which is 32 very nearly; in which 8 is contained four times; and therefore, the squares of the periods of V and U are to one another as the cubes of their distances from x and w, which are the centres of their respective circles. And if the weight in the tower O be four ounces, equal to the square of 2, the distance of V from the centre x; and the weight in the tower P be 10 ounces, nearly equal to the square of 3½, the distance of U from w; it will be found, upon turning the machine by the winch, that the balls U and V will raise their respective weights at very nearly the same instant of time. Which confirms that famous proposition of Kepler, viz. That the squares of the periodical times of the planets round the sun are in proportion to the cubes of their distances from him; and that the sun's attraction is inversely as the square of the distance from its centre: that is, at twice the distance, his attraction is four times less; and thrice the distance, nine times less; at four times the distance, sixteen times less; and so on, to the remotest part of the system.

8. Take off the catgut string E from the great wheel D and the small wheel H, and let the string F remain upon the wheels D and G. Take away also the bearer MX from the whirling-board d, and instead thereof put the machine AB (fig. 4.) upon it, fixing this machine to the centre of the board by the pins c and d, in such a manner, that the end g may rise above the board to an angle of 30 or 40 degrees. In the upper side of this machine there are two glass tubes a and b, close stopped at both ends; and each tube is about three quarters full of water. In the tube a is a little quicksilver, which naturally falls down to the end a in the water, because it is heavier than its bulk of water; and on the tube b is a small cork which floats upon the top of the water at e, because it is lighter; and it is small enough to have liberty to rise or fall in the tube. While the board b with this machine upon it continues at rest, the quicksilver lies at the bottom of the tube a, and the cork floats on the water near the top of the tube b. But, upon turning the winch, and putting the machine in motion, the contents of each tube will fly off towards the uppermost ends (which are farthest from the centre of motion) the heaviest with the greatest force. Therefore, the quicksilver in the tube a will fly off quite to the end f, and occupy its bulk of space there, excluding the water from that place, because it is lighter than quicksilver; but the water in the tube b flying off to its higher end e, will exclude the cork from that place, and cause the cork to descend towards the lowermost end of the tube, where it will remain upon the lowest end of the water near b; for the heavier body having the greater centrifugal force, will therefore possess the uppermost part of the tube; and the lighter body will keep between the heavier and the lowermost part.

This demonstrates the absurdity of the Cartesian doctrine of the planets moving round the sun in vortexes: for, if the planet be more dense or heavy than the bulk of the vortex, it will fly off therein, farther and farther from the sun; if less dense, it will come down to the lowest part of the vortex, at the sun: and the whole vortex itself must be surrounded with something like a great wall, otherwise it would fly off, planets and all together. But while gravity exists, there is no occasion for such vortexes; and when it ceases to exist, a stone thrown upwards will never return to the earth again.

9. If a body be placed upon the whirling board of the machine (fig. 1.) that the centre of gravity of the body be directly over the centre of the board, and the board be put into ever so rapid a motion by the winch B, the body will turn round with the board, but will not remove from the middle of it: for, as all parts of the body are in equilibrium round its centre of gravity, and the centre of gravity is at rest in the centre of motion, the centrifugal force of all parts of the body will be equal at equal distances from its centre of motion; and therefore the body will remain in its place. But if the centre of gravity be placed ever so little out of the centre of motion, and the machine be turned swiftly round, the body will fly off towards that side of the board on which its centre of gravity lies. Thus, if the wire C (fig. 5.) with its little ball B be taken away from the demi-globe A, and the flat side ef of this demi-globe be laid upon the whirling-board of the machine, so as their centres may coincide; if then the board be turned ever so quick by the winch, the demi-globe will remain where it was placed. But if the wire C be screwed in the demi-globe at d, the whole becomes one body, whose centre of gravity is now at or near d. Let the pin c be fixed in the centre of the whirling-board, and the deep groove b cut in the flat side of the demi-globe be put upon the pin, so as the pin may be in the centre of A [See fig. 6., where this groove is represented at b] and let the whirling board be turned by the winch, which will carry the little ball B (fig. 5.) with its wire C, and the demi-globe A, all round the centre-pin c; and then the centrifugal force of the little ball B, which weighs only one ounce, will be so great, as to draw off the demi-globe A, which weighs two pounds, until the end of the groove at e strikes against the pin c, and so prevents the demi-globe A from going any farther; otherwise, the centrifugal force of B would have been great enough to have carried A quite off the whirling-board. Which shews, that if the sun were placed in the very centre of the orbits of the planets, it could not possibly remain there; for the centrifugal forces of the planets would carry them quite off, and the sun with them; especially when several of them happened to be in any one quarter of the heavens. For the sun and planets are as much connected by the mutual attraction that subsists between them, as the bodies A and B are by the wire C which is fixed to them both. And even if there were but one single planet in the whole heavens to go round ever so large a sun in the centre of its orbit, its centrifugal force would soon carry off both itself and the sun. For, the greatest body placed in any part of free space could be easily moved; because if there were no other body to attract it, it could have no weight or gravity of itself; and consequently, though it could have no tendency of itself to remove from that part of space, yet it might be very easily by any other substance. And perhaps it was this consideration which made the celebrated Archimedes say, that if he had a proper place at some distance from the earth whereon to fix his machinery, he could move the whole earth.

10. As the centrifugal force of the light body B will not allow the heavy body A to remain in the centre of motion, even though it be 24 times as heavy as B; let us now take the ball A (fig. 7.) which weighs 6 ounces, and connect it by the wire C with the ball B, which weighs only one ounce; and let the fork E be fixed into the centre of the whirling-board; then, hang the balls upon the fork by the wire C in such a manner that they may exactly balance each other; which will be when the centre of gravity between them, in the wire at d, is supported by the fork. And this centre of gravity is as much nearer to the centre of the ball A, than to the centre of the ball B, as A is heavier than B, allowing for the weight of the wire on each side of the fork. This done, let the machine be put into motion by the winch; and the balls A and B will go round their common centre of gravity d, keeping their balance, because either will not allow the other to fly off with it. For, supposing the ball B to be only one ounce in weight, and the ball A to be six ounces; then, if the wire C were equally heavy on each side of the fork, the centre of gravity d would be six times as far from the centre of the ball B as from the centre of the ball A, and consequently B will revolve with a velocity six times as great as A does; which will give B six times as much centrifugal force as any single ounce of A has: but then, as B is only one ounce, and A six ounces, the whole centrifugal force of A will exactly balance the whole centrifugal force of B; and therefore, each body will detain the other so as to make it keep in its circle. This shews that the sun and planets must all move round the common centre of gravity of the whole system, in order to preserve that just balance which takes place among them. For, the planets being as inactive and dead as the above balls, they could no more have put themselves into motion than these balls can; nor have kept in their orbits, without being balanced at first with the greatest degree of exactness upon their common centre of gravity by the Almighty Hand that made them and put them in motion.

Perhaps it may be here asked, that since the centre of gravity between these balls must be supported by the fork E in this experiment, what prop it is that supports the centre of gravity of the solar system, and consequently bears the weight of all the bodies in it; and by what is the prop itself supported? The answer is easy and plain; for the centre of gravity of our balls must be supported, because they gravitate towards the earth, and would therefore fall to it: but as the sun and planets gravitate only towards one another, they have nothing else to fall to; and therefore have no occasion for anything to support their common centre of gravity: and if they did not move round that centre, and consequently acquire a tendency to fly off from it by their motions, their mutual attractions would soon bring them together; and so the whole would become one mass in the sun: which would also be the case if their velocities round the sun were not quick enough to create a centrifugal force equal to the sun's attraction.

But after all this nice adjustment, it appears evident, that the Deity cannot withdraw his regulating hand from his works, and leave them to be solely governed by the laws laws which he has impressed upon them at first. For if he should once leave them so, their order would in time come to an end; because the planets must necessarily disturb one another's motions by their mutual attractions, when several of them are in the same quarter of the heavens; as is often the case; and then, as they attract the sun more towards that quarter than when they are in a manner dispersed equably around him, if he was not at that time made to describe a portion of a larger circle round the common centre of gravity, the balance would then be immediately destroyed; and as it could never restore itself again, the whole system would begin to fall together, and would in time unite in a mass at the sun.

—Of this disturbance we have a very remarkable instance in the comet which appeared lately; and which, in going last up before from the sun, went so near to Jupiter, and was so affected by his attraction, as to have the figure of its orbit much changed; and not only so, but to have its period altered, and its course to be different in the heavens from what it was last before.

11. Take away the fork and balls from the whirling-board, and place the trough AB (fig. 8.) thereon, fixing its centre to the centre of the whirling-board by the pin H. In this trough are two balls D and E, of unequal weights, connected by a wire f; and made to slide easily upon the wire C stretched from end to end of the trough, and made fast by nut-screws on the outside of the ends. Let these balls be so placed upon the wire C, that their common centre of gravity g may be directly over the centre of the whirling board. Then, turn the machine by the winch ever so swiftly, and the trough and balls will go round their centre of gravity so as neither of them will fly off; because, on account of the equilibrium, each ball detains the other with an equal force acting against it. But if the ball E be drawn a little more towards the end of the trough at A, it will remove the centre of gravity towards that end from the centre of motion; and then, upon turning the machine, the little ball E will fly off, and strike with a considerable force against the end A, and draw the great ball B into the middle of the trough. Or, if the great ball D be drawn towards the end B of the trough, so that the centre of gravity may be a little towards that end from the centre of motion, and the machine be turned by the winch, the great ball D will fly off, and strike violently against the end B of the trough, and will bring the little ball E into the middle of it. If the trough be not made very strong, the ball D will break through it.

12. The reason why the tides rise at the same absolute time on opposite sides of the earth, and consequently in opposite directions, is made abundantly plain by a new experiment on the whirling table. The cause of their rising on the side next the moon every one understands to be owing to the moon's attraction; but why they should rise on the opposite side at the same time, where there is no moon to attract them, is perhaps not so generally understood. For it would seem that the moon should rather draw the waters (as it were) closer to that side, than raise them upon it, directly contrary to her attractive force. Let the circle abcd (fig. 9.) represent the earth, with its side e turned toward the moon, which will then attract the waters so as to raise them from c to g. But the question is, why should they rise as high at that very time on the opposite side, from a to e? In order to explain this, let there be a plate AB (fig. 10.) fixed upon one end of the flat bar DC; with such a circle drawn upon it as abcd (in fig. 9.) to represent the round figure of the earth and sea; and such an ellipse as efgh to represent the swelling of the tide at e and g, occasioned by the influence of the moon. Over this plate AB let the three ivory balls efgh be hung by the silk lines b,i,k, fastened to the tops of the crooked wires H,I,K., in such a manner, that the ball at e may hang freely over the side of the circle e, which is farthest from the moon M (at the other end of the bar;) the ball at f may hang freely over the centre, and the ball at g hang over the side of the circle g, which is nearest the moon. The ball f may represent the centre of the earth, the ball g some water on the side next the moon, and the ball e some water on the opposite side. On the back of the moon M is fixed the short bar N parallel to the horizon, and there are three holes in it above the little weights p,q,r. A silk thread o is tied to the line k close above the ball g, and, passing by one side of the moon M, goes through a hole in the bar N, and has the weight p hung to it. Such another thread n is tied to the line i, close above the ball f, and, passing through the centre of the moon M and middle of the bar N, has the weight q hung to it, which is lighter than the weight p. A third thread m is tied to the line b, close above the ball e, and passing by the other side of the moon M, through the bar N, has the weight r hung to it, which is lighter than the weight q.

The use of these three unequal weights is to represent the moon's unequal attraction at different distances from her. With whatever force she attracts the centre of the earth, she attracts the side next her with a greater degree of force, and the side farthest from her with a less. So, if the weights are left at liberty, they will draw all the three balls towards the moon with different degrees of force; and cause them to make the appearance shewn in (fig. 11.) by which means they are evidently farther from each other than they would be if they hung at liberty by the lines b,i,k; because the lines would then hang perpendicularly. This shews, that as the moon attracts the side of the earth which is nearest her with a greater degree of force than she does the centre of the earth, she will draw the water on that side more than she draws the centre, and so causes it to rise on that side; and as she draws the centre more than she draws the opposite side, the centre will recede farther from the surface of the water on that opposite side, and so leave it as high there as she raised it on the side next to her. For, as the centre will be in the middle between the tops of the opposite elevations, they must of course be equally high on both sides at the same time.

But upon this supposition the earth and moon would soon come together; and to be sure they would, if they had not a motion round their common centre of gravity, to create a degree of centrifugal force sufficient to balance their mutual attraction. This motion they have; for as the moon goes round her orbit every month, at the distance of 240,000 miles from the earth's centre, and of 234,000 miles from the centre of gravity of the earth and moon, so does the earth go round the same centre of gravity every month at the distance of 6000 miles from it; that is, from it to the centre of the earth. Now as the earth is (in round numbers) 8000 miles in diameter, it is plain that its side next the moon is only 2000 miles from the common centre of gravity of the earth and moon; its centre 6000 miles distant therefrom; and its farther side from the moon 10,000. Therefore the centrifugal forces of these parts are as 2000, 6000, and 10,000; that is, the centrifugal force of any side of the earth, when it is turned from the moon, is five times as great as when it is turned towards the moon. And as the moon's attraction (expressed by the number 6000) at the earth's centre keeps the earth from flying out of this monthly circle, it must be greater than the centrifugal force of the waters on the side next her; and consequently, her greater degree of attraction on that side is sufficient to raise them; but as her attraction on the opposite side is less than the centrifugal force of the water there, the excess of this force is sufficient to raise the water just as high on the opposite side.—To prove this experimentally, let the bar DC (fig. 10.) with its furniture be fixed upon the whirling-board of the machine (fig. 2.) by pushing the pin P into the centre of the board; which pin is in the centre of gravity of the whole bar with its three balls e, f, g, and moon M. Now, if the whirling-board and bar be turned slowly round by the winch, until the ball f hangs over the centre of the circle, as in fig. 11, the ball g will be kept towards the moon by the heaviest weight p, (fig. 9.) and the ball e, on account of its greater centrifugal force, and the lesser weight r, will fly off as far to the other side as in fig. 12. And so, whilst the machine is kept turning, the balls e and g will hang over the ends of the ellipses f, f, k. So that the centrifugal force of the ball e will exceed the moon's attraction just as much as her attraction exceeds the centrifugal force of the ball g, whilst her attraction just balances the centrifugal force of the ball f, and makes it keep in its circle. And hence it is evident that the tides must rise to equal heights at the same time on opposite sides of the earth. This experiment, to the best of my knowledge, is entirely new.

From the principles thus established, it is evident that the earth moves round the sun, and not the sun round the earth: for the centrifugal law will never allow a great body to move round a small one in any orbit whatever; especially when we find, that if a small body moves round a great one, the great one must also move round the common centre of gravity between them two. And it is well known, that the quantity of matter in the sun is 227,000 times as great as the quantity of matter in the earth. Now, as the sun's distance from the earth is at least 81,000,000 of miles, if we divide that distance by 227,000, we shall have only 357 for the number of miles that the centre of gravity between the sun and earth is distant from the sun's centre. And as the sun's semidiameter is \( \frac{1}{4} \) of a degree, which, at so great a distance as that of the sun, must be no less than 381,500 miles, if this be divided by 357, the quotient will be 1068\( \frac{2}{3} \), which shews that the common centre of gravity is within the body of the sun, and is only the 1068\( \frac{2}{3} \) part of his semidiameter from his centre toward his surface.

All globular bodies, whose parts can yield, and which do not turn on their axes, must be perfect spheres, because all parts of their surfaces are equally attracted toward their centres. But all such globes which do turn on their axes, will be oblate spheroids; that is, their surfaces will be higher, or farther from the centre, in the equatorial than in the polar regions. For, as the equatorial parts move quickest, they must have the greatest centrifugal force; and will therefore recede farther from the axis of motion. Thus, if two circular hoops AB and CD, (Plate CVII. fig. 1.) made thin and flexible, and crossing one another at right angles, be turned round their axis EF by means of the winch m, the wheel n, and pinion o, and the axis be loose in the pole or intersection e, the middle parts A,B,C,D will swell out so as to strike against the sides of the frame at F and G, if the pole e, in sinking to the pin E, be not kept by it from sinking farther; so that the whole will appear of an oval figure, the equatorial diameter being considerably longer than the polar. That our earth is of this figure, is demonstrable from actual measurement of some degrees on its surface, which are found to be longer in the frigid zones than in the torrid: and the difference is found to be such as prove the earth's equatorial diameter to be 35 miles longer than its axis.—Since then, the earth is higher at the equator than at the poles, the sea, which like all other fluids naturally runs downward (or towards the places which are nearest the earth's centre) would run towards the polar regions, and leave the equatorial parts dry, if the centrifugal force of the water, which carried it to those parts, and so raised them, did not detain and keep it from running back again towards the poles of the earth.

Of the Mechanical Powers.

If we consider bodies in motion, and compare them together, we may do this either with respect to the quantities of matter they contain, or the velocities with which they are moved. The heavier any body is, the greater is the power required either to move it or to stop its motion: and again, the swifter it moves, the greater is its force. So that the whole momentum or quantity of force of a moving body is the result of its quantity of matter multiplied by the velocity with which it is moved. And when the products arising from the multiplication of the particular quantities of matter in any two bodies by their respective velocities are equal, the momenta or entire forces are so too. Thus, suppose a body, which we shall call A, to weigh 40 pounds, and to move at the rate of two-miles in a minute; and another body, which we shall call B, to weigh only four pounds, and to move 20 miles in a minute; the entire forces with which these two bodies would strike against any obstacle would be equal to each other, and therefore it would require equal powers to stop them. For 40 multiplied by 2 gives 80, the force of the body A; and 20 multiplied by 4 gives 80, the force of the body B.

Upon this easy principle depends the whole of mechanics.

chanics: and it holds universally true, that when two bodies are suspended by any machine, so as to act contrary to each other: if the machine be put into motion, and the perpendicular ascent of one body multiplied into its weight, be equal to the perpendicular descent of the other body multiplied into its weight, these bodies, how unequal soever in their weights, will balance one another in all situations: for, as the whole ascent of one is performed in the same time with the whole descent of the other, their respective velocities must be directly as the spaces they move through; and the excess of weight in one body is compensated by the excess of velocity in the other.—Upon this principle it is easy to compute the power of any mechanical engine, whether simple or compound; for it is but only inquiring how much swifter the power moves than the weight does (i.e., how much farther in the same time,) and just so much is the power increased by the help of the engine.

In the theory of this science, we suppose all planes perfectly even, all bodies perfectly smooth, levers to have no weight, cords to be extremely pliable, machines to have no friction; and in short, all imperfection must be set aside until the theory be established, and then proper allowances are to be made.

The simple machines, usually called mechanical powers, are six in number, viz. the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw. They are called mechanical powers, because they help us to raise, weights, move heavy bodies, and overcome resistances, which we could not effect without them.

1. A lever is a bar of iron or wood, one part of which being supported by a prop, all the other parts turn upon that prop as their centre of motion: and the velocity of every part or point is directly as its distance from the prop. Therefore, when the weight to be raised at one end is to the power applied at the other to raise it, as the distance of the power from the prop is to the distance of the weight from the prop, the power and weight will exactly balance or counterpoise each other: and as a common lever has but very little friction on its prop, a very little additional power will be sufficient to raise the weight.

There are four kinds of levers. 1. The common sort, where the prop is placed between the weight and the power; but much nearer to the weight than to the power. 2. When the prop is at one end of the lever, the power at the other, and the weight between them. 3. When the prop is at one end, the weight at the other, and the power applied between them. 4. The bended lever, which differs only in form from the first sort, but not in property. Those of the first and second kind are often used in mechanical engines; but there are few instances in which the third sort is used.

A common balance is a lever of the first kind; but as both its ends are at equal distances from its centre of motion, they move with equal velocities; and therefore, as it gives no mechanical advantage, it cannot properly be reckoned among the mechanical powers.

A lever of the first kind is represented by the bar ABC, (Plate CVII. fig. 2.) supported by the prop D. Its principal use is to loosen large stones in the ground, or raise great weights to small heights, in order to have ropes put under them for raising them higher by other machines. The parts AB and BC, on different sides of the prop D, are called the arms of the lever: the end A of the shorter arm AB being applied to the weight intended to be raised, or to the resistance to be overcome; and the power applied to the end C of the longer arm BC.

In making experiments with this machine, the shorter arm AB must be as much thicker than the longer arm BC, as will be sufficient to balance it on the prop. This supposed, let P represent a power whose intensity is equal to one ounce, and W a weight whose intensity is equal to 12 ounces. Then, if the power be 12 times as far from the prop as the weight is, they will exactly counterpoise; and a small addition to the power P will cause it to descend, and raise the weight W; and the velocity with which the power descends will be to the velocity with which the weight rises, as 12 to 1: that is, directly as their distances from the prop; and consequently, as the spaces through which they move. Hence it is plain, that a man who by his natural strength, without the help of any machine, could support an hundred weight, will by the help of this lever be enabled to support twelve hundred. If the weight be less, or the power greater, the prop may be placed so much the farther from the weight; and then it can be raised to a proportionably greater height. For universally, if the intensity of the weight multiplied into its distance from the prop be equal to the intensity of the power multiplied into its distance from the prop, the power and weight will exactly balance each other; and a little addition to the power will raise the weight. Thus, in the present instance, the weight W is 12 ounces, and its distance from the prop is 1 inch; and 12 multiplied by 1 is 12; the power P is equal to 1 ounce, and its distance from the prop is 12 inches, which multiplied by one is 12 again: and therefore there is an equilibrium between them. So, if a power equal to 2 ounces be applied at the distance of 6 inches from the prop, it will just balance the weight W; for 6 multiplied by 2 is 12, as before. And a power equal to 3 ounces placed at 4 inches distance from the prop would do the same; for 3 times 4 is 12; and so on, in proportion.

The steatara, or Roman steelyard, is a lever of this kind, contrived for finding the weights of different bodies by one single weight placed at different distances from the prop or centre of motion D. For, if a scale hangs at A, the extremity of the shorter arm AB, and is of such a weight as will exactly counterpoise the longer arm BC; if this arm be divided into as many equal parts as it will contain, each equal to AB, the single weight P (which we may suppose to be 1 pound) will serve for weighing any thing as heavy as itself, or as many times heavier as there are divisions in the arm BC, or any quantity between its own weight and that quantity. As for example, if P be 1 pound, and placed at the first division 1 in the arm BC, it will balance 1 pound in the scale at A: if it be removed to the second division at 2, it will balance 2 pounds in the scale; if to the third, 3 pounds; and so on to the end of the arm BC. If each of these integral divisions be subdivided into as many equal parts as a pound contains ounces, and the weight... P be placed at any of these subdivisions, so as to counterpoise what is in the scale, the pounds and odd ounces therein are by that means ascertained.

To this kind of lever may be reduced several sorts of instruments, such as scissors, pinchers, snuffers; which are made of two levers acting contrary to one another; their prop or centre of motion being the pin which keeps them together.

In common practice, the longer arm of this lever greatly exceeds the weight of the shorter; which gains great advantage, because it adds so much to the power.

A lever of the second kind has the weight between the prop and the power. In this, as well as the former, the advantage gained is as the distance of the power from the prop to the distance of the weight from the prop: for the respective velocities of the power and weight are in that proportion; and they will balance each other when the intensity of the power multiplied by its distance from the prop is equal to the intensity of the weight multiplied by its distance from the prop. Thus, if AB (fig. 3.) be a lever on which the weight W of 6 ounces hangs at the distance of 1 inch from the prop G, and a power P equal to the weight of one ounce hangs at the end B, 6 inches from the prop, by the cord CD going over the fixed pulley E, the power will just support the weight; and a small addition to the power will raise the weight 1 inch for every 6 inches that the power descends.

This lever shews the reason why two men carrying a burden upon a stick between them, bear unequal shares of the burden in the inverse proportion of their distances from it. For it is well known, that the nearer any of them is to the burden, the greater share he bears of it: and if he goes directly under it, he bears the whole. So, if one man be at G, and the other at P, having the pole or stick AB resting on their shoulders; if the burden or weight W be placed five times as near the man at G, as it is to the man at P, the former will bear five times as much weight as the latter. This is likewise applicable to the case of two horses of unequal strength, to be so yoked, as that each horse may draw a part proportional to his strength; which is done by dividing the beam so, that the point of traction may be as much nearer to the stronger horse than to the weaker, as the strength of the former exceeds that of the latter.

To this kind of lever may be reduced oars, rudders of ships, doors turning upon hinges, cutting knives which are fixed at the point of the blade, and the like.

If in this lever we suppose the power and weight to change places, so that the power may be between the weight and the prop, it will become a lever of the third kind; in which, that there may be a balance between the power and the weight, the intensity of the power must exceed the intensity of the weight, just as much as the distance of the weight from the prop exceeds the distance of the power from it. Thus, let E (fig. 4.) be the prop of the lever AB, and W a weight of 1 pound, placed 3 times as far from the prop, as the power P acts at F, by the cord C going over the fixed pulley D; in this case, the power must be equal to three pounds, in order to support the weight.

To this sort of lever are generally referred the bones of a man's arm: for when we lift a weight by the hand, the muscle that exerts its force to raise that weight, is fixed to the bone about one tenth part as far below the elbow as the hand is. And the elbow being the centre round which the lower part of the arm turns, the muscle must therefore exert a force ten times as great as the weight that is raised.

As this kind of lever is a disadvantage to the moving power, it is never used but in cases of necessity; such as that of a ladder, which, being fixed at one end, is by the strength of a man's arms reared against a wall; and in clock-work, where all the wheels may be reckoned levers of this kind, because the power that moves every wheel, except the first, acts upon it near the centre of motion by means of a small pinion, and the resistance it has to overcome acts against the teeth round its circumference.

The fourth kind of lever differs nothing from the first, but in being bended for the sake of convenience. ACB (fig. 5.) is a lever of this sort, bended at C, which is its prop, or centre of motion. P is a power acting upon the longer arm AC at F, by means of the cord DE going over the pulley G; and W is a weight or resistance acting upon the end B of the shorter arm BC. If the power be to the weight as BC is to CF, they are in equilibrium. Thus, suppose W to be 5 pounds acting at the distance of one foot from the centre of motion C, and P to be 1 pound acting at F, five feet from the centre C, the power and weight will just balance each other. A hammer drawing a nail is a lever of this sort.

2. The second mechanical power is the wheel and axle, in which the power is applied to the circumference of the wheel, and the weight is raised by a rope which coils about the axle as the wheel is turned round. Here it is plain, that the velocity of the power must be to the velocity of the weight, as the circumference of the wheel is to the circumference of the axle: and consequently, the power and weight will balance each other, when the intensity of the power is to the intensity of the weight as the circumference of the axle is to the circumference of the wheel. Let AB (fig. 6.) be a wheel, CD its axle, and suppose the circumference of the wheel to be 8 times as great as the circumference of the axle; then a power P equal to 1 pound hanging by the cord I, which goes round the wheel, will balance a weight W of 8 pounds hanging by the rope K which goes round the axle. And as the friction on the pivots or gudgeons of the axle is but small, a small addition to the power will cause it to descend, and raise the weight: but the weight will rise with only an eighth part of the velocity wherewith the power descends, and consequently through no more than an eighth part of an equal space, in the same time. If the wheel be pulled round by the handles S, S, the power will be increased in proportion to their length. And by this means, any weight may be raised as high as the operator pleases.

To this sort of engine belong all cranes for raising great weights; and in this case, the wheel may have cogs all around it instead of handles, and a small lantern or trundle may be made to work in the cogs, and be turned by a winch; which will make the power of the engine to exceed the power of the man who works it, as much as the number of revolutions of the winch exceed those of the axle D, when multiplied by the excess of the length of the winch above the length of the semidiameter of the axle, added to the semidiameter or half-thickness of the rope K, by which the weight is drawn up. Thus, suppose the diameter of the rope and axle taken together to be 12 inches, and consequently half their diameters to be 6 inches; so that the weight W will hang at 6 inches perpendicular distance from below the centre of the axle. Now, let us suppose the wheel AB, which is fixed on the axle, to have 80 cogs, and to be turned by means of a winch six inches long, fixed on the axis of a trundle of 8 slaves or rounds, working in the cogs of the wheel.—Here it is plain, that the winch and trundle would make 10 revolutions for one of the wheel AB, and its axis D, on which the rope K winds in raising the weight W; and the winch being no longer than the sum of the semi-diameters of the great axle and rope, the trundle could have no more power on the wheel, than a man could have by pulling it round by the edge, because the winch would have no greater velocity than the edge of the wheel has, which we here suppose to be ten times as great as the velocity of the rising weight; so that, in this case, the power gained would be as 10 to 1. But if the length of the winch be 12 inches, the power gained will be as 20 to 1: if 18 inches (which is long enough for any man to work by) the power gained would be as 30 to 1; that is, a man could raise 30 times as much by such an engine, as he could do by his natural strength without it, because the velocity of the handle of the winch would be 30 times as great as the velocity of the rising weight; the absolute force of any engine being in proportion of the velocity of the power to the velocity of the weight raised by it.—But then, just as much power or advantage as is gained by the engine, so much time is lost in working it. In this sort of machines it is requisite to have a ratchet-wheel G on one end of the axle, with a catch H to fall into its teeth; which will at any time support the weight, and keep it from descending, if the workman should, through inadvertency or carelessness, quit his hold whilst the weight is raising. And by this means, the danger is prevented which might otherwise happen by the running down of the weight when left at liberty.

3. The third mechanical power or engine consists either of one moveable pulley, or a system of pulleys; some in a block or case which is fixed, and others in a block which is moveable, and rises with the weight. For tho' a single pulley, that only turns on its axis, and rises not with the weight, may serve to change the direction of the power, yet it can give no mechanical advantage thereto; but is only as the beam of a balance, whose arms are of equal length and weight. Thus, if the equal weights W and P (fig. 7.) hang by the cord BB upon the pulley A, whose block b is fixed to the beam HI, they will counterpoise each other, just in the same manner as if the cord were cut in the middle, and its two ends hung upon the hooks fixed in the pulley at A and A, equally distant from its centre.

But if a weight W hangs at the lower end of the move-

able block p of the pulley D, and the cord GF goes under the pulley, it is plain that the half G of the cord bears half of the weight W, and the half F the other; for they bear the whole between them. Therefore, whatever holds the upper end of either rope, fulfils one half of the weight; and if the cord at F be drawn up so as to raise the pulley D to C, the cord will then be extended to its whole length, all but that part which goes under the pulley: and consequently, the power that draws the cord will have moved twice as far as the pulley D with its weight W rises: on which account, a power whose intensity is equal to one half of the weight will be able to support it, because if the power moves (by means of a small addition) its velocity will be double the velocity of the weight; as may be seen by putting the cord over the fixed pulley C (which only changes the direction of the power, without giving any advantage to it) and hanging on the weight P, which is equal only to one half of the weight W; in which case there will be an equilibrium, and a little addition to P will cause it to descend, and raise W through a space equal to one half of that through which P descends.—Hence, the advantage gained will be always equal to twice the number of pulleys in the moveable or undermost block. So that, when the upper or fixed block u contains two pulleys, which only turn on the axes, and the lower or moveable block U contains two pulleys, which not only turn upon their axes, but also rise with the block and weight; the advantage gained by this is as 4 to the working power. Thus, if one end of the rope KMOQ be fixed to a hook at I, and the rope passes over the pulleys N and R, and under the pulleys L and P, and has a weight T, of one pound, hung to its other end at T, this weight will balance and support a weight W of four pounds hanging by a hook at the moveable block U, allowing the said block as a part of the weight. And if as much more power be added as is sufficient to overcome the friction of the pulleys, the power will descend with four times as much velocity as the weight rises, and consequently through four times as much space.

The two pulleys in the fixed block X, and the two in the moveable block Y, are in the same case with those last mentioned; and those in the lower block give the same advantage to the power.

As a system of pulleys has not great weight and lies in a small compass, it is easily carried about; and can be applied, in a great many cases, for raising weights, where other engines cannot. But they have a great deal of friction, on three accounts: 1. Because the diameters of their axes bear a very considerable proportion to their own diameters; 2. Because in working they are apt to rub against one another, or against the sides of the block; 3. Because of the stiffness of the rope that goes over and under them.

4. The fourth mechanical power is the inclined plane; and the advantage gained by it is as great as its length exceeds its perpendicular height. Let AB (fig. 8.) be a plane parallel to the horizon, and CD a plane inclined to it; and suppose the whole length CD to be three times as great as the perpendicular height G/F; in this case, the cylinder E will be supported upon the plane CD, and kept kept from rolling down upon it, by a power equal to a third part of the weight of the cylinder. Therefore, a weight may be rolled up this inclined plane with a third part of the power which would be sufficient to draw it up by the side of an upright wall. If the plane was four times as long as high, a fourth part of the power would be sufficient; and so on, in proportion. Or, if a pillar was to be raised from a floor to the height GF, by means of the engine ABDC, (which would then act as a half wedge, where the resistance gives way only on one side) the engine and pillar would be in equilibrium when the power applied at GF was to the weight of the pillar as GF to GD; and if the power be increased, so as to overcome the friction of the engine against the floor and pillar, the engine will be driven, and the pillar raised; and when the engine has moved its whole length upon the floor, the pillar will be raised to the whole height of the engine, from G to F.

The force wherewith a rolling body descends upon an inclined plane, is to the force of its absolute gravity, by which it would descend perpendicularly in a free space, as the height of the plane is to its length. For, suppose the plane AB (fig. 9.) to be parallel to the horizon, the cylinder C will keep at rest upon any part of the plane where it is laid. If the plane be so elevated, that its perpendicular height D (fig. 10.) is equal to half its length AB, the cylinder will roll down upon the plane with a force equal to half its weight; for it would require a power (acting on the direction of AB) equal to half its weight, to keep it from rolling. If the plane AB (fig. 11.) be elevated, so as to be perpendicular to the horizon, the cylinder C will descend with its whole force of gravity, because the plane contributes nothing to its support or hindrance; and therefore, it would require a power equal to its whole weight to keep it from descending.

Let the cylinder C (fig. 12.) be made to turn upon slender pivots in the frame D, in which there is a hook e, with a line G tied to it: let this line go over the fixed pulley H, and have its other end tied to a hook in the weight I. If the weight of the body I, be to the weight of the cylinder C, added to that of its frame D, as the perpendicular height of the plane LM is to its length AB, the weight will just support the cylinder upon the plane, and a small touch of a finger will either cause it to ascend or descend with equal ease: then, if a little addition be made to the weight I, it will descend, and draw the cylinder up the plane. In the time that the cylinder moves from A to B, it will rise through the whole height of the plane ML; and the weight will descend from H to K, through a space equal to the whole length of the plane AB.

If the plane be made to move upon rollers or friction-wheels, and the cylinder be supported upon it; the same power will draw the plane under the cylinder, which before drew the cylinder up the plane, provided the pivots of the axes of the friction-wheels be small, and the wheels themselves be pretty large. For, let the machine ABC (fig. 13.) (equal in length and height to ABM, fig. 12.) be provided with four wheels, whereof two appear at D and E, and the third under C, whilst the fourth is hid from sight by the horizontal board a. Let the cylinder F be laid upon the lower end of the inclined plane CB, and the line G be extended from the frame of the cylinder, about six feet parallel to the plane CB; and, in that direction, fixed to a hook in the wall; which will support the cylinder, and keep it from rolling off the plane. Let one end of the line H be tied to a hook at C in the machine, and the other end to a weight K, the same as drew the cylinder up the plane before. If this line be put over the fixed pulley I, the weight K will draw the machine along the horizontal plane L, and under the cylinder F; and when the machine has been drawn the whole length CB, the cylinder will be raised to d, equal to the perpendicular height AB above the horizontal part at A.

To the inclined plane may be reduced all hatchets, chisels, and other edge tools which are chamfered only on one side.

5. The fifth mechanical power or engine is the wedge, which may be considered as two equally inclined planes DEF and CEF, joined together at their bases EF: then DC (Plate CVIII. fig. 1.) is the whole thickness of the wedge at its back ABCD, where the power is applied; EF is the depth or height of the wedge; DF the length of one of its sides, equal to CF the length of the other side; and OF is its sharp edge, which is entered into the wood intended to be split by the force of a hammer or mallet striking perpendicularly on its back. Thus, ABB (fig. 2.) is a wedge driven into the cleft CDE of the wood FG.

When the wood does not cleave at any distance before the wedge, there will be an equilibrium between the power impelling the wedge downward, and the resistance of the wood acting against the two sides of the wedge; if the power be to the resistance, as half the thickness of the wedge at its back is to the length of either of its sides; that is, as Aa to Ab, or Ba to Bb (fig. 2.) And if the power be increased, so as to overcome the friction of the wedge and the resistance arising from the cohesion or stickage of the wood, the wedge will be drove in, and the wood split asunder.

But, when the wood cleaves at any distance before the wedge (as it generally does) the power impelling the wedge will not be to the resistance of the wood, as half the thickness of the wedge is to the length of one of its sides; but as half its thickness is to the length of either side of the cleft, estimated from the top or acting part of the wedge. For, if we suppose the wedge to be lengthened down from b to the bottom of the cleft at E, the same proportion will hold; namely, that the power will be to the resistance, as half the thickness of the wedge is to the length of either of its sides: or, which amounts to the same thing, as the whole thickness of the wedge is to the length of both its sides.

In order to prove what is here advanced concerning the wedge, let us suppose the wedge to be divided lengthwise into two equal parts: and then it will become two equally inclined planes; one of which, as abc, (Plate CVII. fig. 14.) may be made use of as a half wedge for separating the moulding cd from the wainscot AB. It is evident, that when this half-wedge has been driven its whole length ac between the wainscot and moulding, its side ac will be at ed; and the moulding will be separated

to \( \frac{1}{2} \) from the waincot. Now, from what has been already proved of the inclined plane, it appears, that to have an equilibrium between the power impelling the half-wedge and the resistance of the moulding, the former must be to the latter as \( ab \) to \( ac \); that is, as the thickness of the back which receives the stroke is to the length of the side against which the moulding acts. Therefore, since the power upon the half wedge is to the resistance against its side, as the half back \( ab \) is to the whole side \( ac \), it is plain, that the power upon the whole wedge (where the whole back is double the half back) must be to the resistance against both its sides, as the thickness of the whole back is to the length of both the sides, supposing the wedge at the bottom of the cleft; or as the thickness of the whole back to the length of both sides of the cleft, when the wood splits at any distance before the wedge. For, when the wedge is driven quite into the wood, and the wood splits at ever so small a distance before its edge, the top of the wedge then becomes the acting part, because the wood does not touch it anywhere else. And since the bottom of the cleft must be considered as that part where the whole stickage or resistance is accumulated, it is plain, from the nature of the lever, that the farther the power acts from the resistance, the greater is the advantage.

Some writers have advanced, that the power of the wedge is to the resistance to be overcome, as the thickness of the back of the wedge is to the length only of one of its sides; which seems very strange: for, if we suppose \( AB \) (Plate CVIII. fig. 3.) to be a strong inflexible bar of wood or iron fixed into the ground at \( CB \), and \( D \) and \( E \) to be two blocks of marble lying on the ground on opposite sides of the bar, it is evident that the block \( D \) may be separated from the bar to the distance \( d \), equal to \( ab \), by driving the inclined plane or half wedge \( ab \) down between them; and the block \( E \) may be separated to an equal distance on the other side, in like manner, by the half wedge \( edo \). But the power impelling each half wedge will be to the resistance of the block against its side, as the thickness of that half wedge is to the length of its acting side. Therefore the power to drive both the half wedges is to both the resistances, as both the half backs is to the length of both the acting sides, or as half the thickness of the whole back is to the length of either side. And, if the bar be taken away, the blocks put close together, and the two half wedges joined to make one; it will require as much force to drive it down between the blocks, as is equal to the sum of the separate powers acting upon the half wedges when the bar was between them.

To confirm this by an experiment, let two cylinders, as \( AB \) (fig. 4.) and \( CD \), be drawn towards one another by lines running over fixed pulleys, and a weight of 40 ounces hanging at the lines belonging to each cylinder; and let a wedge of 40 ounces weight, having its back just as thick as either of its sides is long, be put between the cylinders, which will then act against each side with a resistance equal to 40 ounces, whilst its own weight endeavours to bring it down and separate them. And here, the power of the wedge's gravity impelling it downward, will be to the resistance of both the cylinders against the wedge, as the thickness of the wedge is to the length of both its sides; for there will then be an equilibrium between the weight of the wedge and the resistance of the cylinders against it, and it will remain at any height between them; requiring just as much power to push it upward as to pull it downward.—If another wedge of equal weight and depth with this, and only half as thick, be put between the cylinders, it will require twice as much weight to be hung at the ends of the lines which draw them together, to keep the wedge from going down between them. That is, a wedge of 40 ounces, whose back is only equal to half the length of one of its sides, will require 80 ounces to each cylinder, to keep it in an equilibrium between them: and twice 80 is 160, equal to four times 40. So that the power will be always to the resistance, as the thickness of the back of the wedge is to the length (not of its one side, but) of both its sides.

The best way, though perhaps not the neatest, for making a wedge with its appurtenances for such experiments, is as follows. Let \( IKLM \) (fig. 4.) and \( LMNO \) be two flat pieces of wood, each about fifteen inches long and three or four in breadth, joined together by a hinge at \( LM \); and let \( P \) be a graduated arch of brass, on which the said pieces of wood may be opened to any angle not more than 60 degrees, and then fixed at the given angle by means of the two screws \( a \) and \( b \). Then, \( IKNO \) will represent the back of the wedge, \( LM \) its sharp edge which enters the wood, and the outsides of the pieces \( IKLM \) and \( LMNO \) the two sides of the wedge against which the wood acts in cleaving. By means of the said arch, the wedge may be opened so as to adjust the thickness of its back in any proportion to the length of either of its sides, but not to exceed that length: and any weight, as \( p \), may be hung to the wedge upon the hook \( M \), which weight, together with the weight of the wedge itself, may be considered as the impelling power; which is all the same in experiment, whether it be laid upon the back of the wedge to push it down, or hung to its edge to pull it down.

—Let \( AB \) and \( CD \) be two wooden cylinders, each about two inches thick, where they touch the outsides of the wedge; and let their ends be made like two round flat plates, to keep the wedge from slipping off endwise between them. Let a small cord with a loop on one end of it go over a pivot in the end of each cylinder, and the cords \( S \) and \( T \) belonging to the cylinder \( AB \) go over the fixed pulleys \( W \) and \( X \), and be fastened at their other ends to the bar \( wx \), on which any weight, as \( Z \), may be hung at pleasure. In like manner, let the cords \( Q \) and \( R \) belonging to the cylinder \( BC \) go over the fixed pulleys \( U \) and \( V \) to the bar \( uv \), on which a weight \( Y \) equal to \( Z \) may be hung. These weights, by drawing the cylinders towards one another, may be considered as the resistance of the wood acting equally against opposite sides of the wedge; the cylinders themselves being suspended near and parallel to each other, by their pivots in loops on the lines \( E,F,G,H \); which lines may be fixed to hooks in the ceiling of the room. The longer these lines are, the better; and they should never be less than four feet each. The further also the pulleys \( W,V \) and \( W,X \) are from the cylinders, the truer will the experiments be; and they may turn upon pins fixed into the wall.

In this machine, the weights \( Y \) and \( Z \), and the weight \( p \), may be varied at pleasure, so as to be adjusted in proportion portion of the length of the wedge's side to the thickness of its back; and when they are so adjusted, the wedge will be in equilibrium with the resistance of the cylinders.

The wedge is a very great mechanical power, since not only wood but even rocks can be split by it; which would be impossible to effect by the lever, wheel and axle, or pulley: for the force of the blow, or stroke, shakes the cohering parts, and thereby makes them separate the more easily.

6. The sixth and last mechanical power is the screw; which cannot properly be called a simple machine, because it is never used without the application of a lever or winch to assist in turning it: and then it becomes a compound engine of a very great force either in pressing the parts of bodies close together, or in raising great weights. It may be conceived to be made by cutting a piece of paper ABC (fig. 5) into the form of an inclined plane or half wedge, and then coiling it round a cylinder AB (fig. 6). And here it is evident, that the winch E must turn the cylinder once round before the weight or resistance D can be moved from one spiral winding to another, as from d to e: therefore, as much as the circumference of a circle described by the handle of the winch is greater than the interval or distance between the spirals, so much is the force of the screw. Thus, supposing the distance between the spirals to be half an inch, and the length of the winch to be twelve inches; the circle described by the handle of the winch where the power acts will be 76 inches nearly, or about 152 half inches, and consequently 152 times as great as the distance between the spirals: and therefore, a power at the handle, whose intensity is equal to no more than a single pound, will balance 152 pounds acting against the screw; and as much additional force, as is sufficient to overcome the friction, will raise the 152 pounds; and the velocity of the power will be to the velocity of the weight, as 152 to 1. Hence it appears, that the longer the winch be made, and the nearer the spirals are to one another, so much the greater is the force of the screw.

A machine for shewing the force or power of the screw may be contrived in the following manner. Let the wheel C (fig. 7.) have a screw ab on its axis, working in the teeth of the wheel D, which suppose to be 48 in number. It is plain, that for every time the wheel C and screw ab are turned round by the winch A, the wheel D will be moved one tooth by the screw; and therefore, in 48 revolutions of the winch, the wheel D will be turned once round. Then, if the circumference of a circle described by the handle of the winch be equal to the circumference of a groove e round the wheel D, the velocity of the handle will be 48 times as great as the velocity of any given point in the groove. Consequently; if a line G (above number 48) goes round the groove e, and has a weight of 48 pounds hung to it below the pedestal EF, a power equal to one pound at the handle will balance and support the weight. To prove this by experiment, let the circumferences of the grooves of the wheels C and D be equal to one another; and then if a weight H of one pound be suspended by a line going round the groove of the wheel C, it will balance a weight of 48 pounds hanging by the line G; and a small addition to the weight H will cause it to descend, and so raise up the other weight.

If the line G, instead of going round the groove e of the wheel D, goes round its axle I; the power of the machine will be as much increased, as the circumference of the groove e exceeds the circumference of the axle: which, supposing it to be six times, then one pound at H will balance 6 times 48, or 288 pounds hung to the line on the axle: and hence the power or advantage of this machine will be as 288 to 1. That is to say, a man, who by his natural strength could lift an hundred weight, will be able to raise 288 hundred, or 14½ ton weight by this engine.

But the following engine is still more powerful, on account of its having the addition of four pulleys: and in it we may look upon all the mechanical powers combined together, even if we take in the balance. For as the axis D (fig. 8.) of the bar AB is in its middle at C, it is plain that if equal weights are suspended upon any two pins equidistant from the axis C, they will counterpoise each other.—It becomes a lever by hanging a small weight P upon the pin n, and a weight as much heavier upon either of the pins b, c, d, e, or f, as is in proportion to the pins being so much nearer the axis. The wheel and axle FG is evident; so is the screw E, which takes in the inclined plane, and with it the half wedge. Part of a cord goes round the axle, the rest under the lower pulleys K, m, over the upper pulleys L, n, and then it is tied to a hook at m in the lower or moveable block, on which hangs the weight W.

In this machine, if the wheel F has 30 teeth, it will be turned once round in thirty revolutions of the bar AB, which is fixed on the axis D of the screw E: if the length of the bar is equal to twice the diameter of the wheel, the pins a and n at the ends of the bar will move 60 times as fast as the teeth of the wheel do: and consequently, one ounce at P will balance 60 ounces hung upon a tooth at q in the horizontal diameter of the wheel. Then, if the diameter of the wheel F is 10 times as great as the diameter of the axle G, the wheel will have 10 times the velocity of the axle; and therefore one ounce P at the end of the lever AC will balance 10 times 60 or 600 ounces hung to the rope H which goes round the axle. Lastly, if four pulleys be added, they will make the velocity of the lower block K, and weight W, four times less than the velocity of the axle: and this being the last power in the machine, which is four times as great as that gained by the axle, it makes the whole power of the machine 4 times 600, or 2400. So that a man who could lift one hundred weight in his arms, by his natural strength, would be able to raise 2400 hundred weight by this engine.—But it is here as in all other mechanical cases; for the time lost is always as much as the power gained, because the velocity with which the power moves will ever exceed the velocity with which the weight rises, as much as the intensity of the weight exceeds the intensity of the power.

The friction of the screw itself is very considerable; and there are few compound engines, but what, upon account of the friction of the parts against one another, will will require a third part of more power to work them when loaded, than what is sufficient to constitute a balance between the weight and the power.

Of Mills, Cranes, Wheel-carriages, and the Engine for driving Piles.

As these machines are so universally useful; it would be ridiculous to make any apology for describing them.

In a common breaf-mill, where the fall of water may be about ten feet, AA (Plate CVIII. fig. 9.) is the great wheel, which is generally about 17 or 18 feet in diameter, reckoned from the outermost edge of any float-board at a to that of its opposite float at b. To this wheel the water is conveyed through a channel; and so, falling upon the wheel, turns it round.

On the axis BB of this wheel, and within the mill-house, is a wheel D, about 8 or 9 feet diameter, having 61 cogs, which turn a trundle E containing ten upright staves or rounds; and when these are the number of cogs and rounds, the trundle will make 6½ revolutions for one revolution of the wheel.

The trundle is fixed upon a strong iron axis called the spindle, the lower end of which turns in a brafs foot, fixed at F, in the horizontal beam ST called the bridge-tree; and the upper part of the spindle turns in a wooden bush fixed into the nether millstone which lies upon beams in the floor YY. The top part of the spindle above the bush is square, and goes into a square hole in a strong iron crofs abcd (see fig. 3.) called the rynd; under which, and close to the bush, is a round piece of thick leather upon the spindle, which it turns round at the same time as it does the rynd.

The rynd is let into grooves in the under surface of the running millstone G (fig. 2.) and so turns it round in the same time that the trundle E is turned round by the cog-wheel D. This millstone has a large hole quite through its middle, called the eye of the stone, through which the middle part of the rynd and upper end of the spindle may be seen; whilst the four ends of the rynd lie hid below the stone in their grooves.

The end T of the bridge-tree TS (which supports the upper millstone G upon the spindle) is fixed into a hole in the wall; and the end S is let into a beam QR called the brayer, whose end R remains fixed in a mortise; and its other end Q hangs by a strong iron rod P which goes through the floor YY, and has a screw-nut on its top at O; by the turning of which nut, the end Q of the brayer is raised or depressed at pleasure, and consequently the bridge-tree TS and upper millstone. By this means, the upper millstone may be set as close to the under one, or raised as high from it, as the miller pleases. The nearer the millstones are to one another, the finer they grind the corn; and the more remote from one another, the coarser.

The upper millstone G is inclosed in a round box H, which does not touch it anywhere; and is about an inch distant from its edge all around. On the top of this box stands a frame for holding the hopper kk, to which is hung the shoe I by two lines fastened to the hind-part of it, fixed upon hooks in the hopper, and by one end of the crook-string K fastened to the fore-part of it at i; the other end being twisted round the pin L. As the pin is turned one way, the string draws up the shoe closer to the hopper, and so lessens the aperture between them; and as the pin is turned the other way, it lets down the shoe, and enlarges the aperture.

If the shoe be drawn up quite to the hopper, no corn can fall from the hopper into the mill; if it be let a little down, some will fall; and the quantity will be more or less, according as the shoe is more or less let down. For the hopper is open at bottom, and there is a hole in the bottom of the shoe, not directly under the bottom of the hopper, but forwarder towards the end i, over the middle of the eye of the millstone.

There is a square hole in the top of the spindle, in which, is put the feeder e (fig. 10.) This feeder (as the spindle turns round) jogs the shoe three times in each revolution, and so causes the corn to run constantly down from the hopper, through the shoe, into the eye of the millstone, where it falls upon the top of the rynd, and is, by the motion of the rynd and the leather under it, thrown below the upper stone, and ground between it and the lower one. The violent motion of the stone creates a centrifugal force in the corn going round with it, by which means it gets farther and farther from the centre, as in a spiral, in every revolution, until it be thrown right out; and, being then ground, it falls through a spout M, called the mill-eye, into the trough N.

When the mill is fed too fast, the corn bears up the stone, and is ground too coarse; and besides, it clogs the mill so as to make it go too slow. When the mill is too slowly fed, it goes too fast, and the stones by their attrition are apt to strike fire against one another. Both which inconveniences are avoided by turning the pin L backwards or forwards, which draws up or lets down the shoe; and so regulates the feeding as the miller sees convenient.

The heavier the running millstone is, and the greater the quantity of water that falls upon the wheel, so much the fatter will the mill bear to be fed; and consequently so much the more it will grind. And on the contrary, the lighter the stone, and the less the quantity of water, so much slower must the feeding be. But when the stone is considerably wore, and become light, the mill must be fed slowly at any rate; otherwise the stone will be too much borne up by the corn under it, which will make the meal coarse.

The quantity of power required to turn a heavy millstone is but a very little more than what is sufficient to turn a light one: for as it is supported upon the spindle by the bridge-tree ST, and the end of the spindle that turns in the brafs foot therein being but small, the odds arising from the weight is but very inconsiderable in its action against the power or force of the water. And besides, a heavy stone has the same advantage as a heavy fly; namely, that it regulates the motion much better than a light one.

In order to cut and grind the corn, both the upper and under millstones have channels or furrows cut into them, proceeding obliquely from the centre towards the circumference. And these furrows are each cut perpendicularly on one side and obliquely on the other into the stone, which which gives each furrow a sharp edge, and in the two stones they come, as it were, against one another like the edges of a pair of scissors; and so cut the corn, to make it grind the easier when it falls upon the places between the furrows. These are cut the same way in both stones when they lie upon their backs, which makes them run cross ways to each other when the upper stone is inverted by turning its furrowed surface towards that of the lower. For, if the furrows of both stones lay the same way, a great deal of the corn would be driven onward in the lower furrows, and so come out from between the stones without ever being cut.

When the furrows became blunt and shallow by wearing, the running stone must be taken up, and both stones new dress with a chisel and hammer. And every time the stone is taken up, there must be some tallow put round the spindle upon the bush, which will soon be melted by the heat that the spindle acquires from its turning and rubbing against the bush, and so will get in betwixt them; otherwise the bush would take fire in a very little time.

The bush must embrace the spindle quite close, to prevent any shake in the motion; which would make some parts of the stones grate and fire against each other; whilst other parts of them would be too far asunder, and by that means spoil the meal in grinding.

Whenever the spindle wears the bush so as to begin to shake in it, the stone must be taken up, and a chisel drove into several parts of the bush; and when it is taken out, wooden wedges must be driven into the holes; by which means the bush will be made to embrace the spindle close all around it again. In doing this, great care must be taken to drive equal wedges into the bush on opposite sides of the spindle; otherwise it will be thrown out of the perpendicular, and so hinder the upper stone from being set parallel to the under one, which is absolutely necessary for making good work. When any accident of this kind happens, the perpendicular position of the spindle must be restored by adjusting the bridge-tree ST by proper wedges put between it and the brayer QR.

It often happens, that the rynd is a little wrenched in laying down the upper stone upon it; or is made to sink a little lower upon one side of the spindle than on the other; and this will cause one edge of the upper stone to drag all around upon the other, whilst the opposite edge will not touch. But this is easily set to rights, by raising the stone a little with a lever, and putting bits of paper, cards, or thin chips, betwixt the rynd and the stone.

The diameter of the upper stone is generally about six feet, the lower stone about an inch more; and the upper stone when new contains about 22½ cubic feet, which weighs somewhat more than 1900 pounds. A stone of this diameter ought never to go more than 60 times round in a minute; for if it turns falter, it will beat the meal.

The grinding surface of the under stone is a little convex from the edge to the centre, and that of the upper stone a little more concave; so that they are farthest from one another in the middle, and come gradually nearer towards the edges. By this means, the corn at its first entrance between the stones is only bruised; but as it goes farther on towards the circumference or edge, it is cut smaller and smaller; and at last finely ground just before it comes out from between them.

The water-wheel must not be too large, for if it be, its motion will be too slow; nor too little, for then it will want power. And for a mill to be in perfection, the floats of the wheel ought to move with a third part of the velocity of the water, and the stone to turn round once in a second of time.

Such a mill as this, with a fall of water about 7½ feet, will require about 32 hogsheads every minute to turn the wheel with a third part of the velocity with which the water falls, and to overcome the resistance arising from the friction of the gears and attrition of the stones in grinding the corn.

The greater fall the water has, the less quantity of it will serve to turn the mill. The water is kept up in the mill-dam, and let out by a sluice called the penstock, when the mill is to go. When the penstock is drawn up by means of a lever, it opens a passage through which the water flows to the wheel; and when the mill is to be stopped, the penstock is let down, which stops the water from falling upon the wheel.

A less quantity of water will turn an overshot-mill (where the wheel has buckets instead of float boards) than a breast-mill where the fall of the water seldom exceeds half the height AB of the wheel. So that, where there is but a small quantity of water, and a fall great enough for the wheel to lie under it, the bucket (or overshot) wheel is always used. But where there is a large body of water, with a little fall, the breast or float-board wheel must take place. Where the water runs only upon a little declivity, it can act but slowly upon the under part of the wheel at b; in which case the motion of the wheel will be very slow; and therefore, the floats ought to be very long, though not high, that a large body of water may act upon them; so that what is wanting in velocity may be made up in power; and then the cog-wheel may have a greater number of cogs in proportion to the rounds in the trundle, in order to give the millstone a sufficient degree of velocity.

They who have read what is said in the first section, concerning the acceleration of bodies falling freely by the power of gravity acting constantly and uniformly upon them, may perhaps ask, Why should the motion of the wheel be equable, and not accelerated, since the water acts constantly and uniformly upon it? The plain answer is, That the velocity of the wheel can never be so great as the velocity of the water that turns it; for, if it should become so great, the power of the water would be quite lost upon the wheel, and then there would be no proper force to overcome the friction of the gears and attrition of the stones. Therefore, the velocity with which the wheel begins to move, will increase no longer than till its momentum or force is balanced by the resistance of the machine; and then the wheel will go on with an equable motion.

[If the cog-wheel D be made about 18 inches diameter, with 30 cogs, the trundle as small in proportion with 10 slaves, and the millstones be each about two feet in diameter, and the whole work be put into a strong frame of wood, as represented in the figure, the engine will be a hand-mill for grinding corn or malt in private families. And then, it may be turned by a winch, instead of the wheel AA; the millstone making three revolutions for every one of the winch. If a heavy fly be put upon the axle B, near the winch, it will help to regulate the motion.

If the cogs of the wheel and rounds of the trundle could be put in as exactly as the teeth are cut in the wheels and pinions of a clock, then the trundle might divide the wheel exactly; that is to say, the trundle might make a given number of revolutions for one of the wheel, without a fraction. But as any exact number is not necessary in mill-work, and the cogs and rounds cannot be set so truly as to make all the intervals between them equal; a skilful mill-wright will always give the wheel what he calls a hunting cog; that is, one more than what will answer to an exact division of the wheel by the trundle. And then, as every cog comes to the trundle, it will take the next staff or round behind the one which it took in the former revolution: and by that means, will wear all the parts of the cogs and rounds which work upon one another equally, and to equal distances from one another in a little time; and so make a true uniform motion throughout the whole work. Thus, in the above water-mill, the trundle has 10 staves, and the wheel 61 cogs.

Sometimes, where there is a sufficient quantity of water, the cog-wheel AA (Plate CIX. fig. 1.) turns a large trundle BB, on whose axis C is fixed the horizontal wheel D, with cogs all round its edge, turning two trundles E and F at the same time; whose axes or spindles G and H turn two millstones I and K, upon the fixed stones L and M. And when there is not work for them both, either may be made to lie quiet, by taking out one of the staves of its trundle, and turning the vacant place towards the cog-wheel D. And there may be a wheel fixt on the upper end of the great upright axle C for turning a couple of bolting-mills, and other work for drawing up the sacks, fanning and cleaning the corn, sharpening of tools, &c.

If, instead of the cog-wheel AA and trundle BB, horizontal levers be fixed into the axle C, below the wheel D; then, horses may be put to these levers for turning the mill; which is often done where water cannot be had for that purpose.

The working parts of a wind-mill differ very little from those of a water mill; only the former is turned by the action of the wind upon four sails, every one of which ought (as is generally believed) to make an angle of $54\frac{3}{4}$ degrees with a plane perpendicular to the axis on which the arms are fixt for carrying them; it being demonstrable, that when the sails are set to such an angle, and the axis turned end-ways toward the wind, the wind has the greatest power upon the sails. But this angle answers only to the case of a vane or sail just beginning to move: for, when the vane has a certain degree of motion, it yields to the wind; and then that angle must be increased to give the wind its full effect.

Again, the increase of this angle should be different, according to the different velocities from the axis to the extremity of the vane. At the axis it should be $54\frac{3}{4}$ degrees, and thence continually increase, giving the vane a twist, and so causing all the ribs of the vane to lie in different planes.

Lastly, these ribs ought to decrease in length from the axis to the extremity, giving the vane a curvilinear form; so that no part of the force of any one rib be spent upon the rest, but all move on independent of each other. All this is required to give the sails of a wind-mill their true form: and we see both the twist and the diminution of the ribs exemplified in the wings of birds.

It is almost incredible to think with what velocity the tips of the sails move when acted upon by a moderate gale of wind. We have several times counted the number of revolutions made by the sails in ten or fifteen minutes; and from the length of the arms from tip to tip, have computed, that if a hoop of that diameter was to run upon the ground with the same velocity that it would move if put upon the sail-arms, it would go upwards of 30 miles in an hour.

As the ends of the sails nearest the axis cannot move with the same velocity that the tips or farther ends do, although the winds act equally strong upon them; perhaps a better position than that of stretching them along the arms directly from the centre of motion, might be to have them set perpendicularly across the farther ends of the arms, and there adjusted lengthwise to the proper angle. For, in that case, both ends of the sails would move with the same velocity; and being farther from the centre of motion, they would have so much the more power: and then, there would be no occasion for having them so large as they are generally made, which would render them lighter, and consequently there would be so much less friction on the thick neck of the axle where it turns in the wall.

A crane is an engine by which great weights are raised to certain heights, or let down to certain depths. It consists of wheels, axles, pulleys, ropes, and a gib or gibbet. When the rope H (fig. 2.) is hooked to the weight K, a man turns the winch A, on the axis whereof is the trundle B, which turns the wheel C, on whose axis D is the trundle E, which turns the wheel F with its upright axis G, on which the great rope HH winds as the wheel turns; and going over a pulley I at the end of the arm d of the gib cede, it draws up the heavy burden K, which being raised to a proper height, as from a ship to the quay, is then brought over the quay by pulling the wheel Z round by the handles z, z, which turns the gib by means of the half wheel b fixt on the gib-post cc, and the strong pinion a fixt on the axis of the wheel Z. This wheel gives the man that turns it an absolute command over the gib, so as to prevent it from taking any unlucky swing, such as often happens when it is only guided by a rope tied to its arm d; and people are frequently hurt, sometimes killed, by such accidents.

The great rope goes between two upright upright rollers i and k, which turn upon gudgeons in the fixed beams f and g: and as the gib is turned towards either side, the rope bends upon the roller next that side. Were it not for these rollers, the gib would be quite unmanageable; for the moment it were turned ever so little towards any side, the weight K would begin to descend, because the rope would be shortened between the pulley I and axis G; and so the gib would be pulled violently to that side, and either be broken to pieces, or break every thing that came in its way. These rollers must be placed so, that the sides of them, round which the rope bends, may keep the middle of the bended part directly even with the centre of the hole in which the upper gudgeon of the gib turns in the beam f. The truer these rollers are placed, the easier the gib is managed, and the less apt to swing either way by the force of the weight K.

A ratchet-wheel Q is fixed upon the axis D, near the trundle E; and unto this wheel falls the catch or click R. This hinders the machine from running back by the weight of the burden K, if the man who raises it should happen to be careless, and so leave off working at the winch A sooner than he ought to do.

When the burden K is raised to its proper height from the ship, and brought over the quay by turning the gib about, it is let down gently upon the quay, or into a cart standing thereon, in the following manner: A man takes hold of the rope t (which goes over the pulley v, and is tied to a hook at S in the catch R) and so disengages the catch from the ratchet wheel Q; and then, the man at the winch A turns it backward, and lets down the weight K. But if the weight pulls too hard against this man, another lays hold of the handle V, and by pulling it downward, draws the gripe U close to the wheel Y, which, by rubbing hard against the gripe, hinders the too quick descent of the weight; and not only so, but even stops it at any time, if required. By this means, heavy goods may be either raised or let down at pleasure, without any danger of hurting the men who work the engine.

When part of the goods are craned up, and the rope is to be let down for more, the catch R is first disengaged from the ratchet-wheel Q, by pulling the cord t; then the handle q is turned half round backward, which, by the crank nn in the piece o, pulls down the frame h between the guides m and m (in which it slides in a groove) and so disengages the trundle B from the wheel C; and then, the heavy hook β at the end of the rope H descends by its own weight, and turns back the great wheel F with its trundle E, and the wheel C; and this last wheel acts like a fly against the wheel E and hook β, and so hinders it from going down too quick; whilst the weight X keeps up the gripe U from rubbing against the wheel Y, by means of a cord going from the weight, over the pulley w to the hook W in the gripe; so that the gripe never touches the wheel, unless it be pulled down by the handle V.

When the crane is to be set at work again, for drawing up another burden, the handle q is turned half round forwards: which, by the crank nn, raises up the frame h, and causes the trundle B to lay hold of the wheel C; and then, by turning the winch A, the burden of goods K is drawn up as before.

The crank nn turns pretty stiff in the mortise near o, and stops against the farther end of it when it has got just a little beyond the perpendicular; so that it can never come back of itself; and therefore the trundle B can never come away from the wheel C, until the handle q be turned half round.

The great rope runs upon rollers in the lever LM, which keep it from bending between the axle at G and the pulley I. This lever turns upon the axis N by means of the weight O, which is just sufficient to keep its end L up to the rope; so that, as the great axle turns, and the rope coils round it, the lever rises with the rope, and prevents the coilings from going over one another.

The power of this crane may be estimated thus: Suppose the trundle B to have 12 staves or rounds, and the wheel C to have 78 spur cogs; the trundle E to have 14 staves, and the wheel F 56 cogs: then, by multiplying the staves of the trundles, 13 and 14, into one another, their product will be 182; and by multiplying the cogs of the wheels, 78 and 56, into one another, their product will be 4368; and dividing 4368 by 182, the quotient will be 24; which shews that the winch A makes 24 turns for one turn of the wheel F and its axle G on which the great rope or chain HIH winds. So that, if the length or radius of the winch A were only equal to half the diameter of the great axle G, added to half the thickness of the rope H; the power of the crane would be as 24 to 1: but the radius of the winch being double the above length, it doubles the said power, and so makes it as 48 to 1: in which case a man may raise 48 times as much weight by this engine as he could do by his natural strength without it, making proper allowance for the friction of the working parts.—Two men may work at once, by having another winch on the opposite end of the axis of the trundle under B; and so make the power still double.

If this power be thought greater than what may be generally wanted, the wheels may be made with fewer cogs in proportion to the staves in the trundles: and so the power may be of whatever degree is judged to be requisite. But if the weight be so great as will require yet more power to raise it (suppose a double quantity) then the rope H may be put under a moveable pulley, as P, and the end of it tied to a hook in the gib at ε; which will give a double power to the machine, and so raise a double weight hooked to the block of the moveable pulley.

When only small burdens are to be raised, this may be quickly done by men pushing the axle G round by the hand-spokes y,y,y,y; having first disengaged the trundle B from the wheel C: and then, this wheel will only act as a fly upon the wheel F; and the catch R will prevent its running back, if the men should inadvertently leave off pushing before the burden be unhooked from β.

Lastly, when very heavy burdens are to be raised, which might endanger the breaking of the cogs in the wheel F; their force against these cogs may be much abated by men pushing round the hand-spokes y,y,y,y, whilst the man at A turns the winch.

We have only shewn the working parts of this crane, without the whole of the beams which support them; knowing that these are easily supposed, and that if they had been drawn, they would have hid a great deal of the working parts from sight, and also confused the figure.

Another very good crane is made in the following manner. AA (fig. 3.) is a great wheel turned by men walking within it at H. On the part C, of its axle BC, the great rope D is wound as the wheel turns; and this rope draws up goods in the same way as the rope HH does in the above-mentioned crane, the gib-work here being supposed to be of the same sort. But these cranes are very dangerous to the men in the wheel; for, if any of the men should chance to fall, the burden will make the wheel run back and throw them all about within it; which often breaks their limbs, and sometimes kills them.

The late ingenious Mr Padmore of Bristol, whose contrivance the forementioned crane is, observing this dangerous construction, contrived a method for remedying it, by putting cogs all around the outside of the wheel, and applying a trundle E to turn it; which increases the power as much as the number of cogs in the wheel is greater than the number of slaves in the trundle; and by putting a ratchet-wheel F on the axis of the trundle, (as in the above-mentioned crane, with a catch to fall into it,) the great wheel is kept from running back by the force of the weight, even if all the men in it should leave off walking. And by one man working at the winch I, or two men at the opposite winches when needful, the men in the wheel are much assisted, and much greater weights are raised, than could be by men only within the wheel. Mr. Padmore put also a gripe-wheel G upon the axis of the trundle, which being pinched in the same manner as described in the former crane, heavy burdens may be let down without the least danger. And before this contrivance, the lowering of goods was always attended with the utmost danger to the men in the wheel; as every one must be sensible of, who has seen such engines at work.

And it is surprising that the masters of wharfs and cranes should be so regardless of the limbs, or even lives of their workmen, that, excepting the late Sir James Creed of Greenwich, and some gentlemen at Bristol, there is scarce an instance of any who has used this safe contrivance.

The structure of wheel-carriages is generally so well known, that it would be needless to describe them. And therefore, we shall only point out some inconveniences attending the common method of placing the wheels, and loading the waggons.

In coaches, and all other four-wheeled carriages, the fore wheels are made of a less size than the hind ones, both on account of turning short, and to avoid cutting the braces: otherwise, the carriage would go much easier if the fore-wheels were as high as the hind ones; and the higher the better, because their motion would be so much the slower on their axles, and consequently the friction proportionably taken off. But carriers and coachmen give another reason for making the fore-wheels much lower than the hind-wheels; namely, that when they are so, the hind-wheels help to push on the fore ones: which is too unphilosophical and absurd to deserve a refutation; and yet for their satisfaction, we shall shew by experiment that it has no existence but in their own imaginations.

It is plain, that the small wheels must turn as much oftener round than the great ones, as their circumferences are less. And therefore, when the carriage is loaded equally heavy on both axles, the fore-axle must endure as much more friction, and consequently wear out as much sooner, than the hind-axle, as the fore-wheels are less than the hind ones. But the great misfortune is, that all the carriers to a man do obstinately persist, against the clearest reason and demonstration, in putting the heavier part of the load upon the fore-axle of the waggon; which not only makes the friction greatest where it ought to be least, but also presseth the fore-wheels deeper into the ground than the hind wheels, notwithstanding the fore-wheels, being less than the hind ones, are with so much the greater difficulty drawn out of a hole or over an obstacle, even supposing the weights on their axles were equal. For the difficulty, with equal weights, will be as the depth of the hole or height of the obstacle is to the semidiameter of the wheel. Thus, if we suppose the small wheel D (fig 4.) of the waggon AB to fall into a hole of the depth EF, which is equal to the semidiameter of the wheel, and the waggon to be drawn horizontally along; it is evident, that the point E of the small wheel will be drawn directly against the top of the hole; and therefore, all the power of horses and men will not be able to draw it out, unless the ground gives way before it. Whereas, if the hind wheel C falls into such a hole, it sinks not near so deep in proportion to its semidiameter; and therefore, the point G of the large wheel will not be drawn directly, but obliquely, against the top of the hole; and so will be easily got out of it. Add to this, that since a small wheel will often sink to the bottom of a hole, in which a great wheel will go but a very little way, the small wheels, ought in all reason to be loaded with less weight than the great ones; and then the heavier part of the load would be less jolted upward and downward, and the horses tired so much the less as their draught raised the load to less heights.

It is true, that when the waggon road is much up hill, there may be danger in loading the hind part much heavier than the fore part; for then the weight would overhang the hind axle, especially if the load be high, and endanger tilting up the fore-wheels from the ground. In this case, the safest way would be to load it equally heavy on both axles; and then, as much more of the weight would be thrown upon the hind-axle than upon the fore one, as the ground rises from a level below the carriage. But as this seldom happens, and when it does, a small temporary weight laid upon the pole between the horses would overbalance the danger; and this weight might be thrown into the waggon when it comes to level ground; it is strange that an advantage so plain and so obvious as would arise from loading the hind-wheels heaviest, should not be laid hold of, by complying with this method.

To confirm these reasonings by experiment, let a small model of a waggon be made, with its fore-wheels 2½ inches in diameter, and its hind-wheels 4½; the whole model weighing about 20 ounces. Let this little carriage be loaded any how with weights, and have a small cord tied to each of its ends, equally high from the ground it rests upon; and let it be drawn along a horizontal board, first by a weight in a scale hung to the cord at the fore part; the cord going over a pulley at the end of the board to facilitate the draught, and the weight just sufficient to draw it along. Then, turn the carriage, and hang the scale and weight to the hind cord, and it will be found to move along with the same velocity as at first: which shews, that the power required to draw the carriage is all the same, whether the great or small wheels are foremost; and therefore the great wheels do not help in the least to push on the small wheels in the road.

Hang the scale to the fore cord, and place the fore-wheels (which are the small ones) in two holes, cut three eighth-parts of an inch deep into the board; then put a weight of 32 ounces into the carriage, over the fore-axle, and an equal weight over the hind axle: this done, put 44 ounces into the scale, which will be just sufficient to draw out the fore-wheels: but if this weight be taken out of the scale, and one of 16 ounces put into its place, if the hind-wheels are placed in the holes, the 16 ounces weight will draw them out; which is little more than a third part of what was necessary to draw out the fore-wheels. This shews, that the larger the wheels are, the less power will draw the carriage, especially on rough ground.

Put 64 ounces over the axle of the hind-wheels, and 32 over the axle of the fore ones, in the carriage; and place the fore-wheels in the holes: then, put 38 ounces into the scale, which will just draw out the fore wheels; and when the hind ones come to the hole, they will find but very little resistance, because they sink but a little way into it.

But shift the weights in the carriage, by putting the 32 ounces upon the hind axle, and the 64 ounces upon the fore one; and place the fore wheels in the holes; then, if 76 ounces be put into the scale, it will be found no more than sufficient to draw out these wheels; which is double the power required to draw them out, when the lighter part of the load was put upon them: which is a plain demonstration of the absurdity of putting the heaviest part of the load in the fore part of the waggon.

Every one knows what an outcry was made by the generality, if not the whole body, of the carriers, against the broad-wheel act; and how hard it was to persuade them to comply with it, even though the government allowed them to draw with more horses, and carry greater loads, than usual. Their principal objection was, that as a broad wheel must touch the ground in a great many more points than a narrow wheel, the friction must of course be just so much the greater; and consequently, there must be so many more horses than usual, to draw the waggon. It is believed that the majority of people were of the same opinion, not considering, that if the whole weight of the waggon and load in it bears upon a great many points, each fulfils a proportionably less degree of weight and friction, than when it bears only upon a few points; so that what is wanting in one, is made up in the other; and therefore will be just equal under equal degrees of weight, as may be shewn by the following plain and easy experiment.

Let one end of a piece of packthread be fastened to a brick, and the other end to a common scale for holding weights: then, having laid the brick edgewise on a table, and let the scale hang under the edge of the table, put as much weight into the scale as will just draw the brick along the table. Then taking back the brick to its former place, lay it flat on the table, and leave it to be acted upon by the same weight in the scale as before, which will draw it along with the same ease as when it lay upon its edge. In the former case, the brick may be considered as a narrow wheel on the ground; and in the latter, as a broad wheel. And since the brick is drawn along with equal ease, whether its broad side or narrow edge touches the table, it shews that a broad wheel might be drawn along the ground with the same ease as a narrow one (supposing them equally heavy) even though they should drag, and not roll, as they go along.

As narrow wheels are always sinking into the ground, especially when the heaviest part of the load lies upon them, they must be considered as going constantly up hill, even on level ground; and their edges must sustain a great deal of friction by rubbing against the sides of the ruts made by them. But both these inconveniences are avoided by broad wheels; which, instead of cutting and ploughing up the roads, roll them smooth, and harden them; as experience testifies in places where they have been used, especially either on wetish or sandy ground: though after all it must be confessed, that they will not do in stiff clayey cross-roads; because they would soon gather up as much clay as would be almost equal to the weight of an ordinary load.

If the wheels were always to go upon smooth and level ground, the best way would be to make the spokes perpendicular to the naves; that is, to stand at right angles to the axles; because they would then bear the weight of the load perpendicularly, which is the strongest way for wood. But because the ground is generally uneven, one wheel often falls into a cavity or rut when the other does not; and then it bears much more of the weight than the other does: in which case, concave or dishing wheels are best; because when one falls into a rut, and the other keeps upon high ground, the spokes become perpendicular in the rut, and therefore have the greatest strength when the obliquity of the load throws most of its weight upon them; whilst those on the high ground have less weight to bear, and therefore need not be at their full strength. So that the usual way of making the wheels concave is by much the best.

The axles of the wheels ought to be perfectly straight, that the rims of the wheels may be parallel to each other; for then they will move easiest, because they will be at liberty to go on straight forwards. But in the usual way of practice, the axles are bent downward at their ends; which brings the sides of the wheels next the ground nearer to one another than their opposite or higher sides are: and this not only makes the wheels to drag sidewise as they go along, and gives the load a much greater power of crushing them than when they are parallel to each other, but also endangers the overturning of the carriage when any wheel falls into a hole or rut; or when the carriage goes in a road which has one side lower than the other, as along the side of a hill. Thus (in the hind view of a waggon or cart) let AE and BF (fig. 5.) be the great wheels parallel to each other, on their straight axle K, and HCI the carriage loaded with heavy goods from C to G. Then, as the carriage goes on in the oblique road A-B, the centre of gravity of the whole machine and load will be at C (see p. 35, col. 1.) and the line line of direction CdD falling within the wheel BF, the carriage will not overset. But if the wheels be inclined to each other at the ground, as AE and BF (fig. 6.) are, and the machine be loaded as before, from C to G, the line of direction CdD falls without the wheel BF, and the whole machine tumbles over. When it is loaded with heavy goods (such as lead or iron) which lie low, it may travel safely upon an oblique road so long as the centre of gravity is at C, and the line of direction Cd (fig. 5.) falls within the wheels; but if it be loaded high with lighter goods (such as woolpacks) from C to L, (fig. 7.) the centre of gravity is raised from C to K, which throws the line of direction Kk without the lowest edge of the wheel BF, and then the load oversets the waggon.

If there be some advantage from small fore-wheels, on account of the carriage turning more easily and short than it can be made to do when they are large; there is at least as great a disadvantage attending them, which is, that as their axle is below the level of the horses breasts, the horses not only have the loaded carriage to draw along, but also part of its weight to bear; which tires them sooner, and makes them grow much stiffer in their hams, than they would be if they drew on a level with the fore axle: and for this reason, we find coach-horses soon become unfit for riding. So that on all accounts it is plain, that the fore-wheels of all carriages ought to be so high, as to have their axles even with the breasts of the horses; which would not only give the horses a fair draught, but likewise cause the machine to be drawn by a less degree of power.