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.

CHAP. I. Of Matter and its Properties.

By the word matter is here meant everything 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. Dr. Priestley and some others have indeed denied this property to matter; and supposed, that, if a sufficient degree of force was applied to two bodies, they might actually exist in the same place at the same moment; but such arbitrary speculations cannot be of any service in mechanics, the very foundation of which is built on the opposite principle, and necessarily implies the impenetrability or solidity of matter.

A second property of matter is inactivity, or passivity; by which it always endeavours to continue in the state that it is in, whether of rest or motion. And there... 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. A great deal of this inactivity, however, we are assured, arises from gravity; for in those cases wherein gravity is not opposed, a very small body will set a very large one in motion.

But that matter can never put itself into motion is allowed by all men. For they see that 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 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 velocity it is retarded every moment till it falls.

A bowl moves but a short way upon a bowling-green; because the roughness and unevenness of the grassy surface soon creates friction enough to stop it. But if the green were perfectly level, and covered with polished glass, and the bowl were perfectly hard, round, and smooth, it would go a great way farther, 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 track, 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 is 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, or the force of gravity which acts upon all terrestrial bodies.

The fourth property of matter is divisibility, of which there can be no end. For, since matter 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 is laid on a table, the uppermost side of it will be further from the table than the undermost side. Moreover, it would be absurd to say that the greatest mountain on earth has more halves, quarters, or tenth parts, than the smallest particle of matter has.

We have many surprising instances of the smallness to which matter can be divided by art: of which the two following are very remarkable.

1. If a pound of silver be fused with a single grain of gold, the gold will be equally diffused thro' the whole silver; so that taking one grain from any part of the mass (in which there can be no more than the 576th part of a grain of gold) and dissolving it in aqua fortis, the gold will fall to the bottom.

2. The gold-beaters can extend a grain of gold into a leaf containing 50 square inches; and this leaf may be divided into 500000 parts. For an inch in length can be divided into 100 parts, every one of which will be visible to the bare eye: consequently a square inch can be divided into 10000 parts, and 50 square inches into 500000. And if one of these parts be viewed with a microscope that magnifies the diameter of an object only 10 times, it will magnify the area 100 times; and then the 100th part of a 500000th part of a grain (that is, the 50 millionth part) will be visible. Such leaves are commonly used in gilding; and they are so very thin, that if 124500 of them were laid upon one another, and pressed together, they would not exceed one inch in thickness.

Yet all this is nothing in comparison of the lengths that nature goes in the division of matter. For Mr Leeuwenhoek tells us, that there are more animals in the milt of a single cod fish, than there are men upon the whole earth: and that, by comparing these animals in a microscope with grains of common sand, it appeared that one single grain is bigger than four millions of them. Now each animal must have a heart, arteries, veins, muscles, and nerves, otherwise they could neither live nor move. How inconceivably small then must the particles of their blood be, to circulate through the smallest ramifications and jointings of their arteries and veins! It has been found by calculation, that a particle of their blood must be as much smaller than a globe of the tenth part of an inch in diameter, as that globe is smaller than the whole earth; and yet, if these particles be compared with the particles of light, they will be found to exceed them as much in bulk as mountains do single grains of sand. For, the force of any body striking against an obstacle is directly in proportion to its quantity of matter multiplied into its velocity: and since the velocity of the particles of light is demonstrated to be at least a million times greater than the velocity of a cannon-ball, it is plain, that if a million of these particles were as big as a single grain of sand, we durst no more open our eyes to the light, than we durst expose them to a cannon shot point-blank from a cannon.

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, any where toward 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 AR, 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.

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, is 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 noways 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-fugar 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 are 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 lodgment 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, are 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 is moved, it will draw the other after it. If two corks of unequal weights are 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 are rolled in dust, they will not run together, because the particles of dust keep them out of the sphere of each other's attraction.

When 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, a fluid when divided easily unites again. But if glass, or any other hard substance, is 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 in such a manner as not to separate again. If a ball of light wood is 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 in such a manner as to occupy (in comparison) an infinitely greater portion of space than it did before.

Attraction of gravitation is that power by which dissimilar 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, towards the centre: 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.

As the attraction of any large body, this earth, for instance, consists of the united attractions of all its parts, it thence follows, that if a body descends from the surface towards the centre of the earth, it would continually become lighter and lighter, the parts above attracting it, as well as those below; in which case it is demonstrated by mathematicians, that the gravity would decrease in the same proportion with the distance from the centre. Thus, let there be a body, plate as P, placed anywhere within a concave sphere as CIXVII. AB; and let us suppose it divided into an infinite number of thin concentric surfaces; the body P will be attracted equally each way by any one of these; for instance, the interior circle HIKLM. Let there be lines, as ILHK, &c. drawn through any point of the body P, in such a manner as to form the surface of two similar figures, suppose cones; the diameters of whose bases may be IH, KL, which let us suppose infinitely small. These bases being as the squares of the lines IH, KL, (2 El. 12.) will be directly as the squares of their distances from P; for the triangles IPH, KPL, being infinitely small, are similar. But those bases include all the particles of matter in the interior surface that are opposite to each other: the opposite attractions are therefore in the same ratio with those bases; that is, as the squares of the distances KP, PI. But the attraction is inversely as the squares of the distances of the attracting bodies; that is, inversely as the squares of the same distances PK,PI: these two ratios therefore destroying each other, other, it is evident, that if the concavity of the sphere was filled with matter, that alone which lies nearer the centre than the body can affect it; the respective actions of all the parts that are more distant being equal, and in contrary directions; since the same is demonstrable of any of the remaining concentric surfaces.

Let us see then what effect that which lies nearer the centre than the body will have upon it, which may be considered as a sphere on whose surface the body is placed. The distances of each particle of matter from the body, (taken collectively,) will be as the diameter of the sphere, or as the radius, i.e., as the distance of the body from the centre; their action therefore upon the body will be inversely as the square of that distance; but the quantity of matter will be as the cube of that distance, (18 El. 12.) : the attraction therefore will be inversely as that proportion. Now, these two ratios being compounded, the attraction will be only as the distance from the centre.

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 is 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 is 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 20 or 30 pounds weight with the same degree of swiftness that a bullet of 10 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 square distance multiplied by itself increases: that is, a body of the distance 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 16th 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.

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 consist 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 proportionably 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, though 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 have each 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 \( \frac{1}{2} = \text{square of } \frac{1}{2} \) In 2 minutes \( \frac{1+3}{4} = \text{square of } 2 \) In 3 minutes \( \frac{1+3+5}{9} = \text{square of } 3 \) In 4 minutes \( \frac{1+3+5+7}{16} = \text{square of } 4 \), &c.

Of this proposition Mr Rowning gives a mathematical demonstration from the following theorem, viz. that the space passed over by a body with an uniform power of motion, is in a ratio compounded of the time and equal velocity. For the longer a body continues to move uniformly, the more space it passes over; and the faster it moves during any interval of time, the farther it goes: therefore the space is in a ratio compounded of both; that is, it is had by multiplying the one into the other.

Hence may be deduced the following corollary, namely, That the area of a rectangle, one of whose sides represents the celerity with which a body moves, and the other the time of its motion, will express the space it moves through.

Let now the line \( AB \) represent the time a body takes up in falling, and let \( BC \) express the celerity acquired by its fall: farther, let the line \( AB \) be divided into an indefinite number of small portions, \( ei, im, mp, \) and let \( ef, ik, mn, pq, \) &c. be drawn parallel to the base. Now, as the height from which bodies can be let fall is so small, in proportion to their distance from the centre of the earth, that it cannot sensibly alter their gravity, which therefore may be conceived as acting constantly and uniformly upon them during the whole time of their fall; it follows, that they must acquire at every instant an equal degree of velocity.

Hence, the velocities being as the times in which they are acquired, it is plain, that the lines \( ef, ik, mn, pq, \) &c. being to each other (4 El. 6.) as the lines \( Ae, Ai, Am, Ap, \) &c. will represent the celerities in the times represented by these: that is, \( ef \) will be as the velocity of the body in the small portion of time \( ei, \) and \( ik \) will be as the velocity in the portion of time \( im; \) in like manner \( pq \) will be as the velocity in the portion of time \( po; \) which portions of time being taken infinitely small, the velocity of the body may be supposed the same during any whole portion; and consequently the space run over in the time \( ei, \) with the velocity \( ef, \) may be represented by the rectangle \( ef. \) In like manner the space run over in the time \( im, \) with the celerity \( ik, \) may be represented by the rectangle \( ik; \) and that run over with the celerity \( mn, \) in the time \( mp, \) by the rectangle \( pn; \) and so of the rest. Therefore the space run over in all those times will be represented by the sum of all the rectangles; that is, by the triangle \( ABC; \) for those little triangular deficiencies at the end of each rectangle would have vanished, had the lines \( ei, im, mp, \) &c. been infinitely short, as the times they were supposed to represent. Now, as the space the body describes in the time \( AB \) is represented by the triangle \( ABC, \) for the same reason the space passed over in the time \( Ao \) may be represented by the triangle \( Aor; \) but these triangles being similar, are to each other as the squares of their homologous sides \( AB, \) and \( Ao. \) (20 El. 6.) that is, the spaces represented by the triangles are to each other as the squares of the times represented by the sides Q.E.D.

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 \( D \) in rolling down the inclined plane \( AB, \) will acquire such a velocity by the time it arrives at \( B, \) as will carry it up 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 resistances, 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 \( CDKJ, \) 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.

Hence appears the absurdity of people's rising hastily in a coach or boat when it is likely to overturn; for by that means they raise the centre of gravity so far as to endanger throwing it quite out of the base; whereas, had they clapt down to the bottom, they would have brought the line of direction, and consequently the centre of gravity, farther within the base, and by that means might have saved themselves.

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 charge of position being sufficient to remove the line of direction out of the base in the latter 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 is inclined on which the heavy body is placed, that body will slide down upon the plane whilst the line of direction falls within the base; but it will tumble or roll down when the line falls without the base. Thus, the body A 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 up stairs: 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. A thousand more instances might be added.

The quantity of matter in all bodies is in exact proportion to their weight, 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 much more porous than is generally imagined.

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 4½, brass 5, tin 6, lead 6½. An iron rod 3 feet long is about one 70th part of an inch longer in summer than in winter.

The expansion of metals by heat is demonstrated by the following machine called PYROMETER.

AA is a flat piece of mahogany, in which are fixed four brass studs 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 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 the 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 10th 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 show 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 where the bar is quite cold, the index will rest at M, when 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. See MAGNETISM.

Several bodies, particularly amber, glass, jet, seal, electricity, 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. See ELECTRICITY.

CHAP. II. 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, 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 stop it.

As all motion is naturally rectilinear, 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 rectilinear 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 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 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 equally 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 LE; 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, 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 it through the space AD, the joint action of both will carry it in the same time thro' 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 rectilinear 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.

From the uniform projectile motion of bodies in straight lines, and the universal power of gravity or the planetary attraction, arises the curvilinear motion of all the heavenly bodies. If the body A 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 for ever with the same velocity, and in the same direction. But if, at the same moment the pro- Mechanics

The curves described by bodies revolving about their common centre of gravity.

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 to be the sun, and the earth going round him without any moon; and their moving forces regulated as above. In this case, whilst the earth goes round the sun in the dotted circle RT'UWX, &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 SS 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 s, 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 S13, 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 R1T, the earth the curve R5f, and the moon the curve g14r. 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 T2U, and the earth the curve f6g, and the moon the curve r15s, 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 R1T2U3V4W, the earth the irregular curve R5f6g78i, and the moon the yet more irregular curve g14r15t16t17u; 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 S13, which the earth describes round that centre of gravity (in every course of the moon round her orbit) is 12,000 miles in diameter. Consequently the earth is 12,000 miles nearer the sun at the time of full moon than at the time of new. See the Earth at f and at b.

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 be attracted towards the sun, with such a force as would make it fall from A to B, in fig. ro, 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, force ba, or the latter have carried it from A to F. But, if lance a, the projectile force had been twice as great, that is, quadruple such as would have carried the planet from A to H, gravity, 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.

(a) 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. twice the distance of A from F (b). Thus, a double projectile force will balance a quadruple power of gravity in the same circle; as appears plain by the figure, and shall soon be confirmed by an experiment.

The whirling-table is a machine contrived for throwing 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 Ring 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 WX, and YZ, 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 goes 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 (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 braids 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 fit from the edge to the hole, for allowing it to be slided 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 s 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

(b) Here the arcs AG, AI, must be supposed to be very small; otherwise AE, which is equal to HI, will be more than quadruple to AB, which is equal to FG. part of their respective circles.

4. 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.

5. 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 balls at these distances on their cords, by the screw-nuts at top; and 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 five 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 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 six 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 the centre of motion x, the weight once 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 six times as much weight put into it as the tower P has, the balls will raise their weight exactly 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.

9. If bodies of equal weights revolve in equal double circles with unequal velocities, their centrifugal forces velocity in as the squares of the velocities. To prove this law by the same experiment, let two balls U and V of equal weights balance to a be fixed on their cords at equal distances from their quadruple respective centres of motion w and x; and then let power of the catgut string E be put round the wheel K (whose gravity 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 into 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 Kepler's circles, in such a manner that the squares of the times problem 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 fifth parts; 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; therefore the square of the period or revolution of the ball V, is contained four times in the square of the period 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 (A), viz. that the squares of

(A) This law is of infinite use to astronomers; for if they know the periodical time, that is, the time of the circular revolution of two planets, and the distance of one of them from the centre, they can by this find out the distance of the other, which before was not known. For instance, we know the periodical time of the moon to be 27 days, and the periodical time of the earth to be 365 days. The distance of the moon from the centre of its motion we also know to be 60 semi-diameters of the earth. Now we desire to know the distance of the earth from the centre of its motion, namely, the sun? We know by rule, that the proportion of the squares of the periodical times will give the proportion of the cubes of the distances. Then we find out the squares of the periodical times of the two planets. The periodical time of the moon is 27, and the square of that number 729; the periodical time of the earth is 365, and the square 133225. Then we find the distance of the planet, already known, 60, and cube it, which makes 216000. 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 his 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 e 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 stopt 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 on 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 to revolve, 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 farther 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 vortices: for, if the planet be more dense or heavy than its 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 quite off, planets and all together. But while gravity exists, there is no occasion for such vortices; and when it ceases to exist, a stone thrown upwards will never return to the earth again.

9. If a body be so placed on the whirling-board of the machine (fig. 2.) 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 with its 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 that 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 into 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 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 into 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, tho' it could have no tendency of itself to remove from that part of space, yet it might be very easily moved 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 Fig. 7. six ounces, and connect it by the wire C with the ball B, which weighs only one ounce; and let the fork E hang 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 ball A, and consequently B will revolve with a velocity five times as great as A does; which will give B five 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 any thing 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 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 therein, 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-forews 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 a b c d represent the earth, with its side Fig. 9. e turned toward the moon, which will then attract the waters so as to raise them from e 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 fixed upon one end of the flat bar DC, with such a circle drawn upon Fig. 10. it as a b c d (in Fig. 9.) to represent the round figure of the earth and sea; and such an ellipse as e f g h to represent the swelling of the tide at e and g, occasioned by the influence of the moon. Over this plate AB, let... let the three ivory balls e, f, g, be hung by the silk lines h, 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 s 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 h, 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 shown 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 h, 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 cause 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 to 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 sides 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 with its furniture Fig. 10., 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. 12, the ball g will be kept towards the moon by the heaviest weight p (fig. 10.); 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 ellipse 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 was invented by Mr. Ferguson.

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 ¼ of a degree, which, at so great a distance as that of the sun, must be no less than 38,1500 miles; if this be divided by 357, the quotient will be 1068½; which shews that the common centre of gravity is within the body of the sun, and is only the 1068½ 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, 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 the 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 stopped 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 proves 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 downwards (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.

**Chap. III. 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: 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 unequalsoever 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 chemical 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 imperfections must be set aside until the theory be established, and then proper allowances are to be made.

The simple machines, usually called mechanical pow- The me- ers, are six in number, viz. the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the powers, screws. They are called mechanical powers, because what 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 fort, where the prop is placed between the weight and the power; but much nearer to the weight than 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 fort, 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 fort 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, supported by the prop D. Its principal use is to loosen large stones in the ground, or raise great kind of 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 1 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 defend, and raise the weight W; and the velocity with which the power defends 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 100 weight, will by the help of this lever be enabled to support 1200. If the weight be less, or the power greater, the prop may be placed so much farther from the weight; and then it can be raised to a proportionally 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 1 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 be the same; for 3 times 4 is 12; and so on, in proportion.

The *sistema* or Roman *scales* 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 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 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 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 1 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 defends.

This lever shows 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 proportionable 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 be the prop of the lever AB, and W a weight of 1 pound, placed three 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. The fourth kind of lever differs nothing from the first, but in being bended for the sake of convenience.

ABC 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 pully G; and W is a weight of 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 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 round 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 6 inches long, fixed on the axis of a trundle of 8 flames or rounds, working in the cogs of the wheel. Here it is plain, that the winch and trundle would make 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 semidiameters 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 is gained by the engine, so much time is lost in working it. In this sort of machines it is requisite to have a racket-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. The pulley, some in a block or cage which is fixed, and others in a block which is moveable and rises with the weight. For though a single pulley that only turns on its axis, and rises not without 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 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 moveable 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 one 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, sustains 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 thro' 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 A contains two pulleys, which only turn on their 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 thro' 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 no 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 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 GF: in this case, the cylinder E will be supported upon the plane CD, and 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 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 its plane is to its length. For, suppose the plane AB 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 elevated, that its perpendicular height D 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 in the direction of AB) equal to half its weight, to keep it from rolling. If the plane AB 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 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 the 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 thro' the whole height of the plane ML; and the weight will descend from H to K, thro' 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 (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 I., 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.

From the theory of the inclined plane also may be deduced the doctrine of pendulums; the foundation of which is as follows. A body acquires the same velocity in falling down the oblique side of a plane that it would do if it fell freely through the perpendicular height of it. For, the square of the velocity which a body acquires by falling to C as the space AG is to the space AC, as hath been already shown n° 14; that is, (by 8 El. 6. and Dcf. 10. El. 5.) as AG to AB; consequently the velocity itself at G is to the velocity itself at C, as AG to AB: But since AG is run over in the same time as AB, the velocity in G is also to the velocity in B as AG to AB; and consequently, since the velocities both in C and B bear the same proportion to that in G, they must be equal to each other.

Again: A body takes up the same time in falling through the chord of a circle, whether long or short, that it does in falling perpendicularly thro' the diame- ter of the same circle. For, seeing a body will fall from A to G, on the inclined plane ABC, in the same time another would fall freely to B provided AGB is a right angle, in which case AG (by 31 El. 3.) is a chord of that circle of which AB is the diameter; therefore, a body falls thro' the chord of a circle in the same time that it falls thro' the diameter; for the same demonstration will serve for any other chord.

Hence it follows, that if a pendulum could be made to vibrate in the chord of a circle instead of an arch, all its vibrations would be performed in the same time whether they were large or small.

This may be illustrated by conceiving the last figure inverted (as in fig. 4.) where supposing the ball sus- pended in such a manner as to swing in the right line GA instead of the arch GA, it would always fall through it in the same time, however long or short it was; for the inclination of the line GA to the ho- rizontal line BC, is not altered by inverting the fi- gure.

From hence we see the reason, why the shorter arches a pendulum describes, the nearer its vibrations come to an equality, for small arches differ less from their chords than larger ones. But if the pendulum is made to vibrate in a curve, which mathematicians call a cycloid; each swing will then be performed in the same time, whether the pendulum moves through a larger or lesser space. For the nature of this curve is such, that the tendency of a pendulum towards the lowest point of it is always in proportion to its dis- tance from thence; and consequently let that distance be more or less, it will always be run over by the pen- dulum in the same time.

Upon the right line AB, let the circle CDE be so placed as to touch the line in the point C; then let this circle roll along upon it from C to H, as a wheel upon the ground; then will the point C in one rev- olution of the circle describe the curve CKH, which is called a cycloid. Now suppose two plates of metal bent into the form HK and KC, and placed in the situation LH and LC, in such manner, that the points H and C may be applied to L, and the points answering to K be applied to H and C. This done, if a pendulum as LP, in length equal to LH, be made to vibrate be- tween the plates or cheeks of the cycloid LC and LH, it will swing in the line CKH; and the time of each vibration, whether the pendulum swings through a small or a great part of the cycloid, will be to the time a body takes up in falling perpendicularly through a space equal to IK (half the length of the pendulum), as the circumference of a circle to its diameter, and consequently it will always be the same.

The time of the descent and ascent of a pendulum, supposing it to vibrate in the chord of a circle, is equal to the time in which a body falling freely would de- scend through eight times the length of the pen- dulum.

For the time of the descent alone upon the chord is equal to that in which a body would fall through the diameter of the circle; that is, twice the length of the pendulum: but in twice that time (viz. during a whole vibration) the body would fall four times as far; that is, through eight times the length of the pendulum.

The times, that pendulums of different lengths per- form their vibrations in, are as the square roots of their lengths.

Dem. Let there be two pendulums A and B of different lengths, the time the first vibrates in (sup- pose through a chord) is equal to the time in which a body would fall freely through DA, the diameter of the circle; in like manner the time B vibrates is that in which a body would fall through FB. Now the times in which bodies fall through different spaces are as the square roots of those spaces, that is, of DA and FB, or of their halves CA and CB, i.e. of the lengths of the pendulums.

The centre of oscillation, is a point in which if the whole gravity of a pendulum was collected, the time of its vibration would not be altered thereby.

The rule for finding the centre of oscillation.

If the ball AB be hung by the string CD, whose weight is inconsiderable, the centre of oscillation is found thus: suppose E the centre of the globe; take the line K of such a length, that it shall bear the same proportion to ED as ED to EC; then EH being made equal to \( \frac{1}{2} \) of K, the point H shall be the cen- tre of oscillation.

If the weight of the rod CD be too considerable to be neglected, divide CD in I, so that DI may be equal to \( \frac{1}{2} \) of CD; and make a line as G, in the same pro- portion to CI, that the weight of the rod bears to that of the globe; then having found H the centre of oscillation of the globe, as before, divide IH in L, so that IL may bear the same proportion to LH, as the line CH bears to the line G; then will L be the centre of oscillation of the whole pendulum.

This is the point from whence the length of a pen- dulum is measured, which in our latitude, in a pen- dulum that swings seconds, is 39 inches and two- tenths.

The squares of the times in which pendulums, acted upon by different degrees of gravity, perform their vi- brations in, are to each other, inversely as the gravi- ties.

Dem. The spaces falling bodies descend through are as the squares of the times, when the gravity by which they are impelled is given; and as the gravity, when the time is given (for the sum of the velocities produced in any time will always be as the generating forces;) consequently, when neither is given, they are in a ratio compounded of both; the squares of the times are therefore inversely as the gravities. [For if in 3 quantities a, b, c; a is as b c, then b: \( \frac{a}{c} \), i.e. if a is given, as \( \frac{1}{c} \) or as c inversely.] But if the squares of the times, in which bodies fall thro' given spaces, spaces, are inversely as the gravities by which they are acted upon; then the squares of the times, in which pendulums of equal lengths perform their vibrations, will be also in the same ratio, on account of the constant equality between the time of the vibration of a pendulum, and of the descent of a body through eight times its length.

From whence it follows, that a pendulum will vibrate slower when nearer the equator, than the same when nearer the poles; for the gravity of all bodies is less, the nearer they are to the equator; viz. on account of the spheroidal figure of the earth, and its rotation about its axis. To which we may add the increase of the length of the pendulum occasioned by the heat in those parts: (for we find by experiment, that bodies are enlarged in every dimension in proportion to the degree of heat that is given them;) for which reason the vibrations of the pendulum will also be slower.

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 E F; then, DC 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, AB b 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 A a to A b, or B a to B b. And if the power be increased, so as to overcome the friction of the wedge and the resistance arising from the cohesion or slippage 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 a b c, may be made use of as a half wedge for separating the moulding c d from the wainscot AB. It is evident, that when this half wedge has been driven its whole length a e between the wainscot and moulding, its side a c will be at c d, and the moulding will be separated to f g from the wainscot. 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's a b to a c; 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 a b is to the whole side a c, 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 friction 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 A B to be a strong inflexible bar of wood or iron fixed into the ground at C B, and D 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 a b, by driving the inclined plane or half wedge a b c 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 c d e. 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 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 and LMNO be two flat pieces of wood, each about 15 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 w x, 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 u v, 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 farther also the pulleys WV and WX 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 of the length of the wedge's sides 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 fifth 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 of 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 12 inches; the circle described by the handle of the winch where the power acts will be 76 inches nearly, or about 15½ 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 have a screw a b 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 a b 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 around 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 6 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 144 tons 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 as combined together, even if we take in the balance. For as the axis D 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 s, 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 30 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 100 weight in his arms by his natural strength, would be able to raise 2400 hundred-weight by this engine.

By one or more of these simple powers, all great weights are raised to considerable heights: but in them all, the more they diminish the weight, the more slow they are in their operations, and consequently the more do they retard the workman's dispatch; and universally the more simple they are, the more expeditious. Besides this, their friction or rubbing against each other greatly diminishes their power. The friction in the balance is least, it is more in the lever, increased in the axle and wheel, yet more in the pulley, but most of all in the screw. In general, in combined engines, upon account of this friction, they will require a third part more of power to move them, than the theory allows. For this reason, therefore, it will for ever be impossible to fulfil the boast of Wilkins, who vaunted that he could pull up an oak by the roots with a single horse-hair; for the force requisite to work the machine in pulling it up, would nearly amount to a third part of the force which the machine exerts. The large capstan and pulley, used in launching a man of war, would in theory do it most effectually. A simple lever, drawn a proper length by the imagination, would do it as well; it would even fulfil the great boast of Archimedes, it would remove the earth itself. The learned often amuse themselves with fancies like these; and it was for this that Cicero called Archimedes a trifler.—As the friction of machines, however, forms a very considerable resisting power in mechanics, we shall here subjoin those methods by which it can best be computed.

The doctrine of friction, as ascertained by the latest experiments, may be summed up in the following manner. 1. When one body inflicts on another upon a horizontal plane, it presses it with its whole weight; which being equally reacted on, and consequently the whole effect of its gravity destroyed by the plane, it will be absolutely free to move in any horizontal direction by any the least power applied thereto, provided both the touching surfaces be perfectly smooth.

2. But since we find no such thing as perfect smoothness in the surfaces of bodies, but an evident roughness or unevenness of the parts in their surface, arising from their porosity and peculiar texture, it is easy to understand, that when two such surfaces come together, the prominent parts of one will, in some measure, fall into the concave parts of the other; and therefore, when a horizontal motion is attempted in one, the fixed prominent parts of the other will give more or less resistance to the moving surface, by holding and detaining its parts; and this is what we call friction.

3. Now since any body will require a force proportional to its weight to draw it over a given obstacle, it follows, that the friction arising to the moving body will always be in proportion to its weight only, and not the quantity of the surface by which it bears upon the resisting plane or surface. Thus, if a piece of wood four inches wide and one thick, be laid upon another fixed piece of the same wood, it will require the same weight to draw it along, whether it be laid on its broad or narrow side. 4. For though there be four times the number of touching particles on the broad side (ceteris paribus), yet each particle is pressed with but $\frac{1}{4}$ of the weight that those are on the narrow side; and since four times the number, multiplied by $\frac{1}{4}$ of the weight, is equal to $\frac{1}{4}$ of the number multiplied by four times the weight, it is plain the resistance is equal in both cases, and so requires the same force to overcome it. 5. The reason why friction is proportional to the weight of the moving body, is, because the power applied to move the body must raise it over the prominent parts of the surface on which it is drawn; and this motion of the body, as it is not upright, so it will not require a power equal to its whole weight; but being in the nature of the motion on an inclined plane, it will require only a part of its own weight, which will vary with the various degrees of smoothness and asperity. 6. It is found by experiment, that a body will be drawn along by nearly \( \frac{1}{4} \) of its weight; and if the surfaces be hard and well polished, by less than a third part; whereas, if the parts be soft or ragged, it will require a much greater weight. Thus also the cylinder of wood A B, if very smooth, and laid on two well polished supports C D, (having been first oiled or greased) and then charged with the weight of two pounds in the two equal balls G H, it will require an additional weight \( x \), equal to about a third part of the two pounds, to give motion to or overcome the friction of the said cylinder. 7. Now this additional weight, as it causes a greater weight of the cylinder, will likewise increase the friction; and therefore require the addition of another weight \( y \), equal to the third part of its own weight: for the same reason, the weight \( y \) will require another \( z \), a third part less; and so on ad infinitum. Hence, supposing the friction to be precisely a third of the weight, the first weight with all the additional ones, viz. \( 2, \frac{3}{4}, \frac{5}{9}, \frac{7}{16}, \ldots \), &c., will be a series of numbers in geometrical progression decreasing. Now the sum of all these terms, except the first, is found, by a well-known theorem in arithmetic, to be equal to one pound. So that if the weight of the cylinder be inconsiderable, the readiest way to overcome the friction would be to double the power G, or H, at once. 8. But though we may, at a medium, allow a third part of the weight with which any simple machine is charged, for the friction arising from thence; yet this is very precarious, and seldom is the case: for if A B C D be a piece of brass of six ounces, and E F G H be also a plate of brass, and both the surfaces well ground and polished, the weight P of near two ounces will be required to draw along the body A C alone; but if A C be loaded with 6, 8, or 10 lb., then a sixth part of the weight will be sufficient to draw it along the plane. On the other hand, if the plane be covered with a linen or woollen cloth, then a third or half part, and sometimes more, will be requisite to draw it along on the plane. 9. Yet notwithstanding the difficulty and uncertainty attending the estimation of the quantity of friction, it is still a most useful and necessary inquiry, how and by what means the friction of any machine may be diminished. In order to this, we must consider friction mechanically, or as a force acting against a power applied to overcome it. Thus suppose A B an upright stem or shaft, turning freely in the socket B fixed in the table or plane I K L M; and A C, D E, two arms fixed in the said shaft, the latter of which, D E, has three pins going into a socket in the middle of heavy weights, F, G, or H, in such a manner, that when a power applied at C moves the lever A C, it causes the lever D E to protrude or thrust along the weights at F, G, or H, in a circular manner upon the table. 10. Now since we suppose the weight, all the while it is in motion, is freely and wholly supported by the plane, it follows that all the resistance it can give to the power applied to C, is only what arises from its friction on the plane. What this friction is, will be found by applying the weight at G, so that B G be equal to A C; for then the power applied to C, acting in a tangent to the circle C R S, that shall just move the weight G, will be equal to its friction. But if the weight be applied at F, because B F is greater than A C, the same power at C, as before, will not move it, by reason its force is here increased by having a greater velocity than the power; as, on the other hand, if placed at H, a less power at C shall move it, because of its having there less velocity than the power, as is evident from the properties of the lever. 11. Hence we understand, that though the weight of a machine remains the same; yet the friction may be diminished, by contriving that the parts on which it moves and rubs, shall have less velocity than the power which moves it: thus, if the cylinder A B (fig. 6.) were to move on the two small pins or gudgeons E, F, the friction would be abated in the proportion of the diameter of the cylinder to that of the pins. 12. The friction on these gudgeons is still farther diminished by causing them to move on the circumference of a wheel: thus, let F be the gudgeon of the cylinder, revolving on the wheel C D E (fig. 9.), the velocity of the wheel's circumference will be the same with that of the gudgeon; but the velocity of the wheel's axis A B (which is now to be considered as the rubbing part) is less than that of the wheel, in proportion as its diameter is less than that of the wheel: for example, if the friction of the cylinder moving on its surface be \( \frac{1}{4} \) part of the weight, and the gudgeon be to the cylinder as 1 : 10, they will reduce the friction to \( \frac{1}{40} \) part; and if, again, the axis of the wheel be to the wheel as 1 : 10, the wheel will reduce the friction to \( \frac{1}{400} \) part; and if the axis of this wheel be laid on the perimeter of another wheel, the friction will be reduced to a still lesser part of the weight; so that you may proceed in this manner to diminish the friction ad infinitum; and wheels applied in this manner are called friction-wheels. 13. Besides what has been already said, somewhat farther is necessary to diminish the friction of wheel-carriages. It was before observed, that friction arose chiefly by lifting the body over the prominent parts of the plane on which it is moved: now if we can contrive to move the body along without lifting or sustaining its weight, we shall move it without much friction; and this may be done by laying the body on any moveable circular subject, as rollers, wheels, &c.: thus let A B (fig. 10.) be the section of any heavy body, laid on a roller E F, upon the plane C D, and drawn by the power P; it is evident, when A B moves, the alperties of its surface will lay hold on those of the roller, and move it likewise; and it is as evident, that when the body A B is drawn against the prominent parts of the roller, they immediately give way, and make no resistance: thus the perpendicular diameter a b yields into the situation e f; and c d succeeds in its place. By this circular motion of the roller, its prominent parts below do only descend and move upon or over, and are not drawn against, the fixed prominent parts of the plane, and to receive no resistance from them. Hence the body A B is conveyed along without being lifted up, in the same manner as a wheel is moved by a pinion without any considerable resistance.

**Chap. IV. Of Man, considered as an artificial Machine.**

Man has been considered by anatomists as a system of all the artificial machines united in the human fabric; they have found the lever, the pulley, the axle in the wheel, the wedge, and even the screw, or at least something resembling each of them, in his person: thus, his arms have been likened to levers; his head, turning upon its axle; the digestive muscle that assists his swallowing, to a rope running over its pulley; the glands, as lifting up their fluids in the manner of an artificial water-cistern; and his teeth have been compared to wedges. But some have not stopped here: they have gone on not only to please themselves with the resemblance, but to estimate the force of man through all his vital and involuntary motions, such as the running of the blood through his veins, the drawing his breath, and such like, by the inflexible laws of mechanism. They have even applied geometrical rules to measure objects constantly in change, and built theories upon proportions they were unable to discover. Thus, when Borelli once got the hint of comparing the muscles or fleshly parts to cords, he then readily built this theory, and calculated the human force by considering the thickness of the cords, and the length of the lever. Thus, when another found the similitude between the blood running thro' its channels, and water spouting through pipes, he pursued the speculation, till he at last was taught to believe that vomits would cure a spitting of blood, and bathing in warm water would be a remedy for the dropsy; happy, however, had his theory never been put into practice.

It is as impossible to determine the muscular force of any man by the bare inspection or measurement of his muscles, as it is to measure the swiftness of the circulation of his fluids by the spouting of his blood from a vein. Neither can be done, though Cheyne has pretended to demonstrate, that if we compare the muscular strength of two animals, that animal whose fluids circulate twice as swift will be six times as strong. Freind and Wainwright adopted his demonstration, for he called it a demonstration; and indeed it was drawn up with a sufficient degree of mathematical parade. Mattin, however, in a treatise entitled *De similitudine animalium*, has demonstrated that Cheyne's demonstration was false; but it was in order to establish another demonstration of his own. He asserted, that the force in similar animals was as the cube roots of the fourth powers of the limb put into motion. The learner will not perhaps understand the precise meaning of these words; but it is no matter, for his demonstration is as false as the former.

From the mere dimensions of the muscles in two similar animals, it is impossible to determine their force. The strength of the muscle is generally more in proportion to the exercise it has been employed in, than to its size; the legs of a chairman are stronger; the arms of a smith: in short, to use the words of a bully in a Spanish comedy, who mistook his man and was beaten, we can never know the strength of the muscles till we experience their effects.

But, though we cannot determine, with any precision, of two men which are strongest; yet, in the same man, we can compare the force of his muscles with rather more precision: this at least can be said with great certainty, that those muscles which are inserted into the bone, nearest to the place where it moves upon another, overcome the greatest resistance, and consequently act with the greatest force. But to a learner this wants explanation.

All our flesh is composed of muscles, which (if we may use a vulgar similitude) are like red ribbands, and almost all have one of their ends fixed into one bone, and another of their ends into some other bone. Thus, if we feel the great ham-string, which is made up of many muscles, we shall find that at one end it is fixed into the bones of the leg, just under the knee, and at the other end it runs upwards, partly to be fixed in the great bone of the thigh. The muscles being thus stretched from one bone to another, have a wonderful power of contracting and shortening themselves at pleasure; and when we choose to put them into action, they swell in the middle, somewhat into the shape of a nine-pin. As these muscles thus contract, they must necessarily draw the two bones, into which they are inserted, their own way: the hamstring, when it contracts, for instance, draws the leg backward toward the thigh; when we want to make the limb straight, there are muscles inserted under the fore-part of the knee, that, contracting, answer this purpose; while, in the mean time, the ham-string suffers itself to be relaxed, in order to let the opposing muscles take effect. This being understood, it will follow, that if we consider any one of the bones, the arm-bone for instance, as a beam, and the muscles that raise it and put it into motion, as the power that agitates and works the instrument, the whole will give us the idea of the third kind of lever, where the prop is at one end, the weight to be sustained at the other, and the strength is applied between them both. Thus, for instance, if you stretch out your arm, the prop is in the joint of the shoulder, the weight is the hand, and the raising power is the muscles, which are fixed into the arm-bone near the shoulder, and go from thence to be inserted into the bones of the trunk of the body. Now the nearer the shoulder these muscles are inserted into the arm-bone, it is evident that the longer will be the lever against which they are to set, and consequently the greater will appear the weight which they are to sustain. To make this quite plain: Suppose a ladder were laid flat on the ground; and suppose that a person, standing at one end, take the nearest round of the ladder in both his hands, and thus, pulling back, attempt to raise the farthest end, keeping the nearest end still steady to the ground. Would not this require immense strength to effect? Pretty similar is the force that the muscles of the arm exert in raising the whole length of the arm, and the weight of the hand beside. They are inserted into the bone close to the shoulder, and support the whole length of the arm in the desired direction. But what is more, they do not only act upon the lever at so disadvantageous a distance, but also they act upon it in a direction the most oblique, and consequently at a greater disadvantage. disadvantage still. Suppose one attempt to raise the distant end of the ladder by pulling the round nearest him; this, as was said, will be very disadvantageous; but suppose yet farther, that he should first lie upon his back, and then, by drawing the next round to him of the ladder, he should attempt to raise the distant end; the force that would be capable of effecting this, would be incredible. Yet in this very manner it is that the muscles of the shoulder act, in raising the arm. They are not only inserted at the greatest distance from the weight, but they exert their power the most obliquely. The force they exert in keeping the hand and arm extended is great; the force they exert in keeping it extended, while the hand holds a weight of about 20 pounds, is astonishing. Some say that these muscles, upon equal terms, would lift a weight 10,000 times greater. What has been here said of the muscles of the arm is true, in a greater or less degree, of all the muscles in the body; so that this natural machine, thus fashioned by the Great Workman, is infinitely more powerful than any artificial machine that man could form, though it took up four times the space.

The muscles, as we said, are supported by bones; these make altogether a single pillar or column, which, though not perfectly straight, but with about five different curvatures or bendings; yet, when perfectly balanced upon itself, will actually support weights that would surprise the inexperienced. La Hire and Desaguliers give us several accounts of the amazing weight some people have sustained, when they were able to fix the pillar of their bones directly beneath it. The latter tells of a German who showed several feats of this kind at London, and who performed before the king and a part of the royal family. This man, being placed in a proper situation, with a belt which rolled upon his head and shoulders, and which was fixed below to a cannon of 4000 weight, had the props which supported the cannon taken away, and by fixing the pillar of his bones immovably against the weight, supported it with seeming unconcern. There are few that have not seen those men, who, catching a horse by the tail, and placing themselves in direct opposition to the animal's motion, have thus stopped the horse, though whipped by his rider to proceed. In all such cases, the pillar of the bones is placed in direct opposition to the weight; they support each other, and are prevented from rubbing or cracking by elastic gristles fixed between each bone; these give way a little upon great pressure, and restore themselves almost instantly when that is removed. Besides these, there is a viscid or slimy liquor that is squeezed in, as if from a sponge, between every joint, and keeps these gristles smooth, moist, and pliant. By means of this fluid all the joints move easily, and obey the impulse of the muscles with greater dispatch. This fluid, and the gristles (or cartilages, as anatomists call them), contribute not a little to the strength of the animal; they resist the burden with an elastic force, and conform themselves to the inequality of the pressure. In old age both are diminished, the gristles become hard, and this liquor (which anatomists call the synovia) is squeezed out in less quantities. The man therefore, in old age, becomes more stiff and more weak, chiefly upon this account; though partly because his muscles become then also more rigid, hard, and less fleshy, as it is usually called; as those who have eaten the flesh of old animals know. While we are at rest, this fluid or synovia above-mentioned oozes out between the joints, to fit them for the hour of action; when in exercise, the ends of the bones press against their gristles, and these are separated in some measure by the synovia or fluid; but there is still another liquor of an oily nature, which is pressed at the same time from a small fleshy sponge, placed in every joint; and this mixing with the synovia, makes all supple and fit for burdens. It was said, that the synovia or viscid liquor oozes out between the joints in the hour of rest; it is therefore in greatest quantity between them in the morning, after we have taken our rest the preceding night. So great is the quantity usually separated during sleep between the joints of the back-bone, that some men are an inch taller in the morning than at night; and all men are somewhat taller, as may be quickly found by any who choose to make the experiment upon themselves.

From what has been said it appears, that, in carrying large burdens, the whole art consists in keeping the column of the body as directly under the weight as possible, and the body as upright under the weight as we can. For if the centre of gravity in the burden falls without this column, it will go near to fall; in fact, if the supporter were an inanimate machine, it would fall inevitably; but human power, in some measure, catches the centre while yet beginning to descend, and restores the balance which it had lost the moment before. A man balancing under a weight, resembles one of those people whom we usually see walking upon a wire: they totter from side to side, for a moment lose the centre of gravity; but by throwing toward a limb, or distorting their bodies, they recover it again, to the great amusement of every spectator. It is thus that he who carries a weight is obliged to act; on whatever part of his body the weight is placed, he balances it by throwing as much of his column beneath the load as he can. Could the weight be laid and evenly balanced upon him, standing in his natural posture, he could, as we observed before, support an incredible burden; and though he could not move under what he could thus support, yet he could carry a much greater load than if the burden were laid in any other manner. The weight a man could support, when thus evenly laid upon his shoulders, would break the back of the strongest horse in the world. The reason is obvious. In a man, the whole column of bones supports the weight directly; in an horse, the weight is laid upon the column crosswise. The porters of Constantinople are known to carry each a weight of 900 pounds; they lean upon a staff while loaded, and are unloaded in the same manner. The porters of Marseille in France are found to carry yet more; their manner is this: four of them carry the burden between them, each having a sort of hood that covers the temples and head down to the shoulders; to this are fastened the cords that support the frame or bier, on which the weight is laid. By this contrivance the whole column of the bones acts directly against the load, and an immense weight is thus sustained.

We now therefore at length see the reason why two men carrying a load between them, can sustain a greater weight than what either could separately carry, if it were divided into two equal parts. The reason is, that two men can bear the load each more upright, and with the column of their bones more opposed against it.

As man bears a weight the better the more upright he stands against it, it must follow necessarily, that the more bendings he makes in supporting weights, the less will be his power. There are three principal bendings in the human column; the first at the hams, the second at the hips, and the third along the backbone, which resembles the other in pliancy, though it be stronger than the osk. A man of ordinary stature and strength, upon an average, has been computed to weigh 160 pounds; he can support, as we said before, an immense weight if his column acts directly against it; if he bends a little at the hams, such a man may raise from the ground about 170 pounds, provided the weights are placed to the greatest advantage. If he bends at the hips and back, he will lift 30 pounds less. If a weight be placed upon his head, and he be put between the rounds of a ladder placed horizontally and breast-high, he can lift 30 pounds by the strength of the muscles of his shoulders and neck alone.

From this we see, that human strength is not the fourth part as great when the body is bent, as when it is upright. From this also we see, that if a man draws a load after him, as in that case all his muscles act in an oblique direction, he can exert but very little force, when compared to other animals. Defagulier pretends to say, that an horse can draw as much, upon an average, as five English workmen. The French writers say, Dr Barthes in particular, that an horse can draw as much as six Frenchmen, or seven Dutchmen; but if the load were to be placed upon the shoulders, two men will be found to be as strong as an horse. A London porter should carry 300 weight at the rate of three miles an hour; two chairmen carry 150 pounds each, and walk at the rate of four miles an hour: Whereas a travelling horse seldom carries above 200 weight; and a day's journey with such a load, would be apt to disqualify him from travelling the day following.

Man's greatest force, therefore, is directly upward; if he draws a load, he must act at a disadvantage. A man, however, when obliged to draw a load, a rolling stone for instance, hath two methods of doing this. He may either turn his back to the stone, and pushing the frame with his breast, thus go onward, while the stone rolls after; or he may turn his face to the stone, and go backward, drawing the stone with him. This last method may be the most inconvenient, but it gives the workman much the greatest share of power, and that for two reasons. In the first place, by inclining farther back, he can give a greater column of his body to the draught; and in the next place, a greater number of his muscles come into action; particularly the two great deltoid muscles of the arms, the force of which is very great. It is for this reason that men who row a boat, more usually draw the oar to them, than push it from them.

CHAP. V. Of Wheel-Carriages.

By what we have seen of man considered as a machine, it is easy to observe that his frame is not adapted to drawing carriages; while, on the contrary, in that of an animal upon all fours, the column of whose bodies, and the situation of whose muscles, act almost directly upon bodies placed behind them, they are perfectly fitted by nature for this kind of service. Horses are usually employed in the draught in Britain; mules, oxen, sheep, and other animals, are sometimes used in other parts of the world. It might incur ridicule if we pretended to inform the learner that each of these will draw a weight or carriage in proportion as they are strong. But notwithstanding this is generally the case, yet we are going to mention what will seem a paradox; namely, that two horses may be found, one stronger than the other, and also better skilled in the draught, yet the weaker shall draw a weight with the very same carriage the stronger one could not remove! This will be effected, if the weakest horse be the heaviest; if he exceeds his antagonist more in weight than he is exceeded in strength. It is known, that the weight reacts or pulls back the horse, as much as the horse acts upon the weight to pull it forward. Now the horse has two sources of power in drawing the weight along: his strength, which gives him velocity; and his weight, which added gives force; and it is evident that the horse which hath both in the greatest proportion, will draw the heaviest weights. If we should imagine both horses raising an equal weight from a deep pit, and this weight still increased, so as to overcome their strength, it is plain that the lightest horse would soonest be drawn in. We have several instances in ordinary practice, of the great benefit of increasing the horse's weight to promote his draught; for, in many places, horses employed in turning a mill have a small load laid upon their backs, which, though it takes away something from their velocity, adds to their weight, and consequently increases their force.

But supposing the strength, skill, and weight, of two horses to be the same, all the difference then in their drawing the same weights, will arise from the commodiousness of the machine in which they draw. If the load they are to drag after them be breast-high, they can draw it with much greater ease than if it lay along the ground. They can, for instance, draw much greater draughts, if the weights are laid upon a sledge as high as the horse's shoulders, than if the same weights were laid upon a low sledge on the ground. For, in the first case, the column of their bodies acts directly against the weight; in the latter, it acts obliquely; and we have shown before, that the more directly this column can act, the greater is its force. Even in either going up-hill or down-hill, the sledge breast-high is more commodious than that laid low. For if the low sledge is dragged up an hill, it is plain, that it will be then lower, with respect to the horse, than it was before, and consequently they will be obliged to draw it more obliquely upwards than when they drew it along the plain. If, on the contrary, the low sledge is drawn down an hill, it will then be higher with respect to the horse than when on the plain, and therefore their power of drawing it will be greater; but, in going down an hill, its own gravity confines with the draught, and will also help the load to descend, so that the horses in this case are permitted to exert their greatest power where there is the least necessity; they can draw the low sledge down-hill with with all their power, when, by the natural descending of the load, they are not permitted to exert it. This doctrine, however, simple as it is, is different from what is usually taught by merchants upon this subject.

Sledges were probably the first machines used in carrying loads; we find them thus employed in Homer, we mean in the original, in conveying wood for the funeral pile of Patroclus. There are some countries also that preserve their use to this day. However, men early began to find how much more easily a machine could be drawn upon a rough road, that run upon wheels, than one that thus went with a sliding motion. And indeed, if all surfaces were smooth and even, bodies could be drawn with as much ease upon a sledge as upon wheels; and in Holland, Lapland, and other countries, they use sledges upon the smooth surface of the ice; for as every surface upon which we travel is usually rough, wheels have been made use of, which rub less against the inequalities than sledges would do. In fact, wheels would not turn at all upon ice, if it were perfectly smooth, for the cause of the wheels turning upon a common road is the obstacles it continually meets. For if we suppose the wheels to be lifted from the ground, and carried along in the air, the wheels in this case would not turn at all, for there would be nothing to put any part into motion rather than another; in the same manner, if they were carried along upon perfectly smooth ice, they would meet nothing to give a beginning to the circulatory motion, and all their parts would rest equally alike.

But if we suppose the wheel drawn along a common road, then the parts will receive unequal obstructions, for it meets with obstacles that retard it at bottom; therefore the upper part of the wheel, which is not retarded, will move more swiftly than the lower part, which is; but this it cannot do, unless the wheel moves round. And thus it is that the obstructions in the rough road cause this circulatory motion in the wheel.

This rotation of the wheels about their axle very much diminishes that friction which always attends the weight's being drawn along upon a sledge; and this in so great a proportion, that, according to Hellsham, a carriage drawn by four wheels, will be drawn with five times as small an effort as one that slides upon the same surface as a sledge. Still more to diminish the friction in wheel-carriages, an expedient hath been found out, whereby the axle, contrary to what is usual in most carriages, is made to turn round, and its gudgeons or ends, instead of pressing against the boxes as in common wheels, are made to bear on the circumference of moveable wheels; so that by this contrivance, a number of parts are made to roll one over the other, which glided before; such wheels, from their thus diminishing the friction, are called friction-wheels.

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; and make an observation or two upon the advantages of the use of broad wheels.

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 proportionally 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 greater 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 of the waggon A.B 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 uphill, 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

Chap. V.

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 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 eight parts of an inch deep in the board; then put a weight of 32 ounces into the carriage over the fore-axle, and an equal weight over the hind-one: 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 ounce 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 sustains 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 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 are always to go upon smooth and level ground, the best way would be to make the spokes perpendicular to the axes; 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 axes 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 easier, because they will be at liberty to go on straight forwards. But, in the usual way of practice, the axles are bent downward. ward 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 be the great wheels parallel to each other, on their straight axle K, and HCl 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; and the line of direction C#D falling within the wheel BF, the carriage will not overturn. But if the wheels be inclined to each other on the ground, as AE and BF are, and the machine be loaded as before, from C to G, the line of direction C#D 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 C#D falls within the wheels; but if it be loaded high with lighter goods (such as wool-packs) from C to L, the centre of gravity is raised from C to K, which throws the line of direction K#E 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 breast 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.

By means of wheels, some people have contrived carriages to go without horses, or any other moving power than what was given by the passengers, by the wind, &c. One of these is represented by ABCD. It is moved by the footman behind it; and the fore-wheels, which act as a rudder, are guided by the person who sits in the carriage (A).

Between the hind-wheels is placed a box, in which is concealed the machinery that moves the carriage. AA, (fig. 2.) is a small axis fixed into the box. B is a pulley, over which runs a rope, whose two ends are fastened to the ends of the two levers or treddles CD, whose other ends are fixed in such manner in the piece E, which is joined to the box, that they can easily move up and down. F, F, are two flat pieces of iron that are joined to the treddles, and take the teeth of the two wheels H, H, which are fixed on the same axis with the hind-wheels of the carriage, I, I.

It is evident, that when the footman behind presses down one of the treddles, suppose C, with his foot, he must bring down one of the pieces of iron F, and consequently turn the wheel H that is next to it; and at the same time, by means of the rope that goes over the pulley, he must raise the other treddle D, together with its piece F, which being thrust down will turn the other wheel H; and so alternately: and as the great wheels are fixed on the same axis, they must necessarily move at the same time.

It is easy to conceive, that if the ends of the treddles next E, instead of being placed behind the carriage, were turned the opposite way, so as to come under the feet of the person who sits in it, he might move it with equal, or even greater facility, than the footman, as it would then be charged with the weight of one person only.

A machine of this kind will afford a salutary recreation in a garden or park, or on any plain ground; but in a rough or deep road must be attended with more pain than pleasure.

Another contrivance for being carried without draught, is by means of a sailing chariot, or boat fixed fast, with four wheels, as AB; which is driven before the wind by the sails CD, and guided by the rudder E. In a chariot of this kind, the wheels should be farther apart, and the axle-trees longer, than in other carriages, to prevent overturning.

A machine of this sort was constructed in the last century by Stephanus, at Scheveling in Holland, and is celebrated by many writers. Its velocity with a strong wind is said to be so great, that it would carry eight or ten persons from Scheveling to Putten, which are 42 English miles distant, in two hours.

Carriages of this kind are said to be frequent in China; and in any wide, level country, must be, sometimes, both pleasant and profitable. The great inconvenience attending this machine is, that it can only go in the direction the wind blows, and even not then unless it blow strong; so that, after you have got some way on your journey, if the wind should fail, or change, you must either proceed on foot, or go back. Some remedy for this inconvenience will be found in the next contrivance. The Hollanders have, or had, small vessels, something of this kind, that carry one or two persons on the ice, having a sledge at bottom instead of wheels; and being made in the form of a boat, if the ice break the passengers are secured from drowning.

To sail against the wind: Let ABCD be the body of a sailing chariot; M the mast, to which are fixed land against the wings or sails EFGH; the two first of which, EF, are here supposed to be expanded by the wind; R is the rudder by which it is guided. Therefore, the wind driving the sails round, with the mast M, and the cog-wheel K, take the teeth, placed perpendicular to the sides of the two fore-wheels of the carriage, and consequently keep it in continual motion.

The body of this machine should not be large, nor placed very high, not only to prevent overturning, but that its motion may not be thereby impeded; for

(A) This machine was invented by M. Richard, a physician of Rochelle, and was exhibited at Paris in the last century. It is described by M. Ozanam in his Recreations Mathematiques. the velocity will be in proportion to the force of the wind on the sails, to that on the body of the machine. Therefore, if they be both equal, it will stand still; or if the force on the body be greatest, it will go backwards; unless there be a contrivance to lock the wheels. The upper part of the machine next A may be made to take off when the wind is contrary; and there may be another set of sails placed between the two hind-wheels, which will considerably increase its velocity. But after all, for general use, a common carriage must be preferable: for this cannot be expected to go up a moderate ascent without great difficulty; nor down a declivity, when there is a strong wind, without danger; and even on level ground, if the road be in any degree rough, its progress must be very slow, attended both with difficulty and danger. In an open country, however, where there is a large tract of level and smooth ground, and frequent strong winds, a machine of this sort will certainly be very convenient; and in most countries, when made of a small size, may be useful to young people, by affording them a pleasant and healthful exercise.

A carriage the body of which is incapable of being overturned may be made as follows. The body must consist of a regular hollow globe, as AB, at the bottom of which is to be an immovable weight, and which must be proportioned to the number of persons or the load the machine is intended to carry. Round the globe must go two horizontal iron circles D, E, and two others F, G, that are perpendicular to the former. All these circles must be made exactly to fit the globe, that it may move freely in every direction. The two horizontal circles are to be joined on each side by a perpendicular bar, one of which is expressed in the figure by HI. All these irons should be lined with leather, to prevent unnecessary friction. The body of the carriage may be either of leather or hard wood; but the latter will be most eligible, as least liable to wear. The wheel on each side is to be fastened to the perpendicular bar by means of a handle K, that keeps it steady.

Now, the body of this machine moving freely in the iron circles every way, the centre of gravity will always lie at C; therefore, in whatever position the wheels are, or even if they overturn, the body of the carriage will constantly remain in the same perpendicular direction.

At L is placed a pin, round which is a hollow moveable cylinder: this pin moves up and down in the groove MN, that it may not impede the perpendicular motion of the circles, at the same time that it prevents the body of the machine from turning round in a horizontal direction. O is one of the windows, P the door, and QR the shafts to this machine.

When a carriage of this sort is intended for a single person, or a light weight, it may be hung on pivots, in the same manner as the rolling lamp or the sea-compass, which will make its horizontal motion still more regular: and when it is designed to carry several persons, by adding another perpendicular bar on each side, between the two horizontal circles, it may be placed on four wheels. The body of this machine should be frequently oiled or greased, not only to prevent any disagreeable noise that may arise from its rubbing against the circles, but to prevent unnecessary wear in the several parts.

This carriage is not intended for smooth roads, or a regular pavement; there certainly, those of the common construction are much preferable; nor should a carriage totally free from irregular motion be sought after by those who are in perfect health: but there are many persons, subject to different disorders, who by being obliged to travel over rough roads in the common carriages, suffer tortures of which the healthful have no idea; to all these, therefore, and to every one who is forced to travel through dangerous roads, a carriage of this sort must doubtless be highly desirable.

As this design may appear to some persons, on a superficial view, impracticable, we shall here insert an account of a similar carriage, which we have taken from the first volume of the abridgment of the Philosophical Transactions, by Lowthorp. There is not, however, any description of the manner in which that machine was constructed. The account is as follows:

"A new sort of calash described by Sir R. B. This calash goes on two wheels; carries one person; is light enough. Though it hangs not on braces, yet it is easier than the common coach. A common coach will overturn if one wheel go on a superficies a foot and a half higher than the other; but this will admit of the difference of three feet and one-third in height of the superficies, without danger of overturning. We chose all the irregular banks, and sides of ditches, to run over; and I have this day seen it, at five several times, turn over and over, and the horse not at all disordered. If the horse should be in the least unruly, with the help of one pin you disengage him from the calash without any inconvenience (a contrivance of this sort may be easily added to the foregoing design.) I myself have been once overturned, and knew it not till I looked up and saw the wheel flat over my head: and if a man went with his eyes shut, he would imagine himself in the most smooth way, though at the same time there be three feet difference in the height of the ground of each wheel."

**Chap. VI. Of Mills and Cranes.**

In a common breast-mill, where the fall of water may be about ten feet, AA is the great wheel, which mill 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 thro' a channel; and by 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 flaves 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 brass 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 cross a b c d (see fig.). fig. 10.) 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 that it does the rynd.

The rynd is let into grooves in the under surface of the running millstone G (fig. 9.), 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 k k, 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 quite out; and, being then ground, it falls thro' 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 faster 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 brass 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 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 become blunt and shallow by wearing, the running stone must be taken up, and both stones new dressed 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 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 let 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 $22\frac{1}{2}$ 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 faster, it will heat 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 farther 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 thro' 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 A b 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, tho' 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 recollect what has been said 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 leaves, 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 trundles 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 in 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 leaves, and the wheel 61 cogs.

Sometimes, where there is a sufficient quantity of water, the cog-wheel AA turns a large trundle BB, Plate on whose axis C is fixed the horizontal wheel D, with CLXXI 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 flaves of its trundle, and turning the vacant place towards the cog-wheel D. And there may be a wheel fixed 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-wheel; 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 $54^\circ$ degrees with a plane perpendicular to the axis on which the arms are fixed 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 $54^\circ$ 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 10 or 15 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 farthest ends do, although the wind acts 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 fig. 2, or gibbet. When the rope H 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 fixed on the gib-polt cc, and the strong pinion e fixed 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 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 even 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 everything 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 into 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 tt (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 it 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 f; then the handle q is turned half round backwards, which, by the crank nn in the piece o, pulls down the frame k between the guides m and n (in which it slides in a groove) and so disengages the trundle B from the wheel C; and then, the heavy hook s 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 F and hook s, 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 k, 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 13 flaves or rounds, and the wheel C to have 78 spur-cogs; the trundle E to have 14 flaves, and the wheel F 56 cogs. Then, by multiplying the flaves 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 shows, that the winch A make 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 flaves 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 s, and the end of it tied to a hook in the gib at s, 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 raised, this may be quickly done by men pushing the axle G round by the handspokes j,j,j,j; 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 s.

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 handspokes j,j,j,j, whilst the man at A turns the winch.

We have only thrown 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 is a great wheel turned by men walking within it at H. On the part C, of its axle BC, Plate the great rope D is wound as the wheel turns; and CLXXXII this rope draws up goods in the same way as the rope fig. 3 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 flaves 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 T, 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 matters 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, wich, and some gentlemen at Bristol, there is scarce an instance of any who has used this safe contrivance.

**CHAP. VII. Of the Communication of Motion.**

Before we proceed to explain the laws by which bodies communicate their motion from one to another, it is very necessary to make a distinction between motion and velocity; which ought to be well observed, and is as follows.

By the motion of a body (sometimes called its quantity of motion, sometimes its momentum) is not to be understood the velocity only with which the body moves; but the sum of the motion of all its parts taken together: consequently the more matter any body contains, the greater will be its motion, though its velocity remains the same. Thus, supposing two bodies, one containing ten times the quantity of matter the other does, moving with equal velocity; the greater body is said to have ten times the motion, or momentum, that the other has: for it is evident, that a tenth part of the larger has as much as the other whole body. In short, that quality in moving bodies which philosophers understand by the term momentum or motion, is no other than what is vulgarly called their force, which every one knows to depend on their quantity of matter, as well as their velocity. This is that power a moving body has to affect another in all actions that arise from its motion, and is therefore a fundamental principle in mechanics.

Now, since this momentum, or force, depends equally on the quantity of matter a body contains, and on the velocity with which it moves; the method to determine how great it is, is to multiply one by the other. Thus, suppose two bodies, the first having twice the quantity of matter, and thrice the velocity, which the other has; any two numbers, that are to each other as two to one, will express their quantities of matter (it being only their relative velocities and quantities of matter which we need consider); and any two numbers that are as three to one, their velocities: now multiply the quantity of matter in the first, viz. two, by its velocity three, the product is six; and multiply the quantity of matter in the second by its velocity, viz. one, by one, the product is one; their relative forces therefore or powers will be as six to one, or the moment of one is six times greater than that of the other. Again, if their quantities of matter had been as three to eighty, and their velocities as two to three, then would their moments have been as six to twenty-four, that is, as one to four.

This being rightly apprehended, what follows, concerning the laws of the communication of motion by impulse and the mechanical powers, will be easily understood.

The Communication of Motion.

I. In Bodies not Elastic.

Those bodies are said to be not elastic, which, when they strike against one another, do not rebound, but accompany one another after impact, as if they were joined. This proceeds from their retaining the impression made upon their surfaces, after the impinging force ceases to act. For all rebounding is occasioned by a certain spring in the surfaces of bodies, whereby those parts, which receive the impression made by the stroke, immediately spring back, and throw off the impinging body; now, this being wanting in bodies void of elasticity, there follows no separation after impact.

When one body impinges on another which is at rest, or moving with less velocity the same way, the quantity of the motion or momentum in both bodies taken together remains the same after impact as before; for by the third law of nature, the re-action of one being equal to the action of the other, what one gains, the other must lose.

Thus, suppose two equal bodies, one impinging with 12 degrees of velocity on the other at rest: the quantities of matter in the bodies being equal, their moments and velocities are the same; the sum in both 12; this remains the same after impact, and is equally divided between them; they have therefore six a-piece; that is, the impinging body communicates half its velocity, and keeps half.

When two bodies impinge on each other by moving contrary ways, the quantity of motion they retain after impact is equal to the difference of the motion they had before: for by the third law of nature, that which had the least motion, will destroy an equal quantity in the other; after which they will move together with the remainder, that is the difference.

Thus for instance, let there be two equal bodies moving towards each other, the one with three degrees of velocity, the other with five, the difference of their moments or velocities will be two; this remains the same after impact, and is equally divided between them, they have therefore one a-piece: that is, the body which had five degrees of velocity, loses three, or as much as the other had; communicates half the remainder, and keeps the other half.

From these positions it is easy to reduce a theorem, that shall shew the velocity of bodies after impact in all cases whatever. Let there be two bodies A and B, the velocity of the first a, of the other b; then the moment of A will be expressed by Aa, and of B by Bb; therefore the sum of both will be Aa+Bb; and Aa—Bb will be the difference when they meet. Now these quantities remain the same after impact; but knowing the quantities of motion and quantities of matter, we have the velocity by dividing the former by the latter: therefore $\frac{Aa+Bb}{A+B}$ or $\frac{Aa-Bb}{A+B}$ will in all cases express the velocity of the bodies after impact.

II. In Elastic Bodies.

Bodies perfectly elastic are such as rebound, after elastic impact, with a force equal to that with which they impinge upon one another; those parts of their surfaces, that receive the impression, immediately springing back, and throwing off the impinging bodies with a force equal to that of impact.

From hence it follows, that the action of elastic bodies on each other (that of the spring being equal to that of the stroke) is twice as much as the same in bodies void of elasticity. Therefore, when elastic bodies impinge on each other, the one loses and the other gains twice as much motion as if they had not been elastic; we have therefore an easy way of determining the change of motion in elastic bodies, knowing first what it would have been in the same circumstances, had the bodies been void of elasticity.

Thus, if there be two equal and elastic bodies, the one one in motion with 12 degrees of velocity impinging on the other at rest; the impinging body will communicate twice as much velocity as if it had not been elastic, that is, 12 degrees, or all it had; consequently it will be at rest, and the other will move on with the whole velocity of the former.

It sometimes happens, that in bodies not elastic, the one loses more than half its velocity, in which case, supposing them elastic, it loses more than all; that is, the excess of what it loses, above what it has, is negative, or in a contrary direction. Thus, suppose the circumstances of impact such, that a body, which has but 12 degrees of velocity, loses 16: the overplus four is to be taken the contrary way; that is, the body will rebound with four degrees of velocity. e.g. Let it be required to determine the velocity of a body after impact against an immovable object. Let us first suppose the object and body both void of elasticity: it is evident the impinging body would be stopped or lose all its motion, and communicate none; if they are elastic, it must lose twice as much, and consequently will rebound with a force equal to that of the stroke.

It is sufficient if only one of the bodies is elastic, provided the other be infinitely hard; for then the impression in the elastic body will be double of what it would have been had they both been equally elastic: and consequently the force with which they rebound, will be the same as if the impression had been equally divided between the two bodies.

There are no bodies, that we know of, either perfectly elastic, or infinitely hard: the nearer therefore any bodies approach to perfection of elasticity, so much the nearer do the laws, which they observe in the mutual communication of their motion, approach to those we have laid down.

Sir Isaac Newton made trials with several bodies, and found that the same degree of elasticity always appeared in the same bodies, with whatever force they were struck, so that the elastic power, in all the bodies he made trial upon, exerted itself in one constant proportion to the compressing force. He found the celerity with which balls of wool, bound up very compact, receded from each other, to bear nearly the proportion of five to nine to the celerity wherewith they met; and in steel, he found nearly the same proportion: in cork the elasticity was something less, but in glass much greater; for the celerity, with which balls of that material separated after percussion, he found to bear the proportion of 15 to 16 to the celerity wherewith they met.

We have hitherto supposed the direction, in which bodies impinge upon one another, to be perpendicular to their surfaces: when it is not so, the force of impact will be less, by how much the more that direction varies from the perpendicular; for it is manifest, that a direct impulse is the greatest of all others that can be given with the same degree of velocity.

The force of oblique percussion is to that of direct, as the sine of the angle of incidence to the radius.

Dem. Let there be a plane, as AD, against which let a body impinge in the point D in the direction BD: which line may be supposed to express the force of direct impulse, and may be resolved into two others BC and BA; the one parallel, the other perpendicular to the plane; but that force which is exerted in a direction parallel to the plane can no way affect it: the stroke therefore arises wholly from the other force expressed by the line BA; but this is to the line BD, as the sine of the angle of incidence ADB to the radius; from whence the proposition is clear.

If the surface of the body to be struck is a curve, then let AD be made a tangent to D the point of incidence, and the demonstration will be the same.

This is the case when bodies impel one another by acting upon their surfaces; but in forces, where the surfaces of bodies are not concerned, as in attraction, &c. we must not consider the relation which the direction of the force has to the surface of the body to be moved, but to the direction in which it is to be moved by that force. Here the force of action will be less, by how much the more these two directions vary from each other.

The force of oblique action is to that of direct, as the cosine of the angle comprehended between the direction of the force, and that wherein a body is to be moved thereby, to the radius.

Dem. Let FD represent a force acting up Fig. II. on a body as D, and impelling it towards E; but let DM be the only way in which it is possible for the body to move: the force FD may be resolved into two others FG and FH, or which is equal to GD; but it is evident that only the force GD impels it towards M. Now, FD being the radius, GD is the cosine of the angle FDG comprehended between the two directions FE and GM; from whence the proposition is clear.

The meaning in both cases will be understood from the instance of a ship under sail. The force by which the wind acts upon the sail, will be less, by how much the more the direction of the wind varies from one that is perpendicular to the surface of the sail: but the force of the sail, to move the ship forward, will be less, by how much the more the direction of the ship's course varies from that in which she is impelled by the sail.

To this we may add the following proposition relating to oblique forces, viz. that if a body is drawn or impelled three different ways at the same time by as many forces acting in different directions; and if the quantity of those forces is such, that the body is kept in its place by them: then will the forces be to each other, as the several sides of a triangle drawn respectively parallel to the directions in which they act.

Dem. Let the lines AB, AD, AE, represent the three forces acting upon the body A in those directions, and by that means keeping it at rest in the point A. Then the forces EA and DA will be equivalent to BA, otherwise the body would be put into motion by them. But these forces are also equivalent to AC, consequently AC may be made use of to express the force AB; and EC, which is parallel and equal to AD, may express the force AD, while AE expresses its own: but ACE is a triangle whose sides are all parallel to the given directions; therefore the sides of this triangle will express the relation of the forces by which the body is kept at rest. Q.E.D. The simple mechanic powers are usually reckoned fix; the lever, axle and wheel, pulley, wedge, inclined plane, and forew. I shall consider these machines separately, and explain the nature and property of each of them, by shewing from its structure what weight it will enable any given force to sustain.

The lever is considered as an inflexible line, void of weight, and moveable about a fixed point called its fulcrum or prop. The property of the lever, expressed in the most general terms, is this: "When two weights, or any two forces, act against each other on the arms of a lever, and are in equilibrium, they will be to each other inversely as the perpendicular or shortest distances of their lines of direction from the fulcrum."

This proposition contains two cases; for the directions of the forces may either meet in a point, or be parallel to each other. Most writers begin their demonstration of this proposition with the second case, which seems to be the simplest, and from which the other may be deduced by the resolution of forces. Archimedes, in his demonstration, sets out with a supposition, the truth of which may reasonably be doubted; for he supposes, that if a number of equal weights be suspended from the arm of a lever, and at points equidistant from each other, whether all these points be at the same side of the fulcrum, or some of them on the opposite side, these weights will have the same force to turn the lever as they would have were they all united and suspended from a point which lies in the middle between all the points of suspension, and may be considered as the common centre of gravity of all the separate weights. Mr Huygens, in his Miscellaneous observations on mechanics, says, that some mathematicians have endeavoured, by altering the form of this demonstration, to render its defects less sensible; though without success. He therefore proposed another proof, which is extremely tedious and prolix, and also depends on a postulate, that, I think, ought not to be granted on this occasion; it is this: "When two equal bodies are placed on the arms of a lever, that which is further from the fulcrum will prevail and raise the other up." Now, this is taking it for granted, in other words, that a small weight placed further from the fulcrum, will sustain or raise a greater one. The cause and reason of which fact must be derived from the demonstration that follows, and therefore this demonstration ought not to be founded on the supposed self-evidence of what is partly the thing to be proved.

Sir Isaac Newton's demonstration of this proposition is indeed very concise; but it depends on this supposition, that when from the fulcrum of a lever several arms or radii issue out in different directions, all lying in the same vertical plane, a given weight will have the same power to turn the lever from whichever arm it hangs, provided the distance of its line of direction from the fulcrum remains the same. Now it must appear difficult to admit this supposition, when we consider that the weight can exert its whole force to turn the lever only on that arm which is the shortest, and is parallel to the horizon, and on which it acts perpendicularly; and that the forces which it exerts, or with which it acts perpendicularly, on any one of the oblique arms, must be inversely as the length of that arm, which is evident from the resolution of forces.

Mr Maclaurin, in his View of Newton's philosophy, after giving us the methods by which Archimedes and Newton prove the property of the lever, proposes one of his own, which, he says, appears to be the most natural one for this purpose. From equal bodies sustaining each other at equal distances from the fulcrum, he shows us how to infer that a body of one pound (for instance) will sustain another of two pounds at half its distance from the fulcrum; and from thence that it will sustain one of three pounds at a third part of its distance from the fulcrum: and going on thus, he deduces, by a kind of induction, what the proportion is in general between two bodies that sustain each other on the arms of a lever. But this argument, were it otherwise satisfactory, yet as it cannot be applied when the arms of the lever are incommensurable, it cannot conclude generally, and therefore is imperfect.

There are some writers on mechanics, who, from the composition of forces, demonstrate that case of the general proposition relating to the lever, in which the directions of the forces are oblique to each other, and meet in a point: but I do not find that they have had any other way of proving the second case, in which the directions of the forces are parallel, but by considering these directions as making an angle with each other, though an infinitely small one, or as meeting at an infinite distance; which way of reasoning is not to be admitted in subjects of this kind, where the proof should always shew us, directly from the laws of motion, why the conclusion must be true, in such manner that we might see clearly the force of every step from the first principles down to the conclusion, which we are prevented from doing when any such arbitrary and inconsistent supposition is introduced.

From thus considering the various proofs that have been given of this fundamental proposition in mechanics, we may see the reason why many subsequent writers have appeared dissatisfied with the former demonstrations, and have looked for new ones: I shall now propose two methods of demonstrating it, merely from the composition and resolution of forces. The proposition may be expressed as follows:

"When three forces act upon an inflexible line, whether straight or crooked, and keep it in equilibrium, any two of them will be to each other inversely as the perpendicular distances of their lines of direction from that point to which the third force is applied."

Let the three forces E, G, F, (fig. 2.) act upon three points A, B, D, in an inflexible line; and first let the directions of the forces E and F (which act on the same side of the line) meet in the point C. Then it is evident that the force, which is compounded of these two, must act upon the line A B D, in the direction of a right line that passeth through the point C; consequently the force G, which sustains this compounded force, must be equal thereto, and must act in a contrary direction; therefore the force G must act in the direction of the line C B. From the point B draw B H and B K perpendicular to the directions of the forces E and F, and draw B M and B N parallel to these directions, forming the parallelogram B M C N; then, since these three forces are in equilibrium, they must be to each other respectively as the sides and diagonal of this parallelogram to which their directions are parallel; therefore E is to F as C M to C N or M B, that is, (because the sides of a triangle are as the sines of the opposite angles) as the sine of the angle M B C, or its alternate one B C N, to the sine of the angle B C M; but making C B the radius, B K is the sine of the former angle, and B H of the latter; therefore E is to F as B K to B H; so that the forces E and F are to each other inversely as the perpendicular distances of their lines of direction from the point B, on which the third force G acts. Now to compare the forces F and G together: From the point A, on which the third force acts, draw A B and A L perpendicular to the directions of the forces G and F; then, as was said before, F is to G as M B is to C B; but M B is to C B as A B to A L; because, making C A the radius, A B is the sine of the angle M C B, and A L is the sine of the angle M C N, or C M B its supplement, to two right ones; therefore the forces F and G are to each other inversely as the perpendicular distances of their lines of direction from the point A, on which the third force E acts; and thus the first case of the proposition is proved, in which the forces act against each other in oblique directions.

We must now consider what parts of the forces E and F act against the force G in directions parallel to G C; for it is such parts only that really oppose the force G, and keep it in equilibrium; and from thence we shall see what proportion two forces must have to each other when they are in equilibrium, and act in parallel directions. Let the three forces act upon the points A, B, and D, (fig. 2.) let them be in equilibrium, and Fig. 2., their lines of direction meet in the point C, as in the preceding case: then if the points A, B, and D, are not in a right line, draw the line A D meeting B C in P, and from P draw P N and P M parallel to the directions of the forces E and F; through the points A and D draw lines parallel to B C; and through B draw a perpendicular to these lines, meeting them in H and K; from the point M draw M O parallel to A D, and meeting B C in O. Now the three forces E, G, and F, that are in equilibrium, will be to each other respectively as the sides of the triangle C M P, as in the preceding case; but the force E, which is denoted by the line M C, may be resolved into two forces acting in the directions M O and O C, the former of these only urges the point A towards D, and the latter acts in direct opposition to the force G; in like manner the force F, which is denoted by the line P M, may be resolved into two forces acting in the directions O M and O P, the former of which only urges the point D towards A, and the latter acts in direct opposition to the force G; now it is evident that the force G, which is denoted by the line P C, is sustained only by those parts of the forces E and F which act against it, in directions parallel to B C, and are denoted by the lines O C and O P, which, taken together, are equal to P C; for the other parts of the forces E and F which are denoted by M O, are lost, being equal, and contrary to each other: if, therefore, instead of the forces F and E, we suppose two other forces, R and L, to act on the points... D and A, in directions parallel to BC, and to keep the force G in equilibrium, it follows, from what has been proved, that R and L taken together will be equal to G, and that these three forces will be to each other respectively as the lines PO, OC, and PC; therefore R will be to L as (PO to OC, that is, as AM to MC, or as AP to PD, or) HB to BK, consequently the forces R and L are to each other inversely as the perpendicular distances of their lines of direction from the point B, to which the third force is applied. Now to compare the forces R and G together; since the forces R and L may be denoted by BH and BK, and are both together equal to G, that force will be denoted by the whole line KH, and therefore R will be to G as BH to KH; so that these forces are also to each other inversely as the perpendicular distances of their lines of direction from the line of direction of the third force L; and thus the second case of the proposition is proved, in which the forces act against each other in parallel directions. If the point in the inflexible line, to which one of the forces is applied, should become a fixed point, or fulcrum, round which the line may turn, it is evident that the other two forces will continue in equilibrium, as they were before; and therefore the property of the lever, in all cases, is manifestly proved by this proposition.

The centre of gravity of a body is said to be that point which being sustained, or prevented from descending, the body will continue at rest. From hence it follows, that when a body hangs freely from a single point and continues at rest, its centre of gravity will lie perpendicularly under the point of suspension; for in that situation only it will be sustained, and can descend no lower.

From this property, which agrees likewise to the common centre of gravity of two bodies joined together by an inflexible right line, and which may then be considered as one, I shall shew that their centre of gravity is a point in the line that joins them together, situated that the distances of the two bodies from it are to each other inversely as their weights. This theorem concerning the position of the common centre of gravity of two bodies, which is a very noted one in mechanics, I have never seen demonstrated otherwise than by inferring it from the general property of the lever; but I think the method I shall now propose of deducing it directly from the definition of the centre of gravity, is the most concise as well as the most natural, and besides it will afford us a very easy way of demonstrating the property of the lever.

Let the two bodies A and B be joined by an inflexible right line passing through their centres of gravity, and let them be suspended from the fixed point or pin at P, by the threads AP and BP, so that they may hang freely in such a position as their joint gravity will give them. When these bodies continue at rest, their common centre of gravity must lie directly under the point of suspension, or in the perpendicular line PL; consequently it must be at the point C, the intersection of the lines PL and AB; the position of which point, in the line AB, will be determined by finding out the proportion between the segments CA and CB. If the inflexible line was not interposed between these bodies, they would move till their threads coincided with the perpendicular line PL; since there-

for they are kept asunder by this line, they must urge it with certain forces in opposite directions; and these urging forces must be equal, since the line on which they act continues at rest; and therefore the force with which each body urges the other in the direction of this line, may be denoted by the same letter U, and we may denote the weights of the two bodies respectively by the letters A and B. Now the body A is acted upon by three forces, viz. by its weight A in the direction PC, by the force U with which the other body urges it in the direction CA, and by the reaction of the pin in the direction AP; and since these three forces are in equilibrium, and keep the body at rest, they are to each other respectively as the sides of the triangle PCA; therefore A is to U, as PC to CA. In like manner the body B is urged by three forces, viz. its weight B in the direction PC, the urging force U in the direction CB, and the reaction of the pin in the direction BP, which forces are to each other as the sides of the triangle PCB; therefore U is to B, as CB to PC; and therefore (ex aequo perturbate) A is to B, as CB to CA; consequently the weights of the bodies A and B are to each other inversely as their distances from the point C, which lies directly under the point of suspension, and is therefore their common centre of gravity.

When two bodies are connected by an inflexible line, and this line is supported by a prop so that their centre of gravity cannot descend, the bodies must continue to rest, and will be in equilibrium. Therefore it is easy to see how, from the theorem now demonstrated, we may prove the property of the lever in that case where the directions of the forces are parallel; and from thence the other case, in which the directions are oblique to each other, may be deduced by the resolution of forces, as is usually done. And this is the second method by which I said the general property of the lever might be strictly demonstrated.

The lever is the most simple of all the mechanic powers; and to it may be reduced the balance and the axis in peritrochus, or axle and wheel. Though I do not consider the balance as a distinct mechanic power, because it is evidently no other than a lever fitted for the particular purpose of comparing the weights of bodies, and does not serve for raising great weights or overcoming resistances as the other machines do.

When a weight is to be raised by means of an axle and wheel, it is fastened to a cord that goes round the axle, and the power which is to raise it is hung to a cord that goes round the wheel. If then the power be to the weight as the radius of the axle to the radius of the wheel, it will just support that weight, as will easily appear from what was proved of the lever. For the axle and wheel may be considered as a lever, whose fulcrum is a line passing through the centre of the wheel and middle of the axle, and whose long and short arms are the radii of the wheel and axle which are parallel to the horizon, and from whose extremities the cords hang perpendicularly. And thus an axle and wheel may be looked upon as a kind of perpetual lever, on whose arms the power and weight always act perpendicularly, though the lever turns round its fulcrum. And in like manner, when wheels and axles move each other by means of teeth on their peripheries, such a machine is really a perpetual compound lever. lever; and, by considering it as such, we may compute the proportion of any power to the weight it is able to sustain by the help of such an engine. And since the radii of two contiguous wheels, whose teeth are applied to each other, are as the number of teeth in each, or inversely as the number of revolutions which they make in the same time; we may, in the computation, instead of the ratio of these radii, put the ratio of the number of the teeth on each wheel, or the inverse ratio of the number of revolutions they make in the same time.

Some writers have thought the nature and effects of the pully might be best explained by considering a fixed pully as a lever of the first, and a moveable pully as one of the second kind. But the pully cannot properly be considered as a lever of any kind; for when any power sustains a weight by means of a system of pulleys, that power will sustain the same weight if the pulleys be removed, and the ropes be brought over the axles on which the pulleys turned. And in this case I believe no one would say that these axles could be considered as levers. If the weight was to be raised up, there would in this case be a very great resistance from the friction of the ropes on the axles; and it is merely to avoid this resistance that pulleys are used, which move round the axles with but little friction. I think the best and most natural method of explaining the effects of the pulley (that is, of computing the proportion of any power to the weight it can sustain by means of any system of pulleys), is by considering that every moveable pulley hangs by two ropes equally stretched, which must bear equal parts of the weight; and therefore when one end of the rope goes round several fixed and moveable pulleys, since all its parts on each side of the pulleys are equally stretched, the whole weight must be divided equally amongst all the ropes by which the moveable pulleys hang. And consequently if the power which acts on one rope be equal to the weight divided by the number of ropes, that power must sustain the weight.

Upon this principle the proportion of the power to the weight it sustains by means of any system of pulleys, may be computed in a manner so easy and natural, as must be obvious to every common capacity.

The proportion which any power bears to the resisting force it is able to sustain by means of a wedge, has been laid down differently by different authors as they happened to consider it in particular cases. Without examining their several opinions, I shall endeavour to express this proportion in one general proposition, which may extend to the several cases in which the wedge is applied.

Let the equilateral triangle ABC represent a wedge, whose base or back is AC, and sides are the lines AB and CB, and whose height is the line BP, which bisects the vertical angle ABC, and also the base perpendicularly in P. When a power is applied to the wedge in order to overcome or remove any resisting forces, it acts perpendicularly on the back of the wedge, and the resisting forces act on its sides, and they are always supposed to act in directions that make equal angles with the sides. When the resisting forces and the power that acts on the wedge are in equilibrium, the former will be to the latter, as the height of the wedge to a line drawn from the middle of the base to one side, and parallel to the direction in which the resisting force acts on that side.

Let E and F represent two bodies, or two resisting forces acting on the sides of the wedge perpendicularly, and whose lines of direction EP and FP meet at the middle point of the base, on which the power P acts perpendicularly, then will EP and FP be equal; let the parallelogram ENFP be completed; its diagonals PN and EF will bisect each other perpendicularly in H. Now when these forces (which act perpendicularly on the sides and base of the wedge,) are in equilibrium, they will be to each other as the sides and diagonal of this parallelogram, that is, the sum of the resisting forces will be to the power P, as the sides EP and FP to the diagonal PN, or as one side EP to half the diagonal PH, that is (from the similarity of the right-angled triangles BEP, EHP) as BP, the height of the wedge, to EP the line which is drawn from the middle of the base to the side AB, and is the direction in which the resisting force acts on that side.

From the demonstration of this case, in which the resisting forces act perpendicularly on the sides of the wedge, it appears that the resistance is to the power which sustains it, as one side of the wedge AB is to the half of its breadth AP; because AB is to AP as BP is to EP.

It appears from hence, that if PN be made to denote the force with which the power P acts on the wedge, the lines PE and PF, which are perpendicular to the sides, will denote the force with which the power P protrudes the resisting bodies in directions perpendicular to the sides of the wedge.

Let us now suppose in the second case, that the resisting bodies E and F act upon the wedge in directions parallel to the lines DP and OP, that are equally inclined to its sides, and meet in the point P. Draw the lines EG and FK perpendicular to DP and OP; then making PN denote the force with which the power P acts on the wedge, PE and PF will denote the forces with which it protrudes the resisting bodies in directions perpendicular to the sides of the wedge, as I observed before: now each of these forces may be resolved into two, denoted respectively by the lines PG and GE, PK and KF, of which GE and KF will be lost, as they act in directions perpendicular to those of the resisting bodies; and PG and PK will denote the forces by which the power P opposes the resisting bodies, by protruding them in directions contrary to those in which they act on the wedge; therefore, when the resisting forces are in equilibrium with the power P, the former must be so to the latter, as the sum of the lines PG and PK is to PN, or as PG is to PH, that is, as PB, the height of the wedge, is to PD (a) the line drawn from the middle of the base to one side of

(a) [PG is to PH as PB to PD.] The right-angled triangles PGE and PED are similar, having the angle at P common to both; therefore PG is to PE as PE to PD; so likewise the right-angled triangles PHE and PEB are similar, and therefore PH is to PE as PE to PB; therefore the rectangles PG into PD and PH into PB are equal, each of them being equal to the square of PE; consequently their sides are reciprocally proportional, that is, PG is to PH as PB to PD. the wedge and parallel to the direction in which the resisting force acts on that side.

From what has been demonstrated, we may deduce the proportion of the power to the resistance it is able to sustain in all the cases in which the wedge is applied. First, when in cleaving timber the wedge fills the cleft, then the resistance of the timber acts perpendicularly on the sides of the wedge; therefore in this case, when the power which drives the wedge is to the cohesive force of the timber as half the base to one side of the wedge, the power and resistance will be in equilibrium.

Secondly, when the wedge does not exactly fill the cleft, which generally happens, because the wood splits to some distance before the wedge: Let ELF represent a cleft into which the wedge ABC is partly driven; as the resisting force of the timber must act on the wedge in directions perpendicular to the sides of the cleft, draw the line PD in a direction perpendicular to EL the side of the cleft, and meeting the side of the wedge in D; then the power driving the wedge and the resistance of the timber when they balance, will be to each other as the line PD to PB the height of the wedge.

Thirdly, when a wedge is employed to separate two bodies that lie together on an horizontal plane, for instance two blocks of stone; as these bodies must recede from each other in horizontal directions, their resistance must act on the wedge in lines parallel to its base CA; therefore the power which drives the wedge will balance the resistance when they are to each other as PA, half the breadth of the wedge to PB its height; and then any additional force sufficient to overcome the resistance arising from the friction of the bodies on the horizontal plane will separate them from each other.

The inclined plane is reckoned by some writers among the mechanic powers; and I think with reason, as it may be used with advantage in raising weights.

Let the line AB represent the length of an inclined plane, AD its height, and the line BD we may call its base. Let the circular body GEF be supposed to rest on the inclined plane, and to be kept from falling down it by a string CS tied to its centre C. Then the force with which this body stretches the string will be to its whole weight, as the sine of ABD, the angle of elevation, to the sine of the angle which the string contains with a line perpendicular to AB the length of the plane. For let the radius CE be drawn perpendicular to the horizon, and CF perpendicular to AB, and from E draw EO parallel to the string and meeting CF in O: Then, as the body continues at rest and is urged by three forces, viz. by its weight in the direction CE, by the reaction of the plane in the direction FC, and by the reaction of the string in the direction EO; the reaction of the string, or the force by which it is stretched, is to the weight of the body as EO to CE; that is, as the sine of (the angle ECO, which is equal to) ABD the angle of elevation, to the sine of the angle EOC, equal to SCO, the angle which the string contains with the line CF perpendicular to AB, the length of the plane.

When therefore the string is parallel to the length of the plane; the force with which it is stretched, or with which the body tends down the inclined plane, is to its whole weight, as the sine of the angle of elevation to the radius, or as the height of the plane to the length. And in the same manner it may be shewn, that when the string is parallel to BD the base of the plane, the force with which it is stretched is to the weight of the body as AD to BD, that is, as the height of the plane to its base. If we suppose the string which supports the body GEF, to be fastened at S; and that a force, by acting on the line AD, the height of the plane, in a direction parallel to the base BD, drives the inclined plane under the body, and by that means makes it rise in a direction parallel to AD: Then, from what was proved in the third case of the wedge, it will appear, that this force must be to the weight of the body, as AD to BD, or rather in a proportion somewhat greater; if it makes the plane move on and the body rise.

From this last observation we may clearly shew the nature and force of the screw; a machine of great efficacy in raising weights, or in pressing bodies closely together. For if the triangle ABD be turned round a cylinder whose periphery is equal to BD, then the length of the inclined plane BA will rise round the cylinder in a spiral manner, and form what is called the thread of the screw; and we may suppose it continued in the same manner round the cylinder from one end to the other; and AD the height of the inclined plane will be everywhere the distance between two contiguous threads of this screw, which is called a convex screw. And a concave screw may be formed to fit this exactly, if an inclined plane every way like the former be turned round the inside of a hollow cylinder, whose periphery is somewhat larger than that of the other. Let us now suppose the concave screw to be fixed, and the convex one to be fitted into it, and a weight to be laid on the top of the convex screw: then, if a power be applied to the periphery of this convex screw to turn it round, at every revolution the weight will be raised up through a space equal to the distance between the two contiguous threads, that is, to the line AD the height of the inclined plane BA; therefore, since this power applied to the periphery acts in a direction parallel to BD, it must be to the weight it raises as AD to BD, or as the distance between two contiguous threads to the periphery of the convex screw, which distance between two contiguous threads is to be measured by a line parallel to the length of the screw. If we now suppose that a hand-spike or handle is inserted into the bottom of the convex screw, and that the power which turns the screw is applied to the extremity of this handle, which is generally the case; then as the power is removed farther from the axis of motion, its force will be so much increased, and therefore so much may the power itself be diminished. So that the power which, acting on the end of a handle, sustains a weight by means of a screw, will be to that weight as the distance between two contiguous threads of the screw, to the periphery described by the end of the handle. In this case we may consider the machine as composed of a screw and a lever, or, as Sir Isaac Newton expresses it, *cuneus à vertice impulsus*.

Of any two or more of these simple machines combined together, all other machines, however complicated, are composed. And their powers and manner of acting may therefore be explained from the principles here laid down.