In the strict sense of the word, denotes the method of constructing machines to be set in motion, and to answer some useful purposes, by certain powers, either natural or artificial. According to this definition, the nature of the powers themselves is not the object of mechanical investigation, but rather the effect of them upon the passive bodies which we call machines; and the constructing of these in such a manner, that the powers may act upon them with the least possible obstruction, and produce the intended effect to the greatest advantage, is the perfection of Mechanics.

It is usual, in treatises upon this subject, to begin with an investigation of the properties of matter itself, and of central forces; but the former is not to be investigated by mechanical means, and the latter belong so much to astronomy, that very little needs to be said upon them in this place; for which reason we refer to the articles Astronomy, Matter, and Motion, for a discussion of these subjects. In treating of mechanics, therefore, we shall begin with a description of what are commonly called the mechanic powers; and afterwards consider the various ways in which they may be modified, in order to produce the effects expected from them.

Sect. I. Of Material or Mechanical Power in general.

§ 1. Production of Motion and Rest.

In mechanics every thing is called a power which is capable of acting upon a solid body; and every power which can act upon matter is supposed to be material, without regarding any abstract speculations concerning its nature. Hence the force of gravity, of electricity, of fire, of air, of water, the power of animals, of bodies pressing or impinging with violence upon one another, are all accounted mechanical powers when applied to set machines in motion.

As any single power, when applied to a material body, will set it in motion in proportion to its quantity, so the action of an opposite power upon the same body produces rest. This may be easily conceived; for supposing two men to pull a log of wood with equal degrees of strength in directions exactly opposite to one another, the log will remain immovable. In like manner, if we put in a weight into one scale of a balance, motion will be produced; but rest is the certain consequence of counterpoising it with an equal weight in the opposite scale. When a weight is suspended freely in the air, we are apt to imagine that it is acted upon by no force whatever; but we will soon discover our mistake, by withdrawing the pressure of the air from one side; for the body then, instead of remaining at rest, will move with great violence to one side, and even contrary to the direction of gravity itself, unless it be extremely heavy. Whether rest be in all cases produced by the action of opposite powers upon the same substance, is a speculation to be discussed under the article Motion.

§ 2. Of Resistance.

When any moving power is stopped by a fixed obstacle, so that it can proceed no farther, we say that it is resisted by that obstacle. In this case we are apt to imagine that there is no force exerted by the cause of resisting obstacle; but it is found by experience, that resistance is to all intents and purposes equivalent to a power. Mechanical power equal and contrary to that which is impelled against the resisting obstacle. This is exemplified in the case of a man standing in a boat, and pushing with a pole against the bank of a river or lake. In this case every one knows that the boat will go off in a direction contrary to that in which he pushes; but if the boat be fastened by means of a hook and rope to that part of the pole which is between the man's body and the bank, the boat will remain immovable by reason of the equality betwixt the action of the man upon the pole forward, and of the boat upon the same pole backward. Thus, in fig. 1, when the man pushes with the pole C against the bank D, in the direction CA, the boat B will be carried away from the bank in the direction AC; but if, by means of the rope E, the boat be fastened to the pole AC, the recoil of the boat in the direction AC will be just equal to the push given by the man in the direction CA, so that no motion will ensue let him exert ever so much strength. Hence we see, that by means of a resisting obstacle a power may be made to counteract itself, so that a motion or tendency to it may be produced in any direction; and in this case, as well as in the former, rest is produced by the opposition of two contrary forces.

The very same effect would follow, though we should suppose the man in the boat not to push against the bank or any fixed obstacle, but against another boat fastened by means of a rope to his own. In this case both the boats will recede from each other till the rope be stretched; after which they will both remain immovable, unless they be acted upon by some power external to both. If both boats be at liberty, they will mutually recede from each other till they get beyond the reach of the pole.

Resistance, whatever we may speculate about it, seems ultimately to depend on the power of gravity joined with that of cohesion. Thus a weight of 100 pounds, even when suspended in the freest manner we can imagine, will resist much more than 20 pounds suspended in the same manner; and though hard bodies resist to a great degree, yet unless connected with some very heavy body, they are easily moved out of their place; and the immense gravitation of the whole globe of earth, we may justly suppose to be the source of all resistance whatever to mechanical powers.

On the whole, therefore, we may consider resistance as an active power; but the action of which is confined to a very limited space, or to the single point of contact; though several experiments tend to show, that even before actual contact bodies show a very perceptible degree of resistance.

§ 3. Of the Communication of Power.

This depends entirely upon that property of bodies which is called their attraction of cohesion, and the immobility of their particles among themselves; for if the parts of a body are absolutely moveable among themselves, they can neither communicate motion by impulse nor by pressure. The most common method of communicating motion in the mechanical way is by pressure, which is generally accomplished by means of the six mechanical powers to be afterwards described: collision being employed only in certain particular cases, the most remarkable of which will be pointed out under the article Motion.

The motion, which by means of an hard inflexible body is communicated to any other, may be confined to a single point, or it may be diffused over any affigiable space. Thus, in fig. 2, let us suppose that any assignable power is applied to the point E, urging it from E towards D; the whole of that force will rest upon the point of contact betwixt the ball C and the line BD. The weight, if placed exactly perpendicular to the horizon, will remain upright, without inclining either to one side or other; for the power of resistance in the line BD is exactly equal to the impulse of the weight lying upon E; so that it is in the same situation with the man and boat in the first example, when he had the boat hooked to the pole with which he pushed against the bank. If instead of opposing the end of the resisting body BD to the ball C, we place them in the position represented in fig. 3, then the whole of the power will rest upon any point of the line BD we please. For if we suppose the line EC to drive the line BD before it in the position represented in the figure, it is plain that the whole force of that line will be discharged for a moment upon the line F, or upon any obstacle we choose to put in its way in another part of the line; but if we place two supports or resistances to the moving line BD, as F and G in fig. 4, it is equally plain, that one half of the power will rest upon the one and one half upon the other. For the whole force urging forward the line BD is but a certain and determined quantity; and if divided betwixt two obstacles, each of these must undoubtedly bear one half. In like manner, if, as in fig. 5, the power be opposed by four obstacles, each of them will bear only one-fourth part; and so on if we suppose it opposed by ever so many. For reasons afterwards to be assigned, however, it is absolutely necessary that the force on E act in a line directly perpendicular upon BD; that the obstacles be all at equal distances, comparatively speaking, from C; as H and I, F and G, &c. likewise that they be of exactly the same height; for thus only the pressure, and consequently the motion, can be made uniform in all parts. On this principle depends in a great measure the perfection of printing presses, oil presses, and all other machines intended to produce a violent and uniform pressure upon any broad and flat surface.

As the pressure upon a single point may thus be diffused over a broad surface, so may that upon a broad surface be concentrated upon a single point or a surface of small dimension, as in fig. 6. Here it is plain that whatever pressure is applied to the line AB, or any part of it in a perpendicular direction, must be sustained by the point D; for if there was no resistance, this point would be driven along with the line AB, and the moment it was stopped the power which urged it on must likewise be stopped. It is true, that unless the power act directly perpendicular to the point D, or the line CD be supported that it cannot move either to one side or to another, the impulse will be but momentary; but of this we shall treat at large in the subsequent part of this article. On the principle just mentioned depends in a great degree the force of gimlets, augers, boring gimlets, &c. § 4. Of changing the Direction of a Power into one directly opposite.

This in all cases is only to be accomplished by the application of a power greater than that of which we wish to change the direction. Thus, in fig. 2, suppose we wish to change the direction of the power at E from the direction AC to that of CA, we will find it impossible to do so by any other means than the application of a greater power from D towards C. If the two powers are equal, there will be no motion whatever; and the degree of motion produced at last will only be the difference betwixt the two powers. If it be wanted therefore to produce a power in the direction CA, equal and opposite to that in the direction AC, one must be applied in the direction CA double to the former. This principle is different from that first mentioned, in which motion is produced by pushing against a fixed obstacle while the moving power is not resisted on the opposite side; for here the power of gravity, or whatever we suppose to act upon E, resists according to its quantity, and the whole is in the situation of the boat when hooked to the pole fig. 1. To produce motion, therefore, a new force must be applied, as if a person was to push from the bank D against the hooked pole of the boat in that figure. The principle just now laid down does not militate against the apparent ascent of bodies by the action of gravity, or the repulsion of elastic balls from one another by what is called the power of elasticity. In both cases a greater power is applied than the simple force of gravity, and with the excess of this power the body ascends, as shall be afterwards shown.

§ 5. Of the Motion produced by two or more Powers acting upon a Body in directions oblique to each other.

As the action of two powers in direct opposition to each other is attended with the destruction of both if the powers are equal, and of one of them if they are unequal; so the action of powers directed obliquely upon one another is productive of motions in various directions, according to that of the acting powers.

The motion produced by the action of two powers is always in the diagonal of the parallelogram expressed by these powers. Thus, in fig. 7, let the body A be acted upon at once by two forces, one of which would carry it from A to B in the same time that the other would carry it from A to C. The body will then describe AD, the diagonal of a square, in the time that it would have described one of the sides by a single power applied to it. This is in consequence of its obeying both forces; as it is evident that it has moved as far as from A to B, and likewise from A to C, which is precisely the effect that the two powers would have had upon it separately. In this case the body has acquired a greater power than it would have had from a single power, but less than it would have acquired from the union of the two powers if they had acted directly in concert with each other; because the diagonal of a square is less than the sum of the sides, and the power with which any body moves is exactly proportioned to its velocity. If, instead of supposing the forces equal, we suppose one of them considerably greater than the other, then the greater force will carry the body farther in its own direction than the other, and the whole will be represented by a parallelogram, as in fig. 8. In this case it is evident that the body has moved exactly in conformity to the direction of both powers, viz., the whole length of AB, and the whole length of AC. In this case also the loss of motion is less than in the former; because the length of the oblong parallelogram approaches much nearer to the sum of the sides than the diagonal of a square; and the greater inequality there is betwixt the sides, the less power is lost.

If, instead of acting at right angles to each other, the direction of the powers forms an acute angle, as in figs. 9, 10, 11, the power produced will be considerably greater than either of the original ones; and the more acute the angle is, the greater will be the augmentation, as is evident from an inspection of the figures. The reason of this, though not quite so obvious, is the same with the former. Thus the body A in fig. 9, had it been acted upon by only one power, viz., that denoted by AB, would have been at B, or carried as far forward as E, the half of the diagonal; its oblique direction upward not being taken into the account. Had it been acted upon by the force AC alone, it would have been at C with an obliquity as far down as the other is up. As these obliquities, however, are in contrary directions, they must of necessity destroy one another; and therefore the body moves neither to one side nor another, but proceeds with the sum of the direct forces of the powers, or those by which they move in the straight diagonal. But either of the two powers would have brought it forward as far as E; of consequence both conjoined must carry it on to D, the whole length of the diagonal. Thus it appears, that when a body is acted upon by two powers which partly conspire together, the power produced will be the exact sum of them as far as they do conspire, and the loss arises entirely from the opposition betwixt them; for all powers which do not directly conspire, oppose one another in a certain degree. Hence when the acting forces make an obtuse angle with each other, as in figs. 12, 13, there is then a very great loss of power, because there is such an opposition betwixt them; and it is only that small part of their motion which acts in concert that can produce any in the body acted upon; but this, as in the former case, is exactly double to what it would be if only one of them acted upon the body. Thus, in fig. 12, the whole direction of the powers from E to B, and from B to C, is in absolute opposition to each other; and therefore, supposing them equal, must be totally lost. In the direction AD they conspire; and therefore the body will move twice as far in that direction as is expressed by that of the lines in the figure; that is, from A to D, instead of only from A to E, which is the limit of each of the forces. In cases of this kind, the more obtuse the angle is at which the forces act, the greater is the loss of power, as is evident from an inspection of figs. 12, 13.

Some who are but beginning to the study of mechanics may be embarrassed in their ideas how two forces acting at right angles to each other can in any manner of way oppose each other, as in fig. 7; as we find that a body descending by the force of gravity may... Mechanical may be pushed to a side seemingly by the least force imaginable. But this will easily be understood from fig. 14, which is only a square turned into another posture. Here it is plain that the powers \(AB\) and \(AC\) oppose each other as much as they conspire; that is in the proportion of half the diagonal of the square: this quantity therefore is totally lost, and the body proceeds with the other half; which being doubled on account of each of the powers proceeding with one half of the diagonal, gives the whole diagonal for the total motion produced.

But however plain this may appear from an inspection of the figure, it is by no means so apparent when we come to try it by numbers. Thus, supposing each of the sides \(AB\) and \(AC\) to be 5, the diagonal of the square will be nearly 7.071; but if from the sum of the sides 10 we take this number, or half of it from each number, we will have only 2.919 for the whole motion, instead of the diagonal 7.071 which is the reality. From an inspection of the figure also we plainly see, that if one diagonal is gained by the powers conspiring together or acting in concert, another is lost by their opposition. It is natural therefore to inquire, How can any two powers gain or lose more than their own quantity; for the two powers taken together amount but to ten, but the two diagonals, one of which is gained and the other lost, amount between them to upwards of fourteen? To solve this seeming paradox, we must consider, that as the diagonal of the square \(ABCD\), fig. 14, is generated from the two sides \(AB\) and \(CD\), so these sides themselves may be accounted the diagonals of two other smaller squares \(BAE\) and \(AEC\), fig. 15, each of the sides of which is half the diagonal of the large one. From the sum of the sides of these squares, which to the large square are the source of power, it is evident that a diagonal may be taken and another remain, because each of the sides is half a diagonal.

Hence we not only see that every mechanical power we are acquainted with may be derived from two others, but have a demonstration that it actually is so: not only because this supposition explains the phenomena, but because we are involved in an inexplicable contradiction if we suppose anything else, for no power can lose more than its own quantity; and if it loses more than one half, it can never produce effects equivalent to another half; which we see must be the case, if we suppose any two unopposed powers acting upon one another at right angles, or indeed any other way, though the supposition of their acting at right angles makes the matter more plain than any other. This leads to a very curious speculation concerning the origin of mechanical motion, of which an account is given under the article Motion.

Hitherto we have considered both the powers not only as equal at the beginning, but as continuing so throughout their whole course: but this is a supposition which scarcely exists in nature, unless the powers are kept from exerting themselves otherwise than by simple pressure. Thus, in fig. 16, supposing the body \(A\) pulled in the direction \(AB\) by the weight \(D\) of five pounds put over the fixed pin \(B\), and pulled in the direction \(AC\) by \(C\), another weight of five pounds fastened to it by a string; the whole will be kept in the position represented in the figure by a weight of 7.071 pounds fastened to it by a string, and put over the pin \(F\), situated anywhere in the diagonal line \(FAG\); and let us add ever so much weight, provided it be done to \(D\) and \(C\) in the proportion of five, and to \(E\) in that of 7.071, the body \(A\) will remain suspended in the air without altering its position in the least.

If, instead of making the weights equal, we make one exceed the other in any proportion, the weight necessary to counteract them will never be required equal to both, but will always be in proportion to the diagonal of the parallelogram of which the weights represent the sides. Thus, in fig. 17, if we suppose the body \(A\) pulled in the direction \(AB\) by the weight \(G\) of four pounds, and in the direction \(AC\) by the weight \(H\) of three pounds, it will be kept suspended by the weight \(F\) of five pounds put over the pin \(E\), placed anywhere in the diagonal line \(EAD\). For the diagonal \(AD\) is equal (by Prop. 47, Book I. of Euclid) to the square-root of the sum of the squares of the sides \(AB\) and \(AC\), or \(CD\) and \(BD\). But the square of \(AB\) is \(4 \times 4 = 16\) pounds, and that of \(AC\) is \(3 \times 3 = 9\) pounds by the supposition; and \(16 + 9 = 25\), the square-root of which is 5; and these proportions will be found to hold invariably in whatever way we apply mechanical powers; though, when they act at oblique angles, the diagonals must be calculated by other methods.

If, however, we set any of the powers at liberty, we shall find that none of them will continue the same even for a moment. If we suppose any of them to be the power of gravity, which is the most constant and equable we are acquainted with, this is found to increase prodigiously; and, on the other hand, if we suppose one of them to be a projectile force, as of a stone thrown by the hand, we will find in like manner, that it will be diminished to a great degree in a very little time. In all cases, however, where a body is acted upon by two forces either increasing or decreasing, unless both increase or decrease exactly in proportion to their original quantity, the body acted upon will describe a curve. Thus, in fig. 18, suppose the body \(A\) to be acted upon by two equal powers \(Ab\) and \(Ac\); at the motion the end of the first moment it will be at \(d\), the end of how produced, the diagonal of the small square \(Abcd\); but if now the force \(Ac\) be increased to double what it was in the preceding moment, the body will at the end of the second moment be at \(g\), the extremity of the parallelogram \(defg\); and by another increment of the same power, will be at the end of the third moment at \(k\), and so on. This is similar to the motion of falling bodies, of which we shall treat hereafter; but if one of the powers diminishes instead of increasing, the phenomena will be different. Thus, in fig. 19, supposing the body at \(A\) to be actuated the first moment by the two forces \(Ab\) and \(Ac\); at the end of that moment it will be at the extremity of the diagonal \(Ad\); but next moment, supposing the power \(Ac\) to be diminished one half, the other remaining the same, it will then be at \(g\), and the third moment at \(k\), thus describing another kind of curve. If, while one of the powers decreases the other increases, a third kind of curve will be generated; and by proper management of these powers, the body may be made to describe the segment of a circle, as in fig. 20; where... Mechanical where it is manifest that one power continually decreases while the other increases.

The following machine has been contrived to illustrate the operation of oblique powers upon each other. ABCD is a wooden square, so contrived that the part BEFC may draw out from it or be pushed back at pleasure. To this is joined a pulley H, freely moving upon its 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 upon the wire. A thread m is fixed to this ball, and goes over the pulley to I; by which means 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 when this part is drawn out, the ball must be carried along with it parallel to the bottom of the square DC. Thus the ball may be drawn either perpendicularly upward by pulling the thread m, or moved horizontally 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 L, and will be found at g when the sliding part is drawn out as far as it was before; which then will have caused the thread to draw up the ball to the top of the inside of the square, just as high as it was before when drawn up singly by the thread without moving the sliding part.

If a body is acted upon by three forces, the investigation becomes somewhat more complex, though it is still easily explained on the foregoing principles. Thus, in fig. 22, let the body A be pulled sidewise in opposite directions by the two equal weights G G put over the pins B and C, and directly downward by the weight H, the same with G. In this case it is plain, that each of the weights G and G sustain one half of the weight H; and as both taken together are double in quantity to H, it might be supposed that they would be abundantly able to keep the body A in its position. The case, however, is very different. As each of the weights G sustains only one half of H, it follows that H acts only with one half of its weight upon them. The body A, therefore, is pulled in the direction AC and AB by two powers, each of which is as 2, and in the direction AF by two, which are only as one. With the force AB, therefore, were it to act upon it singly, it would describe the diagonal AD, and with that AC it would describe the diagonal AE. These two diagonals are in truth the forces by which it is now actuated, and the effect is precisely according to the principles already laid down. By each of them taken separately, the body would be brought down to F; their lateral action being in opposite directions destroys itself; and by their conjunct action, the body would be brought down to double the space AF, that is to H, and consequently would describe the diagonal of the small square ADHE; which diagonal is equal to the side of the large one, and the very same mechanical that the body would have described though the two lateral weights had not been present.

Hence it appears, that though we pull a body ever so strongly by strings in a direction opposite to each other, it will still require an equal weight to retain it in equilibrium; that is, supposing the strings to be perfectly flexible. There may indeed be a deception in making an experiment of this kind; for the body will never descend as far as H, nor near that distance; but then it must be observed, that when the strings begin to bend in the middle, the weights G G act in a direction different from what they did originally, and pull the body upwards instead of laterally; in which case, it must either remain at rest, as in fig. 23, or move upwards, as in fig. 24.

When the powers act in the direction AB and AC, fig. 23, one half of the weight H is sustained by each of them. The body is therefore pulled in the directions AB and AC by two powers, each of which is as 2; and in the direction AF by other two, each of which is as 1. By the power AB it would be made to move in the diagonal AD of the parallelogram ABDF; and by the power AC, in the diagonal AE of the parallelogram ACFE; but these diagonals are equal and contrary to each other, and therefore destroy each other; of consequence the body remains at rest.

In fig. 24, the body A with the weight H appended to it is placed nearer to the point B than to C by one-third. Of consequence, as will afterwards be explained, it bears two-thirds of the weight H, while C sustains only one-third. The acting powers, therefore, are now the diagonals of two unequal parallelograms. One power draws the body in the direction AB with a force as 3, while the weight H draws it in the direction AH with a force as 2. By it, therefore, the body would be drawn in the direction of the diagonal AD of the parallelogram BDEA. On the other hand, it is acted upon by the power AC, which is likewise as 3, while the weight H draws it down with a force only as 1. By this, therefore, it would be drawn in the direction of the line AG, the diagonal of the parallelogram ACGF. We must now make these two diagonals the sides of a third parallelogram ADIG; and in the diagonal AI of this parallelogram it will go, for the reasons already given.

If four or more forces act upon a body in different directions, the case becomes very complicated; and if acted upon many powers be employed, it will by no means be easy to determine a priori which way the body will tend. Cases of this kind, however, seldom occur in practical mechanics; and when they do, it will be better to determine them by actual experiment than by a tedious investigation, which, after all, may be liable to a mistake. We forbear, therefore, to give more examples; though, if the reader inclines to exercise his ingenuity, he must proceed upon the plan already laid down, viz. by combining the different powers together; forming diagonals from these parallelograms; combining these diagonals into a third set of parallelograms; and the diagonals thence resulting into a fourth set, &c., until at last a single one is met with prevailing over all the rest, or two destroying each other. If one prevails, the body Mechanical body will move in that direction; but if two destroy each other, the body will remain at rest. It must also be observed, that in making drawings of this kind, the longest line always represents the greatest power, and that without a single exception to the contrary. By mere mechanical construction, therefore, with scale and compasses, we may be able to ascertain the direction of oblique powers to as great accuracy as we can ever have occasion for in practice.

We shall conclude this subject with observing, that as every power certainly is produced by the action of two others, so it may be by innumerable others. Thus, in fig. 25, the power AD may be produced by the two sides of the square ABDC; by the sides of the oblique-angled parallelogram AaDb; by the sides of the smaller parallelogram AaDd; or by the large parallelogram AE, DF, &c. Hence it is easy to produce any power, whether strong or weak, from the action of any two powers whose direction we have at our command, without regard to their quantity. If we make the generating powers conspire together, a strong one will be produced; or if they oppose each other to a certain degree, they will produce a weak one. The strongest that can be produced by any two powers is when they act the same way in a direct line.

§ 6. Of the Relation betwixt Velocity and Power.

Hitherto we have supposed the bodies to be moved not to make any resistance to motion in any direction, unless opposed by a fixed obstacle; in which case, velocity and power would be the same thing; and thus it always appears to be when we represent powers by lines upon paper. But when we come to practice, matters turn out very different. A ball of cork moving with any degree of velocity will not have an equal power with one of wood moving with the same velocity; neither will a wooden ball have the same power with a metallic one. Among the metals themselves, too, there is a difference; for the lighter metals are inferior in power to the heavier ones. Gravity, therefore, must be accounted the power which gives to moving bodies what we call their force or momentum; for according to the weight of a body, so will its impulse always be, and that whether it moves upwards, downwards, sidewise, or in a circle. The absolute power of a body, therefore, must be measured by comparing the gravity of different bodies together, and denoting one of them by unity; making the other 2, 3, 4, &c. according as it is twice, thrice, or four times, the specific gravity of the former. Thus let an hollow ball of metal be filled with water; if the ball be very thin, we may let its weight pass unnoticed, or we may make allowance for it in the calculation. Supposing the weight of this ball then to be 1, if it moves with a velocity of 10 feet in a second, its absolute force or momentum will be $10 \times 1 = 10$; a ball of stone of equal size which weighs three times as much as the former, if moved with an equal velocity, will have a force of $10 \times 3 = 30$; a ball of tin which weighs 7 times as much, will have a momentum of 70; and a ball of gold or platinum would have a momentum of 190 or 200.

This will also hold exactly, by increasing the quantity of matter, where it is deficient in specific gravity. Thus, if the hollow metallic ball be increased in diameter, so that it shall equal in weight the ball of stone or of metal, it will have the very same force with that ball; and in like manner, it might be made to have a momentum equal to the metallic balls, though not without a very considerable increase of size.—Great Mr Atwood observes, that masses of matter will therefore supply the place of wood's obfervations; and hence Mr Atwood observes, that the battering-rams used by the ancients were no less powerful than the modern artillery. "The battering-rams of the ancients (says he) consisted of very large beams of wood terminated by solid bodies of brass or iron; such a mass being suspended as a pendulum, and driven partly by its gravity and partly by the force of men against the walls of a fortification, exerted a force which, in some respects, exceeded the utmost effects of our battering cannon, though in others it was probably inferior to the modern ordnance. To compare the effects of the battering-ram, the metal extremity of which suppose equal to a 24 pounder with a cannon-ball of 24 pounds weight; in order that the two bodies may have the same effect in cutting a wall or making a breach in it, the weight of the aries must exceed that of the cannon ball in the proportion of the square of 1700; the velocity of the ball*, to the square of the velocity with which the battering ram could be made to impinge against the wall expressed in feet. If this may be estimated at 10 feet in a second, the proportion of the weights will be that of about 28,900,000 to 100, or of 28,900 to 1; the weight of the battering ram therefore must be 346 ton. In this case the battering-ram and the cannon-ball, moving with the velocities of 10 and 1700 feet respectively in a second, would have the same effect in penetrating the substance of an opposed obstacle; but it is probable that the weight of the aries never amounted to so much as is above described; and consequently the effects of the cannon-ball to cut down walls by making a breach in them, must exceed those of the ancient battering-rams; but the momentum of these, or the impetus whereby they communicated a shock to the whole building, was far greater than the utmost force of cannon-balls; for if the weight of the battering-ram were no more than 170 times greater than that of a cannon-ball, each moving with its respective velocity, the moments of both would be equal; but as it is certain that the weight of these ancient machines was far more than 170 times our heaviest cannon-balls, it follows, that their moment or impetus to shake or overturn walls, &c. was far superior to that which is exerted by the modern artillery. And since the strength of fortifications will in general be proportioned to the means which are used for their demolition, the military walls of the moderns have been constructed with less attention to their solidity and massy weight than the ancients thought a necessary defence against the aries; that sort of cohesive firmness of texture which resists the penetration of bodies being now more necessary than in ancient times; but it is manifest, that even now solidity or weight in fortifications also is of material consequence to the effectual construction of a wall or battery." This difference between the momentum and force of penetration is exemplified in knives, wedges, or any § 7. Of the Multiplication and Increase of Power.

We have now seen that power, absolutely so called, acts in a kind of double capacity, viz. either when it impresses a great velocity upon a small quantity of matter, or when it impresses a small velocity upon a great quantity of matter. It must, however, be remarked, that the matter we speak of is always supposed subject to the laws of gravity; for what would be the consequence of putting a body in motion which had no gravity we cannot possibly conceive, because we never saw any such body. Philosophers indeed mention the vis inertiae of matter as property distinct from gravity; but the arguments in favour of this property are now generally looked upon to be inconclusive, and gravity and the vis inertiae looked upon to be the same.

The two modes in which absolute power acts, come precisely to the same thing whether the velocity be great or small: for it is evident, that when two pounds move with the velocity of 1, it is the same thing with one pound moving with the velocity of 2; the velocities as well as powers being exactly the same. But there is a third way in which power may be directed, in which it has not the relation to velocity already mentioned; and that is, by simple pressure, where no motion is admitted. Thus may the smallest power be made to augment itself to an inconceivable degree, as in fig. 26. Here, suppose the body A to press directly downwards upon the line AB fastened to the small wheel B, moveable upon an axis. If we suppose the extremity of the line at A to be supported so that it shall not fall to a side, the wheel B will press downward with the whole of the weight A upon the line EF, and consequently the line g must sustain the whole of this weight. But if the line EF be supported so that it cannot move perpendicularly downwards to g, it will then roll along the line EF from B towards F; and this tendency to roll in the direction just mentioned will be exactly equivalent to the weight A. Any body therefore laid on the top of a stick set up at an angle of 45 degrees, will require a power double to its own weight to keep it steady at the foot, abstracting from that which will be necessary to prevent it from falling to a side.

If now we suppose the wheel B to press laterally upon another C; and that other, by means of the line CD and wheel D, to press upon the two obstacles i and k, both of which it touches at an angle of 45 degrees; it is plain that not only each of these obstacles must bear the whole weight of the body A, but the reaction of the wheel D will press down the wheel C in the direction Cb with the very same force that D is pressed upwards. This is entirely similar to the case of the man in the boat represented in fig. 1. Thus the weight A produces a pressure equal to five times its own weight; and by multiplying the wheels and rods, we might increase the pressure as much as we please. The case is similar to that in hydrotaties, where a little quantity of liquid may be made to burst the strongest vessel.

§ 8. Of the Lever.

This is the most simple of all the mechanical first kind powers, and is usually no other than a straight bar of wood or iron supported by a prop, as in fig. 27. The weight to be raised is suspended at the short arm of the lever A; and exactly in the inverse proportion of the distance of the weight from the fulcrum or prop C, is the quantity of the weight at B necessary to keep it in equilibrium. Thus if the weight at A be distant one foot or one inch, it signifies nothing which, from the prop, it will require an equal weight placed at the same distance on the other side, as at 1, to balance it; but if the latter be placed at 2, then only half the weight suspended at A will balance it: if the small weight is placed at 3, then only one third will be necessary; if at 4, only one-fourth, &c. and if, as in the figure, it be removed to 10, then only one-tenth part will be required to make a balance. It must still be remembered, however, that if the lever is put in motion, the small weight must move through a space ten times as great as that through which the large one moves; so that in fact there is not any acquisition of power by means of the lever, though it is one of the instruments most commonly used in mechanics, and very serviceable in loosening stones in quarries, or raising great weights to a small distance from the ground; after which they may be raised to greater heights by other machines.

In making experiments with this kind of lever, it is necessary either to have the short arm much thicker than the long one, so that it may exactly balance the latter, or a weight must be appended to it just sufficient to keep it in equilibrium, otherwise no accuracy can be... Mechanical be expected. This lever is the foundation of balances of all kinds, whether of the common kind or of that called the Roman flatera or steel-yard. The latter is no other than the lever represented in fig. 27. For if a scale is appended to the end A of the lever, and a weight, suppose of one pound, used as a counterpoise to the body which is to be put in the scale, it will show exactly the weight of that body, by putting it at a proper distance from the fulcrum upon the long arm. Thus if the weight when placed at the division 5 counterpoises that placed in the scale, it shows that the body weighs exactly five pounds; if it balances at 6, then it shows that the body weighs six pounds, &c. But for a more particular account of this instrument, see the article Steel-Yard. To this kind of lever may be reduced several kinds of instruments, as scissors, snuffers, pincers, &c.

In levers of this kind, the fulcrum C must support both the weight to be raised and likewise that which raises it; so that the weight upon C must be the greater in proportion as the arm CB of the lever is shorter. Thus, if the arms are both equal, the fulcrum C must bear double the weight at A; if the one arm is double the length of the other, then it has only to bear the weight to be raised, and one half more; because any weight at 2 will balance one double to itself at 1; but if removed to 10, the fulcrum will only have \( \frac{1}{10} \) to bear.

In some cases, the weight to be raised is placed between the acting power and the fulcrum, as in fig. 28. This lever is more powerful than the other, and is likewise more easily supported, because only part of the weight to be raised, and none of that which raises it, lies upon the fulcrum. Thus in fig. 28. let the extremity A of the lever AB rest upon a fulcrum at e, and let the small weight 1, by means of a string put over the wheel or pin C, pull up the other extremity; this weight 1 will then counterpoise the large one 10, and very little additional force will be required to raise it up. It is also plain, that the whole weight to be raised being 10, the fulcrum sustains only 9 of it, for the other 1 is sustained by the string BC. It is plain also, that a lever of this kind only ten feet long will raise as great a weight as another of the former kind eleven feet in length; nevertheless there is not any absolute gain of power, because the small weight 1 must move through ten times as much space as the large one; and thus the quantity of motion is exactly equal in both. To this kind of lever we may reduce oars, doors turning upon hinges, cutting knives fixed at the point of the blade, &c. From it also we see the reason why two men carrying a burden upon a pole may bear unequal shares of the weight; for the nearer any one of them is to the burden, the greater share he bears; and if he goes directly under it, he must bear the whole. Hence, if two persons of unequal strength are to carry a burden in this manner, the weaker should always be placed at the greatest distance from it.

If in this lever the moving power be put in the place of the weight, it acts at a great disadvantage; and a very great power will be requisite to overcome a small weight. The reason of this is plain from an inspection of fig. 28; for it is the same thing whether we suppose the body 10 to be the moving power, or the weight to be raised; in either case, nine-tenths of it are spent upon the fulcrum at 0; and the other tenth part at 10 will be able to do no more than balance the weight 1. Levers of this kind are only made use of when we wish to give a considerable degree of velocity to bodies; and hence the flies of clocks, millstones, &c. may be accounted levers of this kind; for these the moving power is applied to a pinion near the centre of motion, and acts at a great disadvantage; the muscles of the arms or legs of a man, by their insertion near the joints, likewise act as levers of this kind; and hence the power exerted by a muscle is always much greater than the force it has to overcome.

In all cases in which the lever is applied, it is necessary, in order to give it the greatest advantage, that the moving power act in a direction exactly perpendicular to the lever itself. If this be not the case, it will be necessary to lengthen the lever in proportion to the obliquity. Thus in fig. 29, suppose the straight lever AB to rest on the fulcrum C, so that a weight of one pound may counteract 10; if the lever be bent in the direction AD, it will then be necessary to lengthen it somewhat in order to produce the same effect. If bent in the direction CE, it must be farther lengthened, and still farther if bent in the direction CF. The reason of this is, that when the weight acts on the bended lever ACF, ACE, and ACD, a part of its force is spent in giving, or attempting to give, a lateral motion to the fulcrum C; and the part thus lost is exactly equal to the advantage gained by the greater length of the lever. To make a lever of a determinate length act always with the same power, it will be necessary to have some contrivance by which the moving power may act always perpendicularly to it; as by having two circular pieces of wood or other solid matter fastened to the ends of it, round which the ropes may wrap themselves when it is put in motion, such as are represented by ab and BG in the figure.

Fig. 30. shows a kind of lever bent so that one part of it may form a right angle with the other, kind of lever. Here the prop or centre of motion is at the angular point C. P is a power acting upon the longer arm AC at F, by means of the cord DE going over the pulley G; and W is a weight 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 will remain in equilibrium. Thus suppose W to be five pounds acting at the distance of one foot from the centre of motion C, and P to be one 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 kind. In this lever the pressure upon the fulcrum downwards is just equal to the weight to be raised; but there is likewise a lateral pressure equal to the weight P; so that the centre of motion must have a double support, otherwise the whole lever with the weight would be drawn towards the side in the direction BC.

If, as in fig. 31. and 32. the lever be bent so as to form two sides of a square, the weight to be raised will always be equal to that upon the fulcrum, in whatever place the fulcrum may be put; but both will vary according to the distance from the angular point. Thus, if as in fig. 31. the fulcrum be placed at the angular point A, the weight F appended to the extremity B Mechanical of the arm AB will just counterpoise an equal weight E by means of the string CDE put over the pin D, and drawn laterally by the arm AC. But if, as in fig. 32, the fulcrum be placed nearer to the extremity of the arm AC, as at 3, the case will then be very much altered, and one pound suspended at the extremity B of the arm AB will counterpoise four at the extremity C of the other arm: the pressure on the fulcrum will likewise be equal to the weight to be raised. Was the fulcrum placed at 2, then a weight of one pound at B would only counterpoise two pounds acting at C; and if it was placed at 1, then a weight of three pounds at B would be requisite to counterpoise at C.

It is worth notice, that levers of this kind cannot be exactly counterpoised by the power of straight levers. Thus, in fig. 33, let any weight, as C, be appended to B, the extremity of the arm of the bent lever BA4. Let DE be a straight lever, the force of which we design to oppose to that of the crooked lever. For this purpose let another weight F act upon the extremity D of this straight lever by means of a string put over the pin G. Let the two levers be connected together by means of the string b3, and let a piece of wood or iron E4 be put between their two extremities: the two weights being now allowed to act, it is evident that the levers will be pulled in different directions, the string b3 will be tightened, and the extremity E of the straight lever DE will be pressed towards 4, while the extremity 4 of the crooked lever will be pressed towards E; by which means the two levers will oppose one another in every point of their action. There is not, however, any weight whatever applied to the straight lever which can be made to counterbalance that at C, in such a manner as to keep the bent lever steady. Let us first suppose the weights to be each one pound, and the string to be placed as in the figure at b3. In this case the weight C pulls the crooked lever from b towards 3, with a force equal to 4, and the extremity 4 will be pressed towards E with an equal degree of force. But in the straight lever, though the point b be pulled in the direction 3b by a force of four pounds, the extremity E is pressed the contrary way by a force equivalent only to three. Thus the weight C must preponderate, and that at F will ascend. Let us next add to the weight F one third of a pound; by which means the pressure from E towards 4 will be augmented to 4, and the two extremities of the levers will counteract each other: but now the pressure in the direction 3b will be greater by one-third of a pound than it is in the direction b3; and of consequence the weight F will prevail, the arm AB of the crooked lever and the weight appended to it being raised. If we attempt to mend matters by augmenting the weight F by not quite a third part, the extremities of the two levers will not balance each other, the pressure from 4 to E will be greater than from E to 4; and in like manner the pressure from 3 to b will be greater than from b to 3. Hence both levers will be pulled in a direction from D towards G, and the weight F will descend if the weights be properly adjusted without any ascent of the other. In short, let us alter the weights as we will, or let us alter the position of the fulcrum as we will, it is easy to see that there is an absolute impossibility that the two levers can counteract each other; because the pressure upon the fulcrum of the crooked lever will always be equal to that by its extremity 4; but in the straight lever the pressure upon the fulcrum must necessarily be greater than that of the extremity.

These are all the varieties of the lever which can be supposed; it remains now only to show the reason of its action, or why a small weight when at rest should counterpoise a great one; motion or velocity being here to appearance out of the question, as we cannot attribute any degree of motion to two bodies absolutely at rest. To do this in a clear and distinct manner has puzzled some of the greatest mathematicians: that of Dr Hamilton professor of philosophy in Dublin, founded upon the resolution of forces, seems to be the ton's demonstration readily understood, and least liable to objection.

The most noted theorem in mechanics (says he) is this, "When two heavy bodies counterpoise each other by means of any machine, and are then made to move together, the quantities of motion with which one descends and the other ascends perpendicularly will be equal." An equilibrium always accompanying this equality of motions, bears such a resemblance to the case wherein two moving bodies stop each other when they meet together with equal quantities of motion, that many writers have thought that the cause of an equilibrium in the several machines might be immediately assigned, by saying, that since one body always loses as much motion as it communicates to another, two heavy bodies counteracting each other must continue at rest, when they are so circumstanced that one cannot descend without causing the other to ascend at the same time, and with the same quantity of motion. For then, should one of them begin to descend, it must instantly lose its whole motion by communicating it to the other. This argument, however plausible it may seem, I think is by no means satisfactory; for when we say that one body communicates its motion to another, we must necessarily suppose the motion to exist first in the one, and then in the other; but in the present case, where the two bodies are so connected that one cannot possibly begin to move before the other, the descending body cannot be said to communicate its motion to the other, and thereby make it ascend: But whatever we should suppose causes one body to descend, must be also the immediate cause of the other's ascending: since from the connection of the bodies, it must act upon them both together as if they were really but one. And therefore, without contradicting the laws of motion, I might suppose the superior weight of the heavier body, which is in itself more than able to sustain the lighter, would overcome the lighter, and cause it to ascend with the same quantity of motion with which the heavier descends; especially as both their motions, taken together, may be less than what the difference of the weights, which is here supposed to be the moving force, would be able to produce in a body falling freely.

However, as the theorem above-mentioned is a very elegant one, it ought certainly to be taken notice of in every treatise of mechanics, and may serve as a very good index of an equilibrium in all machines: but I do not think that we can from thence, or from any one general principle, explain the nature and effects of all the mechanic powers in a satisfactory manner. Mechanics' manner; because some of these machines differ very much from others in their structures, and the true reason of the efficacy of each of them is best derived from its particular structure.

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 term, 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 furthest 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 which-ever 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 force 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 show 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. 34.) 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 passes through the point C; consequently the force G, which sustains this compounded force, must be equal thereunto, 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 Mechanical are in equilibrio, 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 CM to CN or MB, that is, (because the sides of a triangle are as the lines of the opposite angles) as the sine of the angle MBC, or its alternate one BCN, to the sine of the angle BCM; but making CB the radius, BK is the sine of the former angle, and BH of the latter; therefore E is to F as BK to BH; 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 AB and AL perpendicular to the directions of the forces G and F; then, as was said before, F is to G as MB is to CB; but MB is to CB as AB to AL; because, making CA the radius, AB is the sine of the angle MCB, and AL is the sine of the angle MCN, or CMB 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 GC; for it is such parts only that really oppose the force G, and keep it in equilibrio; and from thence we shall see what proportion two forces must have to each other when they are in equilibrio, and act in parallel directions. Let the three forces act upon the points A, B, and D, (fig. 35.) let them be in equilibrio, and 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 AD meeting BC in P, and from P draw PN and PM parallel to the directions of the forces E and F; through the points A and D draw parallel lines to BC; and through B draw a perpendicular to these lines, meeting them in H and K; from the point M draw MO parallel to AD, and meeting BC in O. Now the three forces E, G, and F, that are in equilibrio, will be to each other respectively as the sides of the triangle CMP, as in the preceding case; but the force E, which is denoted by the line MC, may be resolved into two forces acting in the directions MO and OC, 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 PM, may be resolved into two forces acting in the directions OM and PO, 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 PC, is sustained only by those parts of the forces E and F which act against it, in directions parallel to BC, and are denoted by the lines OC and PO, which, taken together, are equal to PC; for the other parts of the forces E and F, which are denoted by MO, 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 equilibrio, 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 equilibrio, 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 show that their centre of gravity is a point in the line that joins them together, so 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 (fig. 36.) 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 therefore they are kept at rest by this line, they must urge it with certain forces in opposite directions; and these... Mechanical 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 equilibrio, 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 directly 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 at rest, and will be in equilibrio. 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 peritrochic, 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.

Though this demonstration will no doubt be abundantly clear to mathematical readers, yet to others less versed in that science its appearance will no doubt be somewhat obscure and perplexed. The following we subjoin as less intricate.

Let AB, fig. 37, represent a straight rod of wood or iron, fastened at the extremity A, at right angles to another piece of the same, and kept steady by two pins C and D. If a weight be put upon the extremity H of the upright rod AH, it will press down that, and along with it the horizontal rod AB, so that every point in the rod will move with the whole force of the weight. Thus, whether we suppose an obstacle to be placed at the extremity B, at the point 2, or at 1, in the horizontal rod AB, it will have exactly the force of the weight placed at H to overcome.—Supposing then that the weight would make the whole descend from A to E in one second; then it is plain that the whole power exerted by the rod in its descent would be expressed by the parallelogram ABEF. But if, instead of supposing the line AB to be the full length represented in the figure, we suppose it to be only half that length, and cut off at 1, then the power of the weight would be represented by the parallelogram AIEI. Were it still farther shortened, by being cut off at 2, then the power would be represented by the parallelogram AEZ; and each of these parallelograms, however unequal they may be as represented upon paper, would in reality be equal when the experiment was made, because, in no case could the weight descend with a greater force than its own. Suppose next the weight to be taken off from H, and put upon B, and the rod AB to be moveable upon the centre A; the whole power of the weight then would be expressed by the triangle ABG, equal to the parallelogram ABEF; but as every point of the lever must bear the whole impulse of the weight as before, it is plain, that as we approach towards the centre, that power is compressed into less and less space. Thus, when the weight has descended from B to G, though the large triangle ABG be equal to the parallelogram ABEF, yet the smaller triangle AIEI is equal only to one half of the parallelogram AIEI, which represents the power. The whole power being therefore compressed into half the space, must of necessity be double to what it was in the former case. In like manner, the triangle AIEI is only equal to one half of the parallelogram AIEI; and this parallelogram itself is only half the space representing the whole power of the weight. In this case, therefore, the power is confined within one fourth part of the space which it naturally has, and for that reason must be four times as great.

§ 2. Of the Wheel and Axle, or Axis in Peritrochic.

This power acts entirely on the principles of the lever, and has therefore sometimes been called a perpetual lever. In it the power is applied to the circumference of a wheel by means of a rope or otherwise, the weight to be raised being fastened to a rope which winds round the axis. It is represented fig. 38, where AB is the wheel, EDF its axle, P the moving power, and W the weight to be raised by means of the rope K coiling itself about the axis. It is plain then from an inspection of the figure, that when the large wheel has made one revolution, the weight P will have descended through a space equal to the circumference, and as much of the cord L, by which it is suspended, will be wound off. On the other hand, the weight W will have ascended only through a space equal to the circumference of the axle, and just so much of the rope K will be wound up upon it. As the circumference of the wheel, therefore, is to that of the axle, so will the velocity of the moving power be to that of the weight to be raised, and of consequence such will be the force of the machine: thus, if the circumference of the wheel be eight, ten, twelve, or any number of times as large as that of the axle, one pound applied to the circumference will counterbalance eight, ten, twelve, or more pounds, applied to the axle, and a small addition will raise it up.

The engines called cranes, for raising great weights, are no other than wheels of this kind. Sometimes they are moved by handles S, S, &c., placed on the circumference of the wheel, which is turned by mens hands, as is shown fig. 38. Sometimes the wheel is hollow, and furnished. Mechanical furnished with steps, on which a man, who is inclosed in the wheel, continually sets his feet, as if he was ascending a stair; and thus the wheel yielding to his weight turns round, and coils up the rope which raises the weight about its axis. When the crane is to be turned by men's hands, it may advantageously have cogs all round the circumference, in which a small trundle may be made to work and be turned by a winch. Thus the power of the man who works it will be greatly increased; for his strength will be augmented as many times as the number of revolutions of the winch exceeds that 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 the 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 six inches perpendicular distance from below the centre of the axle; let us suppose the wheel A.B., which is fixed on the axle, to have 80 cogs, and to be turned by means of a winch six inches long, fixed on the axis of a trundle of eight staves or rounds, working in the cogs of the wheel. Here it is plain that the winch and trundle would make ten revolutions for one of the wheel A.B., 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 then have no greater velocity than the edge of the wheel has, which is supposed to be ten times the velocity of the rising weight; so that in this case the acquisition of power 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 a sufficient length for any man to work with, the acquisition of power will be as 30 to 1; because the velocity of the handle would be 30 times as great as that of the rising weight, and the absolute force of any machine is exactly in proportion to the velocity of the weight raised by it.

We must always remember, whoever, that just as much time is lost in working the machine as there is power gained by it; for none of the mechanical powers are capable of gaining both power and velocity at the same time.

In all cranes, it is necessary to have a racked wheel, represented by 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 happen to let slip his hold. For want of this precaution, terrible accidents have sometimes happened to people inclosed in cranes, by their inadvertently missing a step.

§ 3. Of the Pulley.

The pulley is a single wheel of wood, brass, or iron, moveable upon an axis, and inclosed in a kind of case called its block, which admits of a rope to pass freely over the circumference of the pulley, in which also there is usually a groove to keep the rope from sliding, the axis being generally fixed in the block.

In some pulleys the block is fixed; in others moveable, and rises with the weight. Both these kinds are represented, fig. 39. AA shows a fixed pulley, with its block b. Over the wheel a string BB passes, to the extremities of which are fixed the two weights W and P. This pulley, however, though it changes the direction of a power, yet does not gain any advantage; for one of the weights must always descend as much as the other ascends, of consequence their velocities must always be equal; and when this is the case, there can be neither increase nor decrease of power. A single fixed pulley, therefore, though it may compare the weight of two bodies together, cannot be accounted in any respect a mechanical power. But if with a fixed pulley we combine a moveable one, or one, in which the block arises along with the wheel, we gain an increase of one half. Thus 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 wheel, it is plain that the half G of the cord bears one half of the weight W, and the half F the other. The hook H, therefore, which fulfils the half G of the cord, must therefore bear one half of the weight; and if the cord at F be drawn up, so that the pulley may be raised from D to C, the string will be extended to its whole length, all but that which goes under the wheel of the pulley D; but the weight or power P by which the string is thus drawn up, will have moved twice as far as the weight W which is drawn up: whence we see that only one pound at P will be requisite to counterpoise two pounds at W. If the upper and fixed block contain two pulleys, and the lower one U contain also two, the advantage gained by this combination will be as 4 to 1. Thus, if one end of the string KMOQ be fixed to a hook at I, and the string passes over the pulleys N and R, and under those L and P, the weight T of one pound will balance a weight W of four pounds, suspended by a hook from the moveable block, making allowance for the weight of the block itself. In like manner will the pulleys give an advantage of 4 to 1 when disposed as at X and Y; but in all cases the same relation between velocity and power is preserved as in the lever and axis in peritrochio, viz. if the power balances twice its own weight, it must move or have a tendency to move through twice the space; if it balances four times its weight, then it must have a tendency to move through four times the space that the other does.

Pulleys are of great use in practical mechanics, as by their means great weights may be raised to any height much more expeditiously than by any other method, and the smallness of their weight makes them very convenient for carriage. At sea they are used for hoisting the sails and yards, straining ropes, &c. Archimedes, by means of a machine composed of pulleys, is said to have drawn a ship along the strand, in the presence of Hiero king of Syracuse; but this is scarcely to be credited, on account of the great friction which attends this kind of machines. The friction arises from three causes: 1. The diameter of the axis bearing a considerable proportion to that of the wheels. 2. Their rubbing against their blocks, or against one another. 3. The stiffness of the rope that goes over and under them. All these causes Mechanical causes must necessarily be augmented, in proportion to the weight we have to overcome: and when we consider the immense resistance which a ship must make, with the strength and stiffness of the ropes necessary to overcome it, we can scarce suppose the strength of any individual equal to the task. Pulleys have often been used by inhuman tyrants, in constructing machines for torturing the objects of their cruelty.

The pulley has by some writers been reduced to the lever as well as the wheel and axis; in which method they consider the fixed pulleys as a lever of the first, and the moveable pulley as one of the second kind: but it is justly observed by Professor Hamilton, that the pulley cannot be with any propriety reduced to a lever; because, though both the moveable and immoveable pulleys should be taken away, the ropes would have to sustain the same weight that they do with the pulleys; nay, the very same advantages would be gained by the mere use of pins, without any wheels, were not the friction very great even upon the smoothest pins that could be made use of. It is, indeed, merely to avoid this resistance on the pins that wheels are made use of at all. The best method of computing the power, and explaining the reason of the effects of pulleys, is by considering that every moveable pulley hangs by two ropes equally stretched, each of which bears one half of the weight; and therefore, when the same rope goes round a number of fixed and moveable pulleys, since all its parts on each side of the pulleys are equally stretched, the whole weight must be equally divided amongst all the ropes by which the moveable pulleys hang; consequently, if the power which acts on one rope be equal to the weight divided by the number of ropes, that power will sustain the weight.

A very considerable improvement in the construction of pulleys has been made by Mr James White, who has obtained a patent for his invention, and of which the following description is given by the inventor.

Plate CLXXXVII. Fig. 68 shows the machine, consisting of two pulleys Q and R, one fixed and the other moveable. Each of these has six concentric grooves, capable of having a line put round them, and thus acting like as many different pulleys, having diameters equal to those of the grooves. Supposing then each of the grooves to be a distinct pulley, and that all their diameters were equal, it is evident that if the weight 144 were to be raised by pulling at S till the pulleys touched each other, the first pulley must receive that length of line as many times as there are parts of the line hanging between it and the lower pulley. In the present case, there are 12 lines, b, d, f, &c., hanging between the two pulleys, formed by its revolution about the fix upper and fix lower grooves. Hence as much line must pass over the uppermost pulley as is equal to twelve times the distance of the two. But, from an inspection of the figure, it is plain, that the second pulley cannot receive the full quantity of line by as much as is equal to the distance betwixt it and the first. In like manner, the third pulley receives less than the first by as much as is the distance between the first and third; and so on to the last, which receives only one twelfth of the whole. For this receives its share of line n from a fixed point in the upper frame, which gives it nothing; while all the others in the same frame receive the line partly by turning p.wen. to meet it, and partly by the line coming to meet them.

Supposing now these pulleys to be equal in size, and to move freely as the line determines them, it appears evident, from the nature of the system, that the number of their revolutions, and consequently their velocities, must be in proportion to the number of suspending parts that are between the fixed point above-mentioned and each pulley respectively. Thus the outermost pulley would go twelve times round in the time that the pulley under which the part n of the line, if equal to it, would revolve only once; and the intermediate times and velocities would be a series of arithmetical proportionals, of which, if the first number were 1, the last would always be equal to the whole number of terms. Since then the revolutions of equal and distinct pulleys are measured by their velocities, and that it is possible to find any proportion of velocity on a single body running on a centre, viz. by finding proportionate distances from that centre, it follows, that if the diameters of certain grooves in the same substance be exactly adapted to the above series (the line itself being supposed inelastic, and of no magnitude), the necessity of using several pulleys in each frame will be obviated, and with that form of the inconveniences to which the use of the pulley is liable.

In the figure referred to, the coils of rope by which the weight is supported are represented by the lines a, b, c, &c.; a is the line of friction, commonly called the fall, which passes over and under the proper grooves, until it is fastened to the upper frame just above n. In practice, however, the grooves are not arithmetical proportionals, nor can they be so; for the diameter of the rope employed must in all cases be deducted from each term; without which the smaller grooves, to which the said diameter bears a larger proportion than to the larger ones, will tend to rise and fall faster than they, and thus introduce worse defects than those which they were intended to obviate.

The principal advantage of this kind of pulley is, that it destroys lateral friction, and that kind of shaking motion which are so inconvenient in the common pulley. "And lest (says Mr White) this circumstance should give the idea of weaknesses, I would observe, that to have pins for the pulleys to run on, is not the only nor perhaps the best method; but that I sometimes use centres fixed to the pulleys, and revolving on a very short bearing in the side of the frame, by which strength is increased, and friction very much diminished; for to the last moment the motion of the pulley is perfectly circular: and this very circumstance is the cause of its not wearing out in the centre as soon as it would, assisted by the ever increasing irregularities of a galled bearing. These pulleys, when well executed, apply to jacks and other machines of that nature with peculiar advantage, both as to the time of going and their own durability; and it is possible to produce a system of pulleys of this kind of fix or eight parts only, and adapted to the pockets, which, by means of a skin of sewing silk, or a clue of common thread, will raise upwards of an hundred weight." § 4. Of the Inclined Plane.

This power is represented fig. 40; and the advantage gained by it is exactly in the proportion of the length of the plane to the perpendicular height of it. Thus, let A.B be a plane parallel to the horizon, and C.D one inclined to it; suppose also the whole length C.D to be three times as great as the perpendicular height G.F; in this case, the cylinder E will be supported upon the plane C.D, and kept from rolling down upon it, by one-third part of its weight. Were the length of the plane four times its height, it would be prevented from rolling down by one-fourth part of its weight. The force with which a rolling body descends upon an inclined plane will be to that with which it would descend by the power of gravity, as the height of the plane is to the length of it.—For, suppose the plane A.B (fig. 41.) to be parallel to the horizon, the cylinder C will keep at rest upon any part of the plane on which it is laid. If the plane be so elevated as in fig. 42. that its perpendicular height D be equal to one half of its length A.B, then the cylinder will roll down with half its weight; for it would require a power (acting in the direction A.B) equal to half its weight to keep it from rolling. If the plane be elevated so as to be perpendicular to the horizon, as in fig. 43. the cylinder C will descend with its whole force of gravity, because the plane contributes nothing to the support or hindrance of it; for which reason, it must require a power equal to the whole force of its gravity to keep it from descending.

If, as in fig. 44. the cylinder C be made to turn upon slender pivots in the frame D, which is furnished with a hook, with a line G fastened to it; if 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 L.M is to its length A.B; the weight will just support the cylinder, and a small force will make it either ascend or descend. In the time that the cylinder moves from A to B, it must rise through the whole height of the plane M.L, and the weight will descend from H to K, through a space equal to the whole length of the plane A.B. If the plane be now made to move upon rollers or wheels as in fig. 45. and the cylinder be supported upon it, the same power will draw the cylinder up the plane, provided the pivots of the wheels be small, and the wheels themselves pretty large. For let the machine A.B.C, equal in height and length to A.B.M, fig. 44. be furnished with four wheels, of which two are seen at D and E, the third being under C, while the fourth is concealed by the board a. Let the cylinder F be laid upon the lower end of the inclined plane C.B, and the line G be extended from the frame of the cylinder about five feet, parallel to the plane C.B, and fixed in that direction to a hook in the wall, which will keep the cylinder 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 to a weight K, the same which drew the cylinder up the plane before. If this line be put over the fixed pulley I, the weight K will draw the machine along the horizontal plane L., and under the cylinder F; and when the machine has been drawn the whole length C.B, the cylinder will be raised to d', equal to the perpendicular height A.B above the horizontal part at A.

The inclined plane, considered as a mechanical power, may easily be reduced to the lever; for the power acquired by it is always in the proportion of the length to the height, in the same manner as the inclined power acquired by a lever is in the proportion of the plane long arm to the short one. To compute, or show the reason of the power of an inclined plane, therefore, we have only to construct a lever, the long arm of which is equal to the length of the plane, and the short arm to the height of it; then, whatever weight put upon the long arm counterpoises another put upon the short one, will also keep the same weight from rolling down the inclined plane.

To the inclined plane belong also the wedge, and all cutting instruments which act as wedges, as knives, hatchets, &c. From the theory of the inclined plane also combined with that of falling bodies, we deduce some of the most remarkable properties of the pendulum. See Pendulum.

§ 5. Of the Wedge.

This may be considered as two equally inclined planes DEF and CEF, fig. 46. joined together at their bases E.F.O.: 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 or other matter to be split, by the force of a hammer or mallet striking perpendicularly upon its back. Thus, A.B fig. 47. is a wedge driven into the cleft C.E.D of the wood F.G.

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 either of its sides, and if the power be increased so as to overcome the friction of the wedge, and the resistance arising from the cohesion of the wood, the wedge will be driven in, and the wood split. But when the wood splits, as it commonly does, before the wedge, the power impelling the latter will not be to the resistance of the wood as half the thickness of the wedge is to one of its sides, but as half its thickness is to the length of the other side of the cleft, estimated from the top or acting part of the wedge; for if we suppose the wedge to be lengthened down 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 is the same thing, as the whole thickness of the wedge is to the length of both its sides.

To prove this, let us suppose the wedge is divided lengthwise into two equal parts; in which case, it will become two equally inclined planes, one of which, as abc fig. 48. may be made use of for separating the moulding cd from the wainscot A.B. It is evident, that when this half wedge has been driven its whole length ac between the wainscot and mouldings, its inside ac will be... Mechanical be at cd, and the moulding will be separated to f from the wainscot. But, from what has been already shown concerning the inclined plane, it appears, that, to have an equilibrium between the power impelling the half wedge and the resistance of the moulding, the former must be to the latter as ab to ac, that is, as the thickness of the back which receives the stroke is to the length of the side against which the moulding acts. Since, therefore, the power upon the half wedge is to the resistance against its side as the half back ab is to the whole side ac, it is plain that the power upon the whole wedge, where the whole thickness is double the half-back, must be to the resistance of both its sides as the thickness of the whole back is 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 latter splits at ever so small a distance before it, 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 the place where the whole resistance is accumulated, it is plain from the nature of the lever, that the farther the power is from the resistance, the greater advantage it acts with.

Some have supposed, that the power of the wedge was in the proportion of the thickness of it to the length of one of its sides; but from what has already been advanced, it is plain that this cannot be the case. The wedge, as has already been shown, is composed of two inclined planes, each of which has a perpendicular height of only one half the thickness of the wedge. As the power of the inclined plane therefore is always as the length to its perpendicular height, it is evident that the power of each of these inclined planes of which the wedge is composed must be as the length of one side to half the thickness; and consequently the power of both must be as the length of both sides is to the whole thickness.

The power of the wedge is exceedingly great, inasmuch that not only wood but rocks may be split by it, which could scarce be done by any of the other powers: but in this it is assisted by percussion of the hammer which drives it, and shatters the stone in a manner that could scarcely be done by any simple pressure.—Wedges as well as pulleys have also been used as instruments of torture.

§ 6. Of the Screw.

This is the strongest of all the mechanical powers, though it cannot be accounted a simple one, as no screw can be made use of without a lever or winch to assist in turning it. We may suppose it made by cutting a piece of paper into the form of an inclined plane or half wedge, and then wrapping it round a cylinder, as in fig. 40. From this figure it is evident, that the winch which turns the cylinder must move once round in the time that the paper describes one spiral; and consequently if any weight or greater power of resistance were applied, the winch must turn once round in the time that the weight would move from one spiral thread to another, from d to c for instance. Hence the force of the screw will be as the circumference of the circle defined, by the lever or winch by which it is turned, is to the distance between the threads of the screw itself. Thus, supposing the distance of the threads to be half an inch, and the length of the winch twelve inches, the circle described by the extremity of it where the power acts will be nearly 76 inches, or about 152 times the distance between the threads; whence a single pound acting at the end of such a winch would balance 152 at the extremity of the forew; and as much more as can overcome the friction would turn the winch and raise up the weight.

Fig. 50 represents a machine for exhibiting the force of the screw. Let the wheel C have upon its axis a screw ab, working in the teeth of the wheel D, which suppose to be 48 in number. It is plain that every time the forew ab and wheel C 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. If then the circumference of a circle described by the handle of the winch A be equal to the circumference of a groove e round the wheel D, the velocity of the handle will be 48 times as great as the velocity of any given point in the groove. Consequently if the line G goes round the groove e, and has a weight of 48 pounds hung to it below the pedestal EF, a power of one pound at the handle will balance that weight. If the line G goes round the axle I instead of the groove of the wheel D, the force of the machine will be as much increased as the circumference of the groove e is greater than that of the axle; which, supposing to be six times, then one pound at H will balance 288 pounds suspended by the line at the axle.

The screw is of very extensive use in mechanics, its great power rendering it more eligible for compressing bodies together than any of the rest, and the great disparity betwixt the velocity of the handle and that of the threads of the screw, rendering it proper for dividing space into an almost infinite number of parts. Hence, in the construction of many mathematical instruments, such as telescopes, where it is necessary to adjust the focus to the eyes of different people, the screw is always made use of in order to move the eyeglass a very little nearer or farther away from the object glass. In the 7th volume of the Philosophical Transactions, a new method of applying the screw, so as to make it act with the greatest accuracy, is delivered by Mr Hunter surgeon. The following are the general principles upon which this method depends.

1. That the strength of the several parts of the engine be adjusted in such a manner to the force they are intended to exert, that they shall not break under the weight they ought to counteract, nor yet encumber the motion by a greater quantity of matter than is necessary to give them a proper degree of strength.

2. That the increase of power by means of the machine be so regulated, that while the force we can exert is thereby rendered adequate to the effect, it may not be retarded in procuring it more than is absolutely necessary.

3. That the machine be as simple as is consistent with other conditions.

4. It ought to be as portable, and as little troublesome as possible in the application.

5. The moving power must be applied in such a manner as to act to the greatest advantage; and that Mechanical the motion ultimately produced may have that direction and velocity which is most adapted to the execution of the ultimate design of the machine.

6. Of two machines, equal in other respects, that deserves the preference in which the friction least diminishes the effect proposed by the whole.

To attain all these advantages in any machine is perhaps impossible; but in the application of the screw, the following method promises to be attended with several of them.

Let \( AB \) (fig. 51.) be a plate of metal, in which the screw \( CD \) plays, having a certain number of threads in an inch, suppose 10. Within the screw \( CD \) there is a female screw*, which receives the smaller screw \( DE \) of 11 threads in an inch. This screw is kept from moving about with the former by means of the apparatus at \( AFGB \). But if the handle \( CLK \) be turned ten times round the screw, \( CD \) will advance an inch upwards; and if we suppose the screw \( DE \) to move round along with \( CD \), the point \( E \) will advance an inch. If we now turn the screw \( DE \) ten times backward, the point \( E \) will move downwards \( \frac{1}{11} \) of an inch, and the result of both motions will be to lift the point \( E \) \( \frac{1}{11} \) of an inch upwards. But if, while the screw \( CD \) is turned ten times round, \( DE \) be kept from moving, the effect will be the same as if it had moved ten times round with \( CD \), and been ten times turned back; that is, it will advance \( \frac{1}{11} \) of an inch. At one turn, therefore, it will advance upwards \( \frac{1}{11} \times \frac{1}{11} = \frac{1}{121} \) of an inch. If now the handle be six inches long, the power to produce an equilibrium must be to the weight as 1 to \( \frac{110 \times 6.2832 \times 6}{4146.912} \). Thus the force of Mr Hunter's screw is greatly superior to that of the common one; for in order to have as great a power on the plan of the latter, it must have 110 threads in an inch, which would render it too weak to resist any considerable violence.

With regard to the second general maxim above laid down, Mr Hunter considers both kinds of screws as equally applicable, only that the more complicated structure, and consequently greater expense of his screw, renders it convenient to use the common screw where only a small increase of power is necessary, and his improved one where a great power is wanted. By shortening the handle also, the whole machine is rendered more portable and less troublesome in the using.

To answer the fifth intention, both seem to be equally proper; but for the sixth, the preference must be given to such as best answer the particular purpose proposed. Thus if the screw \( DE \) be designed to carry an index which must turn round at the same time that it rises upward, the common screw is preferable; though our author also proposes a method by which his screw may answer the same purpose: With this view a still smaller screw ought to play within the screw \( DE \), and be connected with the screw \( CD \), so as to move round along with it. It must have, according to the foregoing proportions, 111 threads in an inch; and they must lie in a contrary direction to those of \( CD \); so that when they are both turned together, and \( CD \) moves upwards, this other may move downwards. At one turn this will move upwards \( \frac{1}{111} \) part of an inch, and at the same time will move in a circular direction; but the accuracy required in constructing such screws, even though made with fewer threads than those just mentioned, would probably be too great for practice. In many cases, however, screws upon Mr Hunter's principles may be of considerable use.

The theory of the screw is easily deduced from that of the inclined plane and lever; for the threads of the screw in fact form a continued inclined plane, the height of which is the distance betwixt the two threads, and the length is the circumference of the cylinder. Hence, without any lever, the screw would have a considerable power, were it not for the great friction of the parts upon one another; and this friction would be much more increased by the perpendicular action of a weight on the top of the cylinder than by the horizontal action of a lever.

§ 7. Other methods of accumulating power, which do not properly come under the denomination of any of the mechanical powers already described.

From the account already given of the six mechanical powers, it is evident, that they can do no more than accumulate, or, if we may use the expression, compress, any degree of velocity into a small space. The velocity thus compressed, becomes what we call power, and is capable of again impressing the original degree of velocity upon a body of an equal or nearly equal size to the first which originally impressed it; but in every case the absolute quantity of motion, or of power, remains the same without a possibility of augmentation or diminution by levers, screws, pulleys, or wedges. It follows, therefore, that if by any method we can preserve for a certain time a small quantity of motion, that will at the end of the time specified amount to an astonishing power, which we could scarce at first have imagined to proceed from so small a cause. Thus, though a man cannot raise a ton weight from the ground at once, he is easily capable of raising 100 pounds at once from the ground, and this for a considerable number of times in succession. It is plain, therefore, that in a very short time a man could in this manner raise the ton weight, if it were divided into 20 parts, as effectually as by a lever or other machine; though the fatigue consequent upon slopping down and raising up his body so often would no doubt make the toil much greater. Even by means of a lever, however, before a man could raise a ton weight one foot from the ground, with the trouble of exerting a force equal to 100 pounds, he must have a lever 20 or 21 feet in length, and exert a constant force of 100 pounds, while he goes up through a space of 20 feet, or pulls down a rope through that space. The lever, therefore, only accumulates the power exerted in pulling or carrying the weight of 100 pounds through 20 feet, and discharges it all upon the space of one foot; whence it is plain, that any other thing which could do this would raise the ton weight as effectually as the lever.

One method of accumulating a great power is by suspending a very heavy body by a chain or strong rope of considerable length. This body may be put in motion by a very small degree of power more than is requisite for bending the rope, and will acquire a vibratory motion like a pendulum; by continuing the impulse as the body returns, it will continually acquire greater and greater force, the arches through which it moves becoming continually larger, until at last it might be made to overcome almost any obstacle: and upon this principle the battering rams of old were constructed, the power of which has been already mentioned; nevertheless the power of one stroke of this engine never could exceed the accumulated power of the impulses given to it in order to produce that stroke, or even quite equal it, because the stiffness of the rope, and the resilience of the air, must always take off something from it.

Another method of accumulating force is by means of a very heavy wheel or cylinder, moveable about an axis. A small force will be sufficient to put this wheel in motion; and, if long continued, will accumulate in such a manner as to produce such effects as raising weights and overcoming resistances, as could not by any means be accomplished by the application of the original moving force. On this subject Mr Atwood has demonstrated, that a force of 20 pounds applied for 37 seconds to the circumference of a cylinder of 10 feet radius, and weighing 4713 pounds, would, at the distance of one foot from the centre, give an impulse to a musket-ball equivalent to what it receives from a full charge of gun-powder. The same effect would be produced in six minutes and ten seconds by a man turning the cylinder with a winch one foot long, in which he constantly exerted a force of 20 pounds. In this case, however, as well as the former, there is not any absolute accumulation of power; for the cylinder has no principle of motion in itself, and cannot have more than it receives.

This accumulation of motion, however, in heavy wheels, is of great service in the construction of machines for various purposes, rendering them greatly more powerful and easy to be worked by animals, as well as more regular and steady, when let in motion by water, or any inanimate power. Hence the use of flies, bull's-wheels, &c. which are commonly supposed to increase the power of a machine, though in reality they take something from it, and act upon a quite different principle.—In all machines in which flies are used, a considerably greater force must at first be applied than what is necessary to move the machine without it, or the fly must have been set in motion some time before it is applied to the machine. It is this superfluous power which is collected by the fly, and serves as a kind of reservoir from which the machine may be supplied when the animal slackens his efforts. This, we must observe, will always be the case with animals, for none are able to exert a great power with absolute constancy; some intervals of rest, even though almost imperceptible, are requisite, otherwise the creature's strength would in a short time be entirely exhausted. When he begins to move the machine he is vigorous, and exerts a great power; in consequence of which he overcomes not only the resistance of the machine itself, but communicates a considerable degree of power to the fly. The machine, when moving, yields for a time to a smaller impulse; during which time the fly itself acts as a moving power, and the animal recovers the strength he had lost. By degrees, however, the motion of the machine decreases, and the animal is obliged to renew his efforts. The velocity of the machine would now be considerably increased, were it not that the fly now acts as a resisting power, and the greatest part of the superfluous motion is lodged in it, so that the increase of velocity in the machine is scarcely perceptible. Thus the animal has time to rest himself until the machine again requires an increased impulse, and so on alternately.—The case is the same with a machine moved by water, or by a weight; for tho' the strength of these does not exhaust itself like that of an animal, yet the yielding of the parts of the machine renders the impulse much less after it begins to move: hence its velocity is accelerated for some time, until the impulse becomes so small that the machine requires an increase of power to keep up the necessary motion. Hence the machine slackens its pace, the water meets with more resistance, and of consequence exerts its power more fully, and the machine recovers its velocity. But when a fly is added to the other parts, this acts first as a power of resistance, so that the machine cannot acquire the velocity it would otherwise do. When it next begins to yield to the pressure of the water, and the impulse of course to slacken, the fly communicates part of its motion to the other parts; so that if the machine be well made, there is very little difference in the velocity perceptible.—The truth of what is here advanced will easily be seen, from considering the inequality of motion in a clock when the pendulum is off, and how very regularly it goes when regulated by the pendulum, which here act as a fly.

Flies are particularly useful in any kind of work which is done by alternate strokes, as the lifting of large pebbles, pumping of water, &c. In this case the flies weight of the wheel employed is a principal object; and the method of calculating this is to compare it with the weight to be raised at each stroke of the machine. Thus, suppose it required to raise a pebble 30 pounds weight to the height of one foot 60 times in a minute: Let the diameter of the fly be seven feet, and suppose the pebble to be lifted once at every revolution of the fly; we must then consider what weight passing through 22 feet in a second will be equivalent to 30 pounds moving through one foot in a second. This will be $30 \div 22$ or $1\frac{4}{11}$ pounds. Were a fly of this kind to be applied, therefore, and the machine set a going, the fly would just be able to lift the pebble once after the moving power was withdrawn; but by increasing the weight of the fly to 10, 12, or 20 pounds, the machine when left to itself would make a considerable number of strokes, and be worked with much less labour than if no fly had been used, though no doubt at the first it would be found a considerable incumbrance to the motion. This is equally applicable to the action of pumps; but the weight which can be most advantageously given to a fly has never yet been determined by mechanics. It is certain, however, that the fly does not communicate any absolute increase of power to the machine; for if a man or other animal is not able to set any mechanical engine in motion without a fly, he will not be able to do it though a fly be applied, nor will he be able to keep it in motion though set a-going with a fly by means of a greater power. This may seem to be contradicted by the example of a common clock; for if the pendulum be once stopped, the weight is not able to set it in motion again, though it will keep it going when once put in motion by an external power. This, however, Mechanical depends not upon any insufficiency of the weight, but on the particular mechanism of the crown wheel; which is such, that when once the pendulum is stopped, it would require a much greater weight than that commonly applied to set it in motion; and if the usual weight was to act fairly, it would be more than sufficient to move all the machinery, and make the pendulum vibrate also with much greater force than it does.

§ 8. Of Friction.

The doctrine of friction, according to Mr Ferguson, may be summed up in the following manner:

1. When one body rests 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 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 an 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 \( \frac{1}{4} \)th of the weight that those are on the narrow side; and since four times the number, multiplied by \( \frac{1}{4} \)th of the weight, is equal to \( \frac{1}{4} \)th 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 \( \frac{1}{4} \)d 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 AB, if very smooth, and laid on two well polished supports CD (having been first oiled or greased), and then charged with the weight of two pounds in the two equal balls GH, 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, \( \frac{2}{3} \), \( \frac{2}{9} \), \( \frac{2}{27} \), &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 ABCD be a piece of brass of six ounces, Fig. 53, and EFGH 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 AC alone; but if AC 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 AB an upright item Fig. 56, or shaft, turning freely in the socket B fixed in the table or plane KLM; and AC, DE, two arms fixed in the said shaft, the latter of which, DE, 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 AC, it causes the lever DE 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 BG be equal to AC; for then the power applied to C, acting in a tangent to the circle CRS, that shall just move the weight G, will be equal to its friction. But if the weight be applied at F, because BF is greater than AC, 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 Mechanical 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 AB (fig. 54.) 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 CDE (fig. 57.), the velocity of the wheel's circumference will be the same with that of the gudgeon; but the velocity of the wheel's axis AB (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 \( \frac{1}{3} \)d part of the weight, and the gudgeon be to the cylinder as 1 : 10, they will reduce the friction to \( \frac{1}{30} \)th part; and if, again, the axis of the wheel be to the wheel as 1 : 10, the wheel will reduce the friction to \( \frac{1}{300} \)th 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.: because the asperities of its surface will lay hold on those of the roller, and move it likewise; and it is evident, that when the body is drawn against the prominent parts of the roller, they immediately give way, and make no resistance. 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 so receive no resistance from them. Hence the body is conveyed along without being lifted up, in the same manner as a wheel is moved by a pinion without any considerable resistance.

Sect. III. Of the Combinations of the Mechanical Powers.

From what has been already laid down concerning the mechanical powers in particular, we have seen that none of them is capable of augmenting the absolute force of any acting substance; and from thence we may justly conclude, that no combination of them can do so. In fact, these combinations are very often detrimental, and occasion a great loss of power by friction. This is the great obstacle in mechanics, and must always be greater in complex than in simple machines; and therefore the latter are always to be preferred, excepting where convenience requires some degree of complication. The lever being the simplest machine, and that attended with least friction, is always to be used where it is requisite to raise weights for a small way. It may likewise be used with propriety where bodies are to undergo a long continued degree of pressure, and where they yield but little. For this purpose the lever ought to be of the second kind, represented fig. 25, where one end being fixed at A, a weight may be put upon the other extremity B, and the body to be pressed put at 1, 2, or any of the intermediate divisions, according to the degree of pressure it is designed to undergo. This has the advantage of giving a long and very equable pressure, and is a very advantageous method of pressing cheese or other things which do not require a very great exertion of force. Where this is requisite we must employ wedges or screws; but both these have the disadvantage of slackening their pressure on the least yielding of the materials to be pressed. Wedges therefore require to be almost constantly driven, and screws to be turned by a lever, in order to produce a constant pressure. In oil mills the pressure is produced by wedges, which are constantly driven by great mallets lifted up by the force of the mill. Oil of sweet almonds is made by apothecaries in a press driven by a screw, and turned by a long lever assisted by a capstan.

Where it is necessary to have a considerable weight raised to some height, the pulley is the most useful power, but the friction is extremely great; the axis in peritrochoid combined with a single pulley will answer the purpose extremely well, and with less friction than any machine composed of pulleys alone. The best machines called cranes are generally combinations of these two; and are very much used, especially by the commercial people, for raising goods out of ships, drawing them up into warehouses, and for lowering them down. In these operations we must observe, that lowering goods is much more dangerous than raising them, on account of the vast increase of velocity which bodies acquire every moment by the power of gravity. In the construction of cranes, therefore, it is absolutely necessary to attend to this circumstance, and to guard against accidents. The following are recommended by Mr Ferguson: Fig. 2 shows one crane well calculated for the purposes just mentioned. 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-post c, and the strong pinion a 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 it not for these rollers, the gib would be quite unmanageable; for the moment it were turned ever so little towards any side, the weight K would begin to descend, because the rope would be shortened between the pulley I and axis G; and so the gib would be pulled violently to that side, and either be broken to pieces or break every thing that came in its way. These rollers must be placed so that the sides of them round which the rope bends may keep the middle of the bended part directly even with the centre of the hole in which the upper gudgeon of the gib turns in the beam f. The truer these rollers are placed, the easier the gib is managed, and the less apt to swing either way by the force of the weight K.

A ratchet-wheel Q is fixed upon the axis D, near the trundle E; and 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 to 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 u (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 backwards, and lets down the weight K. But if the weight pulls too hard against this man, another lays hold of the handle V, and by pulling it downward draws the gripe U close to the wheel Y, which by rubbing hard against the gripe hinders the too quick descent of the weight; and not only so, but even stops it at any time if required. By this means, heavy goods may be either raised or let down at pleasure, without any danger of hurting the men who work the engine.

When part of the goods are craned up, and the rope is to be let down for more, the catch R is first disengaged from the ratchet-wheel Q, by pulling the cord t; then the handle q is turned half round backwards, which, by the crank n n in the piece o, pulls down the frame b between the guides m and m (in which it slides in a groove), and so disengages the trundle B from the wheel C; and then the heavy hook a 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 r, 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 n n, raises up the frame b, 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 n n turns pretty stiff in the mortise near o, and stops again at 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 pp 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 staves or rounds, and the wheel C to have 78 spur-cogs; the trundle E to have 14 staves, and the wheel F 56 cogs: then, by multiplying the staves of the trundles, 13 and 14, into one another, their product will be 182; and by multiplying the cogs of the wheels, 78 and 56, into one another, their product will be 4368; and dividing 4368 by 182, the quotient will be 24: which shows that the winch A makes 24 turns for one turn of the wheel F and its axle G, on which the great rope or chain HIH winds. So that if the length or radius of the winch A were only equal to half the diameter of the great axle G, added to half the thickness of the rope H, the power of the crane would be as 24 to 1: but the radius of the winch being double the above length, it doubles the said power, and so makes it as 48 to 1: in which case, a man may raise 48 times as much weight by this engine as he could do by his natural strength without it, making proper allowance for the friction of the working parts. Two men may work at once, by having another winch on the opposite end of the axis of the trundle under B, and so make the power still double.

If this power be thought greater than what may be generally wanted, the wheels may be made with fewer cogs in proportion to the staves in the trundles; and so the power may be of whatever degree is judged to be requisite. But if the weight be so great as will require yet more power to raise it (suppose a double quantity), then the rope H may be put under a moveable pulley, as s, and the end of it tied to a hook in the gib at t; 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 so raised, this may be quickly done by men pushing the axle G round by the handspokes y, y, y, y; having first disengaged the trundle B from the wheel C; and then, this wheel will only act as a fly upon the wheel F; and the catch R will prevent its running back, if the men should inadvertently leave off pushing before the burden be unhooked from 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 y, y, y, y, whilst the man at A turns the winch.

We have only shown 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: manner: AA (fig. 53) is a great wheel turned by men walking within it at H. On the part C, of its axle BC, the great rope D is wound as the wheel turns; and this rope draws up goods in the same way as the rope HH does in the above-mentioned crane, the gib-work here being supposed to be of the same sort. But these cranes are very dangerous to the men in the wheel; for if any of the men should chance to fall, the burden will make the wheel run back and throw them all about within it; which often breaks their limbs, and sometimes kills them. The late ingenious Mr Padmore of Bristol (whose contrivance the fore-mentioned 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 staves 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 stopped from running back by the force of the weight, even if all the men in it should leave off walking. And by one man working at the winch I, or two men at the opposite winches when needful, the men in the wheel are much assisted, and much greater weights are raised, than could be by men only within the wheel. Mr Padmore put also a gripe-wheel G upon the axis of the trundle, which being pinched in the same manner as described in the former crane, heavy burdens may be let down without the least danger. And before this contrivance, the lowering of goods was always attended with the utmost danger to the men in the wheel; as every one must be sensible of who has seen such engines at work. And it is surprising that the masters of wharfs and cranes should be so regardless of the limbs, or even lives of their workmen, that, excepting the late Sir James Creed of Greenwich, and some gentlemen at Bristol, there is scarce an instance of any who has used this safe contrivance.

We shall describe here four new cranes invented and made by Mr Gottlieb of Houndsditch, London, who communicates them to the public as quite new in their principles, and more simple and useful than any hitherto contrived. Fig. 59 is a representation of a crane adapted for a large warehouse, where heavy goods are wanted to be drawn up from a cart or quay. One of this construction has lately been erected in Mr Camden's sugar-house, Old Gravel-Lane, London. Its operation is as follows: The horse yoked below at A turns the upright axis and the wheel B, which is about 6 feet in diameter; this turns a 3 feet wheel C, having an upright axis D through the floor turning with it, and carrying a 3 feet wheel E with perpendicular cogs. The wheel E turns two pinions F and G, the former of 8 inches in diameter, and the latter of 5 inches diameter, both fixed upon one axis. The pinion G turns a 3 feet wheel H, to which is fixed the barrel I and wheel W. The rope K winds round the barrel, and comes over the sheif-wheel L in the upper story, and the pulley M in the gib-head drawing up the goods suspended at the hook N.

By a mark made upon the rope at I, the man superintending the crane knows when the goods are raised enough for landing into the room; he then immediately pushes aside the upright piece O, disengages the Combina-lever P from it; and by putting it downwards, the action of the quarter pinion at Q raises up the pinion at G, and thereby unconnects it with the wheel H. To prevent the machinery now from running backwards, a ratchet-wheel R is fixed upon the wheel H, into which a click-catch S falls. This effectually prevents the wheels going backwards by the weight at N while the man above is employed in landing the goods. When the goods are brought into the store-room, the hook N is thrown out, and the man below, from the usual call, runs to the handle U, slides the pinion T into the wheel H, then turns back the ratchet-wheel R, and pushes back the click S, then slides back again the pinion T; and the wheel H and barrel I being thus at liberty, the hook N and rope run down by their own gravity, and fresh goods are attached; then again, from the usual call, the man pushes up the lever P, fixes it at O, places the click S into the teeth of the ratchet-wheel; and the whole machinery is again in action from the horse below, that keeps constantly going without being stopped at every short interval of the landing, storing, &c.

When the goods are to be carted off, and required to be let down only, it is performed without the horse, and in the following manner: The pinion G is disengaged from the wheel H by the lever P as before, and the pinion V of the fly-wheel is slipped into the teeth of the 2 feet wheel W. The goods being suspended at N, will act by the rope upon the wheel W and pinion V, thereby turning round the fly-wheel X; while the goods are thus descending, the man presses upon the lever Y; and bears against the wheel, making such a resistance as to be sufficient to allow the goods to descend with as gentle a degree of motion as may be necessary.

The hook N being taken from the goods, the man goes to the wheel W, and with his hands turns it round, which winds up the cord and hook in readiness for more goods, and so on as before. The pinions T' and V in this case are slipped out of the wheels H and W.

As the horse at A may likewise be used to turn other mill-work from a connection made with the main-wheel, and supposing that the crane is not wanted at the same time, it is readily disengaged by turning of the winch at Z; which, by the pinion a below, working into the teeth of the bar, and the wheel C which turns upon it, quite unconnects the wheel C from the crane.

It is therefore evident from what has been described, that this crane can be managed by two men only, and occasionally without a horse, when very heavy goods are not raised. All the necessary beams for fixing the machinery by, could not be represented in the figure without obscurity and confusion; but these being omitted, will not to the most ordinary mechanic render the general construction of the crane difficult to understand.

A new portable cellar crane is represented in fig. 60. New cellars which is very useful to wine-merchants, brewers, &c. in crane, drawing up and letting down casks full of wine, beer, &c. It saves the trouble and inconvenience of horses, and in many places can be used where horses could not. AA are two wooden props about 6 feet in height, and and jointed together like a ruler at E. They are connected to each other by an iron round bar C and wooden bar at the bottom D. The iron prongs EE fasten the uprights steadily to the edge of the cellar; F is the axis round which two ropes are coiled, the ends of which are fastened to the two clamps GG. On the axis F is fixed the iron wheel H of 3 feet in diameter: in the teeth of this works the pinion I of about 6 or 7 inches in diameter, and is turned by the handle at K.

It is evident, by a bare inspection of the figure, that when the two ropes are slung over the ends upon the barrel, either at the top or bottom of the cellar, that by turning of the winch K towards or from you, the barrel can be safely and expeditiously taken out or lowered down.

When the crane is done with, it shuts up by unscrewing the nut at B, taking the wheel and axis away out of the loops at L, and folding the sides at A together like a jointed rule; it may then be taken away in the cart or dray, or taken in the mens hands.

Fig. 61. represents a portable stone-crane mounted in a wooden frame and stage, which is judged to be very useful for loading and unloading carts with large heavy stones. It is moveable to any part of a stonewall or ground; the frame is sufficiently wide for a cart to draw under the crane, and at any time it may be taken to pieces.

The frame AAAA is made of wood, is about 9 or 10 feet high, and about 9 feet square. The wheels BB are of iron, and are about 3 feet in diameter, and the pinion D, that is fixed to the axis of the first wheel B, 8 inches in diameter, on the axis of the second wheel B, the axis round which the rope-coils is fixed.

Now the stones being corded and hooked at the end of the rope, it is very evident that the man at C will either raise or lower them as may be necessary, according as he turns the winch towards or from him, and in a safe and very easy manner.

Fig. 62. is a representation of a crane-carriage which Mr Gottlieb conceives to be very useful in moving large stones in quarries, where carts and horses cannot be conveniently or at all managed. Its principle is evidently clear from a bare view of the figure. It consists only of two sets of crane-wheels applied to the two sets of wheels belonging to the carriage; so that two men, one at each winch AA, turning the pinions and wheels round, shall act upon the carriage-wheels and move it along. By their both turning forwards or backwards, the carriage goes accordingly; but if they turn contrary-ways, the carriage will be turned round, or partly so, as may be wanted.

The pinion B is 6 inches in diameter, which turns the wheel C of 3 feet diameter, on the axis of which is fixed the pinion D of 1 foot diameter, which works into 2 wheels E, E, of 3 feet 6 inches diameter, that are fixed upon the carriage-wheels, and give motion to the whole machine.

The friction of the axle-trees of these machines may be considerably diminished, by applying an improved axle-tree invented by Mr Gottlieb, which he calls the anti-attrition axle-tree, and for which he has a patent. It is formed from a steel-roller, from 4 to 6 inches long, turning within a groove cut in the iron part of the axle; and the advantages discovered by experiments made by Mr Gottlieb will be seen by the small table subjoined. A section of this axle-tree is represented in fig. 65, where a is the axle-tree, b the groove, c the roller, d the cavity between the lower part of the tree and the box e. In figs. 66, 67, f represents the oil-vessel supplying it with oil, g the tube to convey the oil by, h the straps of ditto, i the fastening screws. Figs. 63, 64, give a side view of the axle.

Advantage of the anti-attrition axle-tree.

| Old axle-tree | Anti-attrition | |---------------|--------------| | Coaches | 60 | 19 | | Chariots, post-chaises, &c. | 49 | 17 | | Single horse chaises and chairs | 31 | 6 | | Waggon | 78 | 3 | | Drays for beer | 138 | 4 |

One general maxim to be kept in mind by all mechanics is, that whatever a machine gains in power it loses in time, even supposing friction were entirely out of the question. It must likewise be remembered, that in almost all cases where a machine gains by complication, it will lose one third by mere friction, unless its parts are made with an accuracy not to be expected. In some cases, however, a great power must be had; and in these we must have recourse to the most simple machines, which will lose only time, and but little power by friction; for the complicated ones waste both time and power to a great degree. There is not perhaps a better method of procuring a very great power than by combining a screw with a toothed wheel which acts as an axis in peritrochio, as is represented fig. 50; for by making the threads of the screw pretty close, and the diameter of the wheel large, we may increase the power almost to any degree we please, without any considerable increase of friction. In this case, where it can conveniently be done, it is better to increase the diameter of the wheel than to add another, for this augments the power without any sensible augmentation of the friction; and it is absolutely necessary to have the axle as small as can be made of sufficient strength to bear the weight. Archimedes is said to have boasted, that he could move the earth provided he could find a place to stand on; and Bishop Wilkins, that he could pull the strongest oaks up by the roots by means of a single horse-hair. But abstracting from the impossibility in the case of Archimedes, it does not appear that the bishop could more easily have fulfilled his task, on account of the immense friction of the machine he must have employed, and the stiffness of the great ropes which must have been bent in order to accomplish his purpose. To perform feats of this kind, a lever seems more likely than any thing; but the vast room it takes up, and the excessive length requisite to make it act with sufficient force, together with the vast weight it must necessarily have if made of the requisite strength, must easily convince us that all such extravagant boasts are vain, and that wherever great effects are to be accomplished, a great power must originally be applied. Wheel-carriages in general signify all kinds of machines furnished with wheels, for drawing great weights by means of the strength of animals or otherwise.

It is very probable, that in the infancy of the arts, sledges were used before wheels were invented, or at least before the application of them became very general. Homer mentions them as employed in bringing wood for the funeral of Patroclus; though it is not to be doubted that the Greeks at that time were acquainted with the use of wheels, as the same poet mentions them on all occasions when speaking of the war chariots of his heroes.

It is possible, therefore, that by the country people, for inferior purposes, the sledge might be employed, while wheel-carriages were confined to those of superior rank, or used only for war-chariots. It is not long ago indeed since sledges were used for certain purposes in our own country, notwithstanding the number of wheel-carriages used in it from time immemorial. In some of the cold countries, where ice is met with in great quantity, and the ground is covered with frozen snow for a great part of the year, sledges are still used, and run upon the smooth surfaces of these bodies with as great ease as wheels run upon the ordinary ground. Upon very smooth ice, indeed, or upon any body perfectly smooth, wheels would not turn at all; for the only reason why they turn in the ordinary way, is the continual inequality they meet with. If we suppose the wheels to be carried in the air, it is plain that they would not turn, there being nothing to put any part in motion more than another; and the same would be the case if we could suppose ice, or any other body, to be so smooth that it would give as little resistance as air. On common roads, however, the wheels meet with obstructions at the bottom, which retard that part; and in consequence of this the upper part moves forward, and a circulating motion immediately begins to take place. By means of this circulatory motion the friction becomes very much less than what it would be if the weight were drawn along the ground upon a sledge, inasmuch that, according to the computation of Dr Helsham, a four-wheeled carriage may be drawn with five times as much ease as one that slides upon the same surface as a sledge.

The structure of wheel-carriages is so generally known, that it is needless to describe them. In the construction of them, however, there are several particulars to be observed, which may render one method of construction preferable to another, though there may be a general similarity between one carriage and another. In order to ascertain the most proper method for constructing them, it will first be necessary to consider the obstacles which occur to their motion. These are,

1. The vis inertiae of matter.—This, though for a considerable time supposed to be a principle of mere inactivity, or resistance to any change of state from motion to rest in material bodies, is now almost exploded. Mr Anstice, in a late treatise on wheel-carriages, supposes the philosophers who maintain the existence of such a principle, to have mistaken Sir Isaac Newton and other great men. According to him, they meant no more by the vis inertiae of matter than a mere passiveness in it, by which it was disposed to abide in that state, either of rest or motion, in which it originally was; "whereby it alters not its state but in proportion to the quantity of power exerted against it. Thus, should a body of any given weight or quantity of matter, moving with a certain degree of velocity, strike another body at rest of the same weight, it would communicate half its motion to that body, and they would move together with the same velocity as the first; but this proceeds from no principle of the body at rest to resist motion, it does not destroy in the other more than it receives from it; therefore no motion is lost, it is only divided; and the two after division have a power equal to that of the one before it, with the whole velocity of motion. Indeed when we consider that the least degree of motion in any body, however small, will communicate some degree of it to the largest in the universe; and that, on the contrary, none but an equal degree of impetus can deprive a body of actual motion, and that immediately opposed to it: add to this, that since all matter within the reach of our observation, and by analogy we have reason to think it is in actual and rapid motion, impressed on it by its great Creator, and co-existent with it; we may conclude, that if matter do not affect, it is more liable to motion than to rest."

2. Friction. By this is meant the quantity of motion destroyed by bodies sliding over one another, and which is in proportion to the weights laid upon them. See Sect. II. § 8.

Friction depends not only upon the pressure made on the moving bodies, but on the inequalities on the surfaces upon which they move. For as the surfaces of even the most highly polished bodies have some inequalities, whenever two of them are pressed together, the inequalities of the one must enter, and in some degree accommodate themselves, to those of the other; and as the forms of these inequalities are of infinite variety, it is impossible to give any general description which can exactly answer to every one of them.

Mr Anstice supposes the varieties only to be of two kinds, which he thinks may not be very dissimilar to any that occur.

1. Let us imagine two sliding surfaces, when viewed through a microscope, to present such an appearance as is represented in fig. 69, in which A is the sliding body to be moved in the direction C D over the fixed body B. To effect this, it is evident, that either the teeth must be violently broken off, or a power applied to them sufficient to make them slide upon each other: on the principles of the inclined plane; in which case the friction must always be in proportion to the weight of the slider, and that with which it is loaded, without regard to the length or breadth of the bearing surface: for if only one pound rested upon one tooth, there would be no more but that pound to be lifted. If the pound rested upon two teeth, there would only be half a pound to be lifted over each, and so on to any number; but if we suppose the teeth to be of such a shape, that they cannot act as inclined planes, let them be ever so strong, we must calculate the friction in a different manner.

Let surfaces of this kind be represented by fig. 70. In which case it is evident, that instead of depending on the weight or pressure only, it will be in proportion to the number and strength of the teeth to lock- ed together; or, in other words, on the length and breadth of the rubbing surfaces. On this supposition the weight of the slider will have little or no effect in breaking the teeth, or hindering its being done by the power applied in the longitudinal direction; but if one tooth is to be broken, it will be necessary to apply twice that power to break two, thrice the power to break three, &c.

Hence it is evidently impossible to form any general rule concerning the friction which takes place on this principle. As experience, however, has shown that two bricks, or other bodies of that kind, are almost as easily drawn along a table when placed side by side, as when laid upon each other, it seems probable that such a locking of parts seldom occurs; and when it does, the obstacles are soon broken down. Yet it is certain, that some such thing must take place on all occasions, otherwise the wearing of bodies which rub upon one another could not happen.

From what has been said it must appear plain, that if a slider be laid upon an horizontal plane, it must remain at rest; though by a very small force, such as is barely sufficient to overcome the friction, it will be set in motion: because, on a plane quite horizontal, the motion of any body does not remove it in the least farther from the point to which it is attracted by the force of gravity. If the plane be inclined to the horizon, then, besides the power necessary to overcome the friction, it will be necessary to have one sufficient also to overcome that of gravity, by which it is determined to roll down the plane; the proportion of which is ascertained under Sect. II. § 4. The difficulty of raising great weights in this manner, however, where the ascent is steep, and the ways rough, must necessarily be so great, that sledges could not be used with any advantage, and therefore wheels are indispensable.

The advantage of wheels over sledges may be further understood from the following considerations. 1. A sledge, sliding over a plane, suffers a friction equivalent to the distance through which it moves; but if we apply to it an axle, the circumference of which is six inches, and that of the wheels eighteen feet, it is plain, that moving the carriage eighteen feet over the plane, the wheels will make but one revolution; and as there is no sliding of parts between the plane and the wheels but only a mere change of surface, no friction can take place there, the whole being transferred to the nave acting on the axle, so that the only sliding of parts has been betwixt the inside of the nave and the axle; which, if they fit one another exactly, is no more than six inches; and hence it is plain, that the friction must be reduced in the proportion of one to thirty-six. Another advantage is also gained, by having the surfaces confined to such a small extent; by which means they may be more easily kept smooth, and fitted to each other. The only inconvenience is the height of the wheel, which must, in all cases be added to that of the carriage itself.

It has been a matter of no little consideration, whether the wheels of a carriage ought to be small or large; and this subject Mr Anville has treated in a very particular manner. He observes, that in the overcoming of such obstacles as are commonly met with in roads, wheels act as mechanical powers, and therefore the size of the wheel must be regulated upon the principles of these powers. Thus, let the circle O T A G L, fig. 71, represent a wheel of four feet diameter, placed on the level P Q, and opposed in that line by the obstacle O, which is supposed to be 7.03 inches in height; the line in which the carriage is drawn being C T, parallel to the plane P Q. In this case the effort applied to the carriage is communicated to the nave of the wheel where it touches the axle. This part, therefore, represents the part of the lever to which the power is applied, and is the point C in the figure. As the turning point is that where the wheel touches the obstacle, that must represent the fulcrum of the lever; whence that arm of the lever will be represented by C O, which may be supposed a spoke of the wheel: and as the upright spoke C L is in the line which bears the whole weight from the axle, and in which it is to be lifted; hence that part of the circumference of the wheel which is between the fulcrum and the upright spoke bearing on it, must represent the arm of the lever which is to raise the weight. In this case neither the weight nor the power act at right angles to their respective arms of the lever; so that we must represent their powers by the imaginary lines M O and O N. As the length of O M, therefore, is to that of O N; so is the proportion required to the weight to balance it on the obstacle, when rising over it; and in this case the arms are equal, it is plain that the powers must be so likewise. Every obstacle, therefore, exceeding this height, which is as 7.03 to 48, will require a power acting parallel to the plane greater than the weight drawn; and every obstacle whose height bears a smaller proportion to that of the nave, must be overcome by a smaller power.

Again, let a wheel of four feet diameter be represented by the circle in fig. 72, and supposed to be moved along the plane P Q, and an obstacle of twelve inches height be placed before it, the real lever will then be represented by the lines L O C; which being reduced to the imaginary ones M O N, shows that the power is greater than the weight. By the same rule, if an obstacle of three inches be placed in the way of a wheel, as in fig. 73, the power required to move the wheel will be considerably less than the weight, though it is plain that the proportion of power must always be according to the size of the wheel, the height of the obstacle, and the direction in which the carriage is drawn. For instance, if the line of traction in fig. 73, be raised into the direction C S, the power required to move the carriage over it will be to the real weight as the line C O is to the line O N; and in consequence of thus altering the direction, we gain as much as the length of the line C O exceeds that of C N.

This view of the manner in which the wheels of whatever carriages act, will serve to elucidate the question, whether large or small wheels are preferable for carriages? In all cases, the circle fig. 74, represent a wheel of two feet diameter, and the obstacle in its way 7.03 inches in height; then will the true lever be represented by the lines C O L, to be reduced to the imaginary ones M O N. In this case, the power required to draw the carriage must be to its weight as N O is to O M, which is more than double; and thus the advantage of large wheels over small ones is evident. In this, however, as in all other cases where wheels act as mechanical nical powers, we must remember, that the same doctrine applies to them as to the powers themselves when used in any other manner, viz. that as much as we gain in power we lose in time; and therefore, though a wheel of twice the diameter may be raised over an obstacle of any given height with twice the ease that would be required for one of once the diameter, yet the large wheel would require twice the time to move over it that the small one does.

Hitherto we have considered the carriage as being drawn in a direction parallel, or nearly so, to the plane on which the wheels move, which line is supposed to be horizontal: but the case will be different when we suppose them to move upon an inclined plane; for then, even though the line of traction be parallel to the ascending plane, and though the wheels act as levers, we shall find that the action of the weight will increase with the power gained by the increase of size in the wheels; and consequently, that the increased size of the latter will be of no farther use than that of diminishing the friction, in the same manner as is done upon horizontal planes.

To illustrate this, suppose the larger circle in fig. 75, to represent a wheel of four feet diameter, and the smaller circle a wheel of only two, both of which are made to ascend the inclined plane LM, by powers applied in the directions GI and ES parallel to the elevation of the plane, which is 45 degrees; it will then be found, that by describing the lever as in the former case, though the arm of the lever to which the power is applied be double the length in the large wheel that it is in the small, the other is augmented in the same proportion. Neither will the powers be augmented by varying the direction of the line of traction; for while these are kept parallel to one another, their relative powers must always keep the same proportion to one another. The reason is obvious, viz. that when wheels of any dimension ascend or descend inclined planes of any regular elevation, the fulcrum of the lever contained in the wheels must be determined by that part of the wheel which touches the plane, and which must always be of a proportionate height both in large and small wheels. It is otherwise, however, with the fulcrum marked out by perpendicular or irregular obstacles upon the plane itself; for large wheels will always have the advantage over small wheels when these are presented, for the reasons already given. Indeed, when the wheel impinges perpendicularly upon an obstacle as high as the line of traction, it is plain that it cannot be drawn over it by any power whatever, unless the direction of the latter be altered.

From these considerations, our author draws the following conclusions: 1. That in a carriage placed upon an horizontal plane, nothing more is required to produce motion than to overcome the friction which takes place between it and the plane. 2. By the application of wheels to a carriage, the friction is lessened in the proportion of the diameters of the axles and hollow parts of the naves to those of the wheels. 3. In the draught of a carriage without wheels up a regular plain ascent, the friction must not only be overcome, but there is a power likewise to be applied sufficient to lift such a proportion of the weight of the carriage as the perpendicular part of the ascending plane bears to that portion of the plane. 4. If wheels of any size be applied to the carriage in such circumstances, they have only the advantage of lessening the friction; for though they really act as levers, yet as each arm of the lever is lengthened in proportion to the increase of size in the wheels, the power can be no farther augmented than as the ascent may act as a mechanical power for raising up the wheels, carriage, &c. to the top. 5. Large wheels have the advantage over small ones in overcoming obstacles, because they act as levers in proportion to their various sizes. 6. The line of traction, or that in the direction of which the carriage is drawn, should always, if possible, be parallel to that in which the plane lies; for when this is the case, the arm of the lever to which the power is applied will bear the longest proportion possible to the other. This always takes place when the line of traction is perpendicular to that spoke of the wheel which points to the obstacle. As it may not always be possible, however, to alter the direction of the line of traction to this position, it will be most proper to fix upon some medium betwixt that which commonly occurs and that which requires the greatest exertion to overcome the obstacle; that is, betwixt a level line and one rising perpendicular to the spoke of the wheel which points to the obstacle it is likely to meet with. The greater attention ought to be paid to this last, that all wheels, but especially small ones, are liable to sink into the ground over which they pass, and thus produce a constant obstacle to their own progress. The line of traction, it must also be observed, is not an imaginary one drawn from that part of the animal to which the traces or chains are attached to the axle of the wheel, but the real direction of the traces to whatever part of the carriage they are attached; for the effort will be instantly communicated in the same direction from one part of the carriage to all the rest, by reason of the whole being fastened together and in one piece.

Hitherto we have considered the whole weight of the carriages as bearing perpendicularly against the axles of the wheels: but as this cannot be done in chairs, carts, and other carriages having only two wheels, it will be necessary to have their centres, or transverse lines of gravity, as near to the ground as possible. To understand this, it must be premised, that the centre of gravity is that point of any body which if suspended will keep all the parts of the body at rest, let the body be placed in any situation we please. Thus the centre of gravity in a wheel or circle is the centre of the circumference, provided the substance of it be equably ponderous throughout. In like manner, the real centre of a globe coincides with the centre of gravity, provided the matter of which it is composed be equably ponderous. In a square, whether superficial or solid, the centre of gravity will be a point equally distant from all its sides; so that if the substance be equably heavy, it will be impossible to turn it into any position in which there will not be as much matter upon one side of the centre as upon the other; and in like manner, every figure, however irregular, has some point round which, if it be turned, as much matter will always be upon one side as on the other.

If now any body be supported by a transverse line passing not through the centre of gravity itself, but either either above or below it, the body can only be kept in equipoise while that line remains directly above or below the point; for if the body is moved forwards, as in two-wheeled carriages moving down hill, a greater part of the weight will be thrown forwards over the line of suspension than what remains behind it; and consequently this superfluous part must be borne by the animal which draws it. In ascending any height, just the reverse takes place; for thus a portion of the weight is thrown backwards, and will tend to lift up the animal altogether. The consequence of this is, not only that the creature must proceed with great pain, but that the friction on the nave and axle will be augmented by laying upon them a part of the animal's weight also. If the body be suspended above the centre of gravity, the effect, though the same in the main, will be reversed in the ascent and descent of hill, as long as the body is firmly attached to the shafts; but should the whole weight be suspended under the axle, independent of the shafts altogether, then it will always, whether ascending, descending, or moving horizontally, have the same effect as if hung directly by it.

Our author next proceeds to treat of a generally received opinion, that the disadvantages attending carriages suspended either above or below the centre of gravity are augmented by the height of the wheels. The reason given for this opinion is, that the hinder part of the load in ascending an hill, being thrown back, will overhang that part of a large wheel which touches the plane, much more than when a smaller wheel is used. Mr Antice, however, observes, that all the disadvantage, in either case, is expressed by the weight which, from its action upon the axle, tends to lift the animal, which must always be the same whether the wheels are high or low. Thus, in fig. 76, let a carriage be represented with two wheels of four feet diameter, ascending a plane of 35° elevation from the level L.E. Let fig. 77, represent a carriage exactly in the same circumstances with the former, only that the wheels are six feet in diameter. Let C be the centre of gravity, and SP the line of gravity parallel to the central line AR, the line of support or suspension; in each of these the body is thrown so far back by its position, that the space GS and AR is taken from before the line of gravity, and added to the part behind it. Hence a certain part of the animal's weight must be exerted upon the shafts, in order to balance that of the carriage, which is thus thrown back, and which, as is evident from the figures, must be the same in both carriages, though the wheels of the one so much exceed those of the other in size, and the point T, where the wheel touches the plane, is much farther from the line of suspension in the large wheel than in the small one.

To remedy the inconvenience which must arise from placing the centre of gravity in the carriage low enough with respect to the wheels, it will be best to apply three or four wheels, placing them in such a manner that the line of gravity may always fall between the wheels, in whatever situation the carriage may probably be placed. Thus if the body A, fig. 78, be placed on four wheels, the axes of which are at B and C, it will be entirely supported between them, though more by C than B, even though the carriage should be ascending an hill as steep as HI, viz. 50 degrees, which cannot ever happen in practice. Even in this case the animal would have no occasion to make exertions for preserving the balance of the carriage, though, had it been supported only by the axles of two wheels at S, far the greater part of the weight of the carriage would have been thrown behind, and the equilibrium could not have been preserved without the greatest difficulty. Hence it is plain, that the greater the distance between the axles or three or four wheels applied to a carriage, the less liable will it be to have the line of gravity thrown out of its proper direction; but as this distance greatly augments the difficulty in turning a carriage, some medium is to be observed in this as well as other things.

What has been just now observed with regard to the preserving the balance of a carriage longitudinally, applies equally to the preventing it from being overturned laterally upon uneven roads, or such as have been turned one side much higher than the other. In order to this, we must take care to keep the line of gravity so far within the body of the carriage that it cannot be thrown out of it by any ordinary declivity of the road upon one side more than another. In the present case, however, as the wheels are not moveable on an axle in a lateral direction, we must consider the points of suspension to be those where the wheels touch the ground. Thus, let fig. 79, represent the cross section of a carriage moving upon two wheels; let C be its centre of gravity: it is plain, that in the position there represented, each of the points A and B sustains an equal share of the weight, and must do so as long as the carriage moves upon level ground: but if it be drawn along a road one side of which is higher than the other, such as is represented fig. 80, then the centre of gravity, and consequently the whole weight of the carriage, will bear upon the point of the wheel B, with this additional inconvenience, that the pressure does not lie perpendicularly but somewhat obliquely, by which the wheel is in great danger of being broken. To avoid inconveniences of this kind, the points of bearing upon the wheels are removed to a greater distance than the exact perpendicular, and this is called dishing the wheels; the good effects of which are evident from the figure. The wheels are dished by inserting the spokes into the naves in such a manner that they may decline every way from the carriage. Some disadvantage, however, attends this contrivance, for the carriage thus takes up more room upon the road, which makes it more unmanageable; and when it moves upon plain ground, the spokes not only do not bear perpendicularly, by which means their strength is lessened, but the friction upon the nave and axle is made unequal, and the more so the more that the wheels are dished. To obviate these inconveniences, some have bent downwards the ends of the axles; but thus the good effects of the dish is entirely lost, for the wheels are thereby thrown erect, and the breadth of the dish doubly increased on the upper part of the carriage.

The practice of bending forward the ends of the axle is still worse; for thus the wheels are thrown out of that parallel direction which they should always preserve on the ground, and likewise increases the friction both on the shoulders of the axles, and likewise... wife on the ground; for the wheels, by rolling in this position, would soon come together if not prevented by the shoulders of the axles; whence in every revolution they must rub with considerable force upon the ground.

The power of wheels can only be augmented in two ways. 1. By increasing the length of that arm of the lever to which the power is applied; and, 2. By diminishing the friction betwixt the nave and axle. The former is only a temporary expedient in case of any obstacle which cannot be surmounted in the ordinary way. It is accomplished, by transferring the action of the animal's power from the centre to the upper part of the circumference of the wheel: thus the power of the lever will be nearly doubled, as is shown from fig. 71, for if the power be applied to the wheel at A, then the arm of the lever would be represented by the dotted line AO instead of CO; and the former being nearly twice as long as the latter, their powers must be in the same proportion. It is evident, however, that this mode of applying the animal's power can only be useful in any sudden emergency; for were we to attempt to reduce it into practice by winding a rope or chain about the circumference of the wheel, the animal must move twice as fast as the carriage. See this also exemplified in Plate ccLXXXV., fig. 58, where the moving power is represented by the weight P; the wheel EF turning between two toothed planes AB and CD. Here it is evident, that while one of the small divisions ca, ae, &c., moves forward its own length, the plane A must do the same, while the centre, by the motion of which only that of the wheel can be measured, moves but through half the space.

2. With respect to friction-wheels or rollers, the case is different; and we may apply these in as great numbers, and in as great a variety of ways as we please, without fear of inconvenience. The best method of applying them, according to Mr Anslee, is to have the wheels and axle fixed to one another, so that both may turn together. Two friction wheels a little overlapping each other, must then be fixed on each side of the body of the carriage, so that it may bear on the axle in the intersection of the wheels, as is represented in fig. 81. Here ABCD represents the body of the carriage, the large circle one of the wheels fixed to the axle E. The circumference of each of the friction-wheels F and G is supposed to be three feet, and that of their axles three inches. As the large wheel then revolves by the motion of the carriage, and thus transfers the friction from its circumference to its axle; so the friction of the axle itself is now transferred from the circumference of the friction-wheels to their axles. Every revolution of the great wheel, therefore, during which it passes over 18 feet of ground by means of the motion of the axle, puts the lesser wheels round one sixth part of their circle; and consequently their axles are moved through the same part of their circumference, the friction being thus reduced to that upon this small part; which being no more than half an inch, becomes 432 times less than it would have been on the large wheel without any motion on an axle, and 12 times less by means of the friction-wheels than without them. The axles on both sides indeed are in motion, but the calculation must be made as if only one moved; for the greater number of wheels there are, the more will the friction be divided among them.

An objection of considerable weight arises to this method of fixing the wheels and axles together, that thus the wheels are prevented from moving with different velocities as they ought to do, when the carriage moves out of a right line; but this may be obviated by leaving the friction wheels loose upon their axles, by which means they will be at liberty to move with different velocities, at the same time that they will have the advantages of friction-wheels always as to one wheel of the carriage, and generally as to both.—The whole contrivance, however, seems likely to be entirely superseded by the following one of Mr Gamett of Bristol, who has obtained a patent for it. The general principle on which he proceeds is this. Between the axle and nave a hollow space is left to be filled up by friction-fold equal rollers nearly touching each other. These are furnished with axles inserted into a circular ring at each end, by which their relative distances are preserved; and they are kept parallel by means of wires fastened to the rings between the rollers, and which are rivetted to them.

To understand the effect of this machinery we must consider, that if, when plane surfaces move with a roller between them, if the under one be fixed, the upper plane will put the rollers forward but with half the quantity of its own motion. This is owing to the reaction of the stationary plane, which causes the roller to move backward upon itself as much as the other causes it to move forward upon itself. Thus, let CD, fig. 82, be a fixed surface, and AB a movable one, with a roller E between them; if B be moved forward to G, it will cause the roller to move to F, which is but half the distance that AB has moved; because it has rolled in a retrograde direction as far against the surface BA as it has gone forward upon the other. This is entirely owing to the resistance it meets with from CD; for if it did not touch that surface, but was attached by any other means to AB, it would be carried along with it through the whole space without any rolling motion. Hence it is clear, that if a roller be placed between the axle and nave of a wheel, and the latter be turned round, the roller will move with a retrograde motion upon the axle; and in order to carry it quite round, the nave must be turned, as much beyond a whole revolution as is equal on its inner circumference to the whole circumference of the axle. To exemplify this, let ABCD, fig. 83, represent the nave of the wheel E, the inner circumference of which is 18 inches, and the axle so small that it may be considered as a point. Let F and G be two rollers closely fitted between them: if then the wheel be turned round, the rollers will also be carried along, with it round the point which we consider as an axle; for there can neither be rolling nor friction against a mere point. But if the axle be of any sensible size, for instance one inch circumference, then must each roller move round by the motion of the nave against it, and the resistance of the angle on the opposite side. But in order to do this, it must roll in a retrograde direction upon the nave, and consequently the latter must go as far beyond a revolution as is equal to the circumference of the axle upon it, before the roller can go once round the axle, which in this case is by one one 18th part of the circumference. Should the circumference of the axle be nine inches, and that of the inner part of the nave remain as before, the wheel must perform one revolution and an half before the roller could be moved once round, and so on in the same proportion: but as the circumference of an axle must always be less than the inner part of a nave turning upon it with rollers between them, it never can amount to two revolutions of the wheel round the axle, however nearly it may approach to it; for no segment of a circle can ever be a straight line.

It will now be apparent, that if several rollers be placed all round between the nave and axle, whichever way the wheel be turned there cannot be any real friction, but merely a rolling of the rollers. If likewise these rollers be all of one size, and very nicely fitted to the cavity, they will keep their places without shifting, and very effectually answer the purpose of destroying friction. As such rollers, however, were very liable to be displaced by accident, the use of them was neglected, till Mr Gamett suggested the improvement already mentioned, and which is represented in fig. 84. Here ABCD represents a piece of metal to be inserted into the nave of a wheel, of which E is the axle, and 1, 1, 1, &c. rollers of metal having axes inserted into the brazen circle which passes thro' their centres; and both circles being rivetted together by means of bolts passing between the rollers from one side of the nave to the other; and thus they are always kept separate and parallel. By this method, indeed, some friction unavoidably takes place betwixt the axles of the rollers and their sockets in the brassings; but as the quantity of friction depends principally on the force by which the rubbing surfaces are pressed upon each other, and as in this case there is but the slight pressure occasioned by those accidental circumstances which would bring the rollers together, the friction must be too trifling to be noticed.

Thus far with regard to wheel-carriages in general. We must now make some remarks on the methods of drawing them, and the construction of particular carriages.—Men, by reason of their upright form, are by no means fitted for horizontal draughts; but animals who go upon all fours are remarkably so. In Britain horses are commonly made use of; but mules, oxen, sheep, and dogs, in other parts of the world. In all animals, however, the capacity for drawing a load depends upon their weight as well as their absolute strength. Thus it may happen, that a very heavy horse will draw a load, which a lighter though stronger one could not move; and this will always happen, when the weaker horse exceeds the other in weight more than he is exceeded by him in strength. It is well known that the weight, as far as it goes, reacts upon the horse, and pulls him back as much as he pulls it forward, until the exertions of the muscles of the animal resisted, by the solid ground, overcome the resistance of the load upon the moveable wheels, and it goes forward in proportion to the excess of the one power over the other. If the horse were put upon a moveable plane, and attempted to draw a load upon the solid ground, instead of pulling it forward he would pull himself back.—The horse has two sources of power in drawing a load, viz. his strength and weight. The former is the source of velocity; and as we find the actual power of any inanimate body in motion by multiplying the velocity into its quantity of matter, so do we find the power of a horse to draw a load, by considering his weight as well as absolute strength. There are even many instances in common practice, where it is useful to increase the weight of an horse or other animal; and therefore when horses are employed to draw mills, it is usual to put a small load upon their backs in order to increase their absolute momentum. Where the animals are equal in strength and momentum, however, the only difference that can take place in the weights they draw must arise from the convenience or inconvenience of the carriages to which they are yoked, or of the roads upon which they walk. A load breast-high is much more easily drawn than one which is dragged along the ground, because the power of the animal is then exerted directly against it; and this holds good whether the horses go up or down hill. In descending, indeed, as the load is then higher with regard to the horse than when it is on a plane, he will consequently pull it with the greater force; but in this case, its own gravity conspires with the draught, and will likewise help the load to descend; so that in this case the animal has an opportunity of exerting his greatest power when there is the least necessity, nay, when it is often inconvenient.

In all carriages with four wheels the two fore ones are made of a much smaller size than the hind ones, both for the sake of turning more easily, and likewise that there may be no danger of cutting the braces; but were both the fore and hind wheels to be of the same height, the carriage would be drawn with much greater ease. It is imagined indeed by the drivers of carriages, that the high hind wheels push on the fore-wheels; but this is evidently absurd; for the fore-wheels must turn as many times round oftener than the large ones as the latter exceed them in size. Thus, if we suppose the circumference of the large wheels to be 18 feet, and that of the small ones only 6, it is evident that the latter must turn round three times for once that the large ones turn round. Supposing the carriage therefore to be loaded equally on both axles, it is plain that by the greater friction upon the fore-axle than the other, it must wear out much sooner, and that as much as the fore-wheels are smaller than the hind ones. But it is the universal practice of those conversant in loading and driving carriages, to put a much greater load upon the fore than the back axle. Thus the friction not only becomes greatest where it ought to be least, but the small wheels must necessarily sink deeper into the ground than the large ones, which they are at any rate inclined to do from their size. The only danger in laying the greatest load upon the hind axle is, when the carriage goes up a very steep ascent; but in the few cases in which this may happen, a small temporary weight laid upon the pole betwixt the horses would prevent all danger of overfetting.

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 anyhow 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 shows 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 shows, 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 to 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 shown by the following plain and easy experiment.

Let one end of a piece of pack-thread be fastened to a brick, and the other end to a common scale for wheel-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 shows 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 constantly sinking into the road, they not only prove very destructive to the highways over which the carriages move, but by reason of wheels, this very sinking, they must be accounted as going continually up hill in some degree, even when drawn upon plain ground. These inconveniences are obviated by the use of broad wheels; and indeed the utility of these is so obvious, that it seems surprising how the use of narrow wheels is on any occasion permitted by the legislature. The wheels ordinarily used for waggons are nine inches broad; but of late a practice has been introduced of using rollers 16 inches broad; by which the inconveniences of the narrow wheels are removed, and the greatest weights may be drawn over the very worst roads, not only without making them worse, but greatly to their improvement. It has been objected, that broad wheels soon accumulate in clayey roads to such matter that it would soon equal an ordinary load; but, not to mention that such roads ought to have no existence in a country where such sums are annually paid for their reparation, it is evident, that passing heavy rollers over them is the only method to give that firmness to clay which is necessary for its supporting the animals who walk over it; and indeed many of the roads in this country, by reason of the continual poaching by wheels and feet of horses, &c. become throughout a great part of the year almost impassable by people on foot. The legislature appear to be very sensible of the advantages derived from these rollers, and accordingly allow such carriages as are furnished with them to go toll-free.

In the transactions of the Royal Irish Academy for 1758, we meet with some curious observations on the subject of wheel-carriages, by Mr Lowell Edgeworth. This gentleman informs us, that he was present in London in 1773, at a set of experiments tried in order to determine the comparative advantages of low and high wheels. The apparatus for these experiments was constructed with the greatest accuracy. The carriages themselves were made by the best workmen in London, and they were drawn along a smooth table by silk strings of small diameters put over a pulley nicely constructed, and fitted up in such a manner as to have scarce any friction. On applying a weight to the end of the string which passed over the pulley, little difference appeared in the velocities with which the carriages passed along the table, whether the wheels were high or low; but what appeared surprising was, that when obstacles were put in their way, sometimes the high and sometimes the low wheels had the advantage, according to the different shapes and sizes of the obstacles. "It appears at first view (says Mr Lovell), that the force which drew these carriages was employed only in overcoming the friction of the axle-tree, or in lifting the weight over the obstacle. But I suspected at the time, and have since been convinced, that an obstruction of another sort existed more considerable than either of these which I have mentioned, and which has not to my knowledge been taken notice of by any writer upon mechanics."

This obstruction is no other than the vis inertiae of matter, which has so much engaged the attention of philosophers, and the non-entity of which, in distinction from the power of gravity, seems now to be pretty generally admitted. The argument used for its existence by Mr Edgeworth is as follows: "After a carriage has once been set in motion upon a smooth road with any given velocity, its motion, so long as that velocity is continued, is neither retarded nor promoted by the vis inertiae; but whenever it passes over any height, not only the weight of the carriage must be lifted up, but the vis inertiae of that weight must be overcome in a new direction; and as much velocity must be communicated to it in that new direction as will enable it to rise to the height of the obstacle while it passes over its base. When an obstacle is of such a size and shape that a wheel of six feet diameter must strike the top of it at once, and not roll from the bottom upwards, and when its shape will permit a smaller wheel to touch it during its whole ascent, as there is more time allowed for overcoming the vis inertiae of its weight in the latter case than in the former, the smaller wheel may be drawn forward by a less power than the larger, notwithstanding the advantage of a lever, which is in favour of the larger wheel."

To determine this, our author made use of an inclined plane five or six feet long and one foot high, placed on a smooth horizontal floor. He then affixed the distance to which the carriage was driven on the floor by the velocity acquired in descending the plane, as a measure of the force with which it could overcome any obstacle placed in its way; and consequently the diminution of the distance was the measure of the resistance itself. Not satisfied with this apparatus, however, he screwed a circle of iron three feet three inches in diameter upon a solid floor. In the centre of this circle he erected an upright axis or roller upon two pivots, one resting in a socket of brass upon the floor, the other in a bridge raised across the machine. Around the axis was wound a small silk cord, with a scale and weights fastened to it, which passed over a pulley into an adjoining hair-cage, and turned the axis with a determined velocity. An horizontal arm of wood extended from the axis to the circumference of the inner circle, and to the extremity of the arm was fastened a piece of steel in form of the axle-tree of a carriage, having a wheel upon it, which by the motion of the axis was carried round upon its edge like the stone of a tanner's mill. The arm was furnished with an hinge, by means of which the wheel could rise up and pass over any obstacle which stood in its way. Above this was another arm, having on its extremity a tin vane, which by its resistance to the air regulated the motion of the machine. On putting weights into the scale, it was found that eight or ten turns were necessary to give the wheel an uniform velocity, which was preserved in all the experiments, any resistance thrown in the way being overcome by an addition of weight, and consequently this addition being always an accurate measure of the resistance.

On loading the wheel so as to weigh about four pounds, it acquired a velocity of ten feet in a second by nearly five pounds and an half; but on placing in its way an obstacle only a quarter of an inch high, six pounds and an half were required to cause the wheel pass over it. Two such obstacles required fourteen and an half pounds; but on substituting two obstacles of the same height, but making an inclined plane three quarters of an inch long, it required only two pounds to overcome their resistance. "The difference therefore (says he) between two and fourteen, must be attributed to the vis inertiae; for the velocities of the carriage and the heights of the obstacles remaining the same, the only difference that exists is, that in the one case the wheel has much more time to surmount the obstacle than in the other, and consequently had much less vis inertiae."

On this piece of reasoning, however, it is impossible to avoid making the following remark, viz. that nothing happens but what ought to do so upon the common principles of mechanics. One obstacle, when upright, required six pounds and an half to overcome it; but when an inclined plane three times the length was added to it, it ought then to have been overcome by a third part of the power, that is, by something more than two pounds; and the reason why something less than the third part was required, seems to have been the advantage the wheel had by acting as a lever; as has been already observed on the principles of Mr Anville. There is not therefore the least occasion to apply to a vis inertiae, or any obscure principle, for a solution of what may so easily be solved upon the common principles of mechanics and gravity.

Mr Edgeworth concludes his observations with some remarks on the use of springs, which are found greatly to facilitate the draught of carriages. "Whatever (says he) permits the load to rise gradually over an obstacle without obstructing the velocity of the carriage, will tend to facilitate its draught; and the application of springs has this effect to a very considerable degree: the same weight of four pounds being drawn over the same obstacles, when springs were put between the load and the carriage, by four pounds instead of 14. This remarkable difference points out the great advantage of springs in rough roads; an advantage which might be obtained for heavy waggons, as well as for other carriages, by a judicious application of the same means.

"It appears from the Memoirs of the French academy, that the idea of applying springs to carriages had occurred to M. Thomas in the year 1793; who has given a drawing of a carriage constructed upon this principle many years before it was attempted to be put in execution. So little expectation had he of success, that he expressly mentions it as a theory which could not be reduced to practice: he had, however, no notion of applying springs to facilitate the draught, but..." but merely for the convenience of the rider; and I apprehend that it is not at present commonly imagined that springs are advantageous for this purpose; nor would it at first sight appear credible, that, upon a rough paved road, such as are common in Cheshire and other parts of England, a pair of horses could draw a carriage mounted upon springs with greater ease and expedition than four could draw the same carriage if the springs and braces were removed, and the carriage bolted fast down to the perch."

Mr Lovell made also some experiments with high and low, long and short, carriages, in order to determine which was the most advantageous, but could not recollect the particular results of each experiment. He was, however, assured, that the preference lately given in England to high carriages is ill-founded; and that, though in smooth roads, the height of the carriage is a matter of indifference, yet in rough roads it is very disadvantageous. The length of carriages also, if their weight be not increased, is a matter of indifference, except in very uneven roads, and where there are deep ruts; long carriages being preferable in the former case, and short ones in the latter.

The reason why springs so much facilitate the draught of carriages seems to be, not only that they allow the wheels to pass more gradually over the obstacles, as Mr Edgeworth says, but that by their elasticity they make the carriage bound upwards every moment for a small way. Thus its gravity is for that moment in a great measure counteracted, and the progressive motion which it has already acquired is at liberty to act more freely in pushing it forward; for were it possible very suddenly to take away the horses from a carriage mounted on springs, and moving with considerable velocity, it would continue for sometime to move of itself; the weight in this case acting as a fly upon any mechanical engine, by means of which the machine accumulates a certain quantity of power, and will keep itself in motion for a considerable time after the hand is taken away from it. The weight of all carriages indeed has some effect of this kind, otherwise the draught would require an intolerable exertion of strength; and it is to be observed, that this tendency to proceed in the direction in which it is once set a-going, is remarkable in all great quantities of matter, and very perceptible even when weights are pulled directly upward; for in raising great weights by a crane, the burden is lifted with considerably more ease when near the top than at bottom, even after making every necessary allowance for the weight of the rope, &c.

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. 86.) 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 a sail as 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 kind 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 is 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 fledge 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 fail 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, the wind are here supposed to be expanded by the wind; R is Fig. 83. the rudder by which it is guided. Therefore the wind driving

(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. 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 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 A.B, at the bottom of which is to be an immoveable 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 H.I. 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 Q.R 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 swivels, in the same manner as the rolling lamp or the sea-com-

pats, 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 everyone 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."

**Sect. V. Of Mills.**

Mill, in the proper sense of the word, signifies a machine for grinding corn, though, in a more general sense, it is applied to all machines which have an horizontal circular motion. Mills are distinguished by particular names, sometimes taken from the powers by which they are moved, and sometimes from the uses to which they are applied. Hence they are called hand-mills, horse-mills, water-mills, fulling-mills, windmills, corn-mills, levigating mills, boring-mills, &c.

The most simple of these is the hand mill, represented fig. 901, where A and B represent the two stones between which the corn is ground, and of which the upper one A turns round, but the lower one (B) remains... The upper stone is five inches thick, and 21 inches broad; the lower one somewhat broader. C is a cog-wheel, having 16 or 18 cogs, which go into the trundle F, having nine spokes fixed to the axis G, the latter being firmly inserted into the upper stone A, by means of a piece of iron. H is the hopper into which the corn is put; I the flake to carry it by little and little through a hole at K, in betwixt the stones, where being ground into meal, it comes out through the eye at L. Both stones are inclosed in a circular wooden case, of such a size as will admit the upper one to run freely within it.—The under surface of the upper stone is cut into grooves, as represented at Q, which enable it to throw the meal out at the eye L more perfectly than could be done if it was quite plain. Neither of them are entirely flat, the upper one being somewhat concave, and the under one convex. They nearly touch at the edges, but are at some distance in the middle, in order to let the corn go in between them. The under stone is supported by strong beams, not represented in the figure; the spindle G stands on the beam MN, which lies upon the bearer O. One end of this bearer rests upon a fixed beam, and the other has a string fixed to it, and going round the pin P, by the turning of which the timbers O and MN may be raised or lowered, and thus the stones put nearer, or removed farther from each other, in order to grind fine or coarse. When the corn is to be ground, it must be put into the hopper by little at a time. A man turns the handle D, and thus the cog-wheel and trundle are carried round also together with the stone A. The axis G is angular at K; and, as it goes round, shakes the shoe I, and makes the corn fall gradually through the hole K. The upper stone going round grinds it, throwing out the meal, as already said, at the eye L. Another handle, if thought proper, may be put at the other end of the handle E. The spindle must go through both stones, in order to reach the beam MN, and the hole through which it passes is fastened with leather or wood, so that no meal can pass through.

Mr Emerson, from whom this account is taken, observes, that "it is a pity some such mills are not made at a cheap rate, for the sake of the poor, who are much distressed by the roguery of the millers."

The construction of a horse-mill differs not from that of the hand-mill just described, excepting that instead of the handle D, the spindle is furnished with a long horizontal lever and cogged wheel, which turns the trundle and stones, as already mentioned.—The stones are much heavier than in the hand-mill.

The mills most commonly in use for grinding corn are water-mills, the construction of which is not essentially different from that of the hand or horse-mills.—The lower mill-stone, as already mentioned, is fixed, but the upper one moveable upon a spindle. The opposite surfaces of the two stones are not flat, but the one convex and the other concave, though in a very small degree. The upper stone, which is six feet in diameter, is hollowed only about an inch in the middle, and the other rises three quarters of an inch. They approach much nearer each other at the circumference, and the corn begins to be ground about two thirds of the radius distant from the circumference, and there it makes the greatest resistance, the space between the two stones being in that place only about two-thirds or three-fourths of the thickness of a grain of corn; but as these stones, as well as those of the hand-mill or horse-mill, can be separated a little from each other, the meal may be made fine or coarse in them, as well as in the two former mills.

In order to cut and grind the corn, both the upper and under stones have furrows cut in them, as is observed in the hand-mill. These are cut perpendicularly on one side, and obliquely upon the other, by which means each furrow has a sharp edge, and by the turning of the stones, the furrows meet like a pair of scissors, and by cutting the corn, make it grind the more easily. They are cut the same way in both stones when they lie upon their backs, by which means they run crosswise to each other when the upper one is inverted and turned round; and this greatly promotes the grinding of the corn, great part of which would be driven onward in the lower furrows, without being ground at all, if both lay the same way.—When the furrow becomes blunt and shallow by wearing, the running stone must be taken off, and the furrows cut deeper in both by means of a chisel and hammer. Thus, however, by having the furrows cut down a great number of times, the thicknesses of both stones are greatly diminished; and it is observed, that in proportion to the diminution of the thickness of the upper stone, the quantity of flour also diminishes.

By means of the circular motion of the upper stone, the corn is brought out of the hopper by jerks, and recedes from the centre towards the circumference by the centrifugal force; and being entirely reduced to flour at the edges when the stones nearly touch one another, it is thrown at last out at the hole called the eye, as already mentioned. In Scotland, it is frequent to have the stones without any furrows, and only irregularly indented with small holes, by means of an iron instrument. Stones of this kind last a much shorter time than those with furrows, the latter being fit for use for 50 or 40 years, while the former seldom or never last more than seven. The under mill-stone is considerably thicker than the upper; and therefore, when both have been considerably worn by use, the lower one is frequently taken up, and the upper one put in its place, the former being converted into a running-stone.

Fig. 9 shows the construction of a common water mill, where AA is the large water-wheel, commonly mills, about 17 or 18 feet diameter from a, the extremity of any float-board, to b the extremity of the opposite one. This wheel is turned round by the falling of the water upon the boards from a certain height, and the greater the height, provided the water runs in an uninterrupted stream, the smaller quantity will be sufficient to turn the mill. This wheel is without the mill-house, but the wheel has an axle BB of considerable length, which passes through a circular hole in the wall, and has upon it a wheel D, of eight or nine feet diameter, having 61 cogs, which turn a trundle E of ten flaves or spokes; by which means the trundle, and consequently the mill-stone, will make six revolutions, and one-tenth for every revolution of the wheel. The odd cog, commonly called the hunting cog, is added, that as every one comes to the trundle it may take the fluff behind that one which it took at the last revolution; Some degree of nicety is requisite in feeding the mill; for if too great a quantity be poured into it, the stones are separated from each other more than they ought to be, and their motion is also impeded; while, on the other hand, if it be fed too slowly, the stone moves with too great velocity, and the attrition of the two is apt to make them strike fire. This matter is regulated by turning the pin L backwards or forwards as the miller thinks proper.

Sometimes, where plenty of water can be had, there are two trundles applied to the cog-wheel by means of a single large one turned immediately by the perpendicular cog-wheel, and carrying round with it an horizontal caged wheel; on each side of which are placed the smaller trundles above-mentioned carrying the stones. In like manner, the water-wheel may be made to drive fanners, bolting-mills, &c., but it must always be remembered, that by complicating machinery to a great degree, it becomes more ready to give way; and the frequent reparation of which it stands in need, will, by the delay of business, be found at last more expensive than if separate machines had been used.

The wind-mill is furnished with an apparatus similar to the water-mill, but necessarily differs in the external apparatus for applying the power. This is done by means of the two arms AB and CD, fig. 93, interfering each other at right angles in E, and passing through the axis EF, and about 32 feet in length. On these yards are placed two sails or vanes, in the shape sometimes of parallelograms, and sometimes of trapeziums, with parallel bases; the greater whereof HI is about six feet, and the length of the smaller FG is determined by radius drawn from the centre E to I and H.

As the direction of the wind is very uncertain, it becomes necessary to have some contrivance for turning the sails towards it, in order to receive its force in whatever way it may turn; and for this purpose two general methods are in use. In the one, the whole machine is sustained upon a moveable arbor or axis, perpendicular to the horizon, and which is supported by a strong stand or foot very firmly fixed in the earth; and thus by means of a lever the whole machine may be turned round as occasion requires. In the other method, only the roof, which is circular, can be turned round by means of a lever and rollers, upon which the circular roof moves. This last kind of wind-mill is always built of stone, in the form of a round turret, having a large wooden ring on the top of it, above which the roof, which must likewise be of wood, moves upon rollers, as has been already mentioned. To effect this motion the more easily, the wooden ring which lies on the top of the building is furnished with a groove, at the bottom of which are placed a number of bars truckles at certain distances, and within the groove is placed another ring, by which the whole roof is supported. The beams ab and ae are connected with the moveable ring, and a rope is fastened to the beam ab in b, which at the other extremity is fitted to a windlass or axis in peritrochion; and this rope being drawn through the iron hook G, and the windlass turned round, the sails and roof will be turned round also, in order to catch the wind in any direction. Both these methods of construction have their advantages and disadvantages. The former is the least expensive, as the whole may be made of wood, wood, and of any form that is thought proper; while the other requires a costly building of stone: and the roof being round, the building must also be so, while the other can be made of any form, but has the inconvenience of being liable to be carried off altogether by a very high wind, of which an instance occurred not long ago in Essex.

Fig. 94 shows the internal mechanism of a windmill. AHO is the upper room; HOZ the lower one; AB the axle-tree passing through the mill; STVV the sails covered with canvas set obliquely to the wind, and turning round in the order of the letters. CD is the cog-wheel, having about 48 cogs a a a, &c., which carry round the lantern EF, having eight or nine trundles c c c, &c., along with the axis GN. IK is the upper mill-stone, LM the lower one; QR is the bridge supporting the axis or spindle GN, which rests upon the beams c d, XY, wedged up at c, d, and X: ZY is the lifting tree, which stands upright; ab and ef are levers, having Z and e as centres of motion; fghi is a cord, with a stone i wound about the pins g and h, and which thus serves as a balance or counterpoise. The spindle t N is fixed to the upper mill-stone IK by means of a piece of iron called the rynd, and fixed in the lower side of the stone, the whole weight of which rests upon a hard stone fixed in the bridge QR at N. The trundle EF and axis G may be taken away; for it rests its lower part by i in a square socket, and the top runs in the edge of the beam w. By bearing down the end of the lever fe we raise b, which raises also Z Y, and this raises YX, which lifts up the bridge QR, with the axis NG, and the upper stone IK; so that by this contrivance the stones may, as in a water-mill, be set at any distance. The lower stone is fixed upon strong beams, and is broader than the upper one; the flour being conveyed through the tunnel n o into a chest. P is the hopper into which the corn is put, and which runs along the spout r into the hole t, and so falls between the stones, where it is ground. The square axis G t shakes the spout r as it turns round, and makes the corn run out; s is a string going round the pin r, which serves to bring the spout nearer or let it go farther from the axis, and thus makes the corn to run faster or slower according to the velocity of the wind. If the wind be very strong, only part of the sails S, T, V, W, J is covered, or perhaps only one half of the two opposite sails. Another cog-wheel B is placed towards the end B of the axle tree, with a trundle and mill-stones like those already described; so that when the wind is strong, the mill may do twice the business it ordinarily does. When only one pair is to grind, the trundle EF and axis G t are taken out from the other: xyf is a girt of pliable wood, fixed at the end x; and the other end f is tied to the lever km, moveable about k; and the end m being put down, draws the girt xyf close to the cog-wheel; and thus the motion of the mill may be flopped at pleasure: pq is a ladder for ascending to the higher part of the mill; and the corn is drawn up by means of a rope rolled about the axis AB.

Besides these mills for grinding corn, one has lately been invented by Mr Winlaw for threshing it out, and for which he has obtained a patent. It is represented fig. 95. AAA represents the frame of the mill, B the cone, C a large iron wheel, D a regulating screw, E a pinion, G the top curb surrounding the nut, H the fly.

Before the corn is put into this mill, it must undergo the operations of combing the bottoms of the sheaves, and stripping the ears from the straw. The former is performed by means of an hand-comb. The use is obvious, viz. to take out all the loose ears, and straw laid irregularly, which would otherwise be lost, or impede the stripping of the ears. The comb for stripping the ears is made in the form of a cross. The teeth are of an angular form, and set at convenient distances, so as to strip the ears clean. If set too wide, they will pass through without effect; and if too near together, they will not admit the straw to go between them.

The grain is separated from the chaff and straw of the ear by the motion of the inner nut within the outward cone. The distance betwixt these is adjusted by the regulating screw D at the bottom; for if this be screwed up too far, the grain will be bruised, if too far lowered down, the grain will not be separated. The dark marked upon the fly shows the direction in which the handle is to be turned, it being pointed as the handle is to be turned.

This mill was tried in the month of June 1785, in the presence of a number of gentlemen, with great satisfaction to the spectators; and since that time has been used by a number of others, though it has not yet come into general use. At the first trial there passed through the mill one bushel of heads per minute, with very moderate labour to the man who turned it; and by experiment it was found, that four bushels of ears yielded one bushel of clean grain. Hence it appears, that the difference betwixt the expedition of the mill and the labour of the thrasher is immensely great; for allowing that a man will thresh six bushels per day at eight hours work, the mill will clear that quantity in 24 minutes, and that to much greater perfection than can be done by the flail, as it separates every grain from the ear, which cannot but be accounted a very great saving; while much corn flies off by the flail, and a great deal is lost by foul threshing, either when performed by task or day-work. But by the use of the mill, all fraudulent practices must be prevented, the straw preserved in its original reed, and thus answer the purposes of thatching, &c., much better than when bruised under the flail; and every other purpose equally well. The ears may also be combed out with great expedition, as a lad without having practised was found to comb out a bushel of ears in 20 minutes, which is at the rate of six bushels of clean corn per day. - The saving by the use of this mill is calculated at 2½d. per bushel. On a smaller scale the mill answers equally well for clover-feed, the flowers being first combed off from the stems; after which it will do as much work in three hours, as a man in the ordinary way can perform in a week; for a man cannot clean much above a bushel in that time, which is the great reason of the high price of clover-feed. The mill will likewise answer for flax, canary, or any other seeds, or for separating the husks from rice, which in the present mode cannot be done without great labour and expense.

In all mills it is necessary that a considerable power be be employed in order to accomplish the intended purpose.—Water is the most common power, and indeed the best, as being the most constant and equable; while wind comes at times with great violence, and at others is totally gone. Mills may also be moved by the force of steam, as were the Albion-mills at London; but the expense of fuel must undoubtedly prevent this mode of constructing mills from ever becoming general. In all cases it is absolutely necessary to make the most of the power that we can, by making it act to the greatest advantage. Hence the best methods of constructing water and wind-mills have been investigated by those who were most conversant in the principles of mechanics; and so difficult has been the investigation, that the principles are not yet settled absolutely without dispute.

The water-mills are of three kinds: Breast-mills, Underfoot-mills, and Overshot-mills. In the former, the water falls down upon the wheel at right angles, to the float-boards or buckets placed all round the wheel to receive it; if float-boards are used, it acts only by its impulse; but if buckets, it acts also by the weight of water in the buckets in the under quarter of the wheel, which is considerable. In the undershot wheel float-boards only are used, and the wheel is turned merely by the force of the current running under it, and striking upon the boards. In the overshot-wheel the water is poured over the top, and thus acts principally by its weight; as the fall upon the upper part of the wheel cannot be very considerable, let it suffice to dash the water out of the buckets. Hence it is evident, that an undershot-mill must require a much larger supply of water than any other; the breast-mill the next, unless the fall is very great; and an overshot mill the least. Dr Desaguliers found, that a well-made overshot mill would perform as much work as an undershot one with one tenth part of the quantity of water required by the other.

In the first volume of the Philosophical Transactions, Mr Smeaton has considered at great length the best methods of constructing all these mills from machines and models made on purpose: but conscious of the inferiority of models to actual practice, did not venture to give his opinion without having seen them actually tried, and the truth of his doctrines established by practice.

Having described the machines and models used for making his experiments, he observes, that, with regard to power, it is most properly measured by the raising of a weight; or, in other words, if the weight raised be multiplied by the height to which it can be raised in a given time, the product is the measure of the power raising it; and, of consequence, all those powers are equal whose products made by such multiplication are equal: for if a power can raise twice the weight to the same height, or the same weight to twice the height in the same time that another can, the former power will be double the latter; but if a power can only raise half the weight to double the height, or double the weight to half the height, in the same time that another can, the two powers are equal. This, however, must be understood only of a slow and equable motion, without acceleration or retardation; for if the velocity be either very quickly accelerated or retarded, the vis inertiae, in our author's opinion, will produce an irregularity.

To compute the effects of water-wheels exactly, it is necessary to know in the first place what is the real velocity of the water which impinges on the wheel. 2. The quantity of water expended in a given time; and, 3. How much of the power is lost by the friction of the machinery.

1. With regard to the velocity of the water, Mr Smeaton determined by experiments with the machinery described in the volume referred to, that with a head of water 15 inches in height, the velocity of the wheel is 8.96 feet in a minute. The area of the head being 405.8 inches, this multiplied by the weight of a cubic inch of water equal to .579 of an ounce avoirdupoise, gives 61.26 ounces for the weight of as much water as is contained in the head upon one inch in depth; and by further calculations derived from the machinery made use of, he computes that 264.7 pounds of water descend in a minute through the space of 15 inches. The power of the water, therefore, to produce mechanical effects in this case will be $264.7 \times 15$, or 3970. From the result of the experiment, however, it appeared that a vast quantity of the power was lost; the effect being only to raise 9.375 pounds to the height of 135 inches; so that the power was to the effect as 3970 to $9.375 \times 135 = 1265$, or as 10 to 3.18.

This, according to our author, must be considered as the greatest single effect of water upon an undershot-wheel, where the water descends from an height of 15 inches; but as the force of the current is not by any means exhausted, we must consider the true proportion betwixt the power and effect to be that betwixt the quantity of water already mentioned and the sum of all the effects producible from it. This remainder of power, it is plain, must be equal to that of the velocity of the wheel itself multiplied into the weight of the water. In the present experiment, the circumference of the wheel moved with the velocity of 3.123 feet in a second, which answers to a head of 1.52 inches (a); and this height being multiplied by 264.7, the quantity of water expended in a minute gives 481 for the power of the water after it has passed the wheel; and hence the true proportion betwixt the power and the effect will be as 3849 to 1266; or as 11 to 4.

As the wheel revolved 86 times in a minute, the velocity of the water must be equal to 86 circumferences of the wheel; which, according to the dimensions of the apparatus used by Mr Smeaton, was as 86 to 30, or as 25 to 7.—The greatest load with which the wheel would move was 9 lb. 6 oz.; and by 12 lb. it was entirely stopped. Whence our author concludes,

(a) These calculations are founded upon the known maxim in hydrostatics, that the velocity of spouting water is nearly the same with that which an heavy body would acquire by falling from an height equal to that of the reservoir, and is proved by the rising of jets nearly to the height of their reservoirs. cludes, that the impulse of the water is more than double of what it ought to be according to theory; but this he accounts for by observing, that in his experiment the wheel was placed not in an open river, where the natural current, after it has communicated its impulse to the float, has room on all sides to escape, as the theory supposes, but in a conduit, to which the float being adapted, the water cannot otherwise escape than by moving along with the wheel. It is observable, that a wheel working in this manner, as soon as the water meets the float, receiving a sudden check, it rises up against the float like a wave against a fixed object, infomuch that when the sheet of water is not a quarter of an inch thick before the float, yet this sheet will act upon the whole surface of a float whose height is three inches: and consequently, was the float no higher than the thickness of the sheet of water, as the theory also supposes, a great part of the force would have been lost by the water dashingly over the float.

Mr Smeaton next proceeds to give tables of the velocities of wheels with different heights of water; and from the whole deduces the following conclusions.

1. The virtual, or effective head, being the same, the effect will be nearly as the quantity of water expended.

2. The expense of water being the same, the effect will be nearly as the height of the virtual or effective head.

3. The quantity of water expended being the same, the effect is nearly as the square of the velocity.

4. The aperture being the same, the effect will be nearly as the cube of the velocity of the water. Hence, if water passes out of an aperture in the same section, but with different velocities, the expense will be proportional to the velocity; and therefore, if the expense be not proportional to the velocity, the section of the water is not the same.

5. The virtual head, or that from which we are to calculate the power, bears no proportion to the head water; but when the aperture is larger, or the velocity of the water less, they approach nearer to a coincidence: and consequently, in the large openings of mills and sluices, where great quantities of water are discharged from moderate heads, the head of water, and virtual head determined from the velocity, will nearly agree, which is also confirmed by experience.

6. The most general proportion between the power and effect is that of 10 to 3; the extremes 10 to 3.2, and 10 to 2.8. But as it is observable, that where the power is greatest, the second term of the ratio is greatest also; whence we may allow the proportion subsisting in great works to be as three to one.

7. The proportion of velocity between the water and wheel is, in general about 5 to 2.

8. There is no certain ratio between the load that the wheel will carry at its maximum, and what will totally stop it; though the proportions are contained within the limits of 20 to 19, and 20 to 15; but as the effect approaches nearest to the ratio of 20 to 15, or of 4 to 3 when the power is greatest either by increase of velocity or quantity of water, this seems to be the most applicable to large works: but as the load that a wheel ought to have, in order to work to the best advantage, can be assigned by knowing the effect that it ought to produce, and the velocity it ought to have in producing it, the exact knowledge of the greatest load it will bear is of the least consequence in practice.

Mr Smeaton, after having finished his experiments on the undershot mills, reduced the number of floats, which were originally 24, to 12; which caused a diminution in the effect, by reason that a greater quantity of water escaped between the floats and the floor than before; but on adapting to it a circular sweep of such a length, that one float entered into the curve before the other left it, the effect came so near that of the former, as not to give any hopes of advancing it by increasing the number of floats beyond 24 in this particular wheel.

Our author next proceeds to examine the power of water when acting by its own gravity in turning an overshot wheel: "In reasoning without experiment (says he,) one might be led to imagine, that however different the mode of application is, yet that, whenever the same quantity of water descends through the same perpendicular space, the natural effective power would be equal, supposing the machinery free from friction, equally calculated to receive the full effect of the power, and to make the most of it: for if we suppose the height of a column of water to be 30 inches, and resting upon a base or aperture of one inch square, every cubic inch of water that departs therefrom will acquire the same velocity or momentum from the uniform pressure of 30 cubic inches above it, that one cubic inch let fall from the top will acquire in falling down to the level of the aperture; one would therefore suppose that a cubic inch of water let fall through a space of 30 inches, and there impinging upon another body, would be capable of producing an equal effect by collision, as if the same cubic inch had descended through the same space with a slower motion, and produced its effects gradually. But however conclusive this reasoning may seem, it will appear in the course of the following deductions, that the effect of the gravity of descending bodies is very different from the effect of the stroke of such as are non-elastic, though generated by an equal mechanical power."

Having made such alterations in his machinery as were necessary for overshot wheels, our author next gives a table of experiments with the apparatus so altered. In these the head was 6 feet inches, and the height of the wheel 24 inches; so that the whole descent was 30 inches: the quantity of water expended in a minute was 96½ pounds; which multiplied by 30 inches, gives the power = 2900: and after making the proper calculations, the effect was computed at 1914; whence the ratio of the power to it comes to be nearly as 3 to 2. If, however, we compute the power from the height of the wheel only, the power will be to the effect nearly as 5 to 4.

From another set of experiments the following conclusions were deduced.

1. The effective power of the water must be reckoned upon the whole descent; because it must be raised to that height in order to be able to produce the same effect a second time. The ratios between the powers powers so estimated and the effects at a maximum, differ nearly from 4 to 3, and from 4 to 2. Where the heads of water and quantities of it expended are the least, the proportion is nearly from 4 to 3; but where the heads and quantities are greatest, it comes nearer to that of 4 to 2; so that by a medium of the whole the ratio is nearly as 3 to 2. Hence it appears, that the effect of overshot wheels is nearly double to that of undershot ones; the consequence of which is, that non-elastic bodies, when acting by their impulse or collision, communicate only a part of their original impulse, the remainder being spent in changing their figure in consequence of the stroke. The ultimate conclusion is, that the effects as well as the powers are as the quantities of water and perpendicular heights multiplied together respectively.

2. By increasing the head, it does not appear that the effects are at all augmented in proportion; for by raising it from 3 to 11 inches, the effect was augmented by less than one-seventh of the increase of perpendicular height. Hence it follows, that the higher the wheel is in proportion to the whole descent, the greater will be the effect; because it depends less upon the impulse of the head, and more upon the gravity of the water in the buckets: and if we consider how obliquely the water issuing from the head must strike the buckets, we shall not be at a loss to account for the little advantage that arises from the impulse thereof, and shall immediately see of how little consequence this is to the effect of an overshot wheel. This, however, as well as other things, must be subject to limitation; for it is necessary that the velocity of the water should be somewhat greater than the wheel, otherwise the latter will not only be retarded by the striking of the buckets against the water, but some of the power will be lost by the dashing of the water over the buckets.

3. To determine the velocity which the circumference of the wheel ought to have in order to produce the greatest effect, Mr Sineaton observes, that the more slowly any body descends by the force of gravity when acting upon any piece of machinery, the more of that force will be spent upon it, and consequently the effect will be the greater. If a stream of water falls into the bucket of an overshot wheel, it will be there retained till the wheel discharges it by moving round; and of consequence, the slower the wheel moves, the more water it will receive; so that what is lost in velocity is gained by the greater pressure of water upon the buckets. From the experiments, however, it appears, that when the wheel made about 20 turns in a minute the effect was greatest; when it made only 18½ the motion was irregular; and when loaded so as not to admit its turning 18 times, the wheel was overpowered with the load. When it made 30 turns, the power was diminished by about one-thirtieth, and when the number of turns was increased to 40, it was diminished by one-fourth. Hence we see, that in practice the velocity of the wheel should not be diminished farther than what will procure some solid advantage in point of power; because, ceteris paribus, the buckets must be larger as the motion is slower; and the wheel being more loaded with water, the fires will be proportionally increased upon every part of the work. The best velocity for practice therefore will be that when the wheel made 30 turns in a minute, which is little more than three feet in a second. This velocity is applicable to the highest overshot wheels as well as the lowest. Experience however determines, that high wheels may deviate further from this rule before they will lose their power, by a given aliquot part of the whole, than low ones can be permitted to do; for a wheel of 24 feet high may move at the rate of 6 feet per second; while our author has seen one of 33 feet high move very steadily and well with a velocity of little more than two feet. The reason of this superior velocity in the 24 feet wheel, may probably be owing to the small proportion that the head requisite to give the proper velocity to the wheel bears to the whole height.

4. The maximum load for an overshot wheel is that which reduces the circumference of the wheel to its proper velocity; which is known by dividing the effect it ought to produce in a given time by the space intended to be described by the circumference of the wheel in the same time; the quotient will be the resistance overcome at the circumference of the wheel, and is equal to the load required, including the friction and resistance of the machinery.

5. The greatest velocity that an overshot wheel is capable of, depends jointly upon the diameter or height of the wheel and the velocity of falling bodies; for it is plain that the velocity of the circumference can never be greater than to describe a semi-circumference, while a body let fall from the top describes the diameter, nor even quite so great; as the difference in point of time must always be in favour of that which falls through the diameter. Thus, supposing the diameter of the wheel to be 16 feet and an inch in diameter, an heavy body would fall through this space in one second; but such a wheel could never arrive at this velocity, or make one turn in two seconds, nor could an overshot wheel ever come near it; because, after it has acquired a certain velocity, great part of the water is prevented from entering the buckets, and part is thrown out again by the centrifugal force; and as these circumstances have a considerable dependence upon the form of the buckets, it is impossible to lay down any general rule for the velocity of this kind of wheels.

6. Though in theory we may suppose a wheel to be made capable of overcoming any resistance whatever, yet as in practice it is necessary to make the wheel and buckets of some certain and determinate size, we always find that the wheel will be stopped by such a weight as is equal to the effort of the water in all the buckets of a semi-circumference put together. This may be determined from the structure of the buckets themselves; but in practice, an overshot wheel becomes unserviceable long before this time; for when it meets with such an obstacle as diminishes its velocity to a certain degree, its motion becomes irregular; but this never happens till the velocity of the circumference is less than the two feet per second, when the resistance is equable.

7. From the above observations, we may easily deduce the force of water upon breast-wheels, &c. But in general, all kinds of wheels where the water cannot descend through a given space unless the wheel moves with it, are to be considered as overshot wheels; and those. those which receive the impulse or shock of the water, whether in an horizontal, oblique, or perpendicular direction, are to be considered as undershots. Hence a wheel in which the water strikes at a certain point below the surface of the head, and after that descends in the arch of a circle, pressing by its gravity upon the wheel, the effect of such a wheel will be equal to that of an undershot whose head is equal to the difference of level between the surface of the water in the reservoir and the point where it strikes the wheel, added to that of an overshot, whose height is equal to the difference of level between the point where it strikes the wheel and the level of the tail-water.

In the 66th volume of the Transactions, our author considers some of the causes which have produced disagreements and disputes among mathematicians upon this subject. He observes, that soon after Sir Isaac Newton had given his definition, "that the quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly," it was controverted by his contemporary philosophers. They maintained, that the measure of the quantity of motion should be estimated by taking the quantity of matter and the square of the velocity conjointly. On this subject he remarks, that from equal impelling powers acting for equal intervals of time, equal augmentations of velocity are acquired by given bodies when they are not resisted by a medium. Thus a body descending one second by the force of gravity, passes through a space of 16 feet and an inch; but at the end of that time it has acquired a velocity of 32 feet 2 inches in a second; at the end of two seconds, it has acquired one that would carry it through 64 feet 4 inches in a second. If, therefore, in consequence of this equal increase of velocity, we define this to be a double quantity of motion generated in a given time in a certain quantity of matter, we come near to Sir Isaac's definition; but in trying experiments upon the effects of bodies, it appears, that when a body is put in motion by whatever cause, the impression it will make upon an uniformly resisting medium, or upon uniformly yielding substances, will be as the mass of matter of the moving body multiplied by the square of its velocity. The question therefore properly is, whether those terms, the quantity of motion, the momenta, or forces of bodies in motion, are to be esteemed equal, double, or triple, when they have been generated by an equable impulse acting for an equal, double, or triple time? or that it should be measured by the effects being equal, double, or triple, in overcoming resistances before a body in motion can be stopped? For according to the meaning we put upon these words, the momenta of equal bodies will be as the velocities or squares of the velocities of the moving bodies.

Though by a proper attention to the terms employed, however, we will find both these doctrines to be true; it is certain that some of the most celebrated writers upon mechanics have fallen into errors by neglecting to attend to the meaning of the terms they make use of. Desaguliers, for instance, after having been at pains to show that the dispute, which in his time had subsisted for 50 years, was a dispute merely about words, tells us, that both opinions may be easily reconciled in the following case, viz., that the wheel of an undershot water-mill is capable of doing quadruple work when the velocity of the water is doubled, instead of double work only: "For (says he) the advantage being the same, we find, that as the water's velocity is double, there are twice the number of particles that issue out, and therefore the ladle-board is struck by twice the matter; which matter moving with twice the velocity that it had in the first case, the whole effect must be quadruple, though the instantaneous stroke of each particle is increased only in a simple proportion of the velocity." In another place, the same author tells us, that though "the knowledge of the foregoing particulars is absolutely necessary for setting an undershot wheel to work, yet the advantage to be reaped from it would be still guesswork; and we should be at a loss to find out the utmost that it could perform, had it not been for an ingenious proposition of that excellent mechanic M. Parent of the Royal Academy of Sciences, who has showed, that an undershot wheel can do the most work when its velocity is equal to the third part of that of the water; because then two-thirds of the water is employed in driving the wheel, with a force proportional to the square of the velocity. By multiplying the surface of the adjutage or opening by the height of the water, we shall have the column of water that moves the wheel. The wheel thus moved will sustain on the opposite side only four-ninths of that weight which will keep it in equilibrium; but what it can move with the velocity it has, is only one third of the equilibrium." This conclusion is likewise adopted by Mr MacLaurin.

Undershot wheels had been greatly preferred by M. Belidor to those of any other construction. He had even concluded, that water applied in this way will do more than six times the work of an overshot wheel, while Dr Defaguliers, in overthrowing Belidor's proposition, determined that an overshot wheel would do ten times the work of an undershot wheel with an equal quantity of water. Between these two celebrated authors, therefore, there is a difference of no less than 60 to 1.

In consequence of such monstrous disagreement, Mr Smeaton began the experiments of which we have already given an account. From them, besides the positions already deduced, it appears, that where the velocity of water is double, the adjutage or aperture of the sluice remaining the same, the effect is eight times; that is, not as the square, but as the cube of the velocity. In the other conclusion of Defaguliers and MacLaurin, the error was no less; for from thence it would follow, that by means of the wheel only ¼ of the water expended would be raised back again to the height of the reservoir from which it descended, exclusive of the friction, which would still diminish the quantity: but from Mr Smeaton's experiments it appears, that in some cases upwards of one-fourth had been raised. In large works the effects had been still greater, approaching in an undershot wheel to one half, and in an overshot one to the whole; which would be the limit, if it were possible, to remove the friction and resistance of the air. The velocity of the wheel also, which, according to the conclusions of M. Parent and Dr Defaguliers, amounted to no more than one-third of the velocity of the wa- ter, varies, according to Mr Smeaton, between one-third and one-half. But in all great works the maximum lieth much nearer to one-half than a third; the former appearing to be the true maximum, if all friction, resistance of the air, and scattering of water, could be avoided.

To make these matters plain to mechanics, and to prevent them from running into practical errors in consequence of a fallacious theory, Mr Smeaton, in the year 1759, instituted another set of experiments; the immediate object of which was, to determine what proportion or quantity of mechanical power is expended in giving the same body different degrees of velocity. Having constructed a proper apparatus for the purpose, and with it made a number of experiments, he concludes, "that time, properly speaking, has nothing to do with the production of mechanical effects otherwise than as by equally flowing it becomes a common measure; so that whatever mechanical effect is found to be produced in a given time, the uniform continuance of the action of the same mechanical power will, in a double time, produce twice that effect. A mechanical power, therefore, properly speaking, is measured by the whole of its mechanical effects produced, whether that effect be produced in a greater or lesser time; thus, having treasured up 1000 tons of water, which I can let out upon the overshot wheel of a mill, and descending through a perpendicular of 20 feet; this power, applied in a proper manner, will grind a certain quantity of corn in an hour; but supposing the mill to be capable of receiving a greater impulse with as great advantage as a lesser; then, if the corn be let out twice as fast, the same quantity of water will be ground in half an hour, the whole of the water being likewise expended in that time.

What time has therefore to do in the case is this: let the rate of doing the business or producing the effect be what it will; if this rate is uniform, when I have found by experiment what is done in a given time, then, proceeding at the same rate, twice the effect will be produced in twice the time, on supposition that I have a supply of mechanic power to go on with. Thus, 1000 tons of water descending through 20 feet perpendicular, being, as has been shown, a given mechanic power, let it be expended at what rate it will; if, when this is expended, we are to wait another hour till an equal quantity can be procured, then we can only expend 12 such quantities in 24 hours. But if, while the thousand tons of water are expending in one hour, the same quantity is renewed, we can then expend 24 such in the 24 hours, or go on without intermission. The product or effect will then be in proportion to time, which is the common measure; but the quantity of mechanic power arising from the flow of the two rivers, compared by taking an equal portion of time, is double in the one to the other; though each has a mill that, when going, will grind an equal quantity of corn in an hour."

Mr Ferguson, in his directions to millwrights, has adopted the maxim which Mr Smeaton condemns as erroneous, viz. that when the velocity of the wheel is but one-third of that of the water, it then acts to the greatest advantage. He adds, that the millstone ought to make about 60 turns in a minute; for when it makes only 40 or 50 turns it grinds too slowly; and when more than 70, it heats the meal too much, and cuts the bran too small, that a part of it mixes with the meal and cannot be separated from it by any means. The utmost perfection of mill-work, therefore, according to this author, lies in making the train so that the mill-stone shall make about 60 turns in a minute, when the wheel moves with one-third of the velocity of the water. To accomplish this he lays down the following rules. 1. Measure the perpendicular height of the fall of water above the middle of the aperture, where it is let out to act by impulse against the floatboards on the lower side of the undershot wheel. 2. Multiply this constant number 64.2382 by the height of the fall in feet, and extract the square-root of the product, which will give the number of feet that the water moves in a second. 3. The velocity of the floats of the wheel is equal to one-third of the velocity of the water just now found. 4. Divide the circumference of the wheel by the velocity of its floats, and the quotient will be the number of seconds in one turn of the great water-wheel, on whose axis the cog-wheel that turns the trundle is fixed. 5. Divide 60 by the number of seconds in a turn of the water-wheel, and the quotient will be the number of turns it makes in a minute. 6. By this number of turns divide 60, the number of times that a mill-stone ought to have in a minute; the quotient is the number of turns that the mill-stone ought to make for every one of the large wheel. 7. Then as the number of turns required of the mill-stone in a minute is to the number of turns of the cog-wheel in a minute; so must the number of cogs in the wheel be to the number of leaves in the trundle on the axis of the mill-stone, in the nearest whole number that can be found.

On these principles Mr Ferguson has constructed the following table, for the sake of such as have occasion to construct mills, and are not willing to take the trouble of particular calculations. ### The Mill-wright's Table

| Height of the fall of water | Velocity of the fall of water per second | Velocity of the wheel per second | Revolutions of the wheel per minute | Revolution of the millstone for one of the wheel | Cogs in the wheel, and staves in the trundle | Revolutions of the millstone per minute by these staves and cogs | |-----------------------------|------------------------------------------|---------------------------------|-------------------------------------|-----------------------------------------------|-------------------------------------------------|--------------------------------------------------| | Feet | Feet | Feet | Parts of a rev. | Parts of a rev. | Cogs, Staves | Revolutions of a rev. | | 1 | 8 | 2 | 67 | 2 | 83 | 42 | 40 | 254 | 6 | 119 | 84 | | 2 | 11 | 3 | 78 | 4 | 91 | 24 | 44 | 196 | 8 | 120 | 28 | | 3 | 13 | 4 | 89 | 5 | 67 | 21 | 16 | 190 | 9 | 119 | 74 | | 4 | 16 | 5 | 94 | 6 | 58 | 18 | 92 | 170 | 9 | 119 | 68 | | 5 | 17 | 6 | 93 | 6 | 55 | 17 | 94 | 156 | 9 | 120 | 30 | | 6 | 19 | 7 | 84 | 7 | 55 | 16 | 96 | 144 | 9 | 120 | 00 | | 7 | 21 | 8 | 75 | 8 | 50 | 14 | 10 | 140 | 10 | 119 | 14 | | 8 | 22 | 9 | 68 | 9 | 45 | 13 | 38 | 134 | 10 | 120 | 18 | | 9 | 24 | 10 | 59 | 10 | 40 | 12 | 76 | 128 | 10 | 120 | 32 | | 10 | 25 | 11 | 52 | 11 | 35 | 10 | 99 | 109 | 10 | 120 | 96 | | 11 | 26 | 12 | 45 | 12 | 30 | 10 | 92 | 110 | 10 | 120 | 20 | | 12 | 27 | 13 | 37 | 13 | 26 | 10 | 82 | 102 | 10 | 119 | 34 | | 13 | 28 | 14 | 30 | 14 | 22 | 11 | 74 | 118 | 10 | 120 | 36 | | 14 | 30 | 15 | 23 | 15 | 18 | 11 | 69 | 112 | 10 | 118 | 72 | | 15 | 31 | 16 | 16 | 16 | 15 | 10 | 65 | 109 | 10 | 120 | 96 | | 16 | 32 | 17 | 10 | 17 | 12 | 10 | 59 | 106 | 10 | 120 | 20 | | 17 | 33 | 18 | 6 | 18 | 10 | 10 | 53 | 102 | 10 | 119 | 34 | | 18 | 34 | 19 | 5 | 19 | 8 | 9 | 48 | 100 | 10 | 120 | 20 | | 19 | 34 | 20 | 4 | 20 | 6 | 9 | 46 | 94 | 10 | 119 | 18 |

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For the practical construction of water-mills, Mr Imifon hath laid down the following rules.

1. To find the velocity or force of any moderate stream of water; let it be obstructed by a dam in such a manner as to force the whole stream into a spout by which it may be conveyed into a large vessel or reservoir. Measure then the quantity of water which falls into the reservoir in one second or minute; and multiplying by the number of seconds or minutes in an hour, we have the whole force of the stream of water per hour.

In streams which are too large to be measured in this way, the velocity is determined (though we must own in a vague manner) by that of straw or other light body floating down it; and calculations may be made accordingly.

Mr Imifon differs very materially from Mr Ferguson in the number of revolutions which a mill-stone ought to make in a minute; the latter, as has been already mentioned, being of opinion, that 60 revolutions of a mill-stone in a minute are sufficient, while Mr Imifon requires 120; though he agrees with him that the velocity of the wheel should be only one-third of that of the water. The mill-stone, according to Mr Ferguson, ought to be five feet in diameter; but Mr Imifon makes it only four feet and an half.

To construct a mill by this table, find the height of the fall of water in the first column, and against that height in the fifth column you have the number of cogs in the wheel and staves in the trundle for causing the millstone 4 feet 6 inches diameter to make about 120 revolutions in a minute, as near as possible, when the wheel goes with one-third part of the velocity of the water. And it appears by the 7th column, that the number of cogs in the wheel and staves in the trundle are so near the truth for the required purpose, that the least number of revolutions of the mill-stone in a minute is 118, and the greatest number never exceeds 121; which, according to our author, is the velocity of the best mills he had seen.

With regard to the mere mechanical part, our author observes, that an overshot wheel acts with greater power than a breast or undershot wheel; so that where there is a considerable descent, and only a small quantity of water, the overshot wheel ought always to be made use of. Where the water runs only upon a little declivity, it can act but slowly upon the under part of... the wheel; in which case, the motion of the wheel will be very slow: the float-boards therefore ought to be very long, though not high, that a large body of water may act upon them; so that what is wanting in velocity may be made up in power; in which case, the cog wheel may have a greater number of cogs in proportion to the staves of the trundle, in order to give the millstone a sufficient degree of velocity.

For the construction of the different parts of mills, Mr. Imison gives the following general directions:

The method for setting out a spur-wheel and wallower.—Draw the pitch lines A1, B1, A2, B2; then divide them into the number of teeth or cogs required, as abc.

Divide one of those distances, as bc, into seven equal parts, as 1, 2, 3, 4, 5, 6, 7; three parts allow for the thickness of the cogs, as 1, 2, 3, in the cog a, and four for the thickness of the stave, of the wallower (one reason for allowing three parts for the cog and four for the stave, is, the wallower is in general of less diameter than the wheel, therefore subject to more wear in proportion of the number of cogs to the number of staves; but if there is the same number of staves as of cogs, they may be of equal thickness), as 1, 2, 3, 4, in the stave m, fig. 97; the height of the cog is equal to four parts; then divide its height into five equal parts, as 1, 2, 3, 4, 5, in the cog C; allow three for the bottom to the pitch line of the cog; the other two parts for epicycloid, so as to fit and bear on the stave equally. The millwrights in general put the point of a pair of compasses in the dot 3, of the cog a, and strike the line de; then remove the point of the compasses to the point d, and strike the curve line df, which they account near enough the figure of the epicycloid.

The method for a face-wheel is thus: Divide the pitch line A B into the number of cogs intended, as abc; divide the distance le into seven equal parts; three of those parts allow for the thickness of the cogs, as 1, 2, 3, in the cog a, four for the height and four for the width, as de, and four for the thickness of the stave m; draw a line through the centre of the cog, as the line AI at S; and on the point 5 describe the line de; remove the compasses to the point A, and draw the line ef, which forms the shape of the cog; then shape the cog on the sides to a cycloid, as defg. But this method of setting out the shape of a cog is variable, according to the cycloid in different diameters of wheels.

In common spur-nuts, divide the pitch line A into twice as many equal parts as you intend teeth, as a, b, c, d, e, fig. 98.; with a pair of compasses opened to half the distance of any of those divisions, from the points a1, c3, e5, draw the semicircles a, c, and e, which will form the ends of the teeth. From the points 2, 4, and 6, draw the semicircles gbi, which will form the hollow curves for the spaces; but if the ends of the teeth were epicycloids, instead of semicircles, they would act much better.

The principle of level gear.—consists in two cones, rolling on the surface of each other, as the cone A and B revolving on their centres ab, ac; if their bases are equal, they will perform their revolutions in one and the same time, or any other two points equally distant from the centre a, as d1, d2, d3, &c., will revolve in the same time as f1, f2, f3, &c. In the like manner, if the cones afde be twice the diameters at the base de, as the cones afde are; then if they turn about their centres, when the cone afde has made one revolution, the cone ade will have made but half a revolution; or when afde has made two revolutions, ade will have made but one, and every part equally distant from the centre a, as f1, f2, f3, &c. will have made two revolutions to e1, e2, e3, &c. and if the cones were fluted, or had teeth cut in them, diverging from the centre a to the bases de, ef, they would then become bevel gear. The teeth at the point of the cone being small and of little use, may be cut off at E and F, figs. 102, 103, as seen by fig. 104, where the upright shaft ad, with the bevel wheel cd, turns the bevel wheel ef with its shaft bg, and the teeth work freely into each other, as at, fig. 105. The teeth may be made of any dimension, according to the strength required; and this method will enable them to overcome a much greater resistance, and work smoother than a face wheel and wallower of the common form can possibly do; besides, it is of great use to convey a motion in any direction, or to any part of a building, with the least trouble and friction.

The method of conveying motion in any direction, and proportioning or shaping the wheels thereto, is as follows: let the line ab represent a shaft coming from a wheel; draw the line ef to intersect the line ab, in the direction that the motion to be conveyed is intended, which will now represent a shaft to the intended motion.

Again, suppose the shaft ed is to revolve three times, whilst the shaft ab revolves once, draw the parallel line ii, at any distance not too great, suppose one foot by a scale; then draw the parallel line kk at three feet distance, after which draw the dotted line wx, through the intersection of the shafts ad and ed, and likewise through the intersection of the parallel lines ii and kk, in the points X and y; which will be the pitch line of the two bevel wheels, or the line where the teeth of the two wheels act on each other, as may be seen fig. 107, where the motion may be conveyed in any direction.

The universal joint, as represented fig. 108, may be applied to communicate motion instead of bevel gear, where the speed is to be continued the same, and where the angle does not exceed 30 or 40 degrees, and the equality of motion is not regarded; for as it recedes from a right line, its motion becomes more irregular. This joint may be constructed by a cross, as represented in the figure; or with four pins fastened at right angles upon the circumference of a hoop or solid ball. It is of great use in cotton-mills, where the tumbling shafts are continued to a great distance from the moving power. But by applying this joint, the shafts may be cut into convenient lengths, by which it will be enabled to overcome greater resistance.

To describe the cycloid and epicycloid, of use in shaping the teeth of wheels, &c.—If a point or pencil a on the circumference of the circle B proceeds along the plane AC in a right line, and at the same time revolves round its centre, it will describe a cycloid.

And if the generating circle D moves along the circumference of another circle E, and at the same time turns round its centre, the point of contact will describe an epicycloid. In the construction of wind mills, Mr Smeaton has been at no less pains to explain the principles than in those which go by water. For this purpose he constructed a machine, of which a particular description is given in the first volume of the Philosophical Transactions. The general principle of this was, that by means of a determinate weight it carried round an axis with an horizontal arm, upon which were four small moveable sails. Thus the sails met with a constant and equable blast of air; and as they moved round, a string with a weight affixed to it was wound about their axis, and thus showed what kind of size or construction of sails answered the purpose best.

With this machine a great number of experiments were made; the results of which were as follow.

1. The sails let at the angle with the axis, proposed as the best by M. Parent and other geometricians, viz., $55^\circ$, was found to be the worst proportion of any that was tried.

2. When the angle of the sails with the axis was increased from $72^\circ$ to $75^\circ$, the power was augmented in the proportion of $34$ to $45$; and this is the angle most commonly in use when the sails are planes.

3. Were nothing more requisite than to cause the sails acquire a certain degree of velocity by the wind, the position recommended by M. Parent would be the best. But if the sails are intended with given dimensions, to produce the greatest effects possible in a given time, we must, if planes are made use of, confine our angle within the limits of $72$ and $75$ degrees.

4. The variation of a degree or two, when the angle is near the best, is but of little consequence.

5. When the wind falls upon concave sails, it is an advantage to the power of the whole, though each part separately taken should not be disposed of to the best advantage.

6. From several experiments on a large scale, Mr Smeaton has found the following angles to answer as well as any. The radius is supposed to be divided into five parts; and $\frac{1}{5}$th, reckoning from the centre, is called 1, the extremity being denoted 6.

| No | Angle with that axis | Angle with the plane of motion | |----|---------------------|-------------------------------| | 1 | $72^\circ$ | $18^\circ$ | | 2 | $71^\circ$ | $19^\circ$ | | 3 | $72^\circ$ | $18$ middle | | 4 | $74^\circ$ | $16$ | | 5 | $77^\circ$ | $12\frac{1}{2}$ | | 6 | $83^\circ$ | $7$ extremity |

7. Having thus obtained the best method of weathering the sails, i.e., the most advantageous manner in which they can be placed, our author's next care was to try what advantage could be derived from an increase of surface upon the same radius. The result was, that a broader sail requires a large angle; and when the sail is broader at the extremity than near the centre, the figure is more advantageous than that of a parallelogram. The figure and proportion of enlarged sails, which our author determines to be most advantageous on a large scale, is that where the extreme bar is one-third of the radius or whip (as the workmen call it), and is divided by the whip in the proportion of $3$ to $5$. The triangular or loading sail is covered with board from the point downward of its height, the rest as usual with cloth. The angles above mentioned are likewise the most proper for enlarged sails; it being found in practice, that the sails should rather be too little than too much exposed to the direct action of the wind.

Some have imagined, that the more sail the greater would be the power of the windmill, and have therefore proposed to fill up the whole area; and by making each sail a sector of an ellipse, according to M. Parent's method, to intercept the whole cylinder of wind, in order to produce the greatest effect possible. From our author's experiments, however, it appeared, that when the surface of all the sails exceeded seven-eighths of the area, the effect was rather diminished than augmented. Hence he concludes, that when the whole cylinder of wind is intercepted, it cannot then produce the greatest effect for want of proper intertices to escape.

"It is certainly desirable (says Mr Smeaton), that the sails of windmills should be as short as possible; but it is equally desirable, that the quantity of cloth should be the least that may be, to avoid damage by sudden squalls of wind. The best structure, therefore, for large mills, is that where the quantity of cloth is the greatest in a given circle that can be: on this condition, that the effect holds out in proportion to the quantity of cloth; for otherwise the effect can be augmented in a given degree by a lesser increase of cloth upon a larger radius, than would be required if the cloth was increased upon the same radius.

8. The ratios between the velocities of windmill sails unloaded, and when loaded to their maximum, turned out very different in different experiments, but the most common proportion was as $3$ to $2$. In general it happened, that where the power was greatest, whether by an enlargement of the surface of the sails, or an increased velocity of the wind, the second term of the ratio was diminished.

9. The ratios between the least load that would stop the sails and the maximum with which they would turn, were confined betwixt that of $10$ to $8$ and $10$ to $9$: being at a medium about $10$ to $8.3$, and $10$ to $9$, or about $6$ to $5$; though on the whole it appeared, that where the angle of the sails or quantity of cloth was greatest, the second term of the ratio was less.

10. The velocity of windmill sails, whether unloaded or loaded, so as to produce a maximum, is nearly as the velocity of the wind, their shape and position being the same. On this subject Mr Ferguson remarks, that it is almost incredible to think with what velocity the tips of the sails move when acted upon by a moderate wind. He has 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, has computed, that if an hoop of the same size was to run upon plain ground with an equal velocity, it would go upwards of $30$ miles in an hour.

11. The load at the maximum is nearly, but somewhat less, than as the square of the velocity of the wind; the shape and position of the sails being the same.

12. The effects of the same sails at a maximum are nearly, but somewhat less, than as the cubes of the velocity of the wind.

13. The load of the same sails at a maximum is nearly nearly as the squares, and the effect as the cubes of their number of turns in a given time.

14. When sails are loaded so as to produce a maximum at a given velocity, and the velocity of the wind increases, the load continuing the same; then the increase of effect, when the increase of the velocity of the wind is small, will be nearly as the squares of these velocities; but when the velocity of the wind is double, the effects will be nearly as 10 to 27\(\frac{1}{2}\); and when the velocities compared are more than double of that where the given load produces a maximum, the effects increase nearly in a simple ratio of the velocity of the wind. Hence our author concludes, that windmills, such as the different species for draining water, &c., lose much of their effect by acting against one invariable opposition.

15. In sails of a similar figure and position, the number of turns in a given time will be reciprocally as the radius or length of the sail.

16. The load at a maximum that sails of a similar figure and position will overcome, at a given distance from the centre of motion, will be as the cube of the radius.

17. The effects of sails of similar position and figure are as the square of the radius. Hence augmenting the length of the sail without augmenting the quantity of cloth, does not increase the power; because what is gained by the length of the lever is lost by the slowness of the motion. Hence also if the sails are increased in length, the breadth remaining the same, the effect will be as the radius.

18. The velocity of the extremities of the Dutch sails, as well as of the enlarged sails, either unloaded or even when loaded to a maximum, is considerably greater than that of the wind itself. This appears plainly from the observations of Mr Fergufon already related concerning the velocity of sails, and is more fully treated of under the article Wind.

19. From many observations of the comparative effects of sails of various kinds, Mr Smeaton concludes, that the enlarged sails are superior to those of the Dutch construction.

Having thus discussed the subject of the common windmills with oblique vanes, our author next proceeds to the consideration of those called horizontal windmills, in which it is attempted to make the wind impinge directly upon the wheel, as in the case of watermills. To set the probable advantage of this scheme in its proper point of view, Mr Smeaton proceeds in the following manner: "Let AB, fig. 111. be the section of a plane, in which let the wind blow in the direction CD, with such a velocity as to describe a given space BE, in a given time, suppose one second; and let AB be moved parallel to itself in the direction CD. Now, if the plane AB moves with the same velocity as the wind; that is, if the point B moves through the space BE in the same time that a particle of air would move through it, it is plain, that in this case there can be no pressure or impulse of the wind upon the plane; but if the plane moves slower than the wind, so that the point B may move to F, while a particle of air setting out from B would reach E, then BF will express the velocity of the plane; and the relative velocity of the wind and plane would be expressed by the line FE. Let the ratio of FE to BE be given, suppose 2 to 3; let the line AB represent the impulse of the wind upon the plane AB when acting with its whole velocity BE; but when acting with its relative velocity FE, let its impulse be denoted by some aliquot part of AB, as for instance \(\frac{2}{3}\); then will \(\frac{2}{3}\)ths of the parallelogram AF represent the mechanical power of the plane, that is, \(\frac{2}{3}\)ths \(AB \times \frac{2}{3}BE\).

"2. Let IN be the section of a plane inclined in such a manner, that the base IK of the right angled triangle IKN may be equal to AB; and the perpendicular NK = BE: let the plane IN be struck by the wind in the direction LM, perpendicular to IK; then, according to the known rules of oblique forces, the impulse of the wind upon the plain IN, tending to move it according to the direction LM or NK, will be denoted by the base IK; and that part of the impulse tending to move it, according to the direction IK, will be expressed by the perpendicular NK. Let the plane IN be moveable in the direction of IK only; that is, the point I in the direction of IK, and the point N in the direction NO parallel thereto. Now it is evident, that if the point I moves through the line IK, while a particle of air, setting forwards at the same time from the point N, moves through the line NK, they will both arrive at the point K at the same time; and consequently there can be no pressure or impulse of the particle of air upon the plane IN. Now let IO be to IK as BF to BE; and let the plane IN move at such a rate, that the point I may arrive at O, and acquire the position OQ, in the same time that a particle of air would move through the space NK; as OQ is parallel to IN, by the properties of similar triangles, it will cut NK in the point P in such a manner, that NP will be equal to BF, and PK to FE. Hence it appears, that the plane IN, by acquiring the position OQ, withdraws itself from the action of the wind, by the same space NP that the plane AB does by acquiring the position FG; and consequently, from the equality of PK to FE, the relative impulse of the wind PK upon the plane OQ will be equal to the relative impulse of the wind upon the plane FG: and since the impulse of the wind upon AB, with the relative velocity FE, in the direction BE, is represented by \(\frac{2}{3}\) \(AB\); the relative impulse of the wind upon the plane IN in the direction NK will in like manner be represented by \(\frac{2}{3}\) IK; and the impulse of the wind upon the plane IN, with the relative velocity PK, in the direction IK will be represented by \(\frac{2}{3}\) NK: and consequently the mechanical power of the plane IN in the direction IK will be represented by \(\frac{2}{3}\) of the parallelogram IQ; that is, \(\frac{2}{3}\)IK \(\times\) \(\frac{2}{3}\)NK: that is, from the equality of IK to AB, and NK to BE, we shall have \(\frac{2}{3}\)IQ = \(\frac{2}{3}\)AB \(\times\) \(\frac{2}{3}\)BE = \(\frac{2}{3}\)AB \(\times\) \(\frac{2}{3}\)BE = \(\frac{2}{3}\) the area of the parallelogram AF.

"Hence we deduce this general proposition; that all planes, however situated, that intercept the same section of the wind, and having the same relative velocity in regard to the wind, when reduced into the same direction, have equal powers to produce the same mechanical effects. For what is lost by the obliquity of the impulse, is gained by the velocity of the motion.

"Hence it appears, that an oblique sail is under no disadvantage in respect of power, compared with a direct one; except what arises from a diminution of its breadth," breadth, in regard to the section of the wind; the breadth IN being by obliquity reduced to IK.

"The disadvantage of horizontal windmills therefore does not consist in this, that each sail, when directly exposed to the wind, is capable of a less power than an oblique one of the same dimensions; but that in an horizontal windmill little more than one sail can be acting at once: whereas, in the common windmill, all the four act together; and therefore, supposing each vane of an horizontal windmill to be of the same size with that of a vertical one, it is manifest that the power of a vertical mill will be four times as great as that of an horizontal one, let the number of vanes be what we will. This disadvantage arises from the nature of the thing; but if we consider the further disadvantage that arises from the difficulty of getting the sails back again against the winds, &c. we need not wonder if this kind of mill is in reality found to have not above one-eighth or one-tenth of the power of the common sort; as has appeared in some attempts of this kind."

Notwithstanding what is here advanced, it seems that the ideas of Mr Smerton have not been very generally received, as premiums are still held forth for the best methods of constructing horizontal windmills. Indeed, considering the charms and perspicuity of the above reasoning, it seems surprising that public encouragement should continue to be given to attempts which must certainly prove abortive. The principal inconvenience in wind-mills is their excessive irregularity and difficulty of being managed when the wind is high, owing to the great extent of the sails and bulk of the machinery. But were it possible to make a number of small wind-mills exert their power upon one object, these would be much more easily managed than one large one. Perhaps if a number of these were to be employed in pumping up water to a certain height from a lake or reservoir, so as to produce a constant stream of water to turn a common mill, it might be more advantageous than to employ them directly. Wind-mills are commonly erected upon eminences for the sake of receiving the wind to more advantage; and there are few eminences which do not afford a small supply of water at no great distance from their summit. This supply being collected in a reservoir, might be drawn up to the top by pumps worked by wind-mills; where being collected in another reservoir, it might be let down to the former, turning a water-mill in its way, and being again drawn up by the pumps as before.

Some projectors, considering the great power of oblique vanes in wind-mills, have attempted to improve water-mills by giving them oblique vanes, but with as little success. The power of the same section of a stream of water is not greater when acting upon an oblique vane than on a direct one; and any advantage which can be made of intercepting a greater section of water, which sometimes may be done in the case of an open river, must be counterbalanced by the superior resistance that such vanes would meet with by moving at right angles to the current: whereas the common floats always move with the water nearly in the same direction.

Mr Smerton concludes his dissertation upon this subject, with giving a reason why one angle should be preferable to another in setting the sails of a wind-mill.

"It is to be observed (says he), that if the breadth of the sail IN is given, the greater the angle KIN, the less will be the base IK; that is, the section of the wind intercepted will be less. On the other hand, the more acute the angle KIN, the less will be the perpendicular KN; that is, the impulse of the wind in the direction IK being less, and the velocity of the sail greater, the resistance of the medium will be greater also. Hence, therefore, as there is a diminution of the section of the wind intercepted on one hand, and an increase of resistance on the other, there is some angle where the disadvantage arising from these causes upon the whole is the least of all; but as the disadvantage arising from resistance is more of a physical than geometrical consideration, the true angle will best be assigned by experiment."

Sect. VI. Of the Motion of Bodies in Straight Lines and Curves; the Acceleration, Accumulation, and Retardation, of Motion in various Circumstances.

To understand this subject, it is necessary to keep in mind what has been said concerning the momentum or quantity of motion in any moving body, viz. that it is compounded of the velocity multiplied into the quantity of matter. Thus, suppose there are two bodies, one containing twice the quantity of matter contained in the other, but moving with thrice its velocity, the quantities of matter will be expressed by any numbers in the proportion of 2 to 1, and their velocities by any others in the proportion of 3 to 1. Multiplying therefore the quantity of matter in the first (2) by its velocity (3), the product is 6; and multiplying the quantity of matter (1) by its velocity (1), the product is only 1; whence it appears that the momenta or absolute forces of these bodies are to one another as 6 to 1.

As some bodies are elastic and others non-elastic, the effect of motion communicated from one to another becomes very different, according to this circumstance. The motion is likewise very different, according to the manner in which one body acts upon another, and according to which it will be driven forward in a rectilinear direction, or describe curves of various kinds, revolving on its axis, &c. These different kinds of motion have been considered by different authors, but by none more particularly than Mr G. Atwood, who has published a large octavo volume upon the rectilinear motion and rotation of bodies. The fundamental laws of motion assumed by this author as axioms are three.

1. Every body perseveres in its state of rest or uniform motion in a right line, until a change is effected by the agency of some external force.

2. Any change effected in the quiescence or motion of a body is in the direction of the force impressed, and is proportional to it in quantity.

3. Action and reaction are equal, and in contrary directions.

From these three simple axioms, the truth of which must, from what has been already said, be abundantly evident, our author proceeds to demonstrate the most difficult problems concerning the impulse and motion of Mechanics

1. Two mathematical quantities of the same kind, as two lines, two surfaces, two angles, &c., constitute a ratio. Thus, suppose one line two feet in length and another four; these are to one another in the ratio of 4 to 2, or of 2 to 1; but a line cannot be said to bear any ratio to a surface, because they are not quantities of the same kind, and therefore cannot be compared.

2. We may compare the ratio of two quantities of one kind with the ratio of two quantities of a different kind. Thus, when two bodies move uniformly, for an equal time, but with different velocities, the ratio of the spaces passed over may be compared with that of the velocities, though space and velocity are accounted different quantities.

3. If any quantity be divided by another of the same kind, the quotient becomes absolute number. Thus, if we divide a velocity of four feet in a second by one of two feet in a second, the quotient will be 2; and in all cases the quotient will be to 1 as the greater quantity is to the lesser.

4. The ratio of any mathematical quantities may be expressed by two numbers, if both terms of the ratio be divided by the consequent or by the antecedent. Thus, let the antecedent be 8 and the consequent 4; let both be divided by 4, and the ratio will then be 2 to 1; or let both be divided by 8, and it will be 1 to 2, which is the same.

5. Any ratio may be represented by a fraction, the numerator of which is the antecedent, and the denominator the consequent. Thus the ratio of 8 to 4 is represented by the fraction $\frac{8}{4}$; and hence we may add and subtract ratios by the addition and multiplication of fractions. Thus, supposing two bodies to move uniformly, one at the rate of 8 feet in a second and the other 4; supposing them also to move the former for four, and the latter for two seconds, the spaces passed over will be 32 and 8, their ratio $\frac{32}{8}$, or at length $32 : 8 = \frac{32}{8}$. Here it is to be observed, that when the mark of equality is interposed between heterogeneous quantities, the only equality meant is that which subsists between the ratios there expressed; and when the mark of multiplication is interposed between heterogeneous quantities, it means the addition of two ratios, the antecedents of which are the terms expressed, and the consequents are unity.

6. If there are three ratios, consisting of variable terms, and the relation of the quantities to each other be such, that when the third ratio becomes unity the other two become equal; or when the second becomes unity, the first and third are equal; then in all cases, whatever be the magnitudes, we have the first ratio = the second × third. Thus let the three ratios be $\frac{4}{7}$, $\frac{7}{9}$, and $\frac{9}{7}$, diminishing the numbers by 1, we have $\frac{4}{6}$, $\frac{6}{8}$, and $\frac{8}{6}$; it is evident that $\frac{4}{6} = \frac{6}{8} \times \frac{8}{6}$; the same will be the case if we place them in a different order, as $\frac{4}{7}$, $\frac{7}{9}$, and $\frac{9}{7}$; for then, diminishing as before, we have $\frac{4}{6}$, $\frac{6}{8}$, and $\frac{8}{6}$, in which case $\frac{4}{6} = \frac{6}{8} \times \frac{8}{6}$.

7. In comparing the ratios which obtain between mathematical quantities of any sort, the standard to which each of those quantities is referred may be taken = 1. Thus, supposing we compare the weight, magnitude, and density, of any substance with water, we may take a cubic inch of that element for a standard, and call the weight, magnitude, and density of it = 1; by which means we may readily compare the weight, magnitude, or density, of any quantity, however large, of another substance with water.

We now proceed to that part of the work which treats directly of the motion of bodies acted upon by any external impulse.

8. Any force acting continually upon a body in the same direction, will produce a continual acceleration or retardation of the motion. Thus, if a body descends by the force of gravity, its motion is continually accelerated; or if it be thrown up against the force of gravity, the motion will be continually retarded until it be totally destroyed.

9. If, while a body moves, equal quantities of motion be communicated to it, or taken from it in equal spaces of time, the force is said to be constant, and equally accelerated or retarded.

10. When unequal velocities are generated or destroyed in equal spaces of time, the force is said to be variable.

11. When a body is acted upon by a constant force, we must consider the space through which it moves, the time it takes to move through it, the velocity it acquires, and the force which produces it; any two of which being given, we may from them find the other two. Here we must observe, that the force mentioned relates only to the communication of the velocity, without any regard to the quantities of matter moved. As it is proportioned to the velocity generated in a given time, it is thence called the accelerating force. That which relates to the quantity of matter moved, as well as the velocity communicated, is called the moving force; being proportional to the quantity of motion produced in a given time.

12. The moving forces which communicate the same velocity in a given time to different bodies, will be as the quantities of matter contained in the bodies moved. This will appear from a consideration of what has already been said concerning the momenta of bodies. For if one body contains ten times the quantity of matter that another does, it will of course require ten times as much force to move it with an equal degree of velocity; for the former is equivalent to ten such bodies, and it is the same thing whether they be separate or altogether.

13. The moving forces which act upon bodies, and the degrees of velocity communicated to them in a certain time, are proportional to the quantities of matter moved and the velocities communicated jointly; for, by the last proposition, when the velocity communicated in a certain time is the same, the moving force is as the quantity of matter moved. Thus, if a ball of ten pounds weight is made to move at the rate of 10 feet in a second, and another of one pound is made to move at the same rate, the moving forces will be in proportion to the quantities of matter; that is, as 10 to 1. Hence we may easily perceive, that when the quantity of matter is given, the moving force will be as the velocity. Thus, if two balls of ten pounds each... Motion of each arc caused to move, one with the velocity of ten feet in a second, and the other with a velocity of five feet in the same time, the forces will be as the velocities; that is, as 10 to 5, or as 2 to 1; and hence, when both the quantities of matter and velocities are different, the moving forces will be according to these jointly. Thus if a ball of ten pounds is moved with a velocity of ten feet in a second, and a ball of one pound moves with a velocity of five feet in a second, the moving forces will be as $10 \times 10$ or 100 to $1 \times 5$, or as 20 to 1.

Here our author takes occasion to deny that there is any such thing as a communication of motion by an instantaneous impulse or stroke, as has commonly been supposed. Every degree of motion, according to him, is the effect of acceleration. "The latter way (says he) viz. the communication by instantaneous impulse, can obtain only in bodies perfectly hard and inflexible, which exist not in nature; and even in the abstract consideration of these as well as of other cases in mechanics, when metaphysical possibilities instead of the natural state of bodies are attended to, difficulties arise hardly explicable by any method of reasoning: but it is certain, that when finite velocity is communicated to any natural body, the time in which it is communicated must be finite also; so that when the body acted upon begins to move from quiescence, it must, during the action of the force, possess all the intermediate degrees of velocity between 0 and the velocity ultimately communicated.

"To exemplify this further, let it be supposed that a soft and flexible ball of clay impinges against another of the same sort, in the direction of a line joining the centres of the balls. At the first instant of the impact, the body struck will begin to move, and will proceed with a velocity inferior to that of the impinging body, the velocity of which will continue to decrease, and that of the other body to increase, as long as the impinging force causes a change in the figure of the two bodies; that is, till they have both acquired a common velocity; at which instant all acceleration ceases if the bodies be perfectly non-elastic. If the bodies be of such a kind, as, after having received impression from any impact, possess a power of restoring their changed figure with a force equal to that of the impact, it is manifest, that whatever velocity was communicated during the change of figure, an equal quantity will be superadded during the restoration of it. In this case, after the acceleration arising from the impact during the change of the figure of the bodies has ceased, the bodies having then acquired a common velocity, a new acceleration will begin, being caused by the elastic force of the balls, which, acting in a direction of the lines joining their centres, tends to separate them, accelerating the ball struck, and retarding the other.

"From these considerations it appears, that in whatever degree the hardness of perfectly elastic bodies may differ, the effects of their impact will be the same, the weights and velocities before the stroke being given. For the figures of the striking and of the other body continually change, till they have acquired a common velocity, which depends only on the velocity of bodies and their impact, and is determined by the rules for the collision of non-elastic bodies. Moreover, the restoration of the changed figures, how great or how small forever may have been the change, must cause an addition of velocity in the ball struck equal to that received from the impact.

"It follows also, that the effect will be the same, whether the bodies be both perfectly elastic, or whether one of them be perfectly elastic and the other perfectly hard; every thing else being given for the figure of the elastic body must change until the bodies have obtained a common velocity, which depends on the weights and velocities before the stroke only; and will be the same as if the bodies were non-elastic: the restoration of the figure will in this, as well as in the former case, cause an increase of velocity in the ball struck, equal to that before communicated. Although no substance in nature possesses perfect elasticity, or is entirely destitute of it, yet there are several elastic and non-elastic bodies subject to experimental trials, wherein the laws relating to collision are found to agree with fact to a considerable degree of exactness."

14. The accelerating forces which communicate velocity to bodies are directly as the moving forces, and inversely as the quantity of matter moved; for since by prop. 11. the accelerating force is as the velocity generated in a given time; and by prop. 13. the moving force is as the quantity of matter and velocity generated in a given time, it follows, that the moving force is as the accelerating force and quantity of matter moved jointly: that is, the accelerating force is as the moving force directly, and the quantity of matter moved inversely. Thus, let a mass of matter, equal to four ounces, be impelled by a force equal to three ounces; then the force which accelerates the mass of four ounces will be three-fourths when the acceleration of gravity is ; or in other words, it will generate, in a given time, three parts in four of the velocity which gravity generates during any given time.

15. In bodies impelled in a rectilinear direction by forces acting uniformly, the velocities generated are as the forces and times in which they act, conjointly. Thus, suppose a force equal to ten acting upon a ball of ten pounds, and another also equal to ten acting upon a ball of equal weight, the former for one second, and the latter for two; it is plain that the velocity generated in the latter will be double to that generated in the former. But if we suppose the latter ball to be acted upon by a force equal only to five, then will both the velocities be equal, though the latter should continue for two seconds and the former only for one. In all practical inquiries of this kind, however, it must be remarked, that a standard velocity is to be obtained from observing what degree of velocity is generated by the force of gravity during a given time; one second, for instance.

16. If a quiescent body be impelled by any constant force acting upon it for a given time, the space described will be to the space described in the same time by a body moving uniformly with the last acquired velocity, in the ratio of one to two. In order to understand this, we must suppose the time to be divided into such small parts that the acceleration during any one of them is imperceptible: then it is evident, that at the end of two moments, the impulse continuing the same, it will have gained double the velocity it did the first moment; and this The constant forces which accelerate bodies from a state of rest, are in a direct duplicate ratio of the velocities generated, and in an inverse ratio of the spaces described. Hence the following corollary is deduced, viz. that the last acquired velocities are in a subduplicate ratio of the accelerating forces, and a subduplicate ratio of the spaces described jointly.

If bodies unequal in quantity of matter be impelled from rest through equal spaces, by the action of moving forces which are constant, these forces are in a duplicate ratio of the last acquired velocities, and the ratio of the quantity of matter jointly.

In his observations on this proposition, our author takes occasion to consider the theory of those who insist, contrary to the opinion of Sir Isaac Newton that the absolute force of bodies is compounded of the quantity of matter and square of the velocity, instead of the velocity itself. "In the experiments (says he) which have been made on the force of bodies, the loss of motion from resistance has been more attended to than the communication of it by acceleration; and the reason probably arose from a want of adequate methods of subjecting accelerating forces, velocities acquired, and quantities of matter moved, to experimental trials; whereas the impact of bodies on substances which they penetrate, by affording convenient opportunity for observing the depths to which bodies sink before all motion is destroyed, regard being had to the velocities of impact, and the weight and form of the impinging body, has seemed a more eligible method, however imperfect, of investigating the principles of motion."

When a body descends for three seconds by the force of gravity, it acquires, by a force of acceleration, a velocity of 96 feet in a second; also, if a body be projected perpendicularly upward, with a velocity of 96 feet in a second, the whole velocity will be destroyed in three seconds; and in like manner, every other property demonstrated concerning accelerated motions is found to belong to retarded ones, provided we attend to the following circumstances: If in any proposition relating to accelerated motion, the force is constant, it follows, that when this is applied to retarded motion, the retardation must also be constant. Moreover, since in accelerated motions the spaces are estimated from quiescence, so in retarded motions the bodies are supposed to move to quiescence; that is, till all motion is destroyed by retardation: in whatever concerns motions of this kind, therefore, we must consider the retarding force to be directly as the force of resistance, and inversely as the quantity of matter.

"In order to illustrate this subject, it is to be observed, that if a body projected with different initial velocities be retarded by any constant given force, the whole spaces which the body describes are in a duplicate ratio of the initial velocities, which follows from what has been already demonstrated; and conversely, since when bodies are impelled by an accelerating force through various spaces, if these spaces are always as the squares of the last acquired velocities, it follows that the force of acceleration is constant: so when a given body is projected with different velocities, and is retarded by a given force, if the whole spaces described be always in a duplicate ratio of the initial velocities, it is concluded, that the force of retardation..." is constant. It is from this argument inferred, that the force whereby blocks of wood, banks of earth, &c., resist the penetration of bodies impinging on them, is constant; for it is observed, that the depths to which military projectiles of a given magnitude and weight, striking against a body of this kind, enter its substance, are in a duplicate ratio of the initial velocities, which has been sufficiently proved by Mr Robins, who first ascertained the velocities of military projectiles, and applied his method, among other useful purposes, to the discovery of the retardation which bodies suffer by passing through resisting substances.

"The forces of resistance, which are opposed to the motion of bodies impinging on substances which they penetrate, being granted constant, the propositions concerning acceleration already demonstrated may be applied to explain the motion of bodies, which, having been projected with given initial velocities, are interrupted by such obstacles as blocks of wood, banks of earth, or others of a similar kind.—For example, it has been demonstrated, that bodies moving from rest by the acceleration of constant forces, describe spaces which are as the accelerating forces and squares of the times jointly. By applying this proposition to retarded motions, we shall have the whole spaces or depths to which bodies impinging on the substances penetrate, as the forces of retardation and squares of the times wherein the bodies move, jointly. Moreover, it has been demonstrated, that if different quantities of matter be impelled from rest through equal spaces, the moving forces will be in a ratio compounded of the duplicate ratio of the velocities last acquired, and the ratio of the quantities of matter moved. It is hence inferred, that in retarded motions also, if different quantities of matter be projected against any of the substances above described, with different initial velocities, and the whole depths to which the bodies penetrate are equal, the forces whereby the bodies resist the progress of the impinging bodies will be in a duplicate ratio of the initial velocities of impact and the quantities of matter jointly.

"By this proposition we may examine some of the experiments concerning the force of moving bodies, and the conclusions deduced from them by Bernoulli, Leibnitz, Poleni, &c., against the measure of force delivered by Sir Isaac Newton, which he described in the following definitions:

"The quantity of motion is measured by the quantity of matter in a moving body and its velocity jointly.

"The moving forces whereby bodies tend towards centres of attraction are as the quantities of motion generated in a given time.

"It follows, then, from these definitions, that the moving forces, acting for a given time, will be proportional to the quantities of matter moved, and velocities generated, jointly; so that if the ratio of the moving forces be known, and we can find by experiment what velocities are generated in given bodies by the action of them for the same time; the quantities of motion generated in the bodies may be estimated according to Sir Isaac Newton's definition. Moreover, since it is allowed that the effects of a resisting force to destroy are the same as those of an equal force to generate motion in a given time; it follows, that if the ratio of two resisting forces be known, the quantities of matter in bodies which impinge on substances, and penetrate them, and the velocities destroyed in a given time, will give the ratio of the quantities of motion destroyed, according to Sir Isaac Newton's definition.

"In many of the experiments alluded to, which have been greatly varied and multiplied, the resisting forces were made equal, by causing spheres equal in magnitude to impinge on a given substance which they penetrated; and the spheres being of given densities, it was observed in experiments, that whenever the densities or weights of those equal spheres were in an inverse duplicate ratio of their velocities, the depths to which they penetrated would be equal. The conclusions were these: the quantities of matter displaced by the moving bodies were equal, the depths to which the equal sphere penetrated were the same. Moreover, the whole motions which had been communicated to the bodies were destroyed; that is, the whole motion of the impinging bodies must have been as the squares of the velocities into the quantities of matter. But it plainly appears, that this conclusion is not applicable to the Newtonian definition, according to which the moving force generates motion in bodies; and it follows by what has preceded, that the resisting force by which the motion of bodies is destroyed, is proportional to the quantities of motion generated or destroyed in a given time respectively; and consequently, to estimate the quantity of motion destroyed, the time wherein resisting forces act should be equal. If, therefore, the times wherein the bodies in the experiment describe the equal spaces can be proved different, this will plainly show that the quantities of motion destroyed cannot be inferred from the experiment, the different times of the bodies describing the depths to which they link not being taken into the account; this will be easily proved, since from proposition 17, it appears, that the spaces described are universally as the velocities and spaces last described jointly; and from what has been said, the converse of this proposition when applied to retarded motions must also be true. The spaces therefore being given as in the experiment, the times will be inversely as the initial velocities; which velocities being unequal from the experiment, it follows that the times are unequal. This being the case, it is manifest that no conclusion can be drawn from these experiments concerning the quantity of motion destroyed, tending to prove any inconsistency between the Newtonian estimation of force and matter of fact.

"It is next to be shown, that the experiments are strictly consistent with the Newtonian measure, and with the theory in general.—It has already been proved, that in accelerated motions the spaces described are in a duplicate ratio of the velocities last acquired, and the quantities of matter moved, and an inverse ratio of the moving forces. This proposition being applied to retarded motions, it will follow, that the whole spaces or depths to which the impinging bodies sink, are in a duplicate ratio of the quantities of matter, and an inverse ratio of the resisting forces; whence also the depths to which the bodies penetrate must be equal when spheres of equal diameters are projected against a given substance, the weights being in an inverse duplicate..." plicate ratio of the initial velocities, which we find entirely correspondent to experiment. It seems indeed rational to suppose, independent of all theory, that, in estimating the quantities of motion generated or destroyed by given moving or resisting forces, regard must be had to the times wherein the forces act; because moving forces, or those of resistance, may be equal, and may generate or destroy quantities of motion varying in any assignable degree. For it is manifest, that a small resistance, opposed to a moving body for a longer time, may destroy more motion than a greater force acting for a shorter time; which sufficiently shows, that the times wherein the moving and resisting forces act, must either be equal, or must be taken into the account in estimating the quantities of motion generated or destroyed.

21. The moving forces which communicate, and the forces of resistance which destroy, the motion of bodies in the same time, will be in a compound ratio of the quantities of matter in the moving bodies and velocities generated or destroyed.—This and the preceding propositions have been fully illustrated and confirmed by experiments. From them we deduce the following facts: 1. When musket-balls, equal in weight and magnitude, impinge on a block of wood with different velocities, the resisting force being constant, we shall have the whole spaces through which the balls move in the wood as the squares of the velocities. 2. If balls of equal diameters, but different weights, impinge against a block with the same velocity, we have the depths to which they penetrate the block as the weights. 3. If balls of the same kind of substance, that is, of the same density, but of different diameters, impinge against a given block of wood or the same bank of earth with equal velocities, the depths to which they penetrate will be directly as the diameters of the balls.

When the force of resistance is not uniform, the same principle obtains in degree, though the laws are then various; for greater bodies always suffer less by retardation than smaller ones of the same density, moving through the same resisting medium, and projected with a given initial velocity; because, though the force of resistance increases with the increase of the body's magnitude, yet the weight in most bodies increases in a greater proportion. Thus, in cannon-balls, and other solid bodies, though the resistance of the air increases as the square of the ball's diameter, yet the weight increases as the cube. Thus, if a ball two inches in diameter is projected from the mouth of a piece, it is resisted by the atmosphere four times less than one four inches in diameter; but the weight of the latter, being eight times greater, makes the resistance less upon the whole in the large ball than in the small one. It is otherwise when the weight does not increase in this manner; for then the smaller the body is, the less resistance it meets with, and the faster it goes. This is manifest from aerostatical experiments; for small air-balloons always outstrip the larger ones; and the same thing is observable in boats; for the smaller ones, if they have the same advantages in proportion to their bulk, will always fail faster than the larger ones.

22. If bodies, projected with the same velocity, be retarded by different constant forces, these forces will be in an inverse ratio of the whole spaces described by the projected bodies, until all motion is destroyed. For example, let a body be projected on an inclined plane, in a direction contrary to that in which gravity acts in the plane, and with a velocity of 144.467 inches in a second. Suppose the body then projected, ascending along the plane, to describe 216 inches before its motion is destroyed; let it be required to ascertain the retarding force which opposes its ascent, that is, the proportion of it to the force of gravity. If the body were projected perpendicularly upward, with the given velocity of 144.467 inches in a second, it would rise only to 27 inches, as follows in Prop. 19. And since it ascends along the plane 216 inches, the retarding force on the plane will be to that of gravity as 27 to 216, or as 1 to 8; which is also the proportion of the height of the plane to the length of it.

From this proposition, having given the depth to which a body impinging against another penetrates it, the proportion of the retarding force of gravity may be determined. For example, Mr. Robins found that a leaden ball of \( \frac{1}{4} \) of an inch, or \( \frac{1}{16} \) of a foot in diameter, impinging on a block of elm with a velocity of 1700 feet in a second, penetrated it to the depth of five inches, or \( \frac{5}{17} \) of a foot; wherefore, since a body projected upwards with a velocity of 1700 feet in a second, would rise, if the atmosphere made no resistance, to the height of 4492 feet, we have the force by which elm retards the ball to the force of gravity as 4492 to \( \frac{5}{17} \); or as 107,813 to 1.

On this theory it may further be observed, that the resistances opposed to spherical bodies, which impinge on a block of wood, a bank of earth, &c., depend not only on the tenacity or density of the parts, of which the penetrated substances are composed, but upon the diameters of the impinging spheres; so that, although the resisting and retarding forces be determined in any substance for a single case; yet when the diameters and weights of the impinging bodies vary, the forces of resistance and retardation opposed to the impact on the same substance will be different. By the preceding proposition, however, we may be enabled, from a single experiment made on the retardation of any substance opposed to a sphere, the weight and diameter of which are known, to infer the retardation in any other case, however the weights and diameters may vary.

23. If spheres of different diameters and different specific gravities, impinge perpendicularly on fixed obstacles, the resisting forces of which are constant, but of different quantities, the forces which retard the progress of the impinging spheres will be in a direct ratio of the absolute forces of resistance, and the joint inverse ratio of the diameters and specific gravities of the spheres. No absolute conclusion can be drawn from this proposition concerning any matter of fact, unless an experiment be first made on the retarding and resisting force of some substance which is to be considered as a standard.

24. The whole spaces or depths to which spheres, impinging on different resisting substances, penetrate, are in the ratio compounded of the duplicate ratio of the velocities of impact, the joint ratios of the diameters and specific gravities of the spheres, and an inverse ratio of the absolute forces whereby the substances resist the pressures of the spheres. Mr Atwood concludes this section with some problems relative chiefly to military projectiles; and in his next section (the 4th) considers the rectilinear motion of bodies acted upon by forces which vary in some ratio of the distances from a fixed point. This section chiefly relates to the powers of gravity and projection, by which the celestial bodies are actuated, and which consequently chiefly regards astronomy and the motion of pendulums; though there are likewise some curious particulars relating to the action of compressed air, the vibration of musical strings, and the undulation of fluids. The fifth section considers the motion of bodies immersed in fluids; but the sixth treats of a subject which properly belongs to mechanics, viz. the communication of motion to bodies revolving round an axis.

In treating this subject Mr Atwood observes, that in the former part of his work he had supposed the accelerating, as well as resisting, forces, to act upon the body in a straight line passing through the centre of gravity of the moving; in which case every particle of the body must partake of the same degree of velocity, being equal to that with which the common centre of gravity moves. "But (says he) it frequently happens, that a body, or system of bodies, is so constituted, that when any force is impressed upon it, no motion can be produced except round a fixed axis; so that the velocity of the particles which compose the system will be greater or less according as these particles are farther from the common axis or nearer it. These circumstances should be attended to, in order to ascertain the motion of revolving bodies; the preceding principles of acceleration being not wholly of themselves sufficient for that purpose.

"In this investigation two things must be attended to. 1. The moving force by which the revolving motion is generated; and, 2. The inertia of the parts of which the system is composed. The moving force exerted on any given particle of the system, as well as its inertia, depends on its distance from the axis of motion, everything else being the same; and if both these be ascertained, the absolute acceleration of the particle will be determined, and consequently the absolute velocity generated in a certain time. The methods therefore of determining these forces in any given circumstance should next be described.

"Let AFGH (fig. 112) represent the circumference of a wheel which turns in its own plane round an horizontal axis, passing through its centre; and let a weight P, fixed at the extremity of a line AP, communicate motion to the wheel. Moreover, let the whole weight of the wheel be Q; and suppose this weight to be collected uniformly into the circumference AFGH; then, during the descent of the weight P, each point of the circumference must move with a velocity equal to that with which P descends; and consequently, since the moving force is the weight P, and the mass moved \(P+Q\), the force which accelerates P in its descent will, by Prop. 14, be that part of the accelerating force of gravity which is expressed by the fraction \(\frac{P}{P+Q}\). The velocity, therefore, which is generated in P in any given time, is found from the rules before demonstrated. Thus, supposing Q to be equal to P, then will \(\frac{P}{P+Q} = \frac{1}{2}\); and the weight P will be accelerated by a power which is to that of gravity as 1 to 2; and since gravity generates in bodies which descend for one second of time near the earth's surface a velocity of 32 feet in a second; it follows, that the weight P will in the same time have acquired a velocity of 16 feet in a second only.

"The parts of the weight Q which are uniformly disposed over the circumference AFGH, balance each other round the common centre of gravity S; their weight therefore is of no effect in accelerating or retarding the descent of P; and this will be the case whenever the axis of motion passes through the common centre of gravity. But in order to render the properties of rotatory motions more obvious, it will be convenient to dispose the parts of the revolving system so that the axis of motion shall not necessarily pass through the common centre of gravity: thus, instead of having the weight Q uniformly disposed over the circumference AFGH, let it be collected into any point Q. Here it is manifest, that if the mass Q be acted upon by gravity, the force which communicates motion to the system round S will be variable, it being the greatest when SQ is horizontal, and gradually diminishing till Q has arrived at its lowest point. But as we should begin with the most simple cases, the moving force must be constant. This will be effected by supposing the mass which is collected in Q to be destitute of weight, and to possess inertia only. It follows therefore, that during the revolution of Q round S as an axis, the moving force will be constantly equal to P, and the mass moved \(P+Q\). Consequently the force which accelerates the descending weight, or any point in the circumference, will be that part of gravity which is expressed by the fraction \(\frac{P}{P+Q}\) as before.

"In these cases, the force which communicates motion to the system has been supposed a weight or body acted upon by the earth's gravity, and consequently constitutes a part of the mass moved, at the same time that it acts as a moving force: but motion may be communicated by a force which shall add nothing to the inertia of the matter moved; and it will be convenient in many demonstrations to assume the force of this kind; and in this case we have not to take the inertia into the account. Thus if any number of bodies without gravity collected into the points F, H, Q, (fig. 112.) are caused to revolve round the axis S, by a moving force P, the force which accelerates these bodies in their revolution will be \(\frac{P}{F+H+Q}\); provided the bodies F, H, Q, be disposed at a distance from the axis of motion equal to the radius of the circle AFGH, at the circumference of which the moving force P is applied.

"In the preceding example, F, H, Q, &c. have been supposed to move with the same velocity; but when bodies revolve at unequal distances from the axis, their velocities of motion being different, other rules will be necessary to determine the force whereby any given point of the system is accelerated. In demonstrating the properties of rotatory motion, the revolving..." Motion of ving system may be supposed to consist of one or more bodies A, B, C; the magnitude of these may be supposed evanescent; because, were the contrary supposition adopted, the particles in each body would be impelled by different moving forces, and exert different degrees of inertia in opposition to the communication of motion. But the force which impels each individual particle, and the effects of its inertia in different circumstances, must be known before the acceleration of the whole system can be determined.

"The bodies A, B, and C, which may be termed, according to the ideas just described, material points, are imagined to be connected together by some perfectly rigid substance, so as always to possess the same situation in respect to each other; and consequently no motion can be produced in any of them, excepting that all revolve at the same time round the common axis of motion.

"All the points in this imaginary substance, by which the parts of the system are connected together, partake of the same angular motion, describing circles round the common axis S. A force P therefore being applied to any point in the plane of its motion, and in the direction of any line in that plane which passes not through the axis, will communicate an equal angular motion to the whole. Thus let B (fig. 113.) represent a material point moveable about an axis of motion passing through S. With the radius SD describe a circle DGH. Now if B be connected with every point in the area of this circle, which is an inflexible substance, no force can be applied to move the circle but what must communicate the same angular motion to B. Let the force be applied at the point D; it is manifest, that in order to render its effects constant, the inclination of its direction to SD must be always the same, and in a given plane; and the most obvious method of effecting this, either in considering the subject theoretically, or in the practical illustration of it, is by applying a thin and flexible line GHDP round the circumference of the circle DGH, and stretching this line by a given moving force P.

Here it is plain, that in whatever part the point D is situated, the effects of the force P will be the same as if it were directly applied to D in the direction of the plane of motion, and perpendicular to SD, and the point B will revolve with the same absolute and angular velocity in both cases.

"Let now ABC (fig. 114.) be a system of bodies of evanescent magnitude and without gravity, moveable about an axis of motion which passes through S; it must be observed, that the imaginary substance by which the parts of the system ABC are connected, must contribute nothing either by its weight or inertia to accelerate or retard the motion of the material points A, B, C, which are caused to revolve by the action of the given and constant force P, applied at the distance from the axis SD. The absolute force of P to move D, or any point of the circumference, will be P; but the communication of motion to this point D is resisted by the inertia of the bodies A, B, C; which being moved with different velocities, and acted on by different moving forces, their inertia will not be estimated by their quantities of matter only, according to the laws observed in rectilinear motion; the force which accelerates D, therefore, cannot be obtained by dividing P by A+B+C; but if an equivalent mass, or a quantity of matter, can be assigned, which being collected into any points of the circumference a, b, c, will cause an inertia or resistance to the motion of D equal to that exerted by the particles A, B, C, when revolving at their respective distances, the force which accelerates the circumference or any point in it D will be determined. Thus, let the mass Q, when collected into a, be such as will be equivalent in its inertia to A, when revolving at the distance SA; also let R be the mass collected into b, which is equivalent to B when revolving at the distance SB; and let T, the mass collected into C, be equivalent to C when revolving at the distance SC; then will the mass moved by the force P be Q+R+T; and the force which accelerates the circumference \( \frac{P}{Q+R+T} \), being equal to that by which the circumference or any point in it is accelerated when the point consists of A and B and C, revolving at the respective distances from the axis of motion SA, SB, SC."

Our author now proceeds particularly to investigate the motion of revolving bodies in almost all possible circumstances, deducing from his propositions many conclusions very useful in practical mechanics. Many of these regard the pendulum, and are therefore taken notice of under that article; others more immediately relate to the parts of mechanics particularly treated of in this article: the principal of which follow.

1. The force which accelerates the centre of gravity of a sphere, while it rolls down an inclined plane, is to the force by which it would be accelerated were it to slide in the ratio of five to seven. As our limits will not admit of inserting at length the demonstration of this and other propositions, we shall in this only observe, that when a wheel or a sphere rolls, the circumference goes backward, while the centre moves forward; which retrograde motion must of necessity make the other slower than it would otherwise be: and this retardation Mr Atwood has determined to be in the proportion above-mentioned.

From this proposition the following corollaries are deduced.

1. The absolute force whereby motion is generated in the circumference of a sphere in such a situation, is expressed by a fraction consisting of twice the weight of the sphere divided by seven, and multiplied into another fraction consisting of the height of the plane divided by its length; that is, suppose the weight of the sphere to be represented by \( w \), the height of the plane by \( h \), and its length by \( l \), the force by which the circumference of the sphere is impelled will be represented by \( \frac{w}{7} \times \frac{h}{l} \).

2. In the same manner, let a cylinder roll down an inclined plane, keeping the axis always horizontal, and the force which accelerates the axis will be represented by the fraction \( \frac{2}{3} \times \frac{h}{l} \).

2. Let AB (fig. 115.) represent a straight lever moveable round an horizontal axis of motion, which passes through S. Let the arms be SB, SA. Suppose a weight W to be affixed to the extremity of the Motion of shorter arm, and to be raised by the weight P applied at the extremity of the longer arm, when the lever is horizontal. Required to determine the time in which W will be raised through any given height, the weight and inertia of the lever itself not being considered.

"When there is an equilibrium (says Mr Atwood) on any mechanic power, the proportion of the weight sustained to the power sustaining it, will, in all cases, be assigned from having given the dimensions of the mechanic power. An equilibrium having been once formed, the smallest addition of weight will cause the body to which it is applied on either side to preponderate. In this case a certain degree of motion is generated; and hence the uses of the mechanic powers are not only to sustain forces in equilibrio, but to raise weights and overcome resistances, it is a problem of principal consequence to assign the absolute quantity of motion generated by a known moving force in given circumstances." The general solution of the problem is as follows:

"Let AB be the lever, W the weight moved by the power P; each acting in a direction perpendicular to the horizon. Let G be the common centre of gravity of the whole system, including the weights P and W with the lever itself; and o the centre of oscillation *, when AB vibrates round the axis S; the force which accelerates B when the lever is horizontal \( \frac{SG}{SB} = \frac{SO}{SO} \) (c). If this be put \( F = \frac{SA}{SB} \times \frac{1 + x^2}{SB^2 - 5} \), wherein P descends through a perpendicular space \( x \), and consequently wherein W ascends through the corresponding space; then \( x \times \frac{SA}{SB} = \frac{\sqrt{x}}{F} \times \frac{1 + x^2}{SB^2 - 5} \), &c.

3. Let ABC (fig. 116.) represent a wheel and axle, its weight \( w \), and let the axis be horizontal; having a given weight Q applied to the circumference of the axle, and P applied to the circumference of the wheel in order to raise Q. Required to assign the space described by the elevated weight Q in any given time. The solution of this problem, without attending to the demonstration, is this. Having found the accelerating power, which here is \( \frac{P \times SD - Q \times SA \times SA}{w \times SR^2} \) (e.), \( P \times SD^2 + Q \times SA^2 \).

All this he puts \( F = \frac{SA}{SB} \times \frac{1 + x^2}{SB^2 - 5} \); and then \( l \) being \( 193 \) inches as before, the space described by Q in any number of seconds will be \( = \) the square of that number of seconds multiplied into \( l/F \). On this proposition our author makes the following observations.

"Whenever motion is communicated to a body, a certain resistance must have been overcome by the moving force. This resistance is of various kinds. 1. The inertia of the mass moved, whereby it endeavours to persevere in its state of quiescence, or of uniform motion in a right line. 2. That of a weight or other absolute force opposed to the action of the moving power. 3. Obstacles upon which the moving body impinging is retarded in its progress; such, for example, is the resistance which arises from the particles of a fluid thro' which a body moves. The estimation of these resistances, and their effects in retarding the motion of bodies acted on by a given force, are deducible from the laws of motion, and constitute a part of the solution of almost all problems relating to the motion of bodies.

"The moving forces also are of various kinds, viz. The power of gravity, muscular power, the impact of bodies, solid and fluid, &c. It has been shown, that the effects of these moving forces which are exerted on bodies in order to create motion, exclusive of the resistance opposed to them, depend on the various circumstances of the time in which they act, and on the spaces through which the bodies moved are impelled, &c.

"These considerations are urged, to show, that from the great variety of undetermined conditions which may enter into mechanical problems, there must of course be various methods of producing the same mechanical effect: and it is a very material part of the art, considered either in a theoretical or practical view, to proportion the means to the end, and to effect this with all the advantages which the nature of the case is capable of. It is the due observation of these particulars which contributes to render mechanic instruments complete, and the neglect of them defective, in their construction. This proper choice of means to produce mechanical effects, is frequently the result of long continued experience independent of all theory; the knowledge of which, however, when applied to practice, would save the artist much time and trouble, as well as would be productive of other advantages, which experience alone must be destitute of."

4. ABC (fig. 116.) is a wheel and axle moveable round an horizontal axis, which passes through S. Suppose a given weight Q, which is applied to the circumference of the axle; let it be required to assign the proportion of the radii of the wheel and axle, so that the time in which the weight Q ascends through any given space shall be the least possible. In this case, supposing the radius of the wheel to be 10 inches, and its weight 20 ounces; let the radius of the axle SA = 1 inch, the weight to be raised thro' any given space to be 100 ounces, the moving force by which it is raised to be 33 ounces; then the distance of the centre of gyration from the axis is \( \sqrt{50} \) inches; and the length of the radius sought is 0.55 inches.—If, instead of raising the weight perpendicularly, it be required to draw it horizontally, and to assign the distance SD, at which, if a given force P be applied, the time of describing a given space shall be the least, and the moment of g the greatest possible, we have the following conclusion. * Let the quantity of matter to be drawn along the plane be four times greater than that which is contained in the moving

(c) This he had formerly proved when treating of pendulums. (l) \( l \) is here put for 193 inches, the supposed velocity of the weight P. (e) R is the centre of gyration of the wheel. Mechanics.

Motion of Bodies.

Mon of ving force; the radius of the axle SA being given; in order that it may be impelled with the greatest velocity possible and with the greatest moment, the radius of the wheel should be double that of the axle when the inertia of the wheel is not considered.

5. Let ARCH (fig. 117.) be a system of bodies moveable round a vertical axis which passes through the common centre of gravity of the system. Suppose DEG to be a wheel, the axis of which is vertical, and coincident with that of the system; let motion be communicated by means of a line going round this wheel, the string DP being stretched by a given weight P; let it be required to assign the radius of the wheel EGD, so that the angular velocity communicated to the system may be the greatest possible. Here, supposing the moving force to be one-fourth of the weight of the system, it should be applied at a distance from the axis equal to twice the distance of the centre of gyration, in order to produce the greatest possible angular velocity in a given time.

"In order (says Mr Atwood) to increase the action of a moving force against a weight to be raised, or resistance to be overcome, a combination of two or more mechanic powers is frequently made use of. Let p (fig. 118.) be a power applied by means of a line to the vertical wheel C; suppose the circumference of the axle K to be in contact with the circumference of any other vertical wheel B; so that the circumference of the wheel B may always move equally fast with that of the axle which belongs to C; let also the axle of B communicate motion to a vertical wheel A, to the axle of which a weight q is suspended, so as to act in opposition to p; moreover, let the ratio of lmn to 1 be the sum of the ratios of the radius of each wheel to that of its axle: then, if plmn = q, the two weights p and q will sustain each other in equilibrium; but if plmn be at all greater than q, the equilibrium will be destroyed;" and our author gives a method of calculating the quantity of motion communicated in certain circumstances.

Our author next goes through a set of similar propositions relating to the pulley and wedge; after which he treats of the accumulation of power in ponderous cylinders, and the use of ballast-wheels in machines, of which mention has already been made; and having discussed these subjects, he next comes to treat of the action of a stream of water upon a wheel revolving round an horizontal axis.

6. Let ABC (fig. 119.) represent a water-wheel which revolves round an horizontal fixed axis, passing through its centre S. Suppose DEF to be the axle of this wheel, and that a weight W is affixed to a line DW; so wound round the axle, that while the wheel is driven round its own plane by the force of the water impinging at I, the weight W may be raised in a vertical line: having given the area of the boards II, against which the stream impinges perpendicularly, and the altitude from which the water descends, it is required to assign the greatest velocity with which the wheel can revolve.

"When a stream of any fluid (says he) impinges perpendicularly against a plain and quiescent surface, the exact quantity of the moving force is equal to the weight of a column of the fluid, the base of which is the area upon which the fluid impinges, and the altitude that from which a body must descend freely from rest by gravity, in order to acquire that velocity. This will be the moving force which impels the body when quiescent or just beginning to move: but after it has acquired some motion, the impulsive force of the body will be diminished; being the same as if the body were quiescent, and the water impinged upon it with the difference of the former velocities. Wherefore the altitude of the column of the fluid, which is equal to its impelling force, will be always as the difference between the velocity of the impact and that of the body itself; and since the altitudes from which bodies fall from rest are in a duplicate ratio of the velocities acquired, it follows, that the force of the impact will be in a duplicate ratio of the difference between the velocity of the wheel and that of the impact." The following is the conclusion drawn by Mr Atwood concerning the velocity: Putting A for the weight of the column of water when the wheel is quiescent; V the velocity with which it impinges on the boards II, &c. and y the velocity of the circumference sought; W the weight of the wheel; then \( y = \frac{V}{A} \times \sqrt{\frac{W}{SD}} \times \frac{SI}{Sf} \).

7. Every other thing remaining the same, let the weight W be varied; and let it be required to assign the weight W, so that when the wheel has acquired its uniform velocity, the moment of W may be the greatest possible. Here the weight \( = \frac{4A \times SI}{9SD} \).

8. Having given a weight W to be raised by the action of the stream of water, the force of which is \( = A \) against a quiescent surface; let it be required to assign what must be the proportion between the radius of the wheel and that of the axle; so that the uniform velocity of the ascending weight may be the greatest possible. Here the length of the radius \( = \frac{2W \times SD}{4A} \). Hence he concludes, that if the velocity with which the water impinges against the boards be doubled, the greatest moment communicated to a weight ascending uniformly, will be increased in the proportion of 8 to 1.

"The force (says Mr Atwood) which communicates motion to water-wheels, and the resistances which are occasioned by friction, tenacity, and various other causes, render the application of the theory of mechanics to practice, in these cases, extremely difficult. It is probably from this reason, that the construction of machines moved by the force of water, &c. has been almost wholly practical, the best improvements having been deduced from continued observation of the results produced in given circumstances; whereby the gradual correction of error, and varied experience of what is most effectual, have supplied the place of a more perfect investigation from the laws of motion.

"This seems to be the best method, as far as regards the practical construction of these machines, the nature of the case will admit of; for although there may be two ways leading ultimately to the same truths, i.e. a direct investigation from the laws of motion and long continued observation, independent of theory, the latter is frequently the most easy and intelligible, although less direct and less scientific; the former being inaccessible to those who possess the elementary..." Motion of mentary parts of mechanics only. It is in vain to attempt the application of the theory of mechanics to the motion of bodies, except every cause which can sensibly influence the moving power and the resistance to motion be taken into account; if any of these be omitted, error and inconsistency in the conclusions deduced must be the consequence. It was at one time supposed, from this inadequate application of the theory, that the same laws of motion would not extend to all branches of mechanics, but that different principles were to be accommodated to different kinds of motion. If this were truly the case, the science of mechanics would fall short of that superior excellence and extent which it is generally allowed to possess.

For it is probable, that there is no kind of motion but what may be referred to three easy and obvious propositions, the truth of which it is impossible to doubt: and if we are not enabled to investigate the effects from the data in all cases, the deficiency must not be imputed to the science of mechanics, but to the want of methods of applying mathematics to it.

This may be illustrated by an example, in order to show that the motion communicated to water-wheels, however complicated the data may be, is equally referable to the laws of motion, with the effects of the most uncompounded force. If a stream of water falls perpendicularly on a plain surface, the moving force arising from the impact only is equal to the weight of a column of water, the base of which is the surface upon which the water impinges, and altitude that through which a body must fall to acquire the velocity of impact. If the inclination of the stream to the surface should be changed, the force exerted in a direction perpendicular to the plane will be diminished in a duplicate ratio of the radius to the sine of inclination; the surface on which the water impinges remaining. Now, when the water falls on the boards of a water-wheel, the direction of the stream makes different angles with the planes of those boards; for since the particles of water descend in curve lines, they will strike any plain surface in the direction of a tangent to the curve on the point of impact. Moreover, the water will strike the higher boards TT with less velocity, and in a direction more inclined to their planes, than the lower ones II; it is also to be considered, that the stream will impinge on the boards at different distances from the axis of motion: all which circumstances must be taken into account, to find the force which tends to communicate motion to the wheel when quiescent; and when motion has been communicated, the force of the stream to turn the wheel will be determined in the manner already mentioned. But this is not the only consideration which affects the moving force: The force hitherto considered has been supposed to proceed from the impact of the particles only; in which case, each particle after it has struck the board is imagined to be of no other effect in communicating motion: but this is not wholly the case; for after the particle has impinged on the board, it will continue some time to operate by its weight; and this time will be longer or shorter according to the different constructions of the wheel. In the overshot wheel, the continuance of the pressure, arising from the weight of the water, will be longer than in the undershot, the force which arises from the impact of the water being nearly the same in each case. The whole moving force, therefore, will consist of the impact determinable as above, and of the weight of the water descending along with the circumference, and communicating additional motion to it: this entire moving force being determined either by theory or experience, may be denoted by A. After the moving force which impels the circumference has been determined, the resistance to this force must be found; for on the proportion between the moving force and the resistance, the acceleration of the machine will depend. This resistance is of various kinds: 1. That of inertia. 2. If the machine is of that kind which raises weights, such as for instance as water; the weight raised, allowing for its mechanical effect on the point of which we desire to know the acceleration, must be subducted from the moving force before found; and this will be a constant quantity. There are other resistances also homogenous to weight, viz. those of friction and tenacity, &c., which are variable in some ratio of the machine's velocity: and in order to proceed with the investigation, the exact quantity of weight which the friction is equal to, when the wheel moves with a given velocity, must be considered, as well as the variation of the resistances in respect to the velocities; which circumstances must be determined by experiment. If the force equivalent to the friction, &c. be subducted from the moving force, the remainder will give the moving power, by which the circumference is impelled upon the whole; this being divided by the inertia of the mass moved, will give the force which accelerates the circumference.

The following apparatus has been invented by Mr. Atwood, for illustrating his doctrines concerning accelerated motion, and has been found to answer the purpose more completely than any other we have heard of; discovering at once the quantity of matter moved, the force which moves it, the space described from rest, the time of description, and the velocity acquired.

1. Of the mass moved.—In order to observe the effects of the moving force, which is the object of any experiment, the interference of all other forces should be prevented; the quantity of matter moved, therefore, considering it before any impelling force has been applied, should be without weight; for although it be impossible to abstract the natural gravity or weight from any substance whatever, yet the weight may be counteracted as to be of no sensible effect in experiments. Thus in the instrument constructed to illustrate this subject experimentally, A, B, fig. 120, represent two equal weights affixed to the extremities of a very fine and flexible silk line; this line is stretched over a wheel or fixed pulley abcd, moveable round an horizontal axis: the two weights A, B, being precisely equal and acting against each other, remain in equilibrium; and when the least weight is superadded to either (setting aside the effects of friction), it will preponderate. When AB are let in motion by the action of any weight m, the sum A+B+m would constitute the whole mass moved, but for the inertia of the materials which must necessarily be used in the communication of motion; these materials consist of:

1. The The wheel \(abcd\), over which the line sustaining A and B passes. 2. The four friction wheels on which the axle of the wheel \(abcd\) rests: the use of these wheels is to prevent the loss of motion, which would be occasioned by the friction of the axle if it revolved on an immovable surface. 3. The line by which the bodies A and B are connected, so as when set in motion to move with equal velocities. The weight and inertia of the line are too small to have sensible effect on the experiments; but the inertia of the other materials just mentioned constitute a considerable proportion of the mass moved, and must be taken into account. Since when A and B are put in motion, they must necessarily move with a velocity equal to that of the circumference of the wheel \(abcd\) to which the line is applied; it follows, that if the whole mass of the wheels were accumulated in this circumference, its inertia would be truly estimated by the quantity of matter moved; but since the parts of the wheels move with different velocities, their effects in resisting the communication of motion to A and B by their inertia will be different; those parts which are furthest from the axis resisting more than those which revolve nearer in a duplicate proportion of these distances. If the figures of the wheels were regular, from knowing their weights and figures, the distances of their centres of gyration from their axes of motion would become known, and consequently an equivalent weight, which being accumulated uniformly in the circumference of the wheel \(abcd\), would exert an inertia equal to that of the wheels in their constructed form. But as the figures are wholly irregular, recourse must be had to experiment, to assign that equivalent quantity of matter, which being accumulated uniformly in the circumference of the wheel \(abcd\), would resist the communication of motion to A in the same manner as the wheels.

In order to ascertain the inertia of the wheel \(abcd\) with that of the friction wheels, the weights AB being removed, the following experiment was made.

A weight of 30 grains was affixed to a silk line (the weight of which was not so much as \(\frac{1}{4}\)th of a grain, and consequently too inconsiderable to have sensible effect in the experiment); this line being wound round the wheel \(abcd\), the weight 30 grains by descending from rest communicated motion to the wheel, and by many trials was observed to describe a space of about 38 inches in 3 seconds. From these data the equivalent mass or inertia of the wheels will be known from this rule:

Let a weight \(P\) (fig. 121.) be applied to communicate motion to a system of bodies by means of a very slender and flexible line going round the wheel \(SLDIM\), through the centre of which the axis passes (G being the common centre of gravity, R the centre of gravity of the matter contained in this line, and O the centre of oscillation). Let this weight descend from rest through any convenient space \(s\) inches, and let the observed time of its descent be \(t\) seconds; then if \(l\) be the space through which bodies descend freely by gravity in one second, the equivalent weight sought

\[ W \times SR \times SO = \frac{P \times t^2}{SI} - P. \]

Here we have \(f = 30\) grains, \(t = 3\) seconds, \(l = 193\) inches, \(s = 38.5\) inches; and \(\frac{P \times t^2}{SI} = \frac{50 \times 9 \times 194}{385} = 132.3\) grains, or \(2\frac{1}{2}\) ounces.

This is the inertia equivalent to that of the wheel \(abcd\), and the friction wheels together: for the rule extends to the estimation of the inertia of the mass contained in all the wheels.

The resistance to motion therefore arising from the wheel's inertia, will be the same as if they were absolutely removed, and a mass of \(2\frac{1}{2}\) ounces were uniformly accumulated in the circumference of the wheel \(abcd\). This being premised, let the boxes A and B fig. 122. be replaced, being suspended by the silk line over the wheel or pulley \(abcd\), and balancing each other: suppose that any weight \(m\) be added to A so that it shall descend, the exact quantity of matter moved, during the descent of the weight A, will be ascertained, for the whole mass will be \(A + B + m + 2\frac{1}{2}\) oz.

In order to avoid troublesome computations in adjusting the quantities of matter moved and the moving forces, some determinate weight of convenient magnitude may be assumed as a standard, to which all the others are referred. This standard weight in the subsequent experiments is \(\frac{1}{4}\) of an ounce, and is represented by the letter \(m\). The inertia of the wheels being therefore \(= 2\frac{1}{2}\) ounces, will be denoted by \(11m\). A and B are two boxes constructed so as to contain different quantities of matter, according as the experiment may require them to be varied: the weight of each box, including the hook to which it is suspended, \(= 1\frac{1}{2}\) oz., or according to the preceding estimation, the weight of each box will be denoted by \(6m\); these boxes contain such weights as are represented by fig. 122, each of which weighs an ounce, so as to be equivalent to \(4m\); other weights of \(4oz.\) \(= 2m\), \(4m\), and aliquot parts of \(m\), such as \(\frac{1}{4}m\), \(\frac{1}{2}m\), may be also included in the boxes, according to the conditions of the different experiments hereafter described.

If \(4\frac{1}{2}oz.\) or \(19m\), be included in either box, this with the weight of the box itself will be \(25m\); so that when the weights A and B, each being \(25m\), are balanced in the manner above represented, their whole mass will be \(50m\), which being added to the inertia of the wheels \(11m\), the sum will be \(61m\). Moreover, three circular weights, such as that which is represented at fig. 123, are constructed; each of which \(= \frac{1}{4}oz.\) or \(m\): if one of these be added to A and one to B, the whole mass will now become \(63m\), perfectly in equilibrium, and moveable by the least weight added to either (setting aside the effects of friction), in the same manner precisely as if the same weight or force were applied to communicate motion to the mass \(63m\), existing in free space and without gravity.

2. The moving Force. Since the natural weight or gravity of any given substance is constant, and the exact quantity of it easily estimated, it will be convenient here to apply a weight to the mass A as a moving force: thus, when the system consists of a mass \(= 63m\), according to the preceding description, the whole being perfectly balanced, let a weight \(4\frac{1}{2}oz.\) or \(m\), such as is represented in fig. 124, be applied on the mass A; this will communicate motion to the whole system: by adding a quantity of matter \(m\) to the former mass \(63m\), the whole quantity of matter moved will now become Sect. VI.

Motion of become $64m$; and the moving force being $=m$, this will give the force which accelerates the descent of A

$$\frac{m}{64m} \text{ or } \frac{1}{64} \text{ part of the accelerating force by which the bodies descend freely towards the earth's surface.}$$

By the preceding construction, the moving force may be altered without altering the mass moved; for suppose the three weights $m$, two of which are placed on A, and one on B to be removed, then will A balance B. If the weights $3m$ be all placed on A, the moving force will now become $3m$, and the mass moved $64m$ as before, and the force which accelerates the descent of A

$$\frac{3m}{64m} = \frac{3}{64} \text{ parts of the force by which gravity accelerates bodies in their free descent to the surface.}$$

Suppose it were required to make the moving force $2m$, the mass moved continuing the same. In order to effect this, let the three weights, each of which $=m$, be removed; A and B will balance each other; and the whole mass will be $61m$: let $\frac{1}{2}m$, fig. 124, be added to A, and $\frac{1}{2}m$ to B, the equilibrium will still be preserved, and the mass moved will be $62m$; now let $2m$ be added to A, the moving force will be $2m$, and the mass moved $64m$, as before; wherefore the force of acceleration $=\frac{1}{32}$ part of the acceleration of gravity. These alterations in the moving force may be made with great ease and convenience in the more obvious and elementary experiments, there being no necessity for altering the contents of the boxes A and B; but the proportion and absolute quantities of the moving force and mass moved may be of any assigned magnitude, according to the conditions of the proposition to be illustrated.

3. Of the space described. The body A, fig. 120, descends in a vertical line; and a scale about 64 inches in length graduated into inches and tenths of an inch is adjusted vertically, and so placed that the descending weight A may fall in the middle of a square stage, fixed to receive it at the end of the descent: the beginning of the descent is eliminated from o on the scale, when the bottom of the box A is on a level with o. The descent of A is terminated when the bottom of the box strikes the stage, which may be fixed at different distances from the point o; so that by altering the position of the stage, the space described from quiescence may be of any given magnitude less than 64 inches.

4. The time of motion is observed by the beats of a pendulum, which vibrates seconds; and the experiments, intended to illustrate the elementary propositions, may be easily constructed so that the time of motion shall be a whole number of seconds: the estimation of the time, therefore, admits of considerable exactness, provided the observer takes care to let the bottom of the box A begin its descent precisely at any beat of the pendulum; then the coincidence of the stroke of the box against the stage, and the beat of the pendulum at the end of the time of motion, will show how nearly the experiment and the theory agree together. There might be various mechanical devices thought of for letting the weight A begin its descent at the instant of a beat of the pendulum W; let the bottom of the box A, when at o on the scale, rest on a flat rod, held in the hand horizontally, its extremity being coincident with o, by attending to the beats of the pendulum; and with a little practice, the rod which supports the box A may be removed at the moment the pendulum beats, so that the descent of A shall commence at the same instant.

4. Of the velocity acquired. It remains only to describe in what manner the velocity acquired by the descending weight A, at any given point of the space through which it has descended, is made evident to the senses. The velocity of A's descent being continually accelerated will be the same in no two points of the space described. This is occasioned by the constant action of the moving force; and since the velocity of A at any instant is measured by the space which would be described by it, moving uniformly for a given time with the velocity it had acquired at that instant, this measure cannot be experimentally obtained, except by removing the force by which the descending body's acceleration was caused.

In order to show in what manner this is affected particularly, let us again suppose the boxes A and B $=25m$ each, so as together to be $=50m$; this with the wheel's inertia $11m$ will make $61m$; now let $m$, fig. 122, be added to A, and an equal weight $m$ to B, these bodies will balance each other, and the whole mass will be $63m$. If a weight $m$ be added to A, motion will be communicated, the moving force being $m$, and the mass moved $64m$. In estimating the moving force, the circular weight $=m$ was made use of as a moving force: but for the present purpose of showing the velocity acquired, it will be convenient to use a flat rod, the weight of which is also $=m$. Let the bottom of the box A be placed on a level with o on the scale, the whole mass being as described above $=63m$, perfectly balanced in equilibrio. Now let the rod, the weight of which $=m$, be placed on the upper surface of A; this body will descend along the scale precisely in the same manner as when the moving force was applied in the form of a circular weight. Suppose the mass A, fig. 123, to have descended by constant acceleration of force of $m$, for any given time, or through a given space; let a circular frame be so affixed to the scale, contiguous to which the weight descends, that A may pass centrally through it, and that this circular frame may intercept the rod $m$ by which the body A has been accelerated from quiescence. After the moving force $m$ has been intercepted at the end of the given space or time, there will be no force operating on any part of the system which can accelerate or retard its motion: this being the case, the weight A, the instant after $m$ has been removed, must proceed uniformly with the velocity which it had acquired that instant: in the subsequent part of its descent, the velocity being uniform will be measured by space described in any convenient number of seconds.

Other uses of the instrument. It is needless to describe particularly, but it may not be improper just to mention the further uses of this instrument; such as the experimental estimation of the velocities communicated by the impact of bodies elastic and inelastic; the quantity of resistance opposed by fluids, as well as for various other purposes. These uses we shall not insist on; but the properties of retarded motion being a part of the present subject, it may be necessary to show

Motion of Bodies.

In what manner the motion of bodies resisted by constant forces are reduced to experiment by means of the instrument above described, with as great care and precision as the properties of bodies uniformly accelerated. A single instance will be sufficient: Thus, suppose the mass contained in the weights A and B, fig. 125, and the wheels to be 61 m, when perfectly in equilibrium; let a circular weight m be applied to B, and let two long weights or rods, each = m, be applied to A, then will A descend by the action of the moving force m, the mass moved being 64 m: suppose that when it has described any given space by constant acceleration, the two rods m are intercepted by the circular frame above described, while A is descending through it, the velocity acquired by that descent is known; and when the two rods are intercepted, the weight A will begin to move on with the velocity acquired, being now retarded by the constant force m; and since the mass moved is 62 m, it follows, that the force of retardation will be \( \frac{1}{3} \) part of that force whereby gravity retards bodies thrown perpendicularly upwards. The weight A will therefore proceed along the graduated scale in its descent with an uniformly retarded motion, and the spaces described, times of motion, and velocities destroyed by the resisting force, will be subject to the same measures as in the examples of accelerated motion above described.

In the foregoing descriptions, two suppositions have been assumed, neither of which are mathematically true: but it may be easily shown that that are so in a physical sense; the errors occasioned by them in practice being insensible.

1. The force which communicates motion to the system has been assumed constant; which will be true only on a supposition that the line, at the extremities of which the weights A and B, fig. 120, are affixed, is without weight. In order to make it evident, that the line's weight and inertia are of no sensible effect, let a case be referred to, wherein the body A descends through 48 inches from rest by the action of the moving force m, when the mass moved is 64 m; the time wherein A describes 48 inches is increased by the effects of the line's weight by no more than \( \frac{1}{3000} \) th parts of a second; the time of descent being 3.986 seconds, when the string's weight is not considered, and the time when the string's weight is taken into account = 4.0208 seconds; the difference between which is wholly insensible by observation.

2. The bodies have also been supposed to move in vacuo, whereas the air's resistance will have some effect in retarding their motion: but as the greatest velocity communicated in these experiments, cannot much exceed that of about 26 inches in a second (suppose the limit 26.28), and the cylindrical boxes being about 1\(\frac{1}{2}\) inches in diameter, the air's resistance can never increase the time of descent in so great a proportion as that of 240 : 241; its effects therefore will be insensible in experiment.

The effects of friction are almost wholly removed by the friction wheels; for when the surfaces are well polished and free from dust, &c. if the weights A and B be balanced in perfect equilibrium, and the whole mass consists of 63 m, according to the example already described, a weight of 1\(\frac{1}{2}\) grains, or at most 2 grains, being added either to A or B, will communicate motion to the whole; which shows that the effects of friction will not be so great as a weight of 1\(\frac{1}{2}\) or 2 grains. In some cases, however, especially in experiments relating to retarded motion, the effects of friction become sensible; but may be very readily and exactly removed by adding a small weight 1\(\frac{1}{2}\) or 2 grains to the descending body, taking that the weight added is such as is in the least degree smaller than that which is just sufficient to set the whole in motion, when A and B are equal and balance each other before the moving force is applied.