plane of oak, the friction was about one-sixteenth of the pressure; and, when it rolled upon a plane of elm, the friction was only 1th of the pressure.

It is evident, therefore, that between the same substances this species of friction is much less than that of sliding.

The string used in these experiments should be so flexible, that its rigidity or stiffness shall produce no sensible effect upon the results.

If a body, having any round figure, be made to revolve while it is pressed with any force against any surface, and at the same time is prevented from rolling along that surface, a species of friction will be produced different from any which we have yet considered.

To explain this friction, and the experiments by which its properties may be determined, let us suppose a solid cylindrical axis, A B (Plate V. fig. 1), inserted in an hollow cylinder, of a diameter, CB, somewhat greater than A B, so as to permit the hollow cylinder BC to turn round it, AB. Let the cylinders be placed with their axes horizontal, and let the hollow cylinder be the centre or box of a wheel DE. Let an extremely flexible string be passed over the edge of this wheel, in a grove formed to receive it, and let scales G, H, be appended to its extremities. In consequence of the form of the axle and hollow cylinder, and the manner in which the weight of the wheel acts, the points of contact of the axle and the cylinder will be in a straight line, formed by the intersection of a vertical plane passing through the axis of the cylinder, with the surface of the cylinder. In fact, if from the point of contact, B, a line be conceived to be drawn perpendicular to the plane of the paper, along the inner surface of the cylinder, the axle and the cylinder will touch in that line, and in no other points. It appears, therefore, that if the hollow cylinder be supposed to revolve round the axle, as happens in a carriage wheel, every part of the surface of the hollow cylinder is successively exposed to the effect of friction; while no part of the axle suffers this effect, except the side which passes through the point, B, of its section. If, on the contrary, as sometimes happens, the axle revolve within the cylinder, the opposite effects are produced. The entire surface of the axle is successively exposed to the effects of friction, while these effects are confined to one line upon the surface of the hollow cylinder.

By loading the dishes GH with any equal weights, the axle may be submitted to any proposed pressure. If, when they are equally loaded, some fine sand be poured into one of the dishes until its weight just gives motion to the wheel, the weight of the sand will be sufficient to determine the quantity of friction.

The preponderating weight is not, however, in this case, the immediate measure of the friction. It is to be considered that the wheel is turned round its centre, I; that the friction which resists this motion acts at B, and therefore with the leverage BI; while the preponderating weight which overcomes the friction acts with the leverage EI. Let the friction be F, and the preponderating weight be W; then by the established properties of the lever we have

that is, the friction is equal to the additional weight which produces the motion, multiplied by the radius of the wheel, and divided by the radius of the hollow cylinder which plays upon the axle.

Thus it appears that the friction is greater than the preponderating weight in the proportion of the radius of the wheel to the radius of the cylinder.

As, in the experiments to determine the friction of rolling, so here also each experiment should be tried in both dishes, and the mean of the results taken.

To determine whether the friction be a uniformly retarding force, a weight must be placed in one of the dishes greater than that which is necessary to overcome the friction. This will cause the dish to descend with an accelerated motion, and, by placing a graduated vertical scale near it, the rate of its acceleration may be ascertained. If it be found that the spaces through which it descends, in one, two, or three seconds, &c., are as the numbers 1, 4, 9, &c.; in other words, if the spaces be as the squares of the times, the motion is uniformly accelerated. Hence it may be inferred that the friction is a uniformly retarding force.

By a series of experiments, conducted as we have described, Coulomb found that, like the other modifications of friction, the law of the proportionality of the friction to the pressure obtained also in this case, subject however to the exception before mentioned, that in very great pressures the friction is somewhat less in proportion.

He also found that, as in the friction of sliding, great advantage was gained by greasing the surfaces. In general, fresh tallow diminishes the friction by one-half. It increases as the grease is wasted away. This effect is, however, more slow than in the friction of sliding.

This species of friction is also a uniformly retarding force, and is therefore independent of the velocity.

Like the other species of friction, the quantity of this depends on the quality of the surfaces. If iron revolve in contact with brass, the friction is one-seventh of the pressure. When both sur-, faces are wood, the friction is one-twelfth of the pressure.

The friction of bodies turning on pivots seems to come within the species we are now considering. This was also examined by Coulomb, and a memoir on the subject was published by him in the Memoirs of the French Academy in 1790. A very succinct and clear account of this is given by Dr. Gregory, in the second volume of his Mechanics, from which we extract the following particulars :

Bodies which are made to turn upon pivots are usually suspended by means of a cheek, socket, or collar, of very hard matter. The collar has its cavity of a conic form, and terminated at its summit by a little concave segment, whose radius of curvature is very small. The point of the pivot 3 B 2

'After this, Coulomb varied the charge in his experiments, and determined the relative momentum of friction of pivots under different pressures. But, without going further into detail, we may give the following as the principal deductions from the whole.

1. That the friction of pivots is independent of the velocities, being merely as a function of the pressure.

2. That the friction of granite is less than that of glass.

3. That the figure of the point of the pivot, as to acuteness, affects the quantity of friction, in such a manner that when we cause to whirl upon the point of a needle, a body weighing more than five or six drachms, the most advantageous angle for that point appeared to be from 30° to 45°; under a less pressure, the angle might be progressively diminished, without the friction being perceptibly augmented; it may even without great inconvenience be reduced to 10° or 12° with good steel, when the charge does not exceed 100 grains, an important consideration in the suspension of light bodies upon cheeks or sockets.

These rules may be useful to the makers of chronometers.'

Mr. Anstice, the author of a valuable treatise on wheel-carriages, very familiarly illustrates the advantage of wheels over sledges; it is immediately connected with the subject of friction, and may be illustrated by reference to the following considerations.

1. A sledge, in 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 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 1 to 36. 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. Mr. Anstice 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 Thus, let the circle OTA GL, fig. 2, represent a be regulated upon the principles of these powers. wheel of four feet diameter, placed on the level PQ, 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, CT, parallel to the plane PQ. 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 CO, which may be supposed a spoke of the wheel; and as the upright spoke CL is 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 MO and ON. As the length of OM, therefore, is to that of ON, 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. 3, and supposed to be moved along the plane PQ, and an obstacle of twelve inches height be placed before it, the real lever will then be represented by the lines LOC; which, being reduced to the imaginary ones MON, 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. 4, 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. 4 be raised into the direction CS, the power required to move the carriage over it will be to the real weight as the line CO is to the line ON; and in consequence of thus altering the direction we gain as much as the length of the line CO exceeds that of C N.

This view of the manner in which the wheels of carriages act will serve to elucidate the question, whether large or small wheels are preferable for carriages? Let the circle, fig. 5, 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 COL, to be reduced to the imaginary ones MO N. In this case, the power required to draw the carriage must be to its weight as NO 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 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. 6 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 GT and ES parallel to the elevation of the plane, which is 45°; 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 sinall 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, Mr. Anstice draws the following conclusions. 1. That in a carriage placed upon a 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 plane 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 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 lines; 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 at 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 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 matter of which it is composed be equally 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 equally heavy, it will be impossible to turn it into any position in which there will not be as much matter on one side of the centre as upon the other; and in like manner every figure, however irregular, has some part 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, 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 the 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.

A few illustrations of animal power must close the present article, which will be resumed under the heads of MOTION and MILLWORK.

A horse draws with the greatest advantage when the line of draught is not level with his breast, but inclines upwards, making a small angle with the horizontal plane.

A horse drawing a weight over a single pulley, can draw 200lbs. for eight hours a day, and walking at the rate of two miles and a half in an hour, which is about three feet and a half in a second; and, if the same horse be made to draw 240lbs., he can work but six hours a day, and cannot go quite so fast. To this may be referred the working of horses in all sorts of mills and waterworks, where we ought to know, as near as we can, how much we make every horse draw, that

we may judge of what the effect will be, when proper allowance shall have been made for all the frictions and hindrances, before we can cause any machine to be erected.

When a horse draws in a mill or gin of any kind, great care should be taken that the horsewalk, or circle in which he moves, be large enough in diameter, otherwise the horse cannot exert all his strength; for, in a small circle, the tangent (in which the horse draws) deviates more from the circle in which the horse is obliged to go than in a larger circle. The horse-walk should not be less than forty feet in diameter, when there is room for it. In a walk of nineteen feet diameter it has been calculated that a horse loses two-fifths of his strength.

The worst way of applying the force of a horse is to make him carry or draw up hill; for, if the hill be steep, three men will do more than a horse; each man loaded with 100lbs. wil move up faster than a horse that is loaded with 300lbs. This is owing to the position of the parts of a man's body, which are better adapted for climbing than those of a horse.

As a horse, from the structure of his body, can exert most strength in drawing almost horizontally in a straight line, a man exerts the least strength that way; as, for example, if a man weighing 140lbs., walking by a river or canal side, draws along a boat or barge, by means of a rope coming over his shoulder, or otherwise fastened to his body, he cannot draw above twenty-seven pounds, or about 4th of what a horse can draw in that case. Five men are about equal in strength to one horse, and can with the same ease push round the horizontal beam in a forty feet walk; but three of the same men will push round a beam in a nineteen feet walk, which a horse (otherwise equal to five men) can but draw round.

A man turning a horizontal windlass, by a handle or winch, should not have above thirty pounds weight acting against him, if he is to work ten hours a day, and raise the weight at the rate of three feet and a half in a second. This supposes, however, that the semi-diameter of the windlass is equal to the distance from the centre to the elbow of the handle; for, if there be a mechanical advantage, as there usually is, by having the diameter of the axle on which the rope winds four or five times less than the diameter of the circle described by the hand, then may the weight (taking in also the resistance, on account of the friction and stiffness of the rope) be four or five times greater than thirty pounds; that is, so much as it rises slower than the hand moves.

In this operation, the effect of a man's force varies in every part of the circle described by the handle. The greatest force is, when a man pulls the handle upwards from about the height of his knees; and the least force when (the handle being at top) he thrusts from him horizontally; then again the effect becomes greater as a man lays on his weight to push down the handle; but that action cannot be so great as when he pulls up, because he lays on no more than the whole weight of his body; whereas, in pulling, he can exert his whole strength. Lastly, he has but