# WHEEL

, in Mechanics, a simple machine, consisting of a circular piece of wood, metal, or other matter, that revolves on an axis. This is otherwise called Wheel and Axle, or Axis in Peritrochio, as a mechanical power, being one of the most frequent and useful of any. In this capacity of it, the Wheel is a kind of perpetual lever, and the axis another lesser one; or the radius of the Wheel and that of its axis may be considered as the longer and shorter arms of a lever, the centre of the Wheel being the fulcrum or point of | suspension. Whence it is, that the power of this machine is estimated by this rule, as the radius of the axis is to the radius of the Wheel or of the circumference, so is any given power, to the weight it will sustain.

Wheels, as well as their axes, are frequently dented, or cut into teeth, and are then of use upon innumerable occasions; as in jacks, clocks, mill-work, &c; by which means they are capable of moving and acting on one another, and of being combined together to any extent; the teeth either of the axis or circumference working in those of other Wheels or axles; and thus, by multiplying the power to any extent, an amazing great effect is produced.

To compute the power of a combination of Wheels; the teeth of the axis of every Wheel acting on those in the circumference of the next following. Multiply continually together the radii of all the axes, as also the radii of all the Wheels; then it will be, as the former product is to the latter product, so is a given power applied to the circumference, to the weight it can sustain. Thus, for example, in a combination of five Wheels and axles, to find the weight a man can sustain, or raise, whose force is equal to 150 pounds, the radii of the Wheels being 30 inches, and those of the axes 3 inches. Here 3 X 3 X 3 X 3 X 3 = 243, and 30 X 30 X 30 X 30 X 30 = 24300000, therefore as 243 : 24300000 :: 150 : 15000000 lb, the weight he can sustain, which is more than 6696 tons weight. So prodigious is the increase of power in a combination of Wheels!

But it is to be observed, that in this, as well as every other mechanical engine, whatever is gained in power, is lost in time; that is, the weight will move as much flower than the power, as the force is increased or multiplied, which in the example above is 100000 times flower.

Hence, having given any power, and the weight to be raised, with the proportion between the Wheels and axles necessary to that effect; to find the number of the Wheels and axles. Or, having the number of the Wheels and axles given, to find the ratio of the radii of the Wheels and axles. Here, putting p = the power acting on the last wheel, w = the weight to be raised, r = the radius of the axles, R = the radius of the wheels, n = the number of the wheels and axles; then, by the general proportion, as rn : Rn :: p : w; therefore is a general theorem, from whence may be found any one of these five letters or quantities, when the other four are given. Thus, to find n the number of Wheels: we have first . And to sind R/r, the ratio of the Wheel to the axle; it is .

Wheels of a Clock, &c, are, the crown wheel, contrat wheel, great wheel, second wheel, third wheel, striking wheel, detent wheel, &c.

Wheels of Coaches, Carts, Waggons, &c. With respect to Wheels of carriages, the following particulars are collected from the experiments and observations of Desaguliers, Beighton, Camus, Ferguson, Jacob, &c.

1. The use of Wheels, in carriages, is twofold; viz, that of diminishing or more easily overcoming the resistance or friction from the carriage; and that of more easily overcoming obstacles in the road. In the first case the friction on the ground is transferred in some degree from the outer surface of the Wheel to its nave and axle; and in the latter, they serve easily to raise the carriage over obstacles and asperities met with on the roads. In both these cases, the height of the Wheel is of material consideration, as the spokes act as levers, the top of an obstacle being the fulcrum, their length enables the carriage more easily to surmount them; and the greater proportion of the Wheel to the axle serves more easily to diminish or to overcome the friction of the axle. See Jacob's Observations on Wheel Carriages, p. 23 &c.

2. The Wheels should be exactly round; and the fellies at right angles to the naves, according to the inclination of the spokes.

3. It is the most general opinion, that the spokes be somewhat inclined to the naves, so that the Wheels may be dishing or concave. Indeed if the Wheels were always to roll upon smooth and level ground, it would be best to make the spokes perpendicular to the naves, or to the axles; because they would then bear the weight of the load perpendicularly. But because the ground is commonly uneven, one Wheel often falls into a cavity or rut, when the other does not, and then it bears much more of the weight than the other does; in which case it is best for the Wheels to be dished, because the spokes become perpendicular in the rut, and therefore have the greatest strength when the obliquity of the road throws most of the weight upon them; whilst those on the high ground have less weight to bear, and therefore need not be at their full strength.

4. The axles of the Wheels should be quite straight, and perpendicular to the shafts, or to the pole. When the axles are straight, the rims of the Wheels will be parallel to each other, in which case they will move the easiest, because they will be at liberty to proceed straight forwards. But in the usual way of practice, the ends of the axles are bent downwards; which always keeps the sides of the Wheels that are next the ground nearer to one another than their upper sides are; and this not only makes the Wheels drag sideways as they go along, and gives the load a much greater power of crushing them than when they are parallel to each other, but also endangers the overturning the carriage when a Wheel falls into a hole or rut, or when the carriage goes on a road that has one side lower than the other, as along the side of a hill. Mr. Beighton however has offered several reasons to prove that the axles of Wheels ought not to be straight; tor which see Desaguliers's Exp. Phil. vol. 2, Appendix.

5. Large Wheels are found more advantageous for rolling than small ones, both with regard to their power as a longer lever, and to the degree of friction, and to the advantage in getting over holes, rubs, and | stones, &c. If we consider Wheels with regard to the friction upon their axles, it is evident that small Wheels, by turning oftener round, and swifter about the axles, than large ones, must have much more friction. Again, if we consider Wheels as they sink into holes or soft earth, the large Wheels, by sinking less, must be much easier drawn out of them, as well as more easily over stones and obstacles, from their greater length of lever or spokes. Desaguliers has brought this matter to a mathematical calculation, in his Experim. Philos. vol. 1, p. 171, &c. See also Jacob's Observ. p. 63.

From hence it appears then, that Wheels are the more advantageous as they are larger, provided they are not more than 5 or 6 feet diameter; for when they exceed these dimensions, they become too heavy; or if they are made light, their strength is proportionably diminished, and the length of the spokes renders them more liable to break: besides, horses applied to such Wheels would not be capable of exerting their utmost strength, by having the axles higher than their breasts, so that they would draw downwards; which is even a greater disadvantage than small Wheels have in occasioning the horses to draw upwards.

6. Carriages with 4 Wheels, as waggons or coaches, are much more advantageous than carriages with 2 Wheels, as carts and chaises; for with 2 wheels it is plain the tiller horse carries part of the weight, in one way or other: in going down hill, the weight bears upon the horse; and in going up hill, the weight falls the other way, and lifts the horse, which is still worse. Besides, as the Wheels sink into the holes in the roads, sometimes on one side, sometimes on the other, the shafts strike against the tiller's sides, which destroys many horses: moreover, when one of the Wheels sinks into a hole or rut, half the weight falls that way, which endangers the overturning of the carriage.

7. It would be much more advantageous to make the 4 Wheels of a coach or waggon large, and nearly of a height, than to make the fore Wheels of only half the diameter of the hind Wheels, as is usual in many places. The fore Wheels have commonly been made of a less size than the hind ones, both on account of turning short, and to avoid cutting the braces. Crane-necks have also been invented for turning yet shorter, and the fore Wheels have been lowered, so as to go quite under the bend of the crane-neck.

It is held, that it is a great disadvantage in small Wheels, that as their axle is below the bow of the horses breasts, the horses not only have the loaded carriage to draw along, but also part of its weight to bear, which tires them soon, and makes them grow much stiffer in their hams, than they would be if they drew on a level with the fore axle.

But Mr. Beighton disputes the propriety of fixing the line of traction on a level with the breast of a horse, and says it is contrary to reason and experience. Horses, he says, have little or no power to draw but what they derive from their weight; without which they could not take hold of the ground, and then they must slip, and draw nothing. Common experience also teaches, that a horse must have a certain weight on his back or shoulders, that he may draw the better. And when a horse draws hard, it is observed that he bends forward, and brings his breast near the ground; and then if the Wheels are high, he is pulling the carriage against the ground. A horse tackled in a waggon will draw two or three ton, because the point or line of traction is below his breast, by the lowness of the Wheels. It is also common to see, when one horse is drawing a heavy load, especially up hill, his fore feet will rise from the ground; in which case it is usual to add a weight on his back, to keep his fore part down, by a person mounting on his back or shoulders, which will enable him to draw that load, which he could not move before. The greatest stress, or main business of drawing, says this ingenious writer, is to overcome obstacles; for on level plains the drawing is but little, and then the horse's back need be pressed but with a small weight.

8. The utility of broad Wheels, in amending and preserving the roads, has been so long and generally acknowledged, as to have occasioned the legislature to enforce their use. At the same time, the proprietors and drivers of carriages seem to be convinced by experience, that a narrow-wheeled carriage is more easily and speedily drawn by the same number of horses, than a broad-wheeled one of the same burthen: probably because they are much lighter, and have less friction on the axle.

On the subject of this article, see Jacob's Observ. &c. on Wheel-Carriages, 1773, p. 81. Desagul. Exper. Phil. vol. 1, p. 201. Ferguson's Lect. 4to, p. 56. Martin's Phil. Brit. vol. 1, p. 229.

Blowing Wheel, is a machine contrived by Desaguliers, for drawing the foul air out of any place, or for forcing in fresh, or doing both successively, without opening doors or windows. See Philos. Trans. number 437. The intention of this machine is the same as that of Hales's ventilator, but not so effectual, nor so convenient. See Desag. Exper. Philos. vol. 2, p. 563, 568.—This Wheel is also called a centrifugal Wheel, because it drives the air with a centrifugal force.

Water Wheel, of a Mill, that which receives the impulse of the stream by means of ladle-boards or floatboards. M. Parent, of the Academy of Sciences, has determined that the greatest effect of an undershot Wheel, is when its velocity is equal to the 3d part of the velocity of the water that drives it; but it ought to be the half of that velocity, as is fully shewn in the article Mill, pa. 111. In fixing an undershot Wheel, it ought to be considered whether the water can run clear off, so as to cause no back-water to stop its motion. Concerning this article, see Desagul. Exp. Philos. vol. 2, p. 422. Also a variety of experiments and observations relating to undershot and overshot Wheels, by Mr. Smeaton, in the Philos. Trans. vol. 51, p. 100.

Aristotle's Wheel. See Rota Aristotelica.

Measuring Wheel. See Perambulator.

Orffyreus's Wheel. See Orffyreus.

Persian Wheel. See Persian.

Wheel-Barometer. See Barometer.

WHIRL-POOL, an eddy, vortex, or gulph, where the water is continually turning round.

WHIRLING-TABLE, a machine contrived for | representing several phenomena in philosophy, and nature; as, the principal laws of gravitation, and of the planetary motions in curvilinear orbits.

The figure of this instrument is exhibited fig. 1, pl. 35: where AA is a strong frame of wood; B a winch fixed on the axis C of the wheel D, round which is the catgut string F, which also goes round the small wheels G and K, crossing between them and the great wheel D. On the upper end of the axis of the wheel G, above the frame, is fixed the round board d, to which may be occasionally fixed the bearer MSX. On the axis of the wheel H is fixed the bearer NTZ, and when the winch B is turned, the wheels and bearers are put into a Whirling motion. Each bearer has two wires W, X, and Y, Z, fixed and screwed tight into them at the ends by nuts on the outside; and when the nuts are unscrewed, the wires may be drawn out in order to change the balls U, V, which slide upon the wires by means of brass loops fixed into the balls, and preventing their touching the wood below them. Through each ball there passes a silk line, which is fixed to it at any length from the centre of the bearer to its end, by a nut-screw at the top of the ball; the shank of the screw going into the centre of the ball, and pressing the line against the under side of the whole which it goes through. The line goes from the ball, and under a small pulley sixed in the middle of the bearer; then up through a socket in the round plate (S and T) in the middle of each bearer; then through a slit in the middle of the square top (O and P) of each tower, and going over a small pulley on the top comes down again the same way, and is at last fastened to the upper end of the socket fixed in the middle of the round plate above mentioned. Each of these plates S and T has four round holes near their edges, by which they slide up and down upon the wires which make the corner of each lower. The balls and plates being thus connected, each by its particular line, it is plain that if the balls be drawn outward, or towards the end M and N of their respective bearers, the round plates S and T will be drawn up to the top of their respective towers O and P.

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

For a specimen of the experiments which may be made with this machine, may be subjoined the following.

1. Removing the bearer MX, put the loop of the line b to which the ivory ball a is fastened over a pin in the centre of the board d, and turn the winch B; and the ball will not immediately begin to move with the board, but, on account of its inactivity, endeavour to remain in its state of rest. But when the ball has acquired the same velocity with the board, it will remain upon the same part of the board, having no relative motion upon it. However, if the board be suddenly stopped, the ball will continue to revolve upon it, until the friction thereof stops its motion: so that matter resists every change of state, from that of rest to that of motion, and vice versa.

2. Put a longer cord to this ball; let it down through the hollow axis of the bearer MX and wheel G, and fix a weight to the end of the cord below the machine; and this weight, if left at liberty, will draw the ball from the edge of the Whirling board to its centre. Draw off the ball a little from the centre, and turn the winch; then the ball will go round and round with the board, and gradually fly farther from the centre, raising up the weight below the machine. And thus it appears that all bodies, revolving in circles, have a tendency to fly off from those circles, and must be retained in them by some power proceeding from or tending to the centre of motion. Stop the machine, and the ball will continue to revolve for some time upon the board; but as the friction gradually stops its motion, the weight acting upon it will bring it nearer and nearer to the centre in every revolution, till it brings it quite thither. Hence it appears, that if the planets met with any resistance in going round the sun, its attractive power would bring them nearer and nearer to it in every revolution, till they would fall into it.

3. Take hold of the cord below the machine with one hand, and with the other throw the ball upon the round board as it were at right angles to the cord, and it will revolve upon the board. Then, observing the velocity of its motion, pull the cord below the machine, and thus bring the ball nearer the centre of the board, and the ball will be seen to revolve with an increasing velocity, as it approaches the centre: and thus the planets which are nearest the sun perform quicker revolutions than those which are more remote, and move with greater velocity in every part of their respective circles.

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

5. Take away these equal balls, and put a ball of 6 ounces into the bearer MX, at a 6th part of the distance wz from the centre, and put a ball of one ounce into the opposite bearer, at the whole distance xy = wz; and six the balls at these distances on their cords, by the screw nuts at the top: then the ball U, which is 6 times as heavy as the ball V, will be at only a 6th part of the distance from its centre of motion; and consequently will revolve in a circle of only a 6th part of the circumference of the circle in which V revolves. Let equal weights be put into the towers, and the winch be turned; which (as the catgut-string | is on equal wheels below, will cause the balls to revolve in equal times: but V will move 6 times as fast as U, because it revolves in a circle of 6 times its radius, and both the weights in the towers will rise at once. Hence it appears, that the centrifugal forces of revolving bodies are in direct proportion to their quantities of matter multiplied into their respective velocities, or into their distance from the centres of their respective circles.

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

6. Let two balls, U and V, of equal weights, be sixed on their cords at equal distances from their respective centres of motion w and x; and let the catgut string E be put round the wheel K (whose circumference is only half that of the wheel H or G) and over the pulley s to keep it tight, and let 4 times as much weight be put into the tower P as in the tower O. Then turn the winch B, and the ball V will revolve twice as fast as the ball U in a circle of the same diameter, because they are equidistant from the centres of the circles in which they revolve; and the weights in the towers will both rise at the same instant; which shews that a double velocity in the same circle will exactly balance a quadruple power of attraction in the centre of the circle: for the weights in the towers may be considered as the attractive forces in the centres, acting upon the revolving balls; which moving in equal circles, are as if they both moved in the same circle. Whence it appears that, if bodies of equal weights revolve in equal circles with unequal velocities, their centrifugal forces are as the squares of the velocities.

7. The catgut string remaining as before, let the distance of the ball V from the centre x be equal to 2 of the divisions on its bearer; and the distance of the ball U from the centre w be 3 and a 6th part; the balls themselves being equally heavy, and V making two revolutions by turning the winch, whilst U makes one; so that if we suppose the ball V to revolve in one moment, the ball U will revolve in 2 moments, the squares of which are 1 and 4: therefore, the square of the period of V is contained 4 times in the square of the period of U. But the distance of V is 2, the cube of which is 8, and the distance of U is 3 1/6, the cube of which is 32 very nearly, in which 8 is contained 4 times: and therefore, the squares of the periods V and U are to one another as the cubes of their distances from x and w, the centres of their respective circles. And if the weight in the tower O be 4 ounces, or equal to the square of 2, which is the distance of V from the centre x; and the weight in the tower P be 10 ounces, nearly equal to the square of 3 1/6, the distance of U from w; it will be found upon turning the machine by the winch, that the balls U and V will raise their respective weights at very nearly the same instant of time. This experiment confirms the famous proposition of Kepler, viz, that the squares of the periodical times of the planets round the sun are in propor- tion as the cubes of their distances from him; and that the sun's attraction is inversely as the square of the distance from his centre.

8. Take off the string E from the wheels D and H, and let the string F remain upon the wheels D and G; take away also the bearer MX from the Whirlingboard d, and instead of it put on the machine AB (fig. 2), fixing it to the centre of the board by the pins c and d, so that the end ef may rise above the board to an angle of 30 or 40 degrees. On the upper part of this machine, there are two glass tubes a and b, close stopped at both ends, each tube being about three quarters full of water. In the tube a is a little quicksilver, which naturally falls down to the end a in the water; and in the tube b is a small cork, floating on the top of the water, and small enough to rise or fall in the tube. While the board b with this machine upon it continues at rest, the quicksilver lies at the bottom of the tube a, and the cork floats on the water near the top of the tube b. But, upon turning the winch and moving the machine, the contents of each tube fly off towards the uppermost ends, which are farthest from the centre of motion; the heaviest with the greatest force. Consequently, the quicksilver in the tube a will fly off quite to the end f, occupying its bulk of space, and excluding the water, which is lighter than itself: but the water in the tube b, flying off to its higher end c, will exclude the cork from that place, and cause it to descend toward the lowest end of the tube; for the heavier body, having the greater centrifugal force, will possess the upper part of the tube, and the lighter body will keep between the heavier and the lower part.

This experiment demonstrates the absurdity of the Cartesian doctrine of vortices; for, if a planet be more dense or heavy than its bulk of the vortex, it will fly off in it farther and farther from the sun; if less dense, it will come down to the lowest part of the vortex, at the sun: and the whole vortex itself, unless prevented by some obstacle, would fly quite off, together with the planets.

9. If a body be so placed upon the Whirling-board of the machine (fig. 1.) that the centre of gravity of the body be directly over the centre of the board, and the board be moved ever so rapidly by the winch B, the body will turn round with the board, without removing from its middle; for, as all parts of the body are in equilibrio round its centre of gravity, and the centre of gravity is at rest in the centre of motion, the centrifugal force of all parts of the body will be equal at equal distances from its centre of motion, and therefore the body will remain in its place. But if the centre of gravity be placed ever so little out of the centre of motion, and the machine be turned swiftly round, the body will fly off towards that side of the board on which its centre of gravity lies. Then if the wire C (fig. 3) with its little ball B be taken away from the semi-globe A, and the flat side f of the semiglobe be laid upon the Whirling-board, so that their centres may coincide; if then the board be turned ever so quickly by the winch, the semi-globe will remain where it was placed: but if the wire C be screwed into the semi-globe at d, the whole becomes one body, whose centre of gravity is at or near d. Fix the pin c | in the centre of the Whirling-board, and let the deep groove b cut in the flat side of the semi-globe be put upon the pin, so that the pin may be in the centre of A (see fig. 4) where the groove is to be represented at b, and let the board be turned by the winch, which will carry the little ball B (fig. 3) with its wire C, and the semi-globe A, round the centre-pin c i; and then, the centrifugal force of the little ball B, weighing one ounce, will be so great as to draw off the semi-globe A, weighing two pounds, until the end of the groove at c strikes against the pin c, and so prevents A from going any farther: otherwise, the centrifugal force of B would have been great enough to have carried A quite off the whirling-board. Hence we see that, if the sun were placed in the centre of the orbits of the planets, it could not possibly remain there; for the centrifugal forces of the planets would carry them quite off, and the sun with them; especially when several of them happened to be in one quarter of the heavens. For the sun and planets are as much connected by the mutual attraction subsisting between them, as the bodies A and B are by the wire C fixed into them both. And even if there were but one planet in the whole heavens to go round ever so large a sun in the centre of its orbit, its centrifugal force would soon carry off both itself and the sun; for the greatest body placed in any part of free space could be easily moved; because, if there were no other body to attract it, it would have no weight or gravity of itself, and consequently, though it could have no tendency of itself to remove from that part of space, yet it might be very easily moved by any other substance.

10. As the centrifugal force of the light body B will not allow the heavy body A to remain in the centre of motion, even though it be 24 times as heavy as B; let the ball A (fig. 5) weighing 6 ounces be connected by the wire C with the ball B, weighing one ounce, and let the fork E be fixed into the centre of the Whirling-board; then, hang the balls upon the fork by the wire C in such a manner that they may exactly balance each other, which will be when the centre of gravity between them, in the wire at d, is supported by the fork. And this centre of gravity is as much nearer to the centre of the ball A than to the centre B, as A is heavier than B; allowing for the weight of the wire on each side of the fork. Then, let the machine be moved, and the balls A and B will go round their common centre of gravity d, keeping their balance, because either will not allow the other to fly off with it. For, supposing the ball B to be only one ounce in weight, and the ball A to be six ounces; then, if the wire C were equally heavy on each side of the fork, the centre of gravity d would be 6 times as far from the centre of B as from the centre of A, and consequently B will revolve with a velocity 6 times as great as A does; which will give B 6 times as much centrifugal force as any single ounce of A has; but then as B is only one ounce, and A six ounces, the whole centrifugal force of A will exactly balance that of B; and therefore, each body will detain the other, so as to make it keep in its circle.

Hence it appears, that the sun and planets must all move round the common centre of gravity of the whole system, in order to preserve that just balance which takes place among them.

11. Take away the forks and balls from the Whirling-board, and place the trough AB (fig. 6) thereon, fixing its centre to that of the board by the pin H. In this trough are two balls D and E of unequal weights, connected by a wire f, and made to slide easily upon the wire stretched from end to end of the trough, and made fast by nut screws on the outside of the ends. Place these balls on the wire c, so that their common centre of gravity g, may be directly over the centre of the Whirling-board. Then turn the machine by the winch ever so swiftly, and the trough and balls will go round their centre of gravity, so as neither of them will fly off; because, on account of the equilibrium, each ball detains the other with an equal force acting against it. But if the ball E be drawn a little more towards the end of the trough at A, it will remove the centre of gravity towards that end from the centre of motion; and then, upon turning the machine, the little ball E will fly off, and strike with a considerable force against the end A, and draw the great ball B into the middle of the trough. Or, if the great ball D be drawn towards the end B of the trough, so that the centre of gravity may be a little towards that end from the centre of motion; and the machine be turned by the winch, the great ball D will fly off, and strike violently against the end B of the trough, and will bring the little ball E into the middle of it. If the trough be not made very strong, the ball D will break through it.

12. Mr. Ferguson has explained the reason why the tides rise at the same time on opposite sides of the earth, and consequently in opposite directions, by the following new experiment on the Whirling-table. For this purpose, let a b c d (fig. 7) represent the earth, with its side c turned toward the moon, which will then attract the water so as to raise them from c to g: and in order to shew that they will rise as high at the same time on the opposite side from a to e; let a plate AB (fig. 8) be fixed upon one end of the flat bar DC, with such a circle drawn upon it as a b c d (fig. 7) to represent the round figure of the earth and sea; and an ellipse as e f g h to represent the swelling of the tide at e and g, occasioned by the influence of the moon. Over this plate AB suspend the three ivory balls e, f, g, by the silk lines h, i, k, fastened to the tops of the wires H, I, K, so that the ball at e may hang freely over the side of the circle e, which is farthest from the moon M at the other end of the bar; the ball at f over the centre, and the ball at g over the side of the circle g, which is nearest the moon. The ball f may represent the centre of the earth, the ball g water on the side next the moon, and the ball e water on the opposite side. On the back of the moon M is fixed a short bar N parallel to the horizon, and there are three holes in it above the little weights p, q, r. A silken thread o is tied to the line k close above the ball g, and passing by one side of the moon M goes through a hole in the bar N, and has the weight p hung to it. Such another thread m is tied to the line i, close above the ball f, and, passing through the centre of the moon M and middle of the bar N, has the weight q hung to it which is lighter than the weight p. A third thread m is tied to the line h, close | above the ball e, and, paffing by the other side, of the moon M through the bar N, has the weight r hung to it, which is lighter than the weight q. The use of these three unequal weights is to represent the moon's unequal attraction at different distances from her; so that if they are left at liberty, they will draw all the three balls towards the moon with different degrees of force, and cause them to appear as in fig. 9, in which case they are evidently farther from each other than if they hung freely by the perpendicular lines h, i, k. Hence it appears, that as the moon attracts the side of the earth which is nearest her with a greater degree of force than she does the centre of the earth, she will draw the water on that side more than the centre, and cause it to rise on that side: and as she draws the centre more than the opposite side, the centre will recede farther from the surface of the water on that opposite side, and leave it as high there as she raised it on the side next her. For, as the centre will be in the middle between the tops of the opposite elevations, they must of course be equally high on both sides at the same time.

However, upon this supposition, the earth and moon would soon come together; and this would be the case if they had not a motion round their common centre of gravity, to produce a degree of centrifugal force, sufficient to balance their mutual attraction. Such motion they have; for as the moon revolves in her orbit every month, at the distance of 240000 miles from the earth's centre, and of 234000 miles from the centre of gravity of the earth and moon, the earth also goes round the same centre of gravity every month at the distance of 6000 miles from it, i. e. from it to the centre of the earth. But the diameter of the earth being, in round numbers, 8000 miles, its side next the moon is only 2000 miles from the common centre of gravity of the earth and moon, its centre 6000 miles from it, and its farthest side from the moon 10000 miles. Consequently the centrifugal forces of these parts are as 2000, 6000, and 10000; i. e. the centrifugal force of any side of the earth, when it is turned from the moon, is five times as great as when it is turned toward the moon. And as the moon's attraction, expressed by the number 6000 at the earth's centre, keeps the earth from flying out of this monthly circle, it must be greater than the centrifugal force of the waters on the side next her; and consequently, her greater degree of attraction on that side is sufficient to raise them; but as her attraction on the opposite side is less than the centrifugal force of the water there, the excess of this force is sufficient to raise the water just as high on the opposite side.

To prove this experimentally, let the bar DC with its furniture be fixed on the Whirling-board of the machine (fig. 1.) by pushing the pin P into the centre of the board; which pin is in the centre of gravity of the whole bar with its three balls, e, f, g, and moon M. Now if the Whirling-board and bar be turned slowly round by the winch, till the ball f hangs over the centre of the circle, as in fig. 10, the ball g will be kept towards the moon by the heaviest weight p (fig. 8), and the ball e, on account of its greater centrifugal force, and the less weight r, will fly off as far to the other side, as in fig. 10. And thus, whilst the machine is kept turning, the balls e and g will hang over the ends of the ellipse l f k. So that the centrifugal force of the ball e will exceed the moon's attraction just as much as her attraction exceeds the centrifugal force of the ball g, whilst her attraction just balances the centrifugal force of the ball f, and makes it keep in its circle. Hence it is evident, that the tides must rise to equal heights at the same time on opposite sides of the earth. See Ferguson's Lectures on Mechanics, lect. 2, and Desag. Ex. Phil. vol. 1, lect. 5.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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