gained is in this proportion; but the circumferences of circles are to one another in the same proportion as their radii. Thus, in fig. 6, the centre of the axis and that of the wheel, which is the same point, is the centre of motion, the radius of the wheel is the distance of the power acting at the circumference of the wheel, from that point; and the radius of the axle is the distance of the weight from the same point. Hence the effect of the power is as the radius of the wheel o x; and the effect of the weight is as the radius of the axle o z; so that the two will be in equilibrio, or balance each other, when P:W::oz:ox. In fig. 7, we see in what way the wheel and axle is reduced to practice: A B is a wheel, and CD the axis; they move together, and it is evident that the power applied to the spokes a, b, c, d, &c. will move through as much more ↑ space as the bucket W, in proportion as the circumference of the circle, made by the spokes a, b, c, d, is larger than the circumference of the axle. To this kind of machine, made in a thousand different ways, belong all sorts of cranes for raising great weights, capstans, windlasses, &c.; but one of the safest and best cranes is that made by Mr. James White, a most ingenious mechanic at Gosport, of which there is a full description in the tenth volume of the Transactions of the Society for the Encouragement of Arts, &c.

In calculating the power gained by the wheel and axis, the thickness of the rope, and also the number of coils, when they run over or upon each other, must be taken into consideration, which is done by adding half the rope's thickness to the radius of the axle; or when there are several coils, over one another, as many times the thickness as there are coils wanting one half of one. It may be farther observed, that if the rope, chain, &c. be successively applied to wheels whose diameters continually increase, the axis will be turned with more and more ease, unless at the same time, the intensity of the power be diminished in the same proportion, and if this be the case, the axis will always be drawn with the same degree of force by a power continually diminishing. Of this

principle, watchmakers take advantage; for in their machines, the fusee on which the chain is wound, is so contrived as to present to the chain a series of wheels continually increasing in magnitude, so that in the case where the spring is strongest in its action, that is, immediately after it is wound, it draws the fusee by its smallest wheel, and as it unbends and becomes weaker it draws at the larger wheels; so that the motion of the watch-work, which depends upon a spring continually diminishing in strength, is always equable.

The Pulley is a small wheel turning on an axis, with a rope passing over it. Pullies, though varied in their construction, may be reduced to two kinds, viz. those that are fixed, and those that are moveable, or which rise and fall with the weight. A fixed pulley, as that shewn in fig. 8, serves only to change the direction of the power, and gives no mechanical advantage whatever; but when, besides the upper pullies, which turn round in a fixed frame or block, there is a block of pullies moving equally fast with the weight, the velocity of the weight is to the velocity of the power, as one, to twice the number of pullies in the lower or moveable block; and the power and weight balance each other, when the power is to the weight as one to twice the number of pullies in the moveable block or in fig. 9, as P: W:: 1 :: 4. Another kind of pulley is represented in fig. 11.

In comparing the pulley to the lever with respect to the advantage gained, the fulcrum, fig. 10, must be considered as at A, the weight acts at c, placed between the fulcrum, and power P, acting at D. The power therefore being twice as far from the fulcrum as the weight is, the proportion between the power and the weight, in order to balance one another, must be as two to one. The same thing may be proved differently. Every moveable pulley, AD, hangs by two ropes, equally stretched, and of course which bear equal parts of the weight, but the rope, A B, being made fast at B, half the weight is sustained by it; and the other half of the rope, to which the power is applied, has but half the weight to sup

two to one.

port; consequently the advantage gained by this pulley is as One considerable advantage in the pulley is, that the direction of the acting power may be easily changed: thus a heavy weight on the ground may be raised to the top of a high building, by a person standing on the ground, and vice versa; and by a change in the direction of the acting power, we are able to employ the power to the greatest advantage. A horse for instance cannot draw in a vertical direction, but draws with all his advantage in an horizontal one, changing therefore the direction, OP, by lengthening the rope, and having a fixed pulley at n, he becomes qualified to raise a weight W, from any depth, by moving in the direction n p. In solid block pullies, fig. 12, invented by Mr. James White, the several grooves in the lower block are calculated as different pullies, and the advantage gained is in the same proportion.

It may be farther observed, that the space passed through by the power, in the case of a single moveable pulley, as fig. 10, is double the space that the weight passes through in the case of two moveable pulleys it is four times, and so on; and as the velocities are as the spaces passed through, the momenta of the power and weight, will in the cases above described, be equal; that is, when they balance one another,

On an inclined plane, another of the mechanical powers, a weight raised or lowered, or a resistance overcome, moves only through a space equal to the height of that machine, in the time that a power impels it through a space equal to its whole length. Let AB, fig. 13, be a plane parallel to the horizon, and AD a plane inclined to it; and suppose the whole length, AD, to be three times as great as the perpendicular DB, in which case, the cylinder E will be supported upon the plane DA, by a power equal to a third part of the weight of the cylinder, for in this case P: W::DB: AD. This mechanical power is of great use in rolling up heavy bodies, as casks, wheel-barrows, &c. and to it may be reduced hatchets, chissels, and other edged tools, that are chamferred off on one side only,

A wedge, in the common form, is made up of two inclined planes, joined together at their bases, and the thickness of these planes makes the back of the wedge, to which the power of the hammer or mallet is applied in cleaving of wood: now there will be an equilibrium between the power impelling the wedge downward, and the resistance of the wood, or other substance acting against its sides, when the thickness, DG, fig. 14, of the wedge, is to the length of the two sides, BF and DO, or as half the thickness DE of the wedge, at its back, is to the length of one of its sides, so is the power to the resistance or in this case P: W:: BE: BF. The wedge is a very great mechanical power, since not only wood, but even rocks can be split with it, which could not be effected by any of the other mechanical powers.

The sixth and last mechanical power is the screw, which cannot properly be called a simple machine, because it is never used without a winch to assist in turning it, and it becomes a compound engine of very great force, either in pressing the parts of bodies closer together, or in raising weights. It may be conceived to be formed by cutting a piece of paper, fig. 15, into the form of an inclined plane, ABC, and then wrapping it round a cylinder: the edge of the paper, AB, will form a spiral line round the cylinder, which gives the thread of the screw, see fig. 16.

It is evident that the winch must turn the cylinder once round, before the weight, or resistance can be moved from one spiral to another, as from d to c, therefore as much as the circumference of a circle described by the handle of the winch is greater than the interval or distance between the spirals, so much is the force of the screw. If the spirals be only at the distance of a quarter of an inch from one another, and the length of the winch be twelve inches, the circle described by the winch will be seventy-six inches nearly, or three hundred and four quarter inches, consequently three hundred and four times as great as the distance between the spirals, therefore a power at the handle of 1lb. will balance

more than 300lbs. acting against the screw. Hence it follows, that the longer the winch is, and the nearer the spirals are to one another, so much the greater is the force of the

screw.

After the simple mechanical powers have been described, and their several properties investigated and understood, the pupil will advance to the consideration of compound machines, bearing in his mind when he comes to calculate the effects which they are intended to produce, that there will always be an equilibrium when the sum of the several powers is to the weight, as the sum of the velocities of the weight is to the sum of the velocities of the powers. And he will farther bear it in his mind, that though in the theory of the mechanical powers, all planes and other bodies are supposed to be perfectly smooth, levers to have no weight, cords to be perfectly pliable, and that there is no friction to be overcome; yet in practice all these things are to enter into the consideration, and in almost every compound machine, a full third additional power, must be allowed for moving and working a machine more than what is required to keep it in a state of equilibrio.

On the subject of friction, Mr., now Dr. Vince, has given, in vol. 75 of the Phil. Trans. a number of accurate experiments, of which the object was to determine: 1. Whether friction be an uniformly retarding force. 2. The quantity of friction. 3. Whether the friction varies in proportion to the pressure or weight; and 4. Whether the friction be the same on whichsoever of its surfaces a body moves.

In connexion with the subject of mechanics, a person will be led to consider the communication of motion, by direct and oblique impact: in the investigation of the centres of Percussion, Gyration, and Oscillation, his interest will be excited, and if entered into deeply, his mathematical skill will be tried. He will perhaps make himself acquainted with the different kinds of mill-work, which is of great importance in every large manufacturing country; and with the structure of clocks, watches, and other curious machines, which always increase in