areas of all circles are in exact proportion to the squares of their radii, or half diameters." If, for example, we draw a circle with a pair of compasses whose points are stretched 4 inches asunder, and another with an extent of 8 inches, the large circle is exactly four times the size or area of the small one. For the square of 4 is=16, and the square of 8 is 64, which is four times 16. And as the circumferences of the circles are in proportion to the radii, it will follow, that the length of a string which would go round the curve of the larger circle is exactly double the length of one which would go round the lesser. Mechanics, in recognising such theorems, will meet with many opportunities of reducing them to practice.-Again, there is a figure which Geometricians term a parabola, which is formed every time we pour water forcibly from the mouth of a tea-kettle, or throw a stone forward from the hand. One property of the parabola is, that if a spout of water be directed at half a perpendicular from the ground, or at an angle of elevation of 45 degrees, it will come to the ground at a greater distance than if any other direction had been given it, a slight allowance being made for the resistance of the air. Hence the man who guides the pipe of a fire-engine may be directed how to throw the water to the greatest distance, and he who aims at a mark, to give the projectile its proper direction. To surveyors, navigators, land-measurers, gaugers and engineers a knowledge of the mathematical sciences is so indispensably requisite, that without it, such arts cannot be skilfully exercised.

The physical sciences are also of the greatest utility in almost every department of art. To masons, architects, ship-builders, carpenters and every other class employed in combining materials, raising weights, quarrying stones, building piers and bridges, splitting rocks, or pumping water from the bowels of the earth,-a knowledge of the principles of mechanics and dynamics is of the first importance. By means of these sciences the nature of the lever and other mechanical powers may be learned, and their forces estimated-the force produced by any particular combination of these powers calculated-and the best mode of applying such forces to accomplish certain effects, ascertained. By a combination of the mechanical powers the smallest force may be multiplied to an almost in. definite extent, and with such assistance man has been enabled to rear works and to perform operations which excite astonishment, and which his own physical strength, assisted by all that the lower animals could furnish, would have been altogether inadequate to accomplish. An acquaintance with the experiments which have been made to determine the strength of materials, and the results which have been deduced from them, is of immense importance to every class of mechanics employed in engineering and architectural operations. From such experiments, (which have only been lately attended to on scientific principles) many useful deductions might be made respecting the best form of mortises, joints, beams, tenons, scarphs, &c.; the art of mast making, and the manner of disposing and combining the strength of different substances in naval architecture, and in the rearing of our buildings. For example,-from the experiments now alluded to it has been deduced, that the strength of any piece of material depends chiefly on its depth, or on that dimension which is in the direction of its strain. A bar of timber of one inch in breadth, and two inches in depth is four times as strong as a bar of only one inch deep; and it is twice as strong as a bar two inches broad and one deep, that is, a joint or lever is always strongest when laid on its edge. Hence it follows, that the strongest joist that can be cut out of a round tree is not the one which has the greatest quantity of timber in it, but such that the product of its breadth by the square of its depth shall be the greatest possible.-Again, from the same experiments it is found, that a hollow tube is stronger than a solid rod containing the same quantity of matter. This property of hollow tubes is also accompanied with greater stiffness; and the superiority in strength and stiffness is so much the greater as the surrounding shell is thinner in proportion to its diameter. Hence we find that the bones of men and other animals are formed hollow, which renders them incomparably stronger and stiffer, gives more room for the insertion of muscles,

and makes them lighter and more agile, than if they were constructed of solid matter. In like manner the bones of birds, which are thinner than those of other animals, and the quills in their wings, acquire by their thinness the strength which is necessary, while they are so ight as to give sufficient buoyancy to the animal in its flight through the aerial regions. Our engineers and carpenters have, of late, begun to imitate nature in this respect, and now make their axles and other parts of machinery hollow, which both saves a portion of materials and renders them stronger than if they were solid.*

The departments of hydrostatics and hydrau lics, which treat of the pressure and motion of fluids, and the method of estimating their velocity and force, require to be thoroughly understood by all those who are employed in the construction of common and forcing-pumps, water-mills, fountains, fire-engines, hydrostatical presses, and in the formation of canals, wetdocks, and directing the course of rivers; otherwise they will constantly be liable to commit egregious blunders, and can never rise to eminence in their respective professions. Such principles as the following:-that fluids press equally in all directions,-that they press as much upwards as downwards,-that water, in several tubes that communicate with each other, will stand at the same height, in all of them, whether they be small or great, perpendicular or oblique,-that the pressure of fluids is directly as their perpendicular height, without any regard to their quantity,-and that the quantities of water discharged at the same time, by different apertures, under the same heigth of surface in the reservoir, are to each other nearly as the areas of their apertures,-will be found capable of extensive application to plumbers, engineers, pump-makers, and all who are employed in conducting water over hills or vallies, or in using it as a mechanical power, by a recognition of which they will be enabled to foresee, with certainty, the results to be expected from their plans and operations; for want of which knowledge many plausible schemes have been frustrated, and sums of money expended to no purpose.

The following figures and explanations will tend to illustrate some of the principles now stated: -1. Fluids press in proportion to their perpendicular heights, and the base of the vessel containing them, without regard to the quantity. Thus, if the vessel ABC, fig. 2, has its base BC equal to the base FG of the cylindrical vessel DEFG, fig. 1, but is much smaller at the top A than at the bottom, and of the same height; the pressure upon the bottom BC is as great as

The mechanical reader who wishes particular information on this subject is referred to the article Strength of materials in Ency. Brit. 3d edit. which was written by the late Professor Robison.

the pressure upon the bottom of the vessel DK FG, when they are filled with water, or any other liquid, notwithstanding that there will be a much greater quantity of water in the cylindrical than in the conical vessel; or, in other words, the bottom BC will sustain a pressure equal to what it would be if the vessel were as wide at the top as at the bottom. In like manner, the bottom of the vessel HIKL, fig. 3, sustains a pressure only equal to the column whose base is KL, and height KM, and not as the whole quantity of fluid contained in the vessel; all the rest of the fluid being supported by the sides. The demonstration of these positions would oc cupy too much room, and to many readers would appear too abstract and uninteresting; but they will be found satisfactorily demonstrated in most books which treat of the doctrines of hydrostatics.

2. The positions now stated form the founda

water into it, the water will run into the larger vessel AB, and will stand at the same height C and G in both. If we affix an inclined tube EF, likewise communicating with the large vessel, the water will also stand at E, at the same height 13 in the other two; the perpendicular altitude seing the same in all the three tubes, however mall the one may be in proportion to the other. This experiment clearly proves that the small cobumn of water balances and supports the large column, which it could not do if the lateral pressures at bottom were not equal to each other.

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Whatever be the inclination of the tube EF, still the perpendicular altitude will be the same as that of the other tubes, although the column of water must be much longer than those in the upright tubes. Hence it is evident, that a small quantity of a fluid may, under certain circumstances, counterbalance any quantity of the same fluid. Hence also the truth of the principle in hydrostatics, that "in tubes which have a communication, whether they be equal or unequal, short or oblique, the fluid always rises to the same height." From these facts it follows, that water cannot be conveyed by means of a pipe that is laid in a reservoir to any place that is higher than the reser

voir.

These principles point out the mode of conveying water across valleys without those expensive aqueducts which were erected by the ancients for this purpose. A pipe, conforming to the shape of the valley, will answer every purpose of an aqueduct. Suppose the spring at A, fig. 5, and water is wanted on the other side of the valley to supply the house H, a pipe of lead or iron laid from the spring-head across the valley will convey the water up to the level of the spring-head; and if the house stand a little lower than the spring-head, a constant stream will pour into the cisterns and ponds where it is required, as if the house had stood on the other side of the valley; and, consequently, will save the expense of the arches BB, by which the ancient Romans conducted water from one hill to another. But, if the valley be very deep, the pipes must be made very strong near its bottom, otherwise they will be apt to burst; as the pressure of water increases in the rapid ratio of 1, 3, 5, 7, 9, &c. and is always in proportion to its perpendicular height.

board, and pour more water into the pipe, it will run into the bellows, and raise up the board with all the weights upon it. And though the water in the tube should weigh in all only a quarter of a pound, yet the pressure of this small force upon the water below in the bellows shall support the weights, which are 300 pounds; nor will they have weight enough to make them descend, and conquer the weight of water, by forcing it out of the mouth of the pipe. The reason of this will appear from what has been already stated respecting the pressure of fluids of equal heights, without any regard to the quantities. For, if a hole be made in the upper board, and a tube be put into it, the water will rise in the tube to the same height that it does in the pipe; and it would rise as high (by supplying the pipe) in as many tubes as the board would contain holes. Hence, if a man stand upon the upper board, and blow into the bellows through the pipe, he will raise himself upward upon the board; and the smaller the bore of the pipe is, the easier will he be able to raise himself. And if he put his finger on the top of the pipe he may support himself as long as he pleases.

The uses to which this power may be applied are of great variety and extent; and the branches

of art dependent upon it appear to be yet in their infancy. By the application of this power the late Mr. Bramah formed what is called the Hydrostatic Press, by which a prodigious force is obtained, and by the help of which, hay, straw, wool, and other light substances, may be forced into a very small bulk, so as to be taken in large quantities on board a ship. With a machine, on this principle, of the size of a tea-pot, standing before him on a table, a man is enabled to cut through a thick bar of iron as easily as he could clip a piece of pasteboard with a pair of sheers. By this machine a pressure of 500 or 600 tons may be brought to bear upon any substances which it is wished to press, to tear up, to cut in pieces, or to pull asunder.

Upon the same principle, the tun or hogshead HI, fig. 7, when filled with water, may be burst, by pressing it with some pounds additional weight of the fluid through the small tube KL, which may be supposed to be from 25 to 30 feet in height. From what has been already stated, it necessarily follows, that the small quantity of water which the tube KL, contains, presses upon the bottom of the tun with as much force as if a column of water had been added as wide as the tun itself, and as long as the tube, which would evidently be an enormous weight.

A few years ago, a friend of mine, when in Ireand, performed this experiment to convince an English gentleman, who called in question the principle, and who laid a bet of fifty pounds that it would not succeed. A hogshead, above 3 feet high, and above 2 feet wide, was filled with water; a leaden tube, with a narrow bore, between 20 and 30 feet long, was firmly inserted into the top of the hogshead; a person, from the upper window of a house, poured in a decanter of water into the tube, and, before the decanter was quite emptied, the hogshead began to swell, and, in two or three seconds, burst into pieces, while the water was scattered about with immense force.

Hence, we may easily perceive what mischief may sometimes be done by a very small quantity of water, when it happens to act according to its perpendicular height. Suppose, that in any building, near the foundation, a small quantity of water, only of the extent of a square yard, has settled, and suppose it to have completely filled up the whole vacant space, if a tube of 20 feet long were thrust down into the water, and filled with water from above, a force of more than 5 tons would be applied to that part of the building, which would blow it up with the same force as gunpowder. The same effect may sometimes be produced by rain falling into long narrow chinks, that may have inadvertently been left in building the walls of a house; which shows the importance of filling up every crevice and opening of a building, and rendering the walls as close and compact as possible. Hence, likewise, similar processes in nature, connected with pools of water in the bowels of the earth, may occasionally produce the most dreadful devastations. For, should it happen, that, in the interior of a mountain, two or three hundred feet below the surface, a pool of water thirty or forty square feet in extent, and only an inch or two in depth, was collected, and a small crevice or opening of half an inch in breadth were continued from the surface to the water in the pool; and were this crevice to be filled with rain or melted snow, the parts around the layer of water would sustain a pressure of more than six hundred tons, which might shake the mountain to its centre, and even rend it with the greatest violence. In this way, there is every reason to believe, partial earthquakes have been produced, and large fragments of mountains detached from their bases.

The principles now illustrated are capable of the most extensive application, particularly in all engineering and hydraulic operations. It is on the principle of the lateral and upward pressure of fluids that the water, elevated by the New River water-works, in the vicinity of London, after having descended from a bason in a vertical

See fig. 8. p. 68.

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pipe, and then, after having flowed horizontally in a succession of pipes under the pavement, is raised up again through another pipe, as high as the fountain in the Temple Garden. It is upon the same principle that a vessel may be filled either at the mouth or at the bottom indifferently, provided that it is done through a pipe, the top of which is as high as the top of the vessel to be filled. Hence, likewise, it follows, that when piers, aqueducts, or other hydraulic works for the retention of water, are to be constructed, it becomes necessary to proportion their strength to the lateral pressure which they are likely to sustain, which becomes greater in proportion to the height of the water to be sustained. Walls, likewise, designed to support terraces, ought te be sufficiently strong to resist the lateral pressure of the earth and rubbish which they are to sus➡ tain, since this pressure will be greater as the particles of earth, of which the terraces are composed, are less bound together, and in proportion as the terraces are more elevated. The increase of pressure in proportion to the depth of any fluid likewise shows the necessity of forming the sides of pipes or masonry in which fluids are to be retained, stronger towards the bottom, where the pressure is greatest. If they are no thicker than what is sufficient for resisting the pressure near the top, they will soon give way by the superior pressure near the bottom; and if they are thick enough in every part to resist the great pressure below, they will be stronger than necessary in the parts above, and, consequently, a superfluous expense, that might have been saved, will be incurred in the additional materials and labour employed in their construction. The same principle is applicable to the construction of flood-gates, dams, and banks of every description, for resisting the force of water. When the strength and thickness requisite for resisting the pressure at the greatest depth is once ascertained, the walls or banks may be made to taper upwards, according to a certain ratio founded on the strength of the materials, and the gradual decrease of pressure from the bottom upwards; or, if one side be made perpendicular, the other may proceed in a slanting direction towards the top.

From the principles and experiments now stated, we may also learn the reason why the banks of ponds, rivers, and canals blow up, as it is termed. If water can insinuate itself under a bank or dam, even although the layer of water were no thicker than a half-crown piece, the pressure of the water in the canal or pond will force it up. In fig. 8, let A represent the section of a river or canal, and BB a drain running under one of its banks; it is evident, that, if the bank C is not heavier than the column of water BB, that part of the bank must inevitably give way. This effect may be prevented in artificial canals by making the sides very tight with clay heavily