resistance than the solid cylinder. It is for this reason that the bones of animals, the feathers of birds, the stems of corn and other plants, offer greater resistance than if they were solid, the mass remaining the same.

70. Hardness-Hardness is the resistance which bodies offer to being scratched or worn by others. It is only a relative property, for a body which is hard in reference to one body may be soft in reference to others. The relative hardness of two bodies is ascertained by trying which of them will scratch the other. Diamond is the hardest of all bodies, for it scratches all, and is not scratched by any. The hardness of a body is expressed by referring it to a scale of hardness: that usually adopted is

Thus the hardness of a body which would scratch felspar, but would be scratched by quartz, would be expressed by the number 6·5.

The pure metals are softer than their alloys. Hence it is that for jewellery and coinage, gold and silver, which are soft metals, are alloyed with copper to increase their hardness.

The hardness of a body has no relation to its resistance to compression. Glass and diamond are much harder than wood, but the latter offers far greater resistance to the blow of a hammer. Hard bodies are often used for polishing powders; for example, emery, pumice, and tripoli. Diamond, being the hardest of all bodies, can only be ground by means of its own powder.

71. Ductility.-Ductility is the property in virtue of which a great number of bodies change their forms by the action of traction or pressure.

Certain bodies, such as clay, wax, etc., are so ductile that they can be drawn out, flattened, modelled, between the fingers; others, such as the resins and glass, require the aid of heat. Glass is then so ductile that it can be drawn out into fine threads, which are flexible enough to be woven into cloth.

Several metals, such as gold, silver, copper, are ductile, even at ordinary temperatures, but require the use of powerful agents, such as the draw-plate or the rolling mill.

72. Malleability.-Malleability is that modification of ductility which is exhibited when metals are hammered. This property

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Malleability.

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greatly increases with the temperature; everyone knows, for instance, that iron is easily forged when hot, and not when cold.

Gold is very malleable even at the ordinary temperature. To make the extremely thin plates of gold, known as gold leaf, the gold is first pressed, by means of the rolling mill, into long plates from two to three centimeters in breadth, and about the of an inch in thickness. These plates are then beaten into small squares by means of a hammer; these are then cut and beaten again, and so on. By beating them directly, the operation could not long be continued, for the metal would be torn; hence the plates to be beaten must be placed between plates of a substance which, while thin, affords great resistance. Sheets of vellum and parchment are

first used for this purpose, and afterwards gold beater's skin.

Leaves of gold are thus obtained, which are so thin, that 20,000 superposed are only an inch thick. Silver and copper may also be worked in the same manner. These leaves are used in the arts for gilding on wood, paper, and other materials.

The following is the usual order of the metals under the drawplate, the rolling mill, and the hammer, arranged in reference to their decreasing ductility.

The metals must be pure; if they are alloyed with other metals

they are fragile, and have but little ductility.

BOOK II.

HYDROSTATICS.

CHAPTER I.

PRESSURES TRANSMITTED AND EXERTED BY LIQUIDS.

73. Object of Hydrostatics.-The science of hydrostatics, from two Greek words, signifying equilibrium of water, treats of the conditions of the equilibrium of liquids, and of the pressure they exert, whether within their own mass, or on the sides of the vessels in which they are contained.

74. Special characteristics of liquids.-The essential character of a liquid is the extreme mobility of its molecules, which are displaced by the slightest force. The fluidity of liquids is due to this property; it, however, is not perfect, there is always a sufficient adherence between the molecules to produce a greater or less viscosity.

Another essential property of liquids, and one by which they are distinguished from gases, is their almost entire incompressibility. We have already seen that their compressibility is so small, that for a long time they were regarded as being quite incompressible. It was not before 1823 that Oersted, a Swedish physicist, first proved in an exact manner that liquids are compressible. The apparatus he used for this purpose is called the piezometer (Tiew, I compress, μéтpov, measure). By its means it has been found that a pressure of one atmosphere compresses distilled water by about the 200.000 part of its volume; mercury by the same pressure only undergoes about a tenth as great a diminution, and ether about 24 times as much. Liquids are also porous, elastic, and impenetrable, like all other

1

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Pascal's Law.

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bodies. The proofs of their porosity have been already given, their elasticity is a necessary consequence of their compressibility. Their impenetrability is manifested whenever a solid is immersed in water. For if a vessel be quite filled with water, and any solid body be placed in it which does not absorb the liquid, it will be observed that a volume of water flows over, which is exactly equal to that of the solid immersed.

75. Equality of pressures. Pascal's law.-Liquids have the following remarkable property, which is not possessed by solids. It is often called Pascal's law, for it was first enunciated by that distinguished geometrician.

Pressure exerted anywhere upon a mass of liquid is transmitted undiminished in all directions, and acts with the same force on all equal surfaces and in a direction at right angles to those surfaces.

To get a clearer idea of the truth of this principle, let us conceive a cylindrical vessel, in the sides of which are placed various cylindrical tubulures, all of the same

size, and closed by movable pistons (fig. 57). The vessel being filled with water, or any other liquid, the moment any pressure is applied to the piston A, all the other pistons are pressed outwards, showing that the pressure is not merely transmitted downwards upon the piston D, but laterally upon the pistons E and F, and upwards upon the pistons B and C. If, instead of pressing on the piston A, the pressure be exerted upon

Fig. 57.

B, the same effects are produced; the piston A is then forced upwards.

In these different cases, not only is the pressure transmitted in all directions, but for the same surface it is transmitted with the same intensity. For instance, if the pressure on the piston A is twenty pounds, and its surface is equal to that of the piston B, the upward pressure on the latter is also twenty pounds; but if the surface of the piston B is only a twentieth that of A, the pressure upon B is only one pound. This is the principle of the equality of pressure.

F

76. Consequence and verification of Pascal's principle.—It follows from what has been said, that the pressure transmitted by a

P

Fig. 58.

liquid is proportional to the extent of surface; this is indeed only another enunciation of Pascal's principle.

To verify this, two cylinders are taken of unequal dimensions, joined by a tube (fig. 58). These cylinders contain water, and are provided with pistons which move in them

with gentle friction. Now if the surface of the larger one, P, for instance, is twenty times that of the smaller one, p, it will be found that a weight of a pound placed upon will balance a weight of twenty placed upon P; if these weights are in any other ratio, equilibrium is destroyed.

The principle of the equality of pressures forms the basis of the whole science of hydrostatics, and we shall presently find a very important application of it, in the hydraulic press (82).

77. Pressures resulting from the weights of liquids.—In what has been said, we have considered the pressures transmitted towards the sides of the vessel, when some external force is applied. It is not, however, necessary thus to exert an external pressure on the surface of a liquid to produce internal pressures in its mass, and on the sides of the vessel. The mere weight of the liquid is sufficient to produce pressures which vary with the depth and the density of the liquid.

For suppose any vessel filled with liquid; if we conceive the liquid divided into horizontal layers of equal thickness, it is clear that the second layer supports a pressure equal to the weight of the first; that the third supports the weight of the first and second, and so on; so that the pressure increases with the number of layers, which is expressed by saying that gravity produces in liquids pressures proportional to the depth.

It is obvious moreover, that these pressures are proportional to the density of the liquids; that is, that for the same depth, a liquid which has two or three times the density of another, will exert twice or thrice as much pressure.

It follows from the principle of the equality of pressure in all directions, that the pressure produced by gravity in liquids is exerted