complicated for an elementary book. As a simple example let us suppose a sphere of wood floating on water. The centre of gravity of the solid is the centre of the sphere; and it is a result of geometry that the metacentre is also at the centre of the sphere. Thus the equilibrium is of the kind which we have called neutral in Art. 183. If instead of a whole sphere the floating body is a portion of a sphere cut off by a plane, then whether this portion is greater or less than a hemisphere, the centre of gravity will be below the centre of the sphere, while the metacentre is at the centre of the sphere; hence the body will float in stable equilibrium when the flat part is horizontal and outside the water.

XXXVI. SPECIFIC GRAVITY OF SOLIDS.

431. We have often in the preceding Chapters spoken of one body as heavier than another, bulk for bulk; thus gold is more than nineteen times as heavy as water bulk for bulk. In other words a cubic inch of gold is more than nineteen times as heavy as a cubic inch of water; and so for a cubic foot. When we speak of one body as heavier than another we may mean heavier bulk for bulk; in this sense gold is heavier than iron. Or we may mean that one assigned body is heavier than another, as that a certain iron bar is heavier than a certain gold coin. It is always plain from the circumstances in which of these senses we use the word heavier; the former is usually the sense required in the present work. We sometimes use the words heavy and light as if there were no comparison intended between the body with which we are concerned and other bodies. Thus we may say that lead is heavy and that cork is light. But some comparison is really intended; we mean that lead is heavier, bulk for bulk, than most objects with which we are familiar; and that cork is lighter, bulk for bulk, than most objects, or at least than most kinds of wood, with which we are familiar.

432. We have already in Art. 403 defined specific gravity as the proportion of the weight of any substance to the weight of an equal volume of the standard substance;

and we have stated that the standard substance is usually water. But we must now be a little more precise with respect to this standard substance. Water as obtained from springs or rivers is not always the same thing; it contains various substances mixed with it in greater or less degree, and hence the condition is added that the water must be pure. Water is made pure by distillation, that is, the water must be boiled and the vapour collected and condensed by cooling: in this way it is found that the substances which common water holds in solution are left behind, and pure water obtained. Moreover the bulk of water changes as the temperature changes, other things being the same. It is found that pure water diminishes in bulk as the temperature diminishes until the temperature is about 40 degrees of Fahrenheit's thermometer, and after that if the temperature is still lowered the bulk increases. Hence the temperature of 40 degrees of Fahrenheit's thermometer is that which it is found convenient to take for the standard. Thus finally we may say that the specific gravity of any substance is the proportion of the weight of the substance to the weight of an equal volume of pure water at the temperature of 40 degrees of Fahrenheit's thermo

meter.

433. The words dense and density are often used in books on Natural Philosophy, and we may here exemplify the meaning of them. We say that water has its greatest density at the temperature of 40 degrees of Fahrenheit's thermometer, or that water is more dense at this temperature than at any other. The simple fact which we have to express is that a cubic foot of water at this temperature weighs more than a cubic foot of water at any other temperature. As a convenient mode of representing this to our imagination we may suppose that the particles of water are closer together at the standard temperature than at any other. The density of a given body then is greater the smaller the volume of that body is; thus if a body is brought by cold or by pressure to occupy half its original space we say that the density is doubled. It would not be easy to double the density of a solid or of a liquid; but the density of a gaseous body can be easily doubled or even still more increased. We sometimes extend the range of the words dense and density, and use them in the com

parison of two bodies of different material; thus we may say that gold is more dense than silver, or has greater density; but in such a case we mean simply that gold is heavier than silver, bulk for bulk.

434. In order to find the specific gravity of any solid which will sink in water we proceed thus: weigh the solid in water and out of it, the difference is the weight of an equal bulk of the water; divide the weight of the solid out of the water by this and the quotient is the specific gravity of the solid. For example, a piece of gold is found to weigh 97 grains, and on being immersed in water to weigh only 92 grains; thus the weight of an equal bulk of water is 5 grains, and therefore the specific gravity of the 97 gold is that is 19. If the body is in the form of 5' small fragments it may be put into a cup, and the whole immersed in a vessel of water, and the weight in water determined. Then the weight of the cup when immersed alone in the water must be determined, and by subtraction we have the weight in water of the collection of small fragments. Their weight out of water can also be found, and then the specific gravity becomes known.

435. If we know the specific gravities of two metals we can determine the specific gravity of a compound formed by melting together known quantities of these metals, assuming that the volume of the compound is equal to the sum of the volumes of the two metals, and also that in the compound the two metals are thoroughly mixed so as to form a compound of the same density throughout. For example, suppose we take 5 cubic inches of gold of which the specific gravity is 194, and combine them with 20 cubic inches of copper of which the specific gravity is 89, and want to know the specific gravity of the compound. We may if we please work with cubic feet instead of cubic inches, and our language will then become more simple.

A cubic foot of water weighs a cubic foot of gold weighs a cubic foot of copper weighs thus five cubic feet of gold weigh and twenty cubic feet of copper weigh

1000 ounces; 19400 ounces;

8900 ounces; 97000 ounces, 178000 ounces.

Therefore the twenty-five cubic feet of the compound weigh 275000 ounces, and therefore one cubic foot weighs 11000 ounces; and the specific gravity of the compound is that is 11.

11000

1000

436. Various other questions may be proposed with respect to compound bodies. Thus we may suppose that we know we have 5 cubic inches of gold in the compound, and 20 cubic inches of some other metal; also we may have found by experiment the specific gravity of the compound, and may wish to know the specific gravity of the other metal in the compound. For example, suppose that the specific gravity of the compound is found to be 11. Then we know that a cubic foot of the compound will weigh 11000 ounces, and therefore 25 cubic feet of it will weigh 275000 ounces. But 5 cubic feet of gold weigh 97000 ounces, and therefore 20 cubic feet of the other metal weigh 178000 ounces; thus one cubic foot of it weighs 8900 ounces, and the specific gravity of the metal is

8900

1000'

that is 89. The

Tables of Specific Gravity shew us then that this metal has just the same specific gravity as copper, so that if we know it to be a simple metal we infer that it is copper.

437. Or again we may have a compound body which we know is made of gold and copper, but how much of each we are not told. Then if we know the specific gravities of gold, of copper, and of the compound, we can find the quantities of gold and of copper in any assigned quantity of the compound. The following is the rule: as the difference of the specific gravities of gold and of the compound is to the difference of the specific gravities of gold and of copper, so is the bulk of copper to the whole bulk. In any case the reader might find the quantities of gold and of copper by this rule and then verify the result by the method of Art. 435. The demonstration of the rule itself is not sufficiently elementary for the present book.

438. A famous story relating to the philosopher Archimedes is always told in books which treat on the subject now before us. Hiero King of Syracuse gave to an artist

a certain weight of gold to be made into a crown. The crown was furnished, and of course of the proper weight, but the king suspected that some of the gold had been replaced by silver, and he wished to settle this point without doing any injury to the crown. He consulted Archimedes, and it is said that the mode in which the problem might be solved flashed across the mind of the philosopher as he was in his bath; and that in a transport of joy he rushed from his chamber exclaiming in Greek, I have found it, I have found it. But the story, as repeated in modern times, seems to ascribe much more to Archimedes than he really then discovered. What he did was probably this: he used the principle that if a solid sinks in a vessel full of water the volume of the water ejected is exactly equal to the volume of the solid. He found in this way the volume of the crown, the volume of an equal weight of gold, and the volume of an equal weight of silver. This would be sufficient to enable him to determine how much gold and how much silver there was in the crown. He used in fact the geometrical principle involved in Chapter XXXI. but not the mechanical principle involved in Chapter XXXII.

439. We assume in the last four Articles, as stated in the beginning of Art. 435, that the volume of the compound is equal to the sum of the volumes of the metals which form it. But in practice this is frequently not the case, and thus the real specific gravity of a compound differs from that assigned by the process of Art. 435. For example, take the specific gravity of copper as accurately 878, and that of zinc as accurately 686; and let 14 pounds of copper be mixed with 7 pounds of zinc. Theoretically the specific gravity of the compound should be 8'14; but by experiment it is found to be about 86. The volume of the compound is thus a little less than the volumes of the copper and zinc.

440. Hitherto we have considered the specific gravity of any substance to be the proportion of the weight of that substance to the weight of an equal volume of the standard substance. But we may shew that this is the same thing as the proportion of the volume of the standard substance to the volume of an equal weight of the substance considered. For example, suppose that the weight of a certain