likely to possess. Nearly the same considerations apply to the atomic weight as the standard of mass. It is impossible to prove that all atoms of the same substance are of equal mass, and some physicists think that they differ, so that the fixity of combining proportions may be due only to the approximate constancy of the mean of countless millions of discrepant weights. But in any case the detection of difference is probably beyond our powers. In a theoretical point of view, then, the magnitudes suggested by Professor Maxwell seem to be the most fixed ones of which we have any knowledge, so that they necessarily become the natural units.

In a practical point of view, as Professor Maxwell would be the first to point out, they are of little or no value, because in the present state of science we cannot measure a vibration or weigh an atom with any approach to the accuracy which is attainable in the comparison of standard metres and kilograms. The velocity of light is not known probably within a thousandth part, and as we progress in the knowledge of light, so we shall progress in the accurate fixation of other standards. All that can be said then, is that it is very desirable to determine the wave-lengths and periods of the principal lines of the solar spectrum, and the absolute atomic weights of the elements, with all attainable accuracy, in terms of our existing standards. The numbers thus obtained would admit of the reproduction of our standards in some future age of the world to a corresponding degree of accuracy, were there need of such reference; but so far as we can see at present, there is no considerable probability that this mode of reproduction would ever be the best mode.

Subsidiary Units.

Having once established the standard units of time, space, and density or mass, we might employ them for the expression of all quantities of such nature. But it is often convenient in particular branches of science to use multiples or submultiples of the original units, for the expression of quantities in a simple manner. We use the mile rather than the yard when treating of the magnitude of the globe, and the mean distance of the earth and

sun is not too large a unit when we have to describe the distances of the stars. On the other hand, when we are occupied with microscopic objects, the inch, the line or the millimetre, become the most convenient terms of expression.

It is allowable for a scientific man to introduce a new unit in any branch of knowledge, provided that it assists precise expression, and is carefully brought into relation with the primary units. Thus Professor A. W. Williamson has proposed as a convenient unit of volume in chemical science, an absolute volume equal to about 112 litres representing the bulk of one gram of hydrogen gas at standard temperature and pressure, or the equivalent weight of any other gas, such as 16 grams of oxygen, 14 grams of nitrogen, &c.; in short, the bulk of that quantity of any one of those gases which weighs as many grams as there are units in the number expressing its atomic weight.1 Hofmann has proposed a new unit of weight for chemists, called a crith, to be defined by the weight of one litre of hydrogen gas at o° C. and o°76 mm., weighing about 00896 gram. Both of these units must be regarded as purely subordinate units, ultimately defined by reference to the primary units, and not involving any new assumption.

Derived Units.

The standard units of time, space, and mass having been once fixed, many kinds of magnitude are naturally measured by units derived from them. From the metre, the unit of linear magnitude follows in the most obvious manner the centiare or square metre, the unit of superficial magnitude, and the litre that is the cube of the tenth part of a metre, the unit of capacity or volume. Velocity of motion is expressed by the ratio of the space passed over, when the motion is uniform, to the time occupied; hence the unit of velocity is that of a body which passes over a unit of space in a unit of time. In physical science the unit of velocity might be taken as one metre per second.

1 Chemistry for Students, by A. W. Williamson. Clarendon Press Series, 2nd ed. Preface p. vi. 2 Introduction to Chemistry, p. 131.

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Momentum is measured by the mass moving, regard being paid both to the amount of matter and the velocity at which it is moving. Hence the unit of momentum will be that of a unit volume of matter of the unit density moving with the unit velocity, or in the French system, a cubic centimetre of water of the maximum density moving one metre per second.

An accelerating force is measured by the ratio of the momentum generated to the time occupied, the force being supposed to act uniformly. The unit of force will therefore be that which generates a unit of momentum in a unit of time, or which causes, in the French system, one cubic centimetre of water at maximum density to acquire in one second a velocity of one metre per second. The force of gravity is the most familiar kind of force, and as, when acting unimpeded upon any substance, it produces in a second a velocity of 980868.. metres per second in Paris, it follows that the absolute unit of force is about the tenth part of the force of gravity. If we employ British weights and measures, the absolute unit of force is represented by the gravity of about half an ounce, since the force of gravity of any portion of matter acting upon that matter during one second, produces a final velocity of 32 1889 feet per second or about 32 units of velocity. Although from its perpetual action and approximate uniformity we find in gravity the most convenient force for reference, and thus habitually employ it to estimate quantities of matter, we must remember that it is only one of many instances of force. Strictly speaking, we should express weight in terms of force, but practically we express other forces in terms of weight.

We still require the unit of energy, a more complex notion. The momentum of a body expresses the quantity of motion which belongs or would belong to the aggregate of the particles; but when we consider how this motion is related to the action of a force producing or removing it, we find that the effect of a force is proportional to the mass multiplied by the square of the velocity and it is convenient to take half this product as the expression required. But it is shown in books. upon dynamics that it will be exactly the same thing if we define energy by a force acting through a space. The

natural unit of energy will then be that which overcomes a unit of force acting through a unit of space; when we lift one kilogram through one metre, against gravity, we therefore accomplish 980868. . units of work, that is, we turn so many units of potential energy existing in the muscles, into potential energy of gravitation. In lifting one pound through one foot there is in like manner a conversion of 321889 units of energy. Accordingly the unit of energy will be in the English system, that required to lift one pound through about the thirty-second part of a foot; in terms of metric units, it will be that required to lift a kilogram through about one tenth part of a metre. Every person is at liberty to measure and record quantities in terms of any unit which he likes. He may use the yard for linear measurement and the litre for cubic measurement, only there will then be a complicated relation between his different results. The system of derived units which we have been briefly considering, is that which gives the most simple and natural relations between quantitative expressions of different kinds, and therefore conduces to ease of comprehension and saving of laborious calculation.

It would evidently be a source of great convenience if scientific men could agree upon some single system of units, original and derived, in terms of which all quantities could be expressed. Statements would thus be rendered easily comparable, a large part of scientific literature would be made intelligible to all, and the saving of mental labour would be immense. It seems to be generally allowed, too, that the metric system of weights and measures presents the best basis for the ultimate system; it is thoroughly established in Western Europe; it is legalised in England; it is already commonly employed by scientific men; it is in itself the most simple and scientific of systems. There is every reason then why the metric system should be accepted at least in its main features.

Provisional Units.

Ultimately, as we can hardly doubt, all phenomena will be recognised as so many manifestations of energy; and, being expressed in terms of the unit of energy, will

be referable to the primary units of space, time, and density. To effect this reduction, however, in any particular case, we must not only be able to compare different quantities of the phenomenon, but to trace the whole series of steps by which it is connected with the primary notions. We can readily observe that the intensity of one source of light is greater than that of another; and, knowing that the intensity of light decreases as the square of the distance increases, we can easily determine their comparative brilliance. Hence we can express the intensity of light falling upon any surface, if we have a unit in which to make the expression. Light is undoubtedly one form of energy, and the unit ought therefore to be the unit of energy. But at present it is quite impossible to say how much energy there is in any particular amount of light. The question then arises,-Are we to defer the measurement of light until we can assign its relation to other forms of energy? If we answer Yes, it is equivalent to saying that the science of light must stand still perhaps for a generation; and not only this science but many others. The true course evidently is to select, as the provisional unit of light, some light of convenient intensity, which can be reproduced from time to time in the same intensity, and which is defined by physical circumstances. All the phenomena of light may be experimentally investigated relatively to this unit, for instance. that obtained after much labour by Bunsen and Roscoe.1 In after years it will become a matter of inquiry what is the energy exerted in such unit of light; but it may be long before the relation is exactly determined.

A provisional unit, then, means one which is assumed and physically defined in a safe and reproducible manner, in order that particular quantities may be compared inter se more accurately than they can yet be referred to the primary units. In reality the great majority of our measurements are expressed in terms of such provisionally independent units, and even the unit of mass, as we have seen, ought to be considered as provisional.

The unit of heat ought to be simply the unit of energy, already described. But a weight can be measured to the

1 Philosophical Transactions (1859), vol. cxlix. p. 884, &c.