pendulum, performed by Newton and Gauss, shows that all kinds of matter equally gravitate, that is, the attractive power of a substance is exactly proportional to its density. Two portions of matter then which are in equilibrium in the balance, may be assumed to possess equal inertia, and their densities will therefore be inversely as their cubic dimensions.

Unit of Mass.

Multiplying the number of units of density of a portion of matter, by the number of units of space occupied by it, we arrive at the quantity of matter, or, as it is usually called, the units of mass, as indicated by the inertia and gravity it possesses. To proceed in the most simple and logical manner, the unit of mass ought to be that of a cubic unit of matter of the standard density. The founders of the French metrical system took as their unit of mass, the cubic centimetre of water, at the temperature of maximum density (about 4° Centigrade). They called this unit of mass the gramme, and constructed standard specimens of the kilogram, which might be readily referred to by all who required to employ accurate weights. Unfortunately, however, the determination of the bulk of a given weight of water at a certain temperature is an operation involving many practical and theoretical difficulties, and it can not be performed in the present day with a greater exactness than that of about one part in 5000, the results of careful observers being sometimes found to differ as much as one part in 1000*.

Weights, on the other hand, can be compared with each other to at least one part in a million. Hence if different specimens of the kilogram be prepared by direct

t Clerk Maxwell's 'Theory of Heat,' p. 79.

weighing against water, they will not agree very closely with each other; and, as a matter of fact, the two principal standard kilograms neither agree with each other, nor with their true definition". The so-called Kilogram des Archives weighs 15432 34874 grains according to Prof. W H. Miller, while the kilogram deposited at the Ministry of the Interior in Paris, as the standard for commercial purposes, weighs 15432 344 grains.

Now since a standard weight constructed of platinum, or platinum and iridium, can be preserved in all probability free from any appreciable alteration, and since it can be very accurately compared with other weights, we shall ultimately attain the greatest exactness in our recorded measurements of weight and mass, by assuming some single standard kilogram as a provisional standard, leaving the determination of its actual mass in units of space and density for future investigation. This is what is practically done at the present day, and thus a unit of mass takes the place of the unit of density, both in the French and the present English systems. The English pound is defined by a certain lump of platinum, carefully preserved at Westminster, and is an entirely arbitrary mass, made to agree as nearly as possible with old English pounds. The gallon, the old English unit of cubic measurement, is defined by the condition that it shall contain exactly ten pounds weight of water at 62° Fahr.; and although it is stated that it has the capacity of about 277 274 cubic inches, this ratio between the cubic and linear system of measurement is not legally enacted, but is left open to investigation from time to time. While the French metric system as originally designed was theoretically perfect, it does not seem to differ practically in this point from the English system.

u Thomson and Tait's "Treatise on Natural Philosophy,' vol. i. P. 325. * Ibid.

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 found convenient in particular branches of science, to use multiples or submultiples of the original units, for the expression of quantities, in a clear and 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 Prof. A. W. Williamson has proposed as a convenient unit in chemical science, an absolute volume equal to about 112 litres, representing the bulk of one gramme of hydrogen gas at standard temperature and pressure, or the equivalent weight of any other gas, such as 16 grammes of oxygen, 14 grammes of nitrogen, &c.; in short, the bulk of that quantity of any one of those gases which weighs as many grammes as there are units in the number expressing its atomic weight. Professor Hofmann has also proposed a new concrete unit for chemists, called a crith, to be defined by the weight of one cubic decimetre or litre of hydrogen gas at o° C. and o°76mm., weighing about o'0896 grammes 2. Both these units if adopted must be regarded as purely subordinate units, ultimately defined by reference to the primary units, and not involving any new assumption.

y Chemistry for Students,' by A. W. Williamson. Clarendon Press Series, 2nd ed. Preface p. vi. z Introd. to Chemistry,' p. 131.

Derived Units.

The standard units of time, space, and mass having been once fixed, it becomes obvious that many kinds of magnitude are naturally measured by units immediately derived from one or more of the three principal ones. From the standard metre of linear magnitude follows in the most obvious manner the centaire or square metre, the unit of superficial magnitude, and the litre or cube of the tenth part of a metre, the standard of capacity or volume. Velocity of motion, again, is expressed by the ratio of the space passed over, when the motion is uniform, to the time occupied; hence the unit velocity will be that of a body which passes over a unit of space in a unit of time, say one metre per second. 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 9.80868 .... 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 presence and approximate uniformity we find in gravity the most convenient force for reference, and thus habitually employ it to estimate quantities of matter or mass, 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 all 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 most 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 certain 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 9.80868 . . . . 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 32-1889 units of energy. Accordingly the unit of energy will be that required to lift a kilogram through about one tenth part of a metre against gravity, or, in the English system, to lift one pound through the thirty-second part of a foot.

Every person is at perfect liberty to measure and record