quantities in terms of any unit which he likes to adopt. 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 relation between quantitative expressions of different kinds, and therefore conduces to ease of comprehension and saving of laborious calculation.

Provisionally Independent 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 mass. 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, 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 fully and accurately 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 almost every other. 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 exactly 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". 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 provisionally independent 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 almost all our measurements are made by such independent units. Even the unit of mass is practically an independent one, as we have seen (p. 373).

Similarly the unit of heat ought to be simply the unit of energy, already described. But a weight can be measured to the one-millionth part, and temperature to less than the thousandth part of a degree Fahrenheit, and to less therefore than the five-hundredth thousandth part of the absolute temperature, whereas the mechanical equivalent of heat is probably not known to the thousandth part. Hence the need of a provisional unit of heat, which is often taken as that requisite to raise a unit weight of water (say one gramme) through one degree Centigrade of temperature, that is from o° to 1°. This quantity of heat is capable of approximate expression in terms of time, space, and mass; for by the natural constant, determined by Dr. Joule, and called the mechanical

a

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

equivalent of heat, we know that the assumed unit of heat is equal to the energy of 423'55 gramme-metres, or that energy which will raise the mass of 423'55 grammes through one metre against 9.80868 absolute units of force. Heat may also be expressed in terms of the quantity of ice at o° Cent., which it is capable of converting into water under an inappreciable pressure.

The science of electricity has lately become so much a matter of quantity, that it is necessary to have some means of accurate expression. When we know exactly the mechanical equivalent of electricity, we can express quantities of electricity in terms of energy, but in the meantime we need some easy available unit. The British Association accordingly have selected as the unit of electrical force that which can just overcome the resistance offered by a piece of pure silver wire 1 metre in length, and I millemetre in diameter. This unit must be regarded as merely a convenient provision for working purposes, to be employed for the easy expression of quantities not yet brought into precise relation with the ultimate standards of time, space, and mass. There may

also be other provisionally independent units employed in electrical science, such as the voltametric unit of current strength, namely, that current which by decomposing water produces one cubic centimetre of detonating gas at o Cent. and 760 mm. of pressure in one minute. The unit of electrical quantity, again, is that quantity which when concentrated in a point and acting on an equal quantity also concentrated in a point at a unit of distance, exerts a repulsion equal to the unit of force. There must also be a unit of electro-magnetic force. All these electrical units must, however, be definitely related to each other, and to the fundamental units, and it is a matter for continual investigation to determine such relations more and more accurately.

Natural Constants and Numbers.

Having acquired accurate measuring instruments, and decided upon the units in which the results shall be estimated and expressed, there remains the question, What use shall be made of our powers of measurement? Our principal object must be to discover general quantitative laws of nature; but a very large amount of preliminary labour is employed in the accurate determination of the dimensions of existing objects, and the numerical relations between diverse forces and phenomena. Step by step every part of the material universe is surveyed and brought into known relations with other parts. Each manifestation of energy is correlated with each other kind of manifestation. Professor Tyndall has described the care with which such operations are conducted b.

'Those who are unacquainted with the details of scientific investigation, have no idea of the amount of labour expended on the determination of those numbers on which important calculations or inferences depend. They have no idea of the patience shown by a Berzelius in determining atomic weights; by a Regnault in determining coefficients of expansion; or by a Joule in determining the mechanical equivalent of heat. There is a morality brought to bear upon such matters which, in point of severity, is probably without a parallel in any other domain of intellectual action.'

Every new natural constant which is recorded brings many fresh inferences within our power. For if n be the number of such constants known, then (n2—n) is the number of ratios which are within our powers of calculation, and this increases with the square of n. We thus gradually piece together a map of nature, in which the lines of inference from one phenomenon to another

b Tyndall's Sound,' 1st ed. p. 26.

rapidly grow in complexity, and the powers of scientific prediction are correspondingly augmented.

с

The late Mr. Babbage proposed the formation of a complete collection of all the constant numbers of nature; but such a collection would be almost coextensive with the whole mass of scientific literature. Almost all numbers occurring in works on Chemistry, Mineralogy, Physics, Astronomy, &c. are natural constants, and it would be impracticable to give in any one work more than a selection of the more important numbers.

Our present object will be to classify these constant numbers roughly, according to their comparative generality and importance, under the following heads :(1) Mathematical constants. (2) Physical constants. (3) Astronomical constants. (4) Terrestrial numbers. (5) Organic numbers.

(6) Social numbers.

Mathematical Constants.

At the head of the list of natural constants must come those which express the necessary relations of numbers to each other. The ordinary Multiplication Table is the most familiar and the most important of such series of constants, and is, theoretically speaking, infinite in extent. Next we must place the Arithmetical Triangle, the significance of which has already been pointed out (p. 206.) Tables of logarithms also contain vast series of natural constants, arising out of the relations of pure numbers. At the base of all logarithmic theory is the mysterious natural constant commonly denoted by E, e, or e, being

equal to the infinite series 1 +

I

I

I

+ +

+

I

I

1.2 1.2.3 1.2.3.4

+ .....

e British Association, Cambridge, 1833. Report, pp. 484-490.