and thus consisting of the sum of the ratios between the numbers of permutations and combinations of o, 1, 2, 3, 4, &c. things.

Tables of prime numbers and of the factors of composite numbers must not be forgotten.

Another vast and in fact infinite series of numerical constants contains those connected with the measurement of angles, and embodied in trigonometrical tables, whether as natural or logarithmic sines, cosines, and tangents. It should never be forgotten that though these numbers find their chief employment in connexion with trigonometry, or the measurement of the sides of a right-angled triangle, yet the numbers themselves arise out of simple numerical relations bearing no special relation to space.

Foremost among trigonometrical constants is the well known number 7, usually employed as expressing the ratio of the circumference and the diameter of a circle; from follows the value of the arcual or natural unit of angular value as expressed in ordinary degrees (see p. 358).

Among other mathematical constants not uncommonly used may be mentioned tables of factorials (p. 202), tables of Bernouilli's numbers, tables of the error function d, which latter are indispensable not only in the theory of probability but also in several other branches of science.

It should also be clearly understood that the mathematical constants and tables of reference already in our possession, although very extensive, are only an infinitely small part of what might be formed. With the progress of science the tabulation of new functions will be continually demanded, and it is worthy of consideration whether public money should not be constantly available d See J. W. L. Glaisher, 'Philosophical Magazine,' 4th Series, vol. xlii.

P. 421.

to reward the enormous labour which must be undertaken in these calculations. Such labours once successfully completed must benefit the whole human race as long as it shall exist. A valuable account of all the chief mathematical tables yet published will be found in De Morgan's article on Tables, in the English Cyclopædia,' Division of Arts and Sciences, vol. vii. p. 976.

Physical Constants.

The second class of constants contains those which refer to the actual constitution of matter. For the most part they depend upon the peculiarities of the chemical substance in question, but we may begin with those which are of the most general character. In a first sub-class we may place the velocity of light or heat undulations, the numbers expressing the relation between the lengths of the undulations, and the rapidity of the undulations, these numbers depending only on the properties of the ethereal medium, and being probably the same in all parts of the universe. The theory of heat gives rise to several numbers of the highest importance, especially Joule's mechanical equivalent of heat, the absolute zero of temperature, the mean temperature of empty space, &c.

Taking into account the diverse properties of the elements we must have tables of the atomic weights, the specific heats, the specific gravities, the refractive powers, not only of the elements, but their almost infinitely numerous compounds. The properties of hardness, elasticity, viscosity, expansion by heat, conducting powers for heat and electricity, must also be determined in immense detail. There are, however, certain of these numbers which stand out prominently because they serve as intermediate units or terms of comparison. Such are, for instance, the absolute coefficients of expansion of air,

water, and mercury, the temperature of the maximum density of water (39°101 Fahr. or 4°0 Cent.), the latent heats of water and steam, the boiling-point of water under standard pressure, the melting and boiling-points of mercury, and so on.

Astronomical Constants.

The third great class consists of numbers possessing far less generality because they refer, not to the universal properties of matter, but to the special forms and distances in which matter has been disposed in the part of the universe open to our examination. We have, first of all, to define the magnitude and form of the earth, its mean density, the constant of aberration of light expressing the relation between the earth's mean velocity in space and the velocity of light. From the earth, as our observatory, we then proceed to lay down the mean distances of the sun, and of the planets from the same centre; all the elements of the planetary orbits, the magnitudes, densities, masses, periods of axial rotation of the several planets are by degrees determined with growing accuracy. The same labours must be gone through for the satellites. Catalogues of comets with the elements of their orbits, as far as ascertainable, must not be omitted.

From the earth's orbit as a new base of observations, we next proceed to survey the heavens and lay down the apparent positions, magnitudes, motions, distances, periods of variation, &c. of the stars. All catalogues of stars from those of Hipparchus and Tycho, are full of numbers expressing rudely the conformation of the visible universe. But there is obviously no limit to the labours of astronomers; not only are millions of distant stars awaiting their first measurements, but those already registered require endless scrutiny as regards their movements in the three

dimensions of space, their periods of revolution, their changes of brilliancy and colours. It is obvious that though astronomical numbers are conventionally called constant, they are in all cases probably subject to more or less rapid variation.

Terrestrial Numbers.

Our knowledge of the globe we inhabit involves many numerical determinations, which have little or no connexion with astronomical theory. The extreme heights of the principal mountains, the mean elevation of continents, the mean or extreme depths of the oceans, the specific gravities of rocks, the temperature of mines, all the host of numbers expressing the meteorological or magnetic conditions of every part of the surface must fall into this class. Many of such numbers are hardly to be called constant, being subject to periodic or even secular changes, but they are no more variable in fact than many which in astronomical science are set down as constant. In many cases quantities which seem most variable may go through rhythmical changes resulting in a nearly uniform average, and it is only in the long progress of physical investigation that we can hope to discriminate successfully between those elemental numbers which are absolutely fixed and those which vary. In the latter case the law of variation becomes the constant relation which is the object of our search.

Organic Numbers.

All the forms and properties of brute nature having been sufficiently defined by the previous classes of numbers, the organic world, both vegetable and animal, remains outstanding, and offers a higher series of phenomena for

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our investigation. All exact knowledge relating to the forms and sizes of living things, their numbers, the quantities of various compounds which they consume, contain, or excrete, their muscular or nervous energy, &c. must be placed apart in a class by themselves. All such numbers are doubtless more or less subject to variation, and but in a minor degree capable of exact determination. Man, so far as he is an animal, and as regards his physical form, must also be treated in this class.

Social Numbers.

Little or no allusion has hitherto been made in this work to the fact that man in his economical, sanitary, intellectual, æsthetic, or moral relations may become the subject of exact sciences, the highest and most useful of all sciences. Every one who is in any degree engaged in statistical inquiry or study must so far acknowledge the possibility of natural laws governing such statistical facts. Hence we must certainly allot a distinct place to all numerical information relating to the numbers, ages, physical and sanitary condition, mortality, of all different peoples, in short, to vital statistics. Economic statistics, comprehending the quantities of commodities produced, existing, exchanged, and consumed, constitute another most extensive body of science. In the progress of reason exact investigation may possibly subdue regions of phenomena which at present defy all analysis and scientific treatment. That scientific method can ever exhaust the phenomena of the human mind is on the other hand incredible.