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 sig nificance of which has already been pointed out (p. 182). 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, or e, being equal to the infinite series I++ + +

I

I

I

1.2 1.2.3 1.2.3.4 and thus consisting of the sum of the ratios between the numbers of permutations and combinations of 0, 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 connection with trigonometry, or the measurement of the sides of a right-angled triangle, yet the numbers themselves arise out of numerical relations bearing no special relation to space. Foremost among trigonometrical constants is the well known number T, usually employed as expressing the ratio of the circumference and the diameter of a circle; from 7 follows the value of the arcual or natural unit of angular value as expressed in ordinary degrees (p. 306).

Among other mathematical constants not uncommonly used may be mentioned tables of factorials (p. 179), tables of Bernouilli's numbers, tables of the error function,1 which latter are indispensable not only in the theory of probability but also in several other branches of science.

1 J. W. L. Glaisher, Philosophical Magazine, 4th Series, vol. xlii.

p. 421.

It should 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 available to reward the severe, long continued, and generally thankless labour which must be gone through in calculating tables. Such labours are a benefit to the whole human race as long as it shall exist, though there are few who can appreciate the extent of this benefit. A most interesting and excellent description of many mathematical tables will be found in De Morgan's article on Tables, in the English Cyclopædia, Division of Arts and Sciences, vol. vii. p. 976. An almost exhaustive critical catalogue of extant tables is being published by a Committee of the British Association, two portions, drawn up chiefly by Mr. J. W. L. Glaisher and Professor Cayley, having appeared in the Reports of the Association for 1873 and 1875.

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 subclass 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, the latent heats of water and steam, the boiling-point of water under standard pressure, the melting and boiling-points of mercury, and so forth.

Astronomical Constants.

The third great class consists of numbers possessing far less generality because they refer not to the 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 brilliance and colour. It is obvious that though astronomical numbers are conventionally called constant, they are probably in all cases 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 connection with astronomical theory. The extreme heights of the principal mountains, the mean elevations of continents, the mean or extreme depths of the oceans, the specific gravities of rocks, the temperature of mines, the host of numbers expressing the meteorological or magnetic conditions of every part of the surface, must fall into this class. Many such numbers are not to be called constant, being subject to periodic or secular changes, but they are hardly more variable in fact than some 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 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.

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 Gutstanding, and offers a higher series of phenomena for 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 allusion need be made in this work to the fact that man in his economic, sanitary, intellectual, æsthetic, or moral relations may become the subject of sciences, the highest and most useful of all sciences. Every one who is engaged in statistical inquiry must acknowledge the possibility of natural laws governing such statistical facts. Hence we must allot a distinct place to numerical information relating to the numbers, ages, physical and sanitary condition, mortality, &c., of different peoples, in short, to vital statistics. Economic statistics, comprehending the quantities of commodities produced, existing, exchanged and consumed, constitute another extensive body of science. In the progress of time exact investigation may possibly subdue regions of phenomena which at present defy all scientific treatment. That scientific method can ever exhaust the phenomena of the human mind is incredible.