figures required in writing it down, without using about 19,729 figures for the purpose.

The successive orders of the powers of two have then the following values:

It may give us a powerful notion of infinity to remember that at this sixth step, having long surpassed all bounds of conception, we have made no approach to the goal. Nay, were we to make a hundred such steps, we should be as far away as ever from actual infinity.

It is well worth observing that our powers of expression rapidly overcome the possible multitude of finite objects which may exist in any assignable space. Archimedes showed long ago, in one of the most wonderful writings of antiquity, that the grains of sand in the world could be numbered, or rather, that if numbered, the result could readily be expressed in arithmetical notation. Let us extend his problem, and ascertain whether we could express the number of atoms which could exist in the visible universe. The most distant stars which can now be seen by telescopes -those of the sixteenth magnitude-are supposed to have a distance of about 33,900,000,000,000,000 miles." Sir W. Thomson, again, has shown reasons for supposing

that there do not exist more than from 3×1024 to 1026 molecules in a cubic centimetre of a solid or liquid substance. Assuming these data to be true, for the sake of argument, a simple calculation enables us to show that the almost inconceivably vast sphere of our stellar system if entirely filled with solid matter, would not contain more than about 68 x 100 atoms, that is to say, a number requiring for its expression 92 places of figures. Now, this number would be immensely less than the fifth order of the powers of two.

In the variety of logical relations, which may exist between a certain number of logical terms, we also meet a case of higher variations. Two terms, as it has been shewn (p. 154), may form four distinct combinations, but the possible selections from these series of combinations will be sixteen in number, or, excluding cases of contradiction, seven. Three terms may form eight combinations, allowing 256 selections, or with exclusion of contradictory cases, 193. Four terms give sixteen combinations, and no less than 65,536 possible selections from those combinations, the nature of which I naturally abstained from exhaustively examining. Five terms give thirty-two combinations, and 4,294,967,296 possible selections; and for six terms the corresponding numbers are sixty-four and 18,446,744,073,709,551,616. Considering that it is the most common thing in the world to use an argument involving six objects or terms, it may excite some surprise that the complete investigation of the relations in which six such terms may stand to each other, should involve an almost inconceivable number of cases. Yet these numbers of possible logical relations belong only to the second order of combinations.

x 'Nature,' vol. i. p. 553.

CHAPTER X.

THEORY OF PROBABILITY.

THE subject upon which we now enter must not be regarded as an isolated and curious branch of speculation. It is the necessary basis of nearly all the judgments and decisions we make in the prosecution of science, or the conduct of ordinary affairs. As Butler truly said, 'Probability is the very guide of life.' Had the science of numbers been developed for no other purposes, it must have been developed for the calculation of probabilities. All our inferences concerning the future are merely probable, and a due appreciation of the degree of probability depends entirely upon a due comprehension of the principles of the subject. I conceive that it is impossible even to expound the principles and methods of induction as applied to natural phenomena, in a sound manner, without resting them upon the theory of probability. Perfect knowledge alone can give certainty, and in nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowledge-knowledge mingled with ignorance, producing doubt.

Almost the greatest difficulty in this subject consists in acquiring a precise notion of the matter treated. What is it that we number, and measure, and calculate in the theory of probabilities? Is it belief, or opinion, or doubt, or knowledge, or chance, or necessity, or want of art?

Does probability exist in the things which are probable, or in the mind which regards them as such? The etymology of the name lends us no assistance: for, curiously enough, probable is ultimately the same word as provable, a good instance of one word becoming differentiated to two opposite meanings.

Chance cannot be the subject of the theory, because there is really no such thing as chance,a regarded as producing and governing events. This name signifies falling, and the notion is continually used as a simile to express uncertainty, because we can seldom predict how a die, or a coin, or a leaf will fall, or when a bullet will hit the mark. But every one knows, on a little reflection, that it is in our knowledge the deficiency lies, not in the certainty of nature's laws. There is no doubt in lightning as to the point it shall strike; in the greatest storm there is nothing capricious; not a grain of sand lies upon the beach, but infinite knowledge would account for its lying there; and the course of every falling leaf is guided by the same principles of mechanics as rule the motions of the heavenly bodies.

Chance then exists not in nature, and cannot co-exist with knowledge; it is merely an expression for our ignorance of the causes in action, and our consequent inability to predict the result, or to bring it about infallibly. In nature the happening of a physical event has been pre-determined from the first fashioning of the universe. Probability belongs wholly to the mind; this indeed is proved by the fact that different minds may regard the very same event at the same time with totally different degrees of probability. A steam-vessel, for instance, is missing and some persons believe that she has sunk in mid-ocean; others think differently. In the

a Dufau, 'De la Méthode d'Observation,' chap. iii.

Q

6

event itself there can be no such uncertainty; the steamvessel either has sunk or has not sunk, and no subsequent discussion of the probable nature of the event can alter the fact. Yet the probability of the event will really vary from day to day, and from mind to mind, according as the slightest information is gained regarding the vessels met at sea, the weather prevailing there, the signs of wreck picked up, or the previous condition of the vessel. Probability thus belongs to our mental condition, to the light in which we regard events, the occurrence or nonoccurrence of which is certain in themselves. Many writers accordingly have asserted that probability is concerned with degree or quantity of belief. De Morgan says,b By degree of probability we really mean or ought to mean degree of belief.' The late Professor Donkin expressed the meaning of probability as 'quantity of belief;' but I have never felt satisfied with such a definition of probability. The nature of belief is not more clear to my mind than the notion it is used to define. But an all-sufficient objection is, that the theory does not measure what the belief is, but what it ought to be. Few minds think in close accordance with the theory, and there are many cases of evidence in which the belief existing is habitually different from what it ought to be. Even if the state of belief in any mind could be measured and expressed in figures, the results would be worthless. The very value of the theory consists in correcting and guiding our belief, and rendering our states of mind and consequent actions harmonious with our knowledge of exterior conditions.

This objection has been clearly perceived by some of those who still used quantity of belief as a definition of probability. Thus De Morgan adds-'Belief is but another name for imperfect knowledge.' Professor Donkin has

b Formal Logic,' p. 172.