the sum of the approximate numbers 34'70, 52.693, 80'1, is 1675 within less than 07. So far as I know Mr. Sandeman is the only mathematician who has traced out the rules of approximate arithmetic, and his directions are worthy of careful attention. Although the accuracy of measurement has so much advanced since the time of Leslie, it is not superfluous to repeat his protest against the unfairness of affecting by a display of decimal fractions a greater degree of accuracy than the nature of the case requires and admits d. I have known a scientific man to register the barometer to a second of time when the nearest quarter of an hour would have been amply sufficient. Chemists often publish results of analysis to the ten-thousandth or even the millionth part of the whole, when in all probability the processes employed cannot be depended on beyond the hundredth part. It is seldom desirable to give more than one place of figures of uncertain amount; but it must be allowed that a nice perception of the degree of accuracy possible and desirable is requisite to save misapprehension and needless computation on the one hand, and to secure all attainable exactness on the other hand.

e Sandeman, Pelicotetics,' p. 214.

d Leslie, 'Inquiry into the Nature of Heat,' p. 505.

CHAPTER XXII.

QUANTITATIVE INDUCTION.

LET it be observed that we have not yet formally considered any processes of reasoning which have for their object to disclose general laws of nature expressed in quantitative formulæ or equations. We have been inquiring into the modes by which a phenomenon may be measured, and, if it be a composite phenomenon, may be resolved, by the aid of several measurements, into its component parts. We have also considered the precautions to be taken in the performance of observations and experiments in order that we may know what phenomena we really do measure and record. In treating of the approximate character of all observations, we have partially entered upon the subject of Quantitative Induction proper, but we must remember that no number of facts and observations can by themselves constitute science or general knowledge. knowledge. Numerical facts, like other facts, are but the raw materials of knowledge, upon which our reasoning faculties must be exerted in order to draw forth the secret principles of nature. It is by an inverse process of reasoning that we can alone discover the mathematical laws to which varying quantities conform. By wellconducted experiments we gain a series of values of a variable, and a corresponding series of values of a variant, and we now want to know what mathematical function the variant is as regards the variable. In the usual progress of a science three questions will have to be answered as regards every important quantitative phenomenon :

(1) Is there any constant relation between the variable and variant?

(2) What is the empirical formula expressing this relation?

(3) What is the rational formula expressing the law of nature involved?

Probable Connexion of Varying Quantities.

We find it stated in Mr. Mill's System of Logica that 'Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation.' This assertion may be considered true when it is interpreted with sufficient caution; but it might otherwise lead us into great errors. There is nothing whatever in the nature of things to prevent the existence of two variations which should apparently follow the same law, and yet have no connexion with each other. One binary star might be going through a revolution which, so far as we could tell, was of apparently equal period with that of another binary star, and according to the above rule the motion of one would be the cause of the motion of the other, which would not be really the case. Two astronomical clocks might conceivably be made so nearly perfect that, for several years, no difference could be detected, and we might then infer that the motion of one clock was the cause or effect of the motion of the other. This matter really requires the most careful discrimination. We must always bear in mind that the continuous quantities of space, time, force, &c., which we measure, are made up of an infinite number of infinitely small units. We may then meet with two variable phenomena which follow

a Book iii. chap. viii, § 6.

laws so nearly the same, that in no part of the variations. open to our observation can any discrepancy be discovered. I grant that if two clocks could be shown to have kept exactly the same time during one year, or any finite interval of time, the probability would become infinitely high that there was a connexion between their motions. But it is apparent that we can never absolutely prove such coincidences to exist. Allow that we may observe a difference of one tenth or one hundredth of a second in their time, yet it is just possible that they were independently regulated so as to go together within less than that quantity of time. In short it would require either an infinitely long time of observation, or infinitely acute powers of measuring a discrepancy to decide positively whether two clocks were or were not in relation with each other.

A similar question actually occurs in the case of the moon's motion. We have absolutely no record that any other portion of the moon was ever visible to men than such as we now see. This fact sufficiently proves that within the historical period the rotation of the moon on its own axis has coincided with its revolutions round the earth. Does this coincidence prove a relation of cause and effect to exist between these motions? The answer must be in the negative, because there might have been so slight a discrepancy between the motions that there has not yet been time to produce any appreciable effect. There may nevertheless be a high probability of con

nexion.

The whole question of the relation of quantities thus resolves itself into one of probability. When we can only rudely measure a quantitative result, we can assign but slight importance to any correspondence. Because the brightness of two stars seems to vary in the same manner there is no appreciable probability that they have any relation with each other. Could it be shown that

their periods of variation were the same even to infinitely small quantities it would be certain, that is infinitely probable, that they were connected, however unlikely this might be on other grounds. The general mode of estimating such probabilities is identical with that applied to other inductive problems. Thus, if the two periods of variation were assigned by pure chance and entirely independently of each other, the probability would be about one in ten million that they would agree to the one tenmillionth part; but if the periods be observed to agree to less than that part then there is a probability of at least ten million to one in favour of the opposite hypothesis of connexion. That any two periods of variation should by chance become absolutely equal is infinitely improbable ; hence if, in the case of the moon or any other change, we could prove absolute coincidence, we should have certainty of connexion b. With approximate measurements, which alone are within our power, we must hope for approximate certainty at the most.

The general principles of inference and probability, according to which we treat causes and effects varying in amount, are exactly the same as those by which we treated simple experiments. Continuous quantity, however, affords us an infinitely more extensive sphere of observation, because every different amount of cause, however little different, ought to be followed by a different amount of effect. If we can measure temperature to the one hundredth part of a degree centigrade, then even between o° and 100° we have 10,000 possible disting trials. If the precision of our measurements is

so that the one thousandth part of a degree reciated, our trials may be increased tenfold. ity of connexion will be proportional to the

our measurements.

System of the World,' transl. by Harte, vol. ii. p. 366.