satellites, and it is only slightly weakened by the fact that some of the outlying satellites are exceptional in direction, there being considerable evidence of an accidental disturbance in the more distant parts of the system. Hardly less remarkable than the uniform direction of motion is the near approximation of the orbits of the planets to a common plane. Daniel Bernoulli roughly estimated the probability of such an agreement arising from accident as I ÷ (12)6 the greatest inclination of any orbit to the sun's equator being 1-12th part of a quadrant. Laplace devoted to this subject some of his most ingenious investigations. He found the probability that the sum of the inclinations of the planetary orbits would not exceed by accident the actual amount (914187 of a right angle' for the ten planets known in 1801) to be (914187),10 or about 00000011235. This probability may be combined with that derived from the direction of motion, and it then becomes immensely probable that the constitution of the planetary system arose out of uniform conditions, or, as we say, from some common cause.1 10 3 If the same kind of calculation be applied to the orbits of comets, the result is very different. Of the orbits which have been determined 489 per cent. only are direct or in the same direction as the planetary motions. Hence it becomes apparent that comets do not properly belong to the solar system, and it is probable that they are stray portions of nebulous matter which have accidentally become attached to the system by the attractive powers of the sun or Jupiter. The General Inverse Problem. In the instances described in the preceding sections, we have been occupied in receding from the occurrence of certain similar events to the probability that there Lubbock, Essay on Probability, p. 14. De Morgan, Encyc. Metrop. art. Probability, p. 412. Todhunter's History of the Theory of Probability, p. 543. Concerning the objections raised to these conclusions by Boole, see the Philosophical Magazine, 4th Series, vol. ii. p. 98. Boole's Laws of Thought, pp. 364-375. 2 Laplace, Essai Philosophique, pp. 55, 56. We must have been a condition or cause for such events. have found that the theory of probability, although never yielding a certain result, often enables us to establish an hypothesis beyond the reach of reasonable doubt. There is, however, another method of applying the theory, which possesses for us even greater interest, because it illustrates, in the most complete manner, the theory of inference adopted in this work, which theory indeed it suggested. The problem to be solved is as follows: An event having happened a certain number of times, and failed a certain number of times, required the probability that it will happen any given number of times in the future under the same circumstances. All the larger planets hitherto discovered move in one direction round the sun; what is the probability that, if a new planet exterior to Neptune be discovered, it will move in the same direction? All known permanent gases, except chlorine, are colourless; what is the probability that, if some new permanent gas should be discovered, it will be colourless? In the general solution of this problem, we wish to infer the future happening of any event from the number of times that it has already been observed to happen. Now, it is very instructive to find that there is no known process by which we can pass directly from the data to the conclusion. It is always requisite to recede from the data to the probability of some hypothesis, and to make that hypothesis the ground of our inference concerning future events. Mathematicians, in fact, make every hypothesis which is applicable to the question in hand; they then calculate, by the inverse method, the probability of every such hypothesis according to the data, and the probability that if each hypothesis be true, the required future event will happen. The total probability that the event will happen is the sum of the separate probabilities contributed by each distinct hypothesis. To illustrate more precisely the method of solving the problem, it is desirable to adopt some concrete mode of representation, and the ballot-box, so often employed by mathematicians, will best serve our purpose. Let the happening of any event be represented by the drawing of a white ball from a ballot-box, while the failure of an event is represented by the drawing of a black ball. Now, in the inductive problem we are supposed to be ignorant of the contents of the ballot-box, and are required to ground all our inferences on our experience of those contents as shown in successive drawings. Rude common sense would guide us nearly to a true conclusion. Thus, if we had drawn twenty balls one after another, replacing the ball after each drawing, and the ball had in each case proved to be white, we should believe that there was a considerable preponderance of white balls in the urn, and a probability in favour of drawing a white ball on the next occasion. Though we had drawn white balls for thousands of times without fail, it would still be possible that some black balls lurked in the urn and would at last appear, so that our inferences could never be certain. On the other hand, if black balls came at intervals, we should expect that after a certain number of trials the black balls would appear again from time to time with somewhat the same frequency. The mathematical solution of the question consists in little more than a close analysis of the mode in which our common sense proceeds. If twenty white balls have been drawn and no black ball, my common sense tells me that any hypothesis which makes the black balls in the urn considerable compared with the white ones is improbable; a preponderance of white balls is a more probable hypothesis, and as a deduction from this more probable hypothesis, I expect a recurrence of white balls. The mathematician merely reduces this process of thought to exact numbers. Taking, for instance, the hypothesis that there are 99 white and one black ball in the urn, he can calculate the probability that 20 white balls would be drawn in succession in those circumstances; he thus forms a definite estimate of the probability of this hypothesis, and knowing at the same time the probability of a white ball reappearing if such be the contents of the urn, he conbines these probabilities, and obtains an exact estimate that a white ball will recur in consequence of this hypothesis. But as this hypothesis is only one out of many possible ones, since the ratio of white and black balls may be 98 to 2, or 97 to 3, or 96 to 4, and so on, he has to repeat the estimate for every such possible hypothesis. To make the method of solving the problem perfectly evident, I will describe in the next section a very simple case of the problem, originally devised for the purpose by Condorcet, which was also adopted by Lacroix,1 and has passed into the works of De Morgan, Lubbock, and others. Simple Illustration of the Inverse Problem. Suppose it to be known that a ballot-box contains only. four black or white balls, the ratio of black and white balls being unknown. Four drawings having been made with replacement, and a white ball having appeared on each occasion but one, it is required to determine the probability that a white ball will appear next time. Now the hypotheses which can be made as to the contents of the urn are very limited in number, and are at most the following five 4 white and o black balls دو 4 The actual occurrence of black and white balls in the drawings puts the first and last hypothesis out of the question, so that we have only three left to consider. If the box contains three white and one black, the probability of drawing a white each time is 3, and a black ; so that the compound event observed, namely, three white and one black, has the probability × × × 1, by the rule already given (p. 204). But as it is indifferent in what order the balls are drawn, and the black ball might come first, second, third, or fourth, we must multiply by four, to obtain the probability of three white and one black in any order, thus getting 27. Taking the next hypothesis of two white and two black balls in the urn, we obtain for the same probability the quantity × × × × 4, or 16, and from the third hypothesis of one white and three black we deduce likewise ×××× 4, or According, then, as we 3 1 Traité élémentaire du Calcul des Probabilités, 3rd ed. (1833), p. 148. 16 64 adopt the first, second, or third hypothesis, the probability that the result actually noticed would follow is 7, 3 , and Now it is certain that one or other of these hypotheses must be the true one, and their absolute probabilities are proportional to the probabilities that the observed events would follow from them (pp. 242, 243). All we have to do, then, in order to obtain the absolute probability of each hypothesis, is to alter these fractions in a uniform ratio, so that their sum shall be unity, the expression of certainty. Now, since 27+ 16 + 3 = 46, this will be effected by dividing each fraction by 46, and multiplying by 64. Thus the probabilities of the first, second, and third hypotheses are respectively The inductive part of the problem is completed, since we have found that the urn most likely contains three white and one black ball, and have assigned the exact probability of each possible supposition. But we are now in a position to resume deductive reasoning, and infer the probability that the next drawing will yield, say a white ball. For if the box contains three white and one black ball, the probability of drawing a white one is certainly; and as the probability of the box being so constituted is 2, the compound probability that the box will be so filled and will give a white ball at the next trial, is Again, the probability is 1 that the box contains two white and two black, and under those conditions the probability is that a white ball will appear; hence the probability that a white ball will appear in consequence of that condition, is From the third supposition we get in like manner the probability Since one and not more than one hypothesis can be true, |