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countenances his objections °, otherwise I should not have thought them of much weight. Certainly Laplace accepts Michell's views P, and if Michell be in error, it is in the methods of calculation, not in the general validity of his conclusions.

Similar calculations might no doubt be applied to the peculiar drifting motions which have been detected by Mr. R. A. Proctor in some of the constellations 9. Against a general tendency of stars to move in one direction by chance, the odds are very great. It is on a similar ground that a considerable proper motion of the sun is found to exist with immense probability, because on the average the fixed stars show a tendency to move apparently from one point of the heavens towards that diametrically opposite. The sun's motion in the contrary direction would explain this tendency, otherwise we must believe that myriads of stars accidentally agree in their direction of motion, or are urged by some common force from which the sun is exempt. It may be said that the rotation of the earth is proved in like manner, because it is immensely more probable that one body would revolve than that the sun, moon, planets, comets, and the whole of the stars of the heavens should be whirled round the earth daily, with a uniform motion superadded to their own peculiar motions. This appears to be nearly the reason which led Gilbert, one of the earliest English Copernicans, and in every way an admirable physicist, to admit the rotation of the earth, while Francis Bacon denied it г.

In contemplating the planetary system, we are struck with the similarity in direction of nearly all its move

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4 'Proceedings of the Royal Society,' 20 January, 1870. 'Philosophical Magazine,' 4th Series, vol. xxxix. p. 381.

r Hallam's Literature of Europe,' 1st ed. vol. ii. p. 464.

ments. Newton remarked upon the regularity and uniformity of these motions, and contrasted them with the eccentricity and irregularity of the cometary orbits 8. Could we, in fact, look down upon the system from the northern side, we should see all the planets moving round from west to east, the satellites moving round their primaries and the sun, planets, and all the satellites rotating in the same direction, with some exceptions on the verge of the system. Now in the time of Laplace eleven planets were known, and the directions of rotation were known for the sun, six planets, the satellites of Jupiter, Saturn's ring, and one of his satellites. Thus there were altogether 43 motions all concurring, namely:

Orbital motions of eleven planets
Orbital motions of eighteen satellites.
Axial rotations .

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The probability that 43 motions independent of each other would coincide by chance is the 42nd power of 12, so that the odds are about 4,400,000,000,000 to 1 in favour of some common cause for the uniformity of direction. This probability, as Laplace observest, is higher than that of many historical events which we undoubtingly believe. In the present day, the probability is much increased by the discovery of additional planets, and the rotation of other 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 uniformity of motion s' Principia,' bk. ii. General scholium.

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Essai Philosophique,' p. 55. Laplace appears to count the rings of Saturn as giving two independent movements.

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is the near approximation of all the orbits of the planets to a common plane. Daniel Bernouilli roughly estimated the probability of such an agreement arising from accident ( 1 2 )a, 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), 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“.

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If the same kind of calculation be applied to the orbits of comets the result is very differenty. Of the orbits which have been determined 48'9 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 become accidently attached to the system by the attractive powers of the sun or Jupiter.

Statement of the General Inverse Problem.

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In the instances described in the preceding sections, we have been occupied in receding from the occurrence u 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 the late Dr. Boole, see the 'Philosophical Magazine,' 4th Series, vol. ii. p. 98. Boole's Laws of Thought,' pp. 364-375.

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of certain similar events to the probability that there must have been a condition or cause for such events. We 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 happenings. 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

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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, I should expect that after a certain number of trials the future results would agree more or less closely with the past

ones.

The mathematical solution of the question consists in nothing 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

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