CHAPTER XII. THE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY OF PROBABILITY.

The probability that 43 motions independent of each other would coincide by chance is the 42nd power of 1/2, 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 observes,‍¹⁴ 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 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 1 ÷ (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 1/10! (·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.‍¹⁵

If the same kind of calculation be applied to the orbits of comets, the result is very different.‍¹⁶ Of the orbits which have been determined 48·9 per cent. only are direct or in the same direction as the planetary motions.‍¹⁷ 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 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