We may thus state the result in general language. If it is certain that one or other of the supposed causes exists, the probability that any one does exist is the probability that if it exists the event happens, divided by the sum of all the similar probabilities. There may seem to be an intricacy in this subject which may prove distasteful to some readers; but this intricacy is essential to the subject in hand. No one can possibly understand the principles of inductive reasoning, unless he will take the trouble to master the meaning of this rule, by which we recede from an event to the probability of each of its possible causes.
This rule or principle of the indirect method is that which common sense leads us to adopt almost instinctively, before we have any comprehension of the principle in its general form. It is easy to see, too, that it is the rule which will, out of a great multitude of cases, lead us most often to the truth, since the most probable cause of an event really means that cause which in the greatest number of cases produces the event. Donkin and Boole have given demonstrations of this principle, but the one most easy to comprehend is that of Poisson. He imagines each possible cause of an event to be represented by a distinct ballot-box, containing black and white balls, in such a ratio that the probability of a white ball being drawn is equal to that of the event happening. He further supposes that each box, as is possible, contains the same total number of balls, black and white; then, mixing all the contents of the boxes together, he shows that if a white ball be drawn from the aggregate ballot-box thus formed, the probability that it proceeded from any particular ballot-box is represented by the number of white balls in that particular box, divided by the total number of white balls in all the boxes. This result corresponds to that given by the principle in question.[152]
Thus, if there be three boxes, each containing ten balls in all, and respectively containing seven, four, and three white balls, then on mixing all the balls together we have fourteen white ones; and if we draw a white ball, that is if the event happens, the probability that it came out of the first box is 7/14; which is exactly equal to 7/10/7/10 + 4/10 + 3/10, the fraction given by the rule of the Inverse Method.
Simple Applications of the Inverse Method.
In many cases of scientific induction we may apply the principle of the inverse method in a simple manner. If only two, or at the most a few hypotheses, may be made as to the origin of certain phenomena, we may sometimes easily calculate the respective probabilities. It was thus that Bunsen and Kirchhoff established, with a probability little short of certainty, that iron exists in the sun. On comparing the spectra of sunlight and of the light proceeding from the incandescent vapour of iron, it became apparent that at least sixty bright lines in the spectrum of iron coincided with dark lines in the sun’s spectrum. Such coincidences could never be observed with certainty, because, even if the lines only closely approached, the instrumental imperfections of the spectroscope would make them apparently coincident, and if one line came within half a millimetre of another, on the map of the spectra, they could not be pronounced distinct. Now the average distance of the solar lines on Kirchhoff’s map is 2 mm., and if we throw down a line, as it were, by pure chance on such a map, the probability is about one-half that the new line will fall within 1/2 mm. on one side or the other of some one of the solar lines. To put it in another way, we may suppose that each solar line, either on account of its real breadth, or the defects of the instrument, possesses a breadth of 1/2 mm., and that each line in the iron spectrum has a like breadth. The probability then is just one-half that the centre of each iron line will come by chance within 1 mm. of the centre of a solar line, so as to appear to coincide with it. The probability of casual coincidence of each iron line with a solar line is in like manner 1/2. Coincidence in the case of each of the sixty iron lines is a very unlikely event if it arises casually, for it would have a probability of only (1/2)60 or less than 1 in a trillion. The odds, in short, are more than a million million millions to unity against such casual coincidence.[153] But on the other hypothesis, that iron exists in the sun, it is highly probable that such coincidences would be observed; it is immensely more probable that sixty coincidences would be observed if iron existed in the sun, than that they should arise from chance. Hence by our principle it is immensely probable that iron does exist in the sun.
All the other interesting results, given by the comparison of spectra, rest upon the same principle of probability. The almost complete coincidence between the spectra of solar, lunar, and planetary light renders it practically certain that the light is all of solar origin, and is reflected from the surfaces of the moon and planets, suffering only slight alteration from the atmospheres of some of the planets. A fresh confirmation of the truth of the Copernican theory is thus furnished.
Herschel proved in this way the connection between the direction of the oblique faces of quartz crystals, and the direction in which the same crystals rotate the plane of polarisation of light. For if it is found in a second crystal that the relation is the same as in the first, the probability of this happening by chance is 1/2; the probability that in another crystal also the direction will be the same is 1/4, and so on. The probability that in n + 1 crystals there would be casual agreement of direction is the nth power of 1/2. Thus, if in examining fourteen crystals the same relation of the two phenomena is discovered in each, the odds that it proceeds from uniform conditions are more than 8000 to 1.[154] Since the first observations on this subject were made in 1820, no exceptions have been observed, so that the probability of invariable connection is incalculably great.
It is exceedingly probable that the ancient Egyptians had exactly recorded the eclipses occurring during long periods of time, for Diogenes Laertius mentions that 373 solar and 832 lunar eclipses had been observed, and the ratio between these numbers exactly expresses that which would hold true of the eclipses of any long period, of say 1200 or 1300 years, as estimated on astronomical grounds. It is evident that an agreement between small numbers, or customary numbers, such as seven, one hundred, a myriad, &c., is much more likely to happen from chance, and therefore gives much less presumption of dependence. If two ancient writers spoke of the sacrifice of oxen, they would in all probability describe it as a hecatomb, and there would be nothing remarkable in the coincidence. But it is impossible to point out any special reason why an old writer should select such numbers as 373 and 832, unless they had been the results of observation.
On similar grounds, we must inevitably believe in the human origin of the flint flakes so copiously discovered of late years. For though the accidental stroke of one stone against another may often produce flakes, such as are occasionally found on the sea-shore, yet when several flakes are found in close company, and each one bears evidence, not of a single blow only, but of several successive blows, all conducing to form a symmetrical knife-like form, the probability of a natural and accidental origin becomes incredibly small, and the contrary supposition, that they are the work of intelligent beings, approximately certain.[155]