ing 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 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 mm., and that each line in the iron spectrum has 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. 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 ()60 or less than I in a trillion. The odds, in short, are more than a million million millions to unity against such casual coincidence. 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 1 Kirchhoff's Researches on the Solar Spectrum. First part, translated by Roscoe, pp. 18, 19. 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; the probability that in another crystal also the direction will be the same is, and so on. The probability that in n I crystals there would be casual agreement of direction is the nth power of 4. 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 1.1 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 1 Edinburgh Review, No. 185, vol. xcii. July 1850, p. 32; Herschel's Essays, p. 421; Transactions of the Cambridge Philosophical Society, vol. i. p. 43. 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 knifelike 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.1 The Theory of Probability in Astronomy. The science of astronomy, occupied with the simple relations of distance, magnitude, and motion of the heavenly bodies, admits more easily than almost any other science of interesting conclusions founded on the theory of probability. More than a century ago, in 1767, Michell showed the extreme probability of bonds connecting together systems of stars. He was struck by the unexpected number of fixed stars which have companions close to them. Such a conjunction might happen casually by one star, although possibly at a great distance from the other, happening to lie on a straight line passing near the earth. But the probabilities are so greatly against such an optical union happening often in the expanse of the heavens, that Michell asserted the existence of some connection between most of the double stars. It has since been estimated by Struve, that the odds are 9570 to 1 against any two stars of not less than the seventh magnitude falling within the apparent distance of four seconds of each other by chance, and yet ninety-one such cases were known when the estimation was made, and many more cases have since been discovered. There were also four known triple stars, and yet the odds against the appearance of any one such conjunction are 173,524 to 1.2 The conclusions of Michell have been Evans' Ancient Stone Implements of Great Britain. London, 1872 (Longmans). 2 Herschel, Outlines of Astronomy, 1849, p. 565; but Todhunter, in his History of the Theory of Probability, p. 335, states that the calculations do not agree with those published by Struve. entirely verified by the discovery that many double stars are connected by gravitation. 1 Michell also investigated the probability that the six brightest stars in the Pleiades should have come by accidents into such striking proximity. Estimating the number of stars of equal or greater brightness at 1500, he found the odds to be nearly 500,000 to I against casual conjunction. Extending the same kind of argument to other clusters, such as that of Præsepe, the nebula in the hilt of Perseus' sword, he says: "We may with the highest probability conclude, the odds against the contrary opinion being many million millions to one, that the stars are really collected together in clusters in some places, where they form a kind of system, while in others there are either few or none of them, to whatever cause this may be owing, whether to their mutual gravitation, or to some other law or appointment of the Creator." The calculations of Michell have been called in question by the late James D. Forbes,2 and Mr. Todhunter vaguely countenances his objections,3 otherwise I should not have thought them of much weight. Certainly Laplace accepts Michell's views, and if Michell be in error it is in the methods of calculation, not in the general validity of his reasoning and 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.5 The odds are very greatly against any numerous group of stars moving together in any one direction by chance. On like grounds, there can be no doubt that the sun has a considerable proper motion 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 thousands of stars accidentally agree in their direction of motion, or are 1 Philosophical Transactions, 1767, vol. lvii. p. 431. 2 Philosophical Magazine, 3rd Series, vol. xxxvii. p. 401, December 1850; also August 1849. 3 History, &c., p. 334. 4 Essai Philosophique, p. 57. Proceedings of the Royal Society; 20 January, 1870; Philosophical Magazine, 4th Series, vol. xxxix. p. 381. 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 mainly 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. In contemplating the planetary system, we are struck with the similarity in direction of nearly all its movements. Newton remarked upon the regularity and uniformity of these motions, and contrasted them with the eccentricity and irregularity of the cometary orbits. 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 satellites rotating in the same direction, with some exceptions on the verge of the system. 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 II 18 14 43 The probability that 43 motions independent of each other would coincide by chance is the 42nd power of, so that the odds are about 4,400,000,000,000 to I in favour of some common cause for the uniformity of direction. This probability, as Laplace observes, 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 1 Principia, bk. ii. General scholium. 2 Essai Philosophique, p. 55. Laplace appears to count the rings of Saturn as giving two independent movements. |