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. [154] *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. 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] [155] Evans’ *Ancient Stone Implements of Great Britain*. London, 1872 (Longmans). *The Theory of Probability in Astronomy.*