We must not indeed suppose that any science will ever entirely cease to be empirical. Multitudes of phenomena have been explained by the undulatory theory of light; but there yet remain many facts to be treated. The natural colours of bodies and the rays given off by them when heated, are unexplained, and yield few empirical coincidences. The theory of electricity is partially understood, but the conditions of the production of frictional electricity defy explanation, although they have been studied for two centuries. I shall subsequently point out that even the establishment of a wide and true law of nature is but the starting-point for the discovery of exceptions and divergences giving a new scope to empirical discovery.
There is probably no science, I have said, which is entirely free from empirical and unexplained facts. Logic approaches most nearly to this position, as it is merely a deductive development of the laws of thought and the principle of substitution. Yet some of the facts established in the investigation of the inverse logical problem may be considered empirical. That a proposition of the form A = BC ꖌ b c possesses the least number of distinct logical variations, and the greatest number of logical equivalents of the same form among propositions involving three classes (p. 141), is a case in point. So also is the fact discovered by Professor Clifford that in regard to statements involving four classes, there is only one example of two dissimilar statements having the same distances (p. 144). Mathematical science often yields empirical truths. Why, for instance, should the value of π, when expressed to a great number of figures, contain the digit 7 much less frequently than any other digit?7 Even geometry may allow of empirical truths, when the matter does not involve quantities of space, but numerical results and the positive or negative character of quantities, as in De Morgan’s theorem concerning negative areas.
Accidental Discovery.
There are not a few cases where almost pure accident has determined the moment when a new branch of knowledge was to be created. The laws of the structure of crystals were not discovered until Haüy happened to drop a beautiful crystal of calc-spar upon a stone pavement. His momentary regret at destroying a choice specimen was quickly removed when, in attempting to join the fragments together, he observed regular geometrical faces, which did not correspond with the external facets of the crystals. A great many more crystals were soon broken intentionally, to observe the planes of cleavage, and the discovery of the internal structure of crystalline substances was the result. Here we see how much more was due to the reasoning power of the philosopher, than to an accident which must often have happened to other persons.
In a similar manner, a fortuitous occurrence led Malus to discover the polarisation of light by reflection. The phenomena of double refraction had been long known, and when engaged in Paris in 1808, in investigating the character of light thus polarised, Malus chanced to look through a double refracting prism at the light of the setting sun, reflected from the windows of the Luxembourg Palace. In turning the prism round, he was surprised to find that the ordinary image disappeared at two opposite positions of the prism. He remarked that the reflected light behaved like light which had been polarised by passing through another prism. He was induced to test the character of light reflected under other circumstances, and it was eventually proved that polarisation is invariably connected with reflection. Some of the general laws of optics, previously unsuspected, were thus discovered by pure accident. In the history of electricity, accident has had a large part. For centuries some of the more common effects of magnetism and of frictional electricity had presented