162 THE PRINCIPLES OF SCIENCE. 



planation, although they have been studied for two cen- 

 turies or more. 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 or slight 

 divergences giving a wide 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 

 principles of substitution. Yet some of the facts esta- 

 blished in the investigation of the inverse logical problem 

 (vol. i. p. 157) may be considered empirical. Mathematical 

 science often yields empirical truths. Why, for instance, 

 should the value of TT, when expressed to a great number 

 of figures, contain the digit 7 much less frequently than 

 any other digits ? 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 con- 

 cerning negative areas. 



Accidental Discovery. 



There are not a few cases where almost pure accident 

 has undoubtedly determined the moment when a new 

 branch of knowledge was to be created. The true laws 

 of the construction of crystals were not discovered until 

 Hauy happened to drop a beautiful crystal of calc-spar 

 upon a stone pavement. His momentary regret, at de- 

 stroying a choice specimen 4 , 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 



s De Morgan's 'Budget of Paradoxes/ p. 291. 



