XXIV.] EMPirJCAL KNOWLEDGE, EXPLANATION, &c. 529 



studied for two centuries. I shall subsequently point ovit 

 that even the establishment of a wide and true law of 

 nature is but the starting-point for the discovery of ex- 

 ceptions 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 -I- l 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 tlie value of tt, when expressed to a great 

 number of figures, contain the digit 7 much less frequently 

 than any other digit?' Even geometry may allow of 

 empirical truths, when the matter does not involve 

 quantities of space, but numerical results and the ])0sitive 

 or negative character of quantities, as in De Morgan's 

 iheorem concerning negative areas. 



Accidental Discovery. 



There are not a few cases where almost pure accident 

 has determined the moment when a new branch of know- 

 ledge was to be created. The laws of the structure of crystals 

 were not discovered until liatiy happened to drop a 

 beautiful crystal of calc-spar upon a stone pavement. His 

 momentary regret at destroying a choice s))ecimen was 

 quickly removed wlieu, 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, 



• De Morgan's Budget of Paradoxes, p. 291. 



M M 



