678 SCIENTIFIC THOUGHT. 



eluding arrangement. The same difference of views can 

 be established with regard to many other things which 

 form the objects of other sciences. In geometry this 

 difference obtrudes itself, as it were, in its naked form. 

 Thus in all the natural, and even the social, sciences we 

 have become accustomed to look first at the constituent 

 elements or parts of things, to count and measure them, 

 then afterwards to look at their possible arrangement, 

 or existence together in the actual world of nature or 

 society. Astronomy, crystallography, chemistry, geology, 

 the natural history sciences, economics and statistics, the 

 doctrine of chances, all furnish, especially in their sys- 

 tematic development during the last hundred or hundred 

 and fifty years, examples of the twofold aspect just re- 

 ferred to. The progress of these sciences, as we have 

 abundantly seen, has depended largely upon the application 

 of mathematical methods. As the analysis into elements 

 or parts, and the possible synthesis of such elements in 

 complicated structures, has become everywhere the order of 

 study, so there must exist in the abstract science of mathe- 

 matics i.e., in the framework of our scientific reasoning 

 not only the theory of measurement and number, but 

 also that of combination, form or arrangement, and order. 

 37. The doctrine of forms in the well-known prob- 



Theory of ' . 



forms. lems of permutations and combinations begins with 

 modern mathematics in the seventeenth century, and 

 received scientific recognition mainly in connection 

 with the doctrine of chances at the hands of James 

 Bernoulli abroad, and of De Moivre in this country. 

 The process of multiplication of binomials and poly- 

 nomials leads to the formation of combinations, and 



