ix.] COMBINATIONS AND PERMUTATIONS. 175 



observed. To determine exactly how many words might 

 exist in the English language under these circumstances, 

 would be an exceedingly complex problem, the solution of 

 which has never been attempted. The number of existing 

 English words may perhaps be said not to exceed one 

 hundred thousand, and it is only by investigating the com- 

 binations presented in the dictionary, that we can learn the 

 Laws of Euphony or calculate the possible number of 

 words. In this example we have an epitome of the work 

 and method of science. The combinations of natural 

 phenomena are limited by a great number of conditions 

 which are in no way brought to our knowledge except so 

 far as they are disclosed in the examination of nature. 



It is often a very difficult matter to determine the num- 

 bers of permutations or combinations which may exist 

 under various restrictions. Many learned men puzzled 

 themselves in former centuries over what were called 

 Protean verses, or verses admitting many variations in 

 accordance with the Laws of Metre. The most celebrated 

 of these verses was that invented by Bernard Bauhusius, 

 as follows : l 



" Tot tibi auat dotes, Virgo, quot sidera ccelo." 



One author, Ericius Puteanus, filled forty-eight pages of a 

 work in reckoning up its possible transpositions, making 

 them only 1022. Other calculators gave 2196, 3276, 2580 

 as their results. Wallis assigned 3096, but without much 

 confidence in the accuracy of his result. 2 It required the 

 skill of James Bernoulli to decide that the number of 

 transpositions was 3312, under the condition that the sense 

 and metre of the verse shall be perfectly preserved. 



In approaching the consideration of the great Inductive 

 problem, it is very necessary that we should acquire correct 

 notions as to the comparative numbers of combinations 

 which may exist under different circumstances. The 

 doctrine of combinations is that part of mathematical 

 science which applies numerical calculation to determine 

 the numbers of combinations under various conditions. 

 It is a part of the science which really lies at the base not 

 only of other sciences, but of other branches of mathe- 



i Montucla, Histoire, &c., vol. iii. p. 388. 

 9 Wallis, Of Combinations, &c., p. 119. 



