COMBINATIONS AND PERMUTATIONS. 197 



one for the case in which all the letters would be 

 absent. But the formation of our vocal organs prevents 

 our using the far greater part of these conjunctions of 

 letters. At least one vowel must be present in each word ; 

 more than two consonants cannot usually be brought to- 

 gether ; and to produce words capable of smooth utterance 

 a number of other rules must be observed. To determine 

 exactly how many words might exist in the English 

 language under these circumstances, would be - an exceed- 

 ingly 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 combinations presented 

 in the dictionary, that we can learn the Laws of Euphony 

 or calculate the possible number. 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 ex- 

 amination of nature. 



It is often a very difficult matter to determine the 

 numbers 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 Latin 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 a : 



( 'Tot tibi sunt dotes, Virgo, quot sidera coelo.' 



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. Dr. Wallis assigned 3096, but without 



a Montucla, ' Histoire,' &c., vol. iii. p. 388. 



