34 ^yALLIS. 



47. A discourse of coinhinations, alternations, and aliquot 

 parts is attached to the English edition of Wallis's Algebra pub- 

 lished in 1685. In the Latin edition of the Algebra, published in 

 1693, this j^art of the work occupies pages 485 — 529. 



In referring to Wallis's Algebra we shall give the pages of the 

 Latin edition ; but in quoting from him we shall adopt his own 

 English version. The English version was reprinted by Maseres in 

 a volume of reprints which was published at London in 1795 under 

 the title of The Doctrine of Permutations and Gomhinations, being 

 an essential and fundamental part of the Doctrine of Chances. 



48. "Wallis's first Chapter is Of the variety of Elections, or 

 Choise, in taking or leaving One or more, out of a certain Num- 

 her of things proposed. He draws up a Table which agrees 

 with Pascal's Arithmetical Triangle, and shews how it may be 

 used in finding the number of combinations of an assigned set 

 of things taken two, three, four, five,... at a time. Wallis does 

 not add any thing to what Pascal had given, to whom however 

 he does not refer ; and Wallis's clumsy parenthetical style con- 

 trasts very unfavourably with the clear bright stream of thought 

 and language which flowed from the genius of Pascal. The 

 chapter closes with an extract from the Arithmetic of Buckley 

 and an explanation of it ; to this we have aU'eady referred in 

 Art. 38. 



49. Wallis's second Chapter is Of Alternations, or the different 

 change of Order, in any Number of things ptroposed. Here he 

 gives some examples of what are now usually called permutations ; 

 thus if there are four letters a, h, c, d, the number of permutations 

 when they are taken all together is 4 x 3 x 2 x 1. Wallis accord- 

 ingly exhibits the 24 permutations of these four letters. He forms 

 the product of the first twenty-four natural numbers, which is the 

 number of the permutations of twenty-four things taken all toge- 

 ther. 



Wallis exhibits the 24 permutations of the letters in the word 

 Roma taken all together ; and then he subjoins, *' Of which (in 

 Latin) these seven are only useful; Roma, ramo,oram,mora, maro, 

 armo, amor. The other forms are useless ; as affording no (Latin) 

 word of known signification." 



