WALLIS. 35 



Wallis then considers the case in which there is some repetition 

 among the quantities of which we require the permutations. He 

 takes the letters which compose the word Messes. Here if there 

 were no repetition of letters the number of permutations of the 

 letters taken all together would^ be 1x2x3x4x5x0, that is 

 720 ; but as Wallis explains, owing to the occurrence of the letter 

 e twice, and of the letter s thrice, the number 720 must be divided 

 by 2 X 2 X 3, that is by 12. Thus the number of permutations is 

 reduced to 60. Wallis exhibits these permutations and then sub- 

 joins, " Of all which varieties, there is none beside messes itself, 

 that affords an useful AnagTam." The chapter closes with Wallis's 

 attempt at determining the number of arrangements of the verse 



Tot tibi sunt dotes, virgo, quot sidera caelo. 



The attempt is followed by these w^ords, " I will not be posi- 

 tive, that there may not be some other Changes : (and then, those 

 may be added to these :) Or, that most of these be twice repeated, 

 (and if so, those are to be abated out of the Number :) But I do 

 not, at present, discern either the one and other." 



Wallis's attempt is a very bad specimen of analysis ; it involves 

 both the errors he himself anticipates, for some cases are omitted 

 and some counted more than once. It seems strange that he 

 should have failed in such a problem considering the extraordinary 

 powers of abstraction and memory which he possessed ; so that 

 as he states, he extracted the square root of a number taken at 

 random wdth 53 figures, in tenebris decumbens, sola fretus 

 memoria. See his Algebra, page 150. 



50. Wallis's third Chapter is Of the Divisors and Aliquot 

 paints, of a Number i^roposed. This Chapter treats of the resolu- 

 tion of a number into its prime factors, and of the number of 

 divisors Avhich a given number has, and of the least numbers 

 which have an assigned number of divisors. 



51. Wallis's fourth Chapter is Monsieur Fermafs Problems con- 

 cerning Divisors and Aliquot Parts. It contains solutions of two 

 problems which Fermat had proposed as a challenge to Wallis and 

 the English mathematicians. The problems relate to what is now 

 called the Theory of Numbers. 



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