64 JA^IES BERNOULLI. 



109. The second part of the Ars Conjectandi occupies pages 

 72 — ] 87 : it contains the doctrine of Permutations and Combina- 

 tions. James Bernoulli says that others have treated this subject 

 before him, and especially Schooten, Leibnitz, Wallis and Prestet ; 

 and so he intimates that his matter is not entirely new. He con- 

 tinues thus, page 73, 



...tametsi qusedam non contemnenda de nostro adjecimus, inprimis 

 demonstrationem generalem et facilem proprietatis numerorum figura- 

 torum, cui csetera pleraque innituntur, et quam nemo quod sciam ante 

 nos dedit eruitve. 



110. James Bernoulli begins by treating on permutations; 

 he proves the ordinary rule for finding the number of permuta- 

 tions of a set of things taken all together, when there are no 

 repetitions among the set of things and also when there are. He 

 gives a full analysis of the number of arrangements of the verse 

 Tot tibi sunt dotes, Virgo, quot sidera coeli ; see Art. 40. He then 

 considers combinations ; and first he finds the total number of ways 

 in which a set of things can be taken, by taking them one at a 

 time, two at a time, three at a time, ...He then proceeds to find 

 what we should call the number of combinations of n things taken 

 r at a time ; and here is the part of the subject in which he 

 added most to the results obtained by his predecessors. He 

 gives a figure which is substantially the same as Pascal's Arith- 

 metical Triangle; and he arrives at two results, one of which 

 is the well-known form for the nth. term of the rth order of 

 figurate numbers, and the other is the formula for the sum of 

 a given number of terms of the series of figurate numbers of a 

 given order ; these results are expressed definitely in the modern 

 notation as we now have them in works on Algebra. The mode of 

 proof is more laborious, as might be expected. Pascal as we have 

 seen in Arts. 22 and 41, employed without any scruple, and indeed 

 rather with approbation, the method of induction : James Bernoulli 

 however says, page 95,... modus demonstrandi per inductionem 

 parum scientificus est. 



James Bernoulli names his predecessors in investigations on 

 figurate numbers in the following terms on his page 95 : 



Multi, ut hoc in transitu notemus, numerorum figuratorum contem- 



