82 MOXTMORT. 



of the history of the Theory of Probability from its origin. He 

 attributes to himself the merit of having explored a subject which 

 had been only slightly noticed and then entirely forgotten for 

 sixty years ; see his page xxx. 



143. The first part of Montmort's work is entitled TraiU des 

 Combinaisons ; it occupies pages 1 — 72. Montmort says, on his 

 page XXV, that he has here collected the theorems on Combina- 

 tions which were scattered over the work in the first edition, and 

 that he has added some theorems. 



Montmort begins by explaining the properties of Pascal's Arith- 

 metical Triangle. He gives the general expression for the term 

 which occupies an assigned place in the Arithmetical Tiiangle. He 

 shews how to find the sum of the squares, cubes, fourth powers, . . . 

 of the first n natural numbers. He refers, on his page 20, to a 

 book called the New introduction to the Mathematics written by 

 M. Johnes, scavant Geometre Anglois. The author here meant is 

 one who is usually described as the father of Sir William Jones. 

 Montmort then investigates the number of permutations of an 

 assigned set of things taken in an assigned number together. 



14-i. Much of this part of Montmort's work would however 

 be now considered to belong rather to the chapter on Chances 

 than to the chapter on Combinations in a treatise on Algebra. 

 We have in fact numerous examples about drawing cards and 

 throwing dice. 



We will notice some of the more interesting points in this 

 part. We may remark that in order to denote the number of 

 combinations of n things taken r at a time, Montmort uses the 

 symbol of a small rectangle with n above it and r below it. 



145. Montmort proposes to establish the Binomial Theorem; 

 see his page 32. He says that this theorem may be demonstrated 

 in various ways. His own method will be seen from an example. 

 Suppose we require (a + 6)^ Conceive that we have four counters 

 each having two faces, one black and one white. Then Montmort 

 has already shewn by the aid of the Arithmetical Triangle that 

 if the four counters are thrown promiscuously there is one way 

 ia which all the faces presented will be black, four ways in which 



