ix.] COMBINATIONS AND PEEMUTATIONS. " 185 



Abacus Panchrestus. Pascal, however, was the first who 

 wrote a distinct treatise on these numbers, and gave them 

 the name by which they are still known. But Pascal did 

 not by any means exhaust the subject, and it remained for 

 James Bernoulli to demonstrate fully the importance of 

 the figurate numbers, as they are also called. In his 

 treatise De Arte Conjectandi, he points out their applica- 

 tion in the theory of combinations and probabilities, and 

 remarks of the Arithmetical Triangle, " It not only con- 

 tains the clue to the mysterious doctrine of combinations, 

 but it is also the ground or foundation of most of the im- 

 portant and abstruse discoveries that have been made in 

 the other branches of the mathematics." l 



The numbers of the triangle can be calculated in a 

 very easy manner by successive additions. We commence 

 with unity at the apex ; in the next line we place a second 

 unit to the right of this ; to obtain the third line of figures 

 we move the previous line one place to the right, and add 

 them to the same figures as they were before removal ; we 

 can then repeat the same process ad infinitum. The 

 fourth line of figures, for instance, contains I, 3, 3, I ; 

 moving them one place and adding as directed we obtain : 



Fourth line . . . I 3 3 i 



I 3_3 i 



Fifth line . . . . I 4 6 4 I 



1464 I 

 Sixth line . . . . I 5 10 10 5 I 



Carrying out this simple process through ten more steps 

 we obtain the first seventeen lines of the Arithmetical 

 Triangle as printed on the next page. Theoretically 

 speaking the Triangle must be regarded as infinite in 

 extent, but the numbers increase so rapidly that it soon 

 becomes impracticable to continue the table. The longest 

 table of the numbers which I have found is in Portia's 

 " Traite des Progressions " (p. 80), where they are given up 

 to the fortieth line and the ninth column. 



1 Bernoulli, De Arte Conjectandi, translated by Francis Maseres. 

 London, 1795, p. 75- 



