COMBINATIONS AND PERMUTATIONS. 207 



cation in the theory of combinations and probabilities, and 

 remarks of the Arithmetical Triangle, ' It not only contains 

 the clue to the mysterious doctrine of combinations, but it 

 is also the ground or foundation of most of the important 

 and abstruse discoveries that have been made in the other 

 branches of the mathematics/ * 



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, and we can then repeat the same process ad 

 infinitum. The fourth line of figures, for instance, con- 

 tains i, 3, 3, i ; moving them one place and adding as 

 directed we obtain : 



Fourth line . . . i .3 3 i 



1331 



Fifth line ....14 6 4 



i 4 6 



Sixth line . . . . i 5 10 10 51 



i 5 10 10 5 



Seventh line ... i 6 15 20 15 6 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 almost 'impracticable to continue the table. The 

 longest table of the numbers which I have found is given 

 in Fortia'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. 



