10$ HFPOTHXMS OF THE DEVELOPMENT OP 



te*Wk u Jargor than we expected by 10,000. The next 

 term i* larger than was anticipated by 30,000, and the 

 exc-Hw of each term above what we had expected, forms 

 the iVSkwing- table : 



10,000 



30,000 



60,000 

 100,000 

 J 60,000 



being, in fact, fate series of triangular numbers* each 

 multiplied by 10,000. 



" If we now continue to observe the numbers presented 

 by the wheel, we shall find that for a hundred, or even 

 for a thousand terms, they continue to follow the new law 

 relating to the triangular numbers ; but after watching 

 them for 2761 terms, we find that this law fails in the 

 case of the 2762d term. 



" If we continue to observe, we shall discover another 

 law then coming into action, which also is dependent, but 

 in a different manner, on triangular numbers. This will 

 continue through about 1430 terms, when a new law is 

 again introduced which extends over about 950 terms, 

 and this, too, like all its predecessors, fails, and gives 

 place to other laws, which appear at different intervals. 



" Now it must be observed that the law that each num- 

 ber presented by the engine is greater by unity than the 

 preceding number, which law the observer had deduced 

 from an induction of a hundred million instances, was 

 not the true law that regulated its action, and that the 

 occurrence of the number, 100,010,002 at the 100,000,002d 

 term was as necessary a consequence of the original ad- 



* The numbers 1, 3, 6, 10, 15, 21, 28, &c, are formed by adding 

 the successive terms of the series of natural numbers thus : 

 1= 1 

 1X2= 3 

 1X2X3= 6 

 1X2x3x4= 10, &c. They are called trian- 

 gular numbers, because a number of points corresponding to an} 

 term can always be placed in the form of a triangle ; for instance— 



10 



