t(& aVPOTM-EM!* OF THE DEVELOPMENT OF 



term u forger than we expected by 10,000. The next 

 term ii? larger than was anticipated by 30,000, and the 

 exce-M* of each term above what we had expected, forms 

 the follcmbg: table : 



10,000 



30,000 



60,000 

 100,000 

 150,000 



being, in fact, ';he 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 anj 

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



6* 



10 



