158 VESTIGES OF THE 



being, in fact, the series of triangular numbers,'^ each 

 multipKecl 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 watch- 

 ing them for 2761 terms we find that this law fails in 

 the case of the 2762nd 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, whicli appear at different 

 intervals. 



"Now it must be observed that the laid that each 

 'uainher ■presented by the emgine 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 regidated its action, and that the occur- 

 rence of the number 100,010,002 at the 100,000,002nd 

 term was as necessary a consequence of the original ad- 



* The numbere, i, 3, 6, 10, 15, 21, 28, &c. are formed by adding 

 the successive terms of the series of natural numbers thus : 



I = I 

 1+2 - 3 

 1+2 + 3 = 6 

 1+2 + 3 + 4 -=io, &c. 

 They are called triangidar numbers, because a niunber of points cor- 

 responding to any term can always be placed in the form of a 

 triande ; for instance — 



