126 Mr Babbage on the Theoretical Principles 
The Series of Cubes. 
The Series Pr< 
1 
1 
8 
8 
27 
* 37 
64 
,74 
125 
135 
216 
* 236 
843 
363 
512 
532 
729 
749 
1000 
* 1030 
1331 
1361 
1728 
1758 
2197 
* 2237 
2744 
2784 
indicating; the number at which the 1 
O 
These, atid other similar series, open a wide field of analytical 
inquiry, — a subject which I shall take some other opportunity 
of resuming. I will, however, mention an unexpected circum- 
stance, as it illustrates, in a striking manner, the connection be- 
tween remote inquiries in mathematics, and as it may furnish a 
lesson to those who are rashly inclined to undervalue the more 
recondite speculations of pure analysis* from an erroneous idea 
of their inapplicability to practical matters. Amongst the singu- 
lar and difficult equations of finite differences to which these se- 
ries led, I recognised one which I had several years since met 
with, in an analytical attempt to solve a problem considered by 
Euler and Vandermonde ; it relates to the knight’s move at 
chess. At that time, I had advanced several steps ; but the 
equation in question proved an obstacle I was then unable to 
surmount. In its present shape, although I have not yet de- 
duced the solution from the equation, yet, as I am in posses- 
sion of the former, it is not too much to anticipate a general 
process applicable to this class of equations ; and should that be 
the case, I shall be able to advance some steps farther in a very 
curious and difficult inquiry, connected with the geometry of si- 
tuation. 
As an erroneous idea has been entertained relative to the na- 
ture of the machinery I have contrived, I will endeavour to state 
