ON BABBAGE’S ANALYTICAL ENGINE. 723 
knowledge, various collateral influences, besides the main and primary 
object attained. 
To return to the executive faculties of this engine: the question must 
arise in every mind, are they really even able to follow analysis in its 
whole extent? No reply, entirely satisfactory to all minds, can be 
_ given to this query, excepting the actual existence of the engine, and 
actual experience of its practical results. We will however sum up 
for each reader’s consideration the chief elements with which the en- 
_ gine works :— 
7 1. It performs the four operations of simple arithmetic upon any 
- numbers whatever. 
2. By means of certain artifices and arrangements (upon which we 
cannot enter within the restricted space which such a publication as 
the present may admit of), there is no limit either to the magnitude of 
the numbers used, or to the number of quantities (either variables or 
constants) that may be employed. 
3. It can combine these numbers and these quantities either alge- 
_ braically or arithmetically, in relations unlimited as to variety, extent, 
or complexity. 
4, It uses algebraic signs according to their proper laws, and deve- 
lopes the logical consequences of these laws. 
5. It can arbitrarily substitute any formula for any other; effacing 
the first from the columns on which it is represented, and making the 
second appear in its stead. 
6. It can provide for singular values. Its power of doing this is re- 
_ ferred to in M. Menabrea’s memoir, page 685, where he mentions the 
passage of values through zero and infinity. The practicability of 
causing it arbitrarily to change its processes at any moment, on the oc- 
currence of any specified contingency (of which its substitution of 
~ CLeos.m +14+ 5 cos.n—1 4) for (cos.4.cos.4) explained in 
ote E., is in some degree an illustration), at once secures this point. 
The subject of integration and of differentiation demands some no- 
_ tice. The engine can effect these processes in either of two ways :— 
First. We may order it, by means of the Operation and of the 
Variable-cards, to go through the various steps by which the required 
limit can be worked out for whatever function is under consideration. 
Secondly. It may (if we know the form of the limit for the function 
_ in question) effect the integration or differentiation by direct * substitu- 
cy r 
X 
“*) 
3 
___ * The engine cannot of course compute limits for perfectly simple and wn- 
| compounded functions, except in this manner. It is obvious that it has no 
_ power of representing or of manipulating with any but finite increments or de- 
-erements; and consequently that wherever the computation of limits (or of any 
| other functions) depends upon the direct introduction of quantities which either 
‘increase or decrease indefinitely, we are absolutely beyond the sphere of its 
powers. Its nature and arrangements are remarkably adapted for taking into 
account all finite increments or decrements (however small or large), and for 
_ developing the true and logical modifications of form or value dependent upon 
differences of this nature. ‘The engine may indeed be considered as including 
the whole Calculus of Finite Differences; many of whose theorems would be 
_ especially and beautifully fitted for development by its processes, and would 
_ offer peculiarly interesting considerations. We may mention, as an example, 
_ the calculation of the Numbers of Bernoulli by means of the Differences of 
Nothing. 
