ON BABBAGE’S ANALYTICAL ENGINE. 713 
_ separate. The engine must in such a case appropriate as many columns 
to results as there are terms to compute. 
_ Thirdly. It may be desired to compute the numerical value of various 
__ subdivisions of each term, and to keep all these results separate. It may 
__ be required, for instance, to compute each coefficient separately from 
__ its variable, in which particular case the engine must appropriate two 
_ result-columns to every term that contains both a variable and coeffi- 
cient. : 
__. There are many ways in which it may be desired in special cases 
_ to distribute and keep separate the numerical values of different parts 
_ of an algebraical formula; and the power of effecting such distribu- 
tions to any extent is essential to the algebraical character of the Ana-~ 
_ lytical Engine. Many persons who are not conversant with mathe- 
_ matical studies, imagine that because the business of the engine is to 
_ give its results in numerical notation, the nature of its processes must 
_ consequently be arithmetical and numerical, rather than algebraical 
_ and analytical. ‘This is an error. The engine can arrange and combine 
_ its numerical quantities exactly as if they were /eéters or any other 
_ general symbols ; and in fact it might bring out its results in algebraical 
_ notation, were provisions made accordingly. It might develope three 
sets of results simultaneously, viz. symbolic results (as already alluded 
to in Notes A. and B.); mwmerical results (its chief and primary ob- 
_ ject); and algebraical results in literal notation, This latter however 
7 not been deemed a necessary or desirable addition to its powers, 
partly because the necessary arrangements for effecting it would in- 
crease the complexity and extent of the mechanism to a degree that 
would not be commensurate with the advantages, where the main ob- 
ject of the invention is to translate into numerical language general 
ormulz of analysis already known to us, or whose laws of formation 
_are known to us. But it would be a mistake to suppose that because 
its results are given in the notation of a more restricted science, its 
processes are therefore restricted to those of that science. The object 
of the engine is in fact to give the utmost practical efficiency to the re- 
sources of numerical interpretations of the higher science of analysis, 
while it uses the processes and combinations of this latter. 
__ To return to the trigonometrical series. We shall only consider the 
four first terms of the factor (A + A,cos@ + &c.), since this will 
_ be sufficient to show the method. We propose to obtain separately 
the numerical value of each coefficient Co, C,, &e. of (1.). The direct 
multiplication of the two factors gives 
BA+BA,cosé+BA, cos26+BA, COS BO ew itty, Fo ase ay eg 9 
+8, Acos é+ B,Ajcos é.cos +B, A; POPPE MET eee Ls -) 
| result which would stand thus on the engine :— 
Variables for Data, 
~~ 
a 
eb aS Ve 
eh SW V, ee ve 
ERI Fa Ee lh al er 
cosé cos24 cos3é cos 6 
? aad zoey 
