‘ON BABBAGE’S ANALYTICAL ENGINE. 727 
for computing B, » — 1, (in the case of m = 4); while the table beneath 
them presents a complete simultaneous view of all the successive 
changes which these columns then severally pass through in order to 
perform the computation. (The reader is referred to Note D, for ex- 
planations respecting the nature and notation of such tables.) 
Six numerical daca are in this case necessary for making the requisite 
combinations. These data are 1, 2,7 (=4), B,, B,,B,. Weren=5, 
the additional datum B, would be needed. Were n=6, the datum B, 
would be needed; and so on. Thus the actual xwmber of data needed 
will always be x + 2, form =; and out of these m + 2 data, (n + 2 — $) 
of them are successive Numbers of Bernoulli. The reason why the 
Bernoulli Numbers used as data, are nevertheless placed on Result- 
columns in the diagram, is because they may properly be supposed to 
have been previously computed in succession by the engine itself; un- 
_ der which circumstances each B will appear as a result, previous to 
being used as a datum for computing the succeeding B. Here then is 
an instance (of the kind alluded to in Note D.) of the same Variables 
filling more than one office in turn. It is true that if we consider our 
computation of B, as a perfectly zsolated calculation, we may conclude 
B,, B,, B, to have been arbitrarily placed on the columns; and it would 
then perhaps be more consistent to put them on V,, V;, V,; as data and 
not results. But we are not taking this view. On the contrary, we 
suppose the engine to be 7x the course of computing the Numbers to an 
indefinite extent, from the very beginning ; and that we merely single 
out, by way of example, one amongst the successive but distinct series’ 
of computations it is thus performing. Where the B’s are fractional, it 
must be understood that they are computed and appear in the nota- 
tion of decimal fractions. Indeed this is a circumstance that should 
be noticed with reference to all calculations. In any of the examples 
_ already given in the translation and in the Notes, some of the data, or 
of the temporary or permanent results, might be fractional, quite as pro- 
bably as whole numbers. But the arrangements are so made, that the 
nature of the processes would be the same as for whole numbers. 
In the above table and diagram we are not considering the signs of 
any of the B’s, merely their numerical magnitude. The engine would 
bring out the sign for each of them correctly of course, but we cannot 
enter on every additional detail of this kind, as we might wish to do. 
The circles for the signs are therefore intentionally left blank in the 
diagram. 
Operation-cards 1, 2, 3, 4, 5,6 prepare — i fA ca Thus, Card 
p » 4) J, F, 0,0 prep Q2°I9n+1'° va 
1 multiplies two into , and the three Receiving Variable-cards be- 
longing respectively to V,, V,, V,, allow the result 2” to be placed on 
each of these latter columns (this being a case in which a triple receipt 
of the result is needed for subsequent purposes) ; we see that the upper 
indices of the two Variables used, during Operation 1, remain unaltered. 
We shall not go through the details of every operation singly, since 
the table and diagram sufficiently indicate them ; we shall merely notice 
some few peculiar cases. 
By Operation 6, a positive quantity is turned into a negative quan- 
tity, by simply subtracting the quantity from a column which has only 
