L. F. MENABREA ON BABBAGE’S ANALYTICAL ENGINE. 673 
similarly, the needle A shall advance one division for every 
stroke of the registering hammer of the dial B. Such is the ge- 
neral disposition of the mechanism. 
This being understood, let us at the beginning of the series 
of operations we wish to execute, place the needle C on the di- 
vision 2, the needle B on the division 5, and the needle A on the 
division 9. Let us allow the hammer of the dial C to strike; it 
will strike twice, and at the same time the needle B will pass 
over two divisions. The latter will then indicate the number 7, 
which succeeds the number 5 in the column of first differences. 
If we now permit the hammer of the dial B to strike in its turn, it 
will strike seven times, during which the needle A will advance 
seven divisions; these added to the nine already marked by it, 
will give the number 16, which is the square number consecu- 
tive to 9. If we now recommence these operations, beginning 
with the needle C, which is always to be left on the division 2, 
we shall perceive that by repeating them indefinitely, we may 
successively reproduce the series of whole square numbers by 
means of a very simple mechanism. 
The theorem on which is based the construction of the ma- 
chine we have just been describing, is a particular case of the fol- 
lowing more general theorem: that if in any polynomial what- 
ever, the highest power of whose variable is m, this same variable 
be increased by equal degrees; the corresponding values of the 
polynomial then calculated, and the first, second, third, &c. 
differences of these be taken (as for the preceding series of 
squares) ; the mth differences will all be equal to each other. So 
that, in order to reproduce the series of values of the polyno- 
mial by means of a machine analogous to the one above de- 
scribed, it is sufficient that there be (m + 1) dials, having the 
mutual relations we have indicated. As the differences may be 
either positive or negative, the machine will have a contrivance 
for either advancing or retrograding each needle, according as 
the number to be algebraically added may have the sign plus 
or minus. 
If from a polynomial we pass to a series having an infinite 
number of terms, arranged according to the ascending powers 
of the variable, it would at first appear, that in order to apply the 
machine to the calculation of the function represented by such 
a series, the mechanism must include an infinite number of dials, 
which would in fact render the thing impossible. But in many 
i ae’ 7 
