672 L. F. MENABREA ON BABBAGE’S ANALYTICAL ENGINE. q 
which has been in part put together, and to which the name 
Difference Engine is applicable, on account of the principle 
upon which its construction is founded. To give some notion | 
of this, it will suffice to consider the series of whole square — 
numbers, 1, 4, 9, 16, 25, 36, 49, 64, &c. By subtracting each 
of these from the succeeding one, we obtain a new series, which 
we will name the Series of First Differences, consisting of the 
numbers 3, 5, 7, 9, 11, 13, 15, &c. On subtracting from each 
of these the preceding one, we obtain the Second Differences, 
which are all constant and equal to 2. We may represent this 
succession of operations, and their results, in the following 
table :— 
y m= | © | From the mode in which the two last 
genet jisst_ | Second | columns Band C have been formed, it 
Numbers. | ences. } ences. | is easy to see that if, for instance, we 
1 desire to pass from the number 5 to the 
es ey ant 9» | Succeeding one 7, we must add to the 
-esteaeel 5 former the constant difference 2; simi- 
Mees. ? asa RY? ad ie larly, if from the square number 9 we 
ee | ee as 2 would pass to the following one 16, we 
Pigs Fleets 0) 2 must add to the former the difference 
Richie 1] 7, which difference is in other words the 
preceding difference 5, plus the con- 
stant difference 2; or again, which comes to the same thing, to 
obtain 16 we have only to add together the three numbers 2, 
5, 9, placed obliquely in the direction a. Similarly, we ob- 
tain the number 25 by summing up the three numbers placed 
in the oblique direction dc: commencing by the addition 2 + 7, 
we have the first difference 9 consecutively to 7; adding 16 to 
the 9 we have the square 25. We see then that the three num- 
bers 2, 5, 9 being given, the whole series of successive square 
numbers, and that of their first differences likewise, may be 
obtained by means of simple additions. 
Now, to conceive how these operations may be reproduced by 
a machine, suppose the latter to have three dials, designated as 
A, B, C, on each of which are traced, say a thousand divisions, 
by way of example, over which a needle shall pass. The two 
dials, C, B, shall have in addition a registering hammer, which 
is to give a number of strokes equal to that of the divisions in- 
dicated by the needle. For each stroke of the registering ham- 
mer of the dial C, the needle B shall advance one division; 
