L. F. MENABREA ON BABBAGE’S ANALYTICAL ENGINE. 683 
that if the digits of the signs are both even, or both odd, their 
sum will be an even number, and consequently will express a po- 
sitive number; but that if, on the contrary, the two digits of the 
signs are one even and the other odd, their sum will be an 
odd number, and will consequently express a negative num- 
ber. In the case of division, instead of adding the digits of 
the discs, they must be subtracted one from the other, which 
“will produce results analogous to the preceding; that is to say, 
_ that if these figures are both even or both uneven, the remain- 
: der of this subtraction will be even; and it will be uneven in 
_ the contrary case. When I speak of mutually adding or sub- 
_ tracting the numbers expressed by the digits of the signs, I 
‘merely mean that one of the sign-discs is made to advance or 
_ retrograde a number of divisions equal to that which is expressed 
by the digit on the other sign-disc. We see, then, from the 
preceding explanation, that it is possible mechanically to com- 
_ bine the signs of quantities so as to obtain results conformable to 
those indicated by algebra*. 
_ The machine is not only capable of executing those numerical 
calculations which depend on a given algebraical formula, but 
_ it is also fitted for analytical calculations in which there are one 
or several variables to be considered. It must be assumed that 
the analytical expression to be operated on can be developed 
~ according to powers of the variable, or according to determinate 
functions of this same variable, such as circular functions, for 
instance ; and similarly for the result that is to be attained. If 
_ we then suppose that above the columns of the store, we have 
inscribed the powers or the functions of the variable, arranged 
according to whatever is the prescribed law of development, the 
coefficients of these several terms may be respectively placed on 
e corresponding column below each. In this manner we shall 
have a representation of an analytical development ; and, suppo- 
sing the position of the several terms composing it to be invari- 
able, the problem will be reduced to that of calculating their co- 
efficients according to the laws demanded by the nature of the 
question. In order to make this more clear, we shall take the 
* Not having had leisure to discuss with Mr. Babbage the manner of intro- 
ducing into his machine the combination of algebraical signs, I do not pretend 
here to expose the method he uses for this purpose; but I considered that I 
ought myself to supply the deficiency, conceiving that this paper would have 
been imperfect if I had omitted to point out one means that might be employed 
for resolving this essential part of the problem in question. 
