the digit on this disc is even or uneven, the number inscribed on the corresponding column
below it will be considered as positive or negative. This granted, we may, in the following
manner, conceive how the signs can be algebraically combined in the machine. When a
number is to be transferred from the store to the mill, and vice versa, it will always be
transferred with its sign, which will be effected by means of the cards, as has been explained in
what precedes. Let any two numbers then, on which we are to operate arithmetically, be placed
in the mill with their respective signs. Suppose that we are first to add them together; the
operation-cards will command the addition: if the two numbers be of the same sign, one of the
two will be entirely effaced from where it was inscribed, and will go to add itself on the column
which contains the other number; the machine will, during this operation, be able, by means of
a certain apparatus, to prevent any movement in the disc of signs which belongs to the column
on which the addition is made, and thus the result will remain with the sign which the two given
numbers originally had. When two numbers have two different signs, the addition commanded
by the card will be changed into a subtraction through the intervention of mechanisms which
are brought into play by this very difference of sign. Since the subtraction can only be effected
on the larger of the two numbers, it must be arranged that the disc of signs of the larger
number shall not move while the smaller of the two numbers is being effaced from its column
and subtracted from the other, whence the result will have the sign of this latter, just as in fact it
ought to be. The combinations to which algebraical subtraction give rise, are analogous to the
preceding. Let us pass on to multiplication. When two numbers to be multiplied are of the same
sign, the result is positive; if the signs are different, the product must be negative. In order that
the machine may act conformably to this law, we have but to conceive that on the column
containing the product of the two given numbers, the digit which indicates the sign of that
product has been formed by the mutual addition of the two digits that respectively indicated the
signs of the two given numbers; it is then obvious 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 positive 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 number. 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 remainder of this subtraction will be even; and it will be uneven
in the contrary case. When I speak of mutually adding or subtracting 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 combine
the signs of quantities so as to obtain results conformable to those indicated by algebra[9].
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 the corresponding column below each. In this manner we shall have a representation
of an analytical development; and, supposing the position of the several terms composing it to
be invariable, the problem will be reduced to that of calculating their coefficients according to
the laws demanded by the nature of the question. In order to make this more clear, we shall
take the following[10] very simple example, in which we are to multiply (a + bx
1
) by (A + B cos
1
x). We shall begin by writing x
0
, x
1
, cos
0
x, cos
1
x, above the columns V
0
, V
1
, V
2
, V
3
; then
since, from the form of the two functions to be combined, the terms which are to compose the
products will be of the following nature, x
0
.cos
0
x, x
0
.cos
1
x, x
1
.cos
0
x, x
1
.cos
1
x, these will be
inscribed above the columns V
4
, V
5
, V
6
, V
7
. The coefficients of x
0
, x
1
, cos
0
x, cos
1
x being
given, they will, by means of the mill, be passed to the columns V
0
, V
1
, V
2
and V
3
. Such are the
primitive data of the problem. It is now the business of the machine to work out its solution, that
is, to find the coefficients which are to be inscribed on V
4
, V
5
, V
6
, V
7
. To attain this object, the