- (a). The number of digits in each constant inserted in the Engine must be without limit.
- (b). The number of constants to be inserted in the Engine must also be without limit.
- (c). The number of operations necessary for arithmetic is only four, but these four may be repeated an unlimited number of times.
- (d). These operations may occur in any order, or follow an unlimited number of laws. {124}
〈ALGEBRAICAL DIFFICULTIES.〉
The following conditions relate to the algebraic portion of the Analytical Engine:—
- (e). The number of litteral constants must be unlimited.
- (f). The number of variables must be without limit.
- (g). The combinations of the algebraic signs must be unlimited.
- (h). The number of functions to be employed must be without limit.
This enumeration includes eight conditions, each of which is absolutely unlimited as to the number of its combinations.
Now it is obvious that no finite machine can include infinity. It is also certain that no question necessarily involving infinity can ever be converted into any other in which the idea of infinity under some shape or other does not enter.
It is impossible to construct machinery occupying unlimited space; but it is possible to construct finite machinery, and to use it through unlimited time. It is this substitution of the infinity of time for the infinity of space which I have made use of, to limit the size of the engine and yet to retain its unlimited power.
(a). I shall now proceed briefly to point out the means by which I have effected this change.
〈LARGER NUMBERS TREATED.〉
Since every calculating machine must be constructed for the calculation of a definite number of figures, the first datum must be to fix upon that number. In order to be somewhat in advance of the greatest number that may ever be required, I chose fifty places of figures as the standard for the Analytical Engine. The intention being that in such a machine two numbers, each of fifty places of figures, might be multiplied together and the resultant product of one hundred places might then be divided by another number of fifty {125} places. It seems to me probable that a long period must elapse before the demands of science will exceed this limit. To this it may be added that the addition and subtraction of numbers in an engine constructed for n places of figures would be equally rapid whether n were equal to five or five thousand digits. With respect to multiplication and division, the time required is greater:—