Thus if a . 1050 + b and a′ . 1050 + b′ are two numbers each of less than a hundred places of figures, then each can be expressed upon two columns of fifty figures, and a, b, a′, b′ are each less than fifty places of figures: they can therefore be added and subtracted upon any column holding fifty places of figures.
The product of two such numbers is—
a a′ 10100 + (a b′ + a′ b) 1050 + b b′.
This expression contains four pair of factors, a a′, a b′, a′ b, b b′, each factor of which has less than fifty places of figures. Each multiplication can therefore be executed in the Engine. The time, however, of multiplying two numbers, each consisting of any number of digits between fifty and one hundred, will be nearly four times as long as that of two such numbers of less than fifty places of figures.
The same reasoning will show that if the numbers of digits of each factor are between one hundred and one hundred and fifty, then the time required for the operation will be nearly nine times that of a pair of factors having only fifty digits.
Thus it appears that whatever may be the number of digits the Analytical Engine is capable of holding, if it is required to make all the computations with k times that number of digits, then it can be executed by the same Engine, but in an amount of time equal to k2 times the former. Hence the {126} condition (a), or the unlimited number of digits contained in each constant employed, is fulfilled.
It must, however, be admitted that this advantage is gained at the expense of diminishing the number of the constants the Engine can hold. An engine of fifty digits, when used as one of a hundred digits, can only contain half the number of variables. An engine containing m columns, each holding n digits, if used for computations requiring k n digits, can only hold m / k constants or variables.
〈OF PUNCHING CARDS.〉
(b). The next step is therefore to prove (b), viz.: to show that a finite engine can be used as if it contained an unlimited number of constants. The method of punching cards for tabular numbers has already been alluded to. Each Analytical Engine will contain one or more apparatus for printing any numbers put into it, and also an apparatus for punching on pasteboard cards the holes corresponding to those numbers. At another part of the machine a series of number cards, resembling those of Jacquard, but delivered to and computed by the machine itself, can be placed. These can be called for by the Engine itself in any order in which they may be placed, or according to any law the Engine may be directed to use. Hence the condition (b) is fulfilled, namely: an unlimited number of constants can be inserted in the machine in an unlimited time.
I propose in the Engine I am constructing to have places for only a thousand constants, because I think it will be more than sufficient. But if it were required to have ten, or even a hundred times that number, it would be quite possible to make it, such is the simplicity of its structure of that portion of the Engine.