Page:Passages from the Life of a Philosopher.djvu/141

Rh 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:—

Thus if $$a.10^{50} + b$$ and $$a^\prime. 10^{50} + b^\prime$$ are two numbers each of less than a hundred places of figures, then each can be expressed upon two columns of fifty figures, and $$a, b, a^\prime, b^\prime$$ 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—

This expression contains four pair of factors, $$aa^\prime, ab^\prime, a^\prime b, bb^\prime$$, 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