The various applications of mathematics to physical questions, or to the transactions of common life, continually require the computation of numerical results. At one time isolated results have to be calculated from particular formulæ; at another it is required to calculate a series of values of the same analytical formula; in other words, to tabulate a function. It is only in the latter case that different instances have so much in common as to permit of the application of general methods irrespective of the particular function to be calculated, But even in the tabulating of functions one or other of two objects may be kept in view. At one time a result may be arrived at expressed in complicated, perhaps transcendental, formula, and the mathematician may desire to know merely the general progress of the function. In such a case it will be sufficient to calculate values at rather wide intervals, and the mode of calculation must depend upon the peculiar function. But at other times functions present themselves which are of such common occurrence, or of such practical importance, that it is desirable to tabulate them for the values of the variable, increasing by small steps. In these cases general methods of interpolation come into use; it is sufficient to perform the calculations directly for comparatively wide intervals of the variable, and the intervening values of the function can be supplied by the mere addition of differences.

It is well known that Mr. Babbage was the first person who conceived the idea of performing all these systems of additions mechanically, and thereby saving both the mental labour and the risk of error attending their calculation in the ordinary way.