The method of differences is, in fact, a method of additions; and as it includes within its means a larger number of results attainable by addition simply, than any other mathematical principle, it was very appropriately selected as the basis on which to construct an Adding Machine, so as to give to the powers of such a machine the widest possible range.
In abstract mathematics, of course operations alter those particular relations which are involved in the considerations of number and space, and the results of operations are those peculiar results which correspond to the nature of the subjects of operation.
The Analytical Engine is an embodying of the science of operations, constructed with peculiar reference to abstract number as the subject of those operations.
Forget this world and all its troubles and if possible its multitudinous Charlatans-- everything in short but the Enchantress of Numbers.
The Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine… Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.