of calculation with signs merely uttered or conceived of as uttered; he must write down his equations and series, and work out painfully, in long rows of figures, his numerical results: for, though all was implied in his first assumption, as evolved according to the unvarying relations of numbers, and the principles of mathematical reasoning, he is unable to grasp the various quantities with his mind, and to follow out unerringly the successive steps of the processes, without recording each as he takes it. It is none the less true, however, that the whole work is a mental one: mathematical quantities are identical neither with the written figures and symbols, nor with the spoken signs; nor is mathematical reasoning dependent for its existence upon the one or the other; both are kindred instrumentalities, whereby the mind is enabled to accomplish what would otherwise be wholly beyond its power.
The main truths which we have to accept as touching the relation of language to thought are, I think, brought out by this illustration. It is, indeed, an extreme illustration on the side of the indispensability of language. For no other class of conceptions are so eminently abstract as are the mathematical, none so wholly dependent upon spoken and written signs and symbols. They are so essentially ideal in their character, so divorcible from concrete objects, that they can be worked with mechanically, can be put together and taken apart without constant reference to real conditions—though only according to rules and methods ultimately founded on concrete exemplification, on immediate synthetic apprehensions which are capable of being grasped by minds lower than human. Yet, even here, the signs are merely the instruments of thought, and created by it. The symbols of the calculus are not more truly the device of the masterminds which, exalted upon the vantage-ground of their own and others' previous studies, apprehended the higher and more recondite relations involved in this new mode of mathematical reasoning, than the whole nomenclature of numbers is the gradually elaborated work of men who saw and felt impelled to signify the simpler and more fundamental relations, those which seem to lie within the reach of every
