in algebraic studies; "monstrum horrendum inform' ingens cui lumen ademtum;" but to that of the learned reader, and to his own eye, it appears as the harmonious and beautiful arrangement of simple elements, each having its due place and force, combining to one noble, important, and useful result.

Further, in geometrical and mathematical reasoning the premises are few; the terms employed are few; and the mind is only engaged in tracing the relations of a few distinct simple ideas, which are fixed by sensible impressions. The whole vocabulary of Euclid may be comprised in a couple of pages. Each book turns upon a few definitions. The whole volume is filled with repetitions of the same terms, with appeals to the same brief premises; attention is more or less frequently recalled to each proposition as it passes in review, and which ranks, when proved, among the foregone premises. The notations of algebra are comparatively few; the letters which stand for unknown quantities derive their meaning solely from connexion with, and relation to, the known quantities, at least in their first.

use; and at last from their relation to each other, in consequence of an extended meaning in the symbols, with which meaning, by habitual contemplation, the mind becomes familiar. Among the figures of arithmetic there are but nine units; after ten you begin with new relations of the first nine; hundreds are combinations of tens, thousands of hundreds, and so on. And with regard to the higher numbers, we can always make clear their value to the senses; for though we could form not the least notion how many men there might be in a field of battle, or how many grains of corn in a sack, by looking at them in the mass, yet divide them into companies of thousands, of hundreds, and tens, and by this arrangement the mind gains a clear and practical sense of the number. It is doubtless by understanding the number and character, and the due arrangement of his forces, that a commander-in-chief is enabled to dispose of them to the best advantage, and form the order of battle.

Our ideas of number and figure are what Locke calls "distinct simple modes;” and however varied in combination or relations,

the same signs or terms are invariably connected with the same conformations of figure, and the same relations of number. Put down a three-sided figure in lines, or any four or more of the Arabic numerals in a line, as 4565, and every human being using the English language would express the relation in the same terms, would pronounce the one a triangle, and read the other four thousand five hundred and sixty-five.

"The idea of two is as distinct from that of one," says Locke, B. II., chap. xiii., as blueness from heat, or either of them from any number; and yet it is made up only of that simple idea of an unit repeated; and repetitions of this kind joined together, make those distinct simple modes of a dozen, a gross, a million."

Thus also he speaks concerning figure, § 6: "The mind having a power to repeat the idea of any length directly stretched out, and join it to another in the same direction, which is to double the length of that straight line, or else join another with what inclination it thinks fit, and so make what sort of

angle it pleases; and being able also to shorten any line it imagines by taking from it one-half, or one-fourth, or what part it pleases, without being able to come to an end of any such division, it can make an angle of any bigness; so also the lines that are its sides of any length it pleases, which joining again to other lines of different lengths, and at different angles, till it has wholly enclosed any space; it is evident that it can multiply figures, both in their shape and capacity, in infinitum; all which are but so many different simple modes of space."

There does not appear any advantage, but the contrary, in the use of the term "mode," and alternating it with "idea," as Locke does in this and in other parts of his Essay; but whether ideas or modes, it is evident they are simple, because they do not admit of being resolved into other ideas or notions still simpler, but result at once from uniformity in the structure and impressions of the senses, which uniformity lays the foundation for language and reasoning.

The simplicity and uniformity of the sen

sible impressions of space, or figure and number, and the comparative fewness of the terms or symbols in use in mathematical reasoning, constantly associated with the same impressions,-terms or symbols which are in fact human contrivances for conveying those impressions from one mind to another,

-these things are to be borne in mind, and duly weighed, in estimating the nature of demonstrative evidence. Nor let any man despise mathematical studies, or think them a mere ringing of changes upon the same set of bells, because the terms employed are few, and the original simple ideas few; otherwise, let him despise the English language, or language in general, because there are only twenty-six letters in the alphabet. For what endless varieties of thought,-what worlds of wisdom,-what vast structures of science, are these twenty-six letters, all-sufficient! And what would human life be without them?

But we have not yet analysed the nature of mathematical reasoning. We have said that mathematical reasoning sets out from