that a term shall not be used in different senses; a real definition of anything belongs to the science or system which is employed about that thing. It is to be noted, that in mathematics (and indeed in all strict sciences) the nominal and the real definition exactly coincide; the meaning of the word and the nature of the thing being exactly the same. This holds good also with respect to logical terms, most legal, and many ethical terms."

Upon the whole we conclude that the definitions of geometry settle the meaning of terms.*

Thirdly, These terms are the signs of our ideas of figure and quantity, including in the latter term number and magnitude (both the

*Pascal, in his Reflexions sur la Géometrie en Général, justly observes, however, that many notions are assumed, and terms are used in mathematics which are not defined. "Cette judicieuse science est bien éloignée de definir ces mots primitifs, espace, temps, mouvement, égalité, majorité, diminution, tout, et les autres que le monde entend de soi-même." And this must be the case in all reasoning; for, as definition is merely explaining one term by many, it is obvious we might go on defining without end, and not advance a step towards any valuable conclusion.

how many and how great, quantus); which ideas or notions come to us, so to speak, originally from without; i.e. they originate in sensible impressions. They are not significant merely of what passes within, or of mental states, like the terms memory, the will, judgement, attention, and desire, unless indeed every sensation, such as of whiteness or blackness, be considered a mental state, idea an affection of the mind.

and every

Here, perhaps, I am treading upon the most doubtful, because metaphysical, ground. Right or wrong, however, in what may be said under this head, it will not invalidate what has been said about definition and its object. It appears to me that mathematical reasoning consists in tracing the relations of our ideas of figure and quantity by means of exactly defined symbols, whether words, diagrams, or other symbols, one with another, in respect of agreement or disagreement, equality, or inequality; and these terms and ideas receive clearness and strength by constant application and reference to external things, or sensible impressions; and also by their

observed, clear, uniform, and well-defined relation to each other.

The subject matter of mathematical reasoning may therefore be considered to be real existencies, with as much justice as the subject matter of any other reasoning. For in all reasoning, what has the mind before it but its own abstractions or notions, and terms affixed to those notions? And who can say that circles, angles, squares, lines, have not as much foundation in, and reference to, things as they exist, as white, blue, black, soft, hard, or other qualities of body, solid, liquid, brittle, or elastic; or the abstract ideas of space, time, beauty, honour, virtue, and so on? Our ideas of number and figure are ideas constantly forced upon us by sensible objects, and all that fills this visible diurnal sphere; the terms significant of these ideas are in constant use and application in ordinary life. They are employed by the humblest in station and education with uniformity of meaning, with clearness and accuracy for their purposes. It is true they may not know anything of the rela

tions and properties of triangles, squares, circles, parallelograms, as traced by the mathematician; but the mathematician's skill and wisdom consist only in having traced and studied these relations by means of his exact definitions, and by his deeper or more frequent meditation on their several connexions and consequences. The ideas or notions of number and figure are common to all minds. Attention and instruction only are necessary to furnish them with the exact definitions and new combinations. In number, it is obvious that the terms or figures are themselves definitions, or their equivalents.

It is because the subject matter of mathematical reasoning consists, in our ideas of figures and magnitudes or quantities, that the reasoning may be carried on by other signs than words, viz. sensible diagrams. The Arabic numerals, and the notations of algebra, are artificial contrivances or abbreviated symbols for tracing the relations of quantity as they are wanted, or as those relations follow from the nature of the contrivances themselves. These diagrams, these

figures and notations, are the signs and instruments of the mathematician's or algebraist's thoughts; and it is because they are always of a clear and certain nature, and bear a uniform, fixed, and definite relation one to another, that the geometrical reasoning, and the arithmetical and algebraic processes are the same to every mind.

Upon this circumstance, namely, the power of fixing the attention and carrying on the reasoning by means or help of sensible diagrams, Locke fastens, as of the first importance, and the great peculiarity in mathematical studies.

"That which has given the advantage to the ideas of quantity, and made them thought more capable of certainty and demonstration, is, first, that they can be set down and represented by visible marks, which have a greater and nearer correspondence with them than any words or sounds whatsoever. Diagrams drawn on paper are copies of the ideas in the mind, and not liable to the uncertainty that words carry in their signification. An angle, circle, or square, drawn in lines,