1. _Discursive Reasoning._--WE have thus seen that our notions of space, time, and their modifications, necessarily involve a certain activity of the mind; and that the conditions of this activity form the foundations of those sciences which have the relations of space, time, and number, for their object. Upon the fundamental principles thus established, the various sciences which are included in the term _Pure Mathematics_, (Geometry, Algebra, Trigonometry, Conic Sections, and the rest of the Higher Geometry, the Differential Calculus, and the like,) are built up by a series of reasonings. These reasonings are subject to the rules of Logic, as we have already remarked; nor is it necessary here to dwell long on the nature and rules of such processes. But we may here notice that such processes are termed _discursive_, in opposition to the operations by which we acquire our fundamental principles, which are, as we have seen, _intuitive_. This opposition was formerly very familiar to our writers; as Milton,-- . . . Thus the soul reason receives, Discursive or intuitive.--_Paradise Lost_, v. 438. For in such reasonings we obtain our conclusions, not by looking at our conceptions steadily in one view, which is _intuition_, but by passing from one view to another, like those who run from place to place (_discursus_). Thus a straight line may be at the same time a side of a triangle and a radius of a circle: and in the first proposition of Euclid a line is considered, first in one of these relations, and then in the other, and thus the sides of a certain triangle are proved to be equal. And by this 'discourse of reason,' as by our older {148} writers it was termed, we set forth from those axioms which we perceive by intuition, travel securely over a vast and varied region, and become possessed of a copious store of mathematical truths. 2. _Technical Terms of Reasoning._--The reasoning of mathematics, thus proceeding from a few simple principles to many truths, is conducted according to the rules of Logic. If it be necessary, mathematical proofs may be reduced to logical forms, and expressed in Syllogisms, consisting of major, minor, and conclusion. But in most cases the syllogism is of that kind which is called by logical writers an _Enthymeme_; a word which implies something existing in the thoughts only, and which designates a syllogism in which one of the premises is understood, and not expressed. Thus we say in a mathematical proof, 'because the point C is the center of the circle AB, AC is equal to BC;' not stating the _major_,--that all lines drawn from the center of a circle to the circumference are equal; or introducing it only by a transient reference to the definition of a circle. But the enthymeme is so constantly used in all habitual forms of reasoning, that it does not occur to us as being anything peculiar in mathematical works.