therefore also the angles CBA, BA C, A CB, are together equal to two right angles. "Therefore if a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three and the three interior angles of An intuition that things which are equal to the same thing are equal to each other; which, as before hinted, is itself known through an intuition of the equality of two relations. An intuition that the relation between lines and angles found to subsist in this triangle, subsists in any triangle, in all triangles—that the relation in every other case is equal to the relation in this case. every triangle are equal to two right angles. Q.E.D." Thus in each step by which the special conclusion is reached, as well as in the step taken from that special conclusion to the general one, the essential operation gone through is the establishment in consciousness of the equality of two relations. And as, in each step, the mental act is undecomposable-as for the assertion that any two such relations are equal, no reason can be assigned save that they are perceived to be so; it is manifest that the whole process of thought is thus expressed. § 283. Perhaps it will be deemed needless to prove that each step in an algebraic argument is of the same nature; since it has been shown that the axiom-Relations which are equal to the same relation are equal to each other, twice involves an intuition of the above-described kind; and since the implication is, that reasoning which proceeds upon this axiom is built up of such intuitions. But it may be well definitely to point out that only in virtue of such intuitions do the successive transformations of an equation become allowable. Unless it is perceived that a certain modification made in the form of the equation, leaves the relation between its two sides the same as before-unless it is seen that each new relation established is equal to the foregoing one, the reasoning is vicious. A convenient mode of showing that the mental act continually repeated in one of these analytical processes is of the kind described, is suggested by an ordinary algebraic artifice. When a simplification may be thereby achieved, it is usual to throw any two forms of an equation into a proportion-a procedure in which the equality of the relations is specifically asserted. Here is an illustration: not such an one as would occur in practice, but one that is simplified to serve present purposes. and if proof be needed that this mode of presenting the facts is legitimate, we at once obtain it by multiplying extremes and means; whence results the truism 2xy=2xy2. This clearly shows that the mental act determining each algebraic transformation, is one in which the relation expressed by the new form of the equation is recognized as equal to the relation which the previous form expresses. CHAPTER IV. IMPERFECT AND SIMPLE QUANTITATIVE REASONING. § 284. ABILITY to perceive equality implies a correlative ability to perceive inequality: neither can exist without the other. But though inseparable in origin, the cognitions of equality and inequality, whether between things or relations, differ in this; that while the one is definite the other is indefinite. There is but one equality; but there are numberless degrees of inequality. To assert an inequality involves the affirmation of no fact, but merely the denial of a fact; and therefore, as positing nothing specific, the cognition of inequality can never be a premiss to any specific conclusion. Hence, reasoning which is perfectly quantitative in its results, proceeds wholly by the establishment of equality between relations, the members of which are either equal or one a known multiple of the other. Conversely, if any of the magnitudes standing in immediate relation are neither directly equal nor the one equal to so many times the other; or if any of the successive relations which the reasoning establishes are unequal; the results are imperfectly quantitative. The truth is illustrated in that class of geometrical theorems in which it is asserted of some thing that it is greater or less than some other; that it falls within or without some other; and the like. Let us take as an example the proposition-" Any two sides of a triangle are together greater than the third side." "Let A B C be a triangle; any two sides of it are, together, greater than the third side; namely, B A, A C, greater than B C; and A B, B C, greater than A C; and B C, C A, greater than A B. "Produce B A to D, and make A D equal to A C; and join D C. therefore B A, A C are greater than B C. "In the same manmay be demon ner it strated that the sides A B, B C are greater than CA, and B C, CA greater than A B." An immediate intuition of the equality of two relations of inequality which have one term in common, and the other terms equal. The relations subsisting in other cases are equal to the relation subsisting in this case. And It will be observed that here, though the magnitudes dealt with are unequal, yet the demonstration proceeds by showing that certain relations among them are equal to certain other relations: though the primary relations (between quantities) are those of inequality, yet the secondary relations (between relations) are those of equality. this holds in the majority of imperfectly-quantitative arguments. Though, as we shall see by and by, there are cases in which both the magnitudes and the relations are unequal, yet they are comparatively rare; and are incapable of any but the simplest forms. § 285. Another species of imperfectly-quantitative reasoning occupies a position in mathematical analysis, like that which the foregoing species does in mathematical synthesis. The ordinary algebraic inequation supplies us with a sample of it. Thus, if it is known that a + 22 √y is less than a+ay, the argument instituted is as follows: |