exercises, because, no doubt, the compilers had not thought the matter out, we who use their collections, being as indolent as they, have contented ourselves with the general probability of, at least, partial success. In the mean time even classical writers have spoken of clearing an equation from radicals, in order to its solution, as a process of course, and which would not in any way affect the conditions. The consequence is, that a habit prevails of talking about equations without any regard to this peculiar case, and therefore in language which when applied to it becomes quite incorrect. The term root of an equation passes for synonymous with any quantity which, being substituted for the unknown, satisfies the conditions; and it is affirmed, and demonstrated, that every equation has at least one root; and that, having one, it must have as many roots as there are units in the greatest index attached to the unknown. It is therefore quite startling, when we are reminded that equations may be proposed ad libitum, whose conditions cannot be satisfied by any quantity, positive, negative, or imaginary; that notwithstanding this, the roots obtained from such equations may be real quantities. Nor is the enigma solved by discovering that the roots obtained from one equation are sure to satisfy the conditions of another, not much unlike it: on the contrary, one is quite displeased at this kind of thimble-rig shuffling, where we were assured of finding truth, the whole truth, and nothing but the truth. A logician of the old school would settle the business by crying "distinguo"; but we should still reply, that it is a lame distinction which clears up only one half of the premises: we know that these are surd equations we are now speaking of, and that just before we were speaking of rational equations, or equations cleared of surds; but the difficulty remains unexplained. If he really knew a little of the subject, he would, perhaps, next try the pass-word ambiguity: "There is always a certain ambiguity adhering to surd expressions." When, however, the most is said that can be said to that purport, it amounts in short to this, that in the reading of formula, when we meet with a radical, we ought not to use the definite but the indefinite article. We have a knack of saying "the", where we ought to say "a", that is all; and if we did but read a square root, a cube root, and so on, we should be certain of finding one that would satisfy the existing conditions. This sounds plausibly, and at least ninetynine out of every hundred of algebraists would inquire no further; but you would perhaps object, that at this rate might be a good equation, unless, with

+ √ x = − √

Lindley Murray, we admit the "the" when the same quantity appears a second time under the same radical; and that without a greater latitude still, we should never be able to prove that x − a x − √ x+a made - √ x2 —a2, and so on. So that the professed ambiguity is subject, after all, to a conventional permanence.

The source of the whole mystery, in my judgement, is to be found in the almost unavoidable imperfection of the manner in which we are taught to transform equations when we are at school. The operations consequent on transposition are correct as far as their principles are resolvable into Euclidian axioms. Beyond that they are liable to fallacy; and, generally speaking, we are infallible in our judgement only as long as every term is on one side. We may then determine satisfactorily in what cases zero is admissible as the aggregate value. An instance of the hazard attending the neglect of this principle is given in my paper in the Lond. and Edinb. Phil. Mag. for September 1834, (vol. v.) p. 189. In the management of surds, instances might easily be accumulated.

And whence this hazard? and why consequent upon transposition? Because, from the nature of analysis, we are continually arguing from the direct to the converse. An equation is formed hypothetically. We trace out certain direct consequences, in the form of equations also, and so on; until an equation is obtained, such that if the first be true, the last is therefore true. But the converse is that which we wished to ascertain. Is the hypothetical equation true, because the resulting equation is so? To determine this, a similar query must be instituted from link to link throughout the chain of reasoning. Is each equation in succession true, because the next in succession is so? If each of these subordinate inquiries admits of a decided affirmative, the reply to the general query is satisfactory; otherwise, it is not. Now in the management of equations, we have been taught, either virtually or in direct terms, to rely upon certain axioms, which for the present purpose will be most effectually stated in pairs, viz.

If equals be added to equals, the wholes are equal;

and,

If equals be subtracted from equals, the remainders

are equal.

If equals be multiplied by equals, the products are

equal; and

If equals be divided by equals, the quotients are

>I.

II.

equal.

If equals be raised to powers denoted by equal exponents, the powers are equal; and

If of equal quantities roots be extracted, which III. are denoted by equal exponents, the roots are

equal.

It has never been my chance, either to hear the validity of any of these principles called in question, or even any caution suggested as necessary in the application of them; and yet, when tested by the combined trial of their direct and reflex action, they will presently appear to be very susceptible of misuse in incautious hands.

The first and second pair, abstractedly considered, afford such entire conviction, that in each of them, if either proposition is granted, the other can be strictly demonstrated by means of it; and the second pair are truly corollaries to the first. No hesitation, no ambiguity, is felt.

The fifth proposition, as a clear corollary to the third, is, in itself, equally satisfactory; but quite otherwise in regard to its reflex effect, as described in the sixth. For, being aware that if unequal quantities (+ a, a) be raised to power, denoted by equal exponents, the powers may nevertheless be equal; we are assured that, conversely, if of equal quantities roots be extracted which are denoted by equal exponents, the roots may nevertheless be unequal.

This remark furnishes a sufficient reason for rejecting the third pair of principles, and consequently the ordinary method of clearing an equation from surds. For, in every instance in which this is effected by transposition and involution, in compliance with the fifth axiom, we tacitly assume that such step can be retraced with equal certainty by means of the sixth; whereas, in any such transit, the consequent equation may be quite true, and yet the antecedent be quite false.

If, however, we attribute the failure of the third pair of axioms to a special ambiguity peculiar to evolution, we shall remain under a delusion, and miss the cause and remedy of the evil. Involution is but a single instance of the erroneous application of the axioms of the third pair; but the use of any of the four unexceptionable axioms is liable to be frustrated by a similar cause, although in some cases the absurdity introduced is so palpable as to occasion a kind of instinctive unconscious avoidance. In other instances, however, even acute minds have failed to observe the fallacy. This I shall now point out, and prove that unless connected with the use of the first pair of axioms, it will be avoided, if no member of the equation is transposed to the zero side.

The origin of the fallacy in question will be rendered more

evident by a course of amusing experiments upon a familiar equation, e. g.

whose roots are 1, 2, -3, -2. Applying the four axioms in succession, we shall perceive how the incautious blending of two truths, by means of rules in themselves unexceptionable, will produce a falsehood.

a false equation, with regard to all the values of x, with the single exception of 1, the value already used. Similar results would accrue from the addition or subtraction of any other divisor of the equation; the result will be false in every value, except those which are also found in the equation added or subtracted. Thus,

whose only correct roots are those also of x2 3x + 2 = 0. 2ndly, The given equation is resolvable into the quadratics 3x + 2 = 0, and x2 + 5x + 6 = 0.

a statement altogether erroneous, not containing a single cor

incorrect, except in respect of the roots of x2-3x+2 = 0. You will clearly perceive, without dwelling upon the distinction of cases, the very simple nature and origin of the paradox. The axioms speak of quantities which are simultaneously equal; but no two roots of an equation, unless they be equal roots, are coexistent: if x = 1 it is not at the same time = 2. Consequently, as in each of the examples x in the upper of the two equations has some values, which substituted in the

lower will render its sides unequal, the results, as far as such values of x are concerned, are no longer coincident with the conditions of the axioms on which the management of equations is founded, but are illustrations of the opposite axioms, viz. that unequals added to, or subtracted from, or multiplied or divided by, equals, produce unequal results, or in algebraic language, false equations.

The reason why this inconvenience, in the use of the second pair of axioms, cannot occur when all the terms are on one side and zero alone on the other, is very evident; although, by another of those paradoxes by which equations are beset, the complete truth appears at first sight to be the result of combining a truth with an error, and equals to result from combining equals with unequals. It is, however, easy to avoid all suspicion of error. Thus, it was said, that the given equation is resolvable into + 3x + 2 = 0, and x2 + 5x + 6 = 0. But as these statements are not simultaneously true, but, on the contrary, any value of r which satisfies one of the quadratics will render the other = A, some numerical quantity differing from 0, we in fact collect the product of 3x + 2 = 0, or A, x2 + 7 x + 12 = A, or 0,

by

x2

-

in finding (3 x + 2) (x2 + 7 x+12)= 0; where the premises being strictly correct, the result is unexceptionable. And the same result arises, although not with equally clear evidence of its truth, when A is superseded by zero.

The same test, of a hypothetical adjustment of one of the two proposed equations, would at once expose the fallacy of each of the conclusions attained in our imaginary experi

ment.

The general propriety of keeping the zero-side of each equation in a chain of argument clear from any transposed terms, is proved therefore by the liberty which it allows to the mind, of conceiving any zero, which happens to be pro tempore incorrect, to be superseded by the correct value, and of perceiving without any embarrassment or additional labour the exact conditions of the final result. But the especial propriety of adhering to this expedient, when surds are to be extricated, appears in the necessity which it imposes of attending to the copula of the argument, the suppression of which in the vulgar process occasions all the obscurity that is complained of. Thus, between the statements