Secondly, we may make an error of the same kind in the details of operation. For instance, suppose we have a + b — c, which it is con venient to exhibit in the form of a altered by one single addition or subtraction. If we assume an addition, and write a + (b—c), we may Le In error; for if b be less than e, the proper alteration is a — (c— b).
It is evident that both species of mistake are precisely of the same kind. Let us call them, for distinction, errors of interpretation and errors of operation, and let us show first that an error of interpretation will produce the error of operation and no other. If, in the first problem, we suppose a—b to be lost where b—a is really gained, and if the problem, for instance, require the result of the proceeding to be annexed to a loss .r, we shall suppose there is altogether a lose of x + (a —6), whereas it should be a loss of only x —(6 —a). Secondly, the error of operation will produce the error of interpretation, when ever any interpretation is made ; for when we look at x + (a —b) as a loss, we shall evidently suppose it to be more of a loss than .r, or that a—b is lost besides; whereas, had we looked at x—(b—a), we should have inferred that there is a less loss than x. Now the first step of the young algebraist, before he attempts any transition from universal arithmetic to algebra, Must be to examine by many instances the effect of both classes of errors upon the subsequent proceedings and result*. We shall here only state the truths at which he will finally arrive, with an example of each. The beginner cannot, as the proficient may do, see a sufficient reason for these results in the common rules of alga bmical operation ; and wo should doubt that anything but a large number of examples would serve to give hint the necessary insight into the conclusions.
1. The mistake of operation, how often soever repeated, and how complicated soever the deductions which may be drawn from it, pro duces no result in any way different from that of the correct process; that is, its result can be reduced to the result of the correct process by the use of uo more than those rules which apply in the rational process.
Thus if x+a— b, wrongly taken to be x+ (a—b), 6 being greater than a, be multiplied by x +p—g, wrongly taken as x+ (p—q), I being greater than p, we find as the (supposed) product x' + (a —b + p—q) x + (a— b) (p — g), to which the application of the common rules gives ex— +p.e— gx+ ap —bp —aq +by,
precisely the same as the product of .r + a —b and x + p—y. The reason of this is as follows all the rational cases of the four operations, a term in the construction of which two signs are used has + before it, if those two signs be alike, and — if they be unlike, as in a+ b— (c—d), or a+ b —(0 + c—d) =a+b—c+d (a — b) (c—d) or (0 +a— b) (0 + c—d) =0+ac —ad —bc bd.
If then a term were subjected to the signs + +, it would make no difference if the same term were subjected to the signs + — —, for the effect of — — is the use of +. If then we take x+ a—b wrongly as x—(b—a), we see that when we come to add this, say to is we have c+ in which a, before it is disengaged, must come under the signs + — or, if the phrase be leas objectionable, under the application of the rules to the signs, successively. But the correct process would give c+ {x+(a—b)} in which a falls under the application of the rules to + + ; and such application to + + gives the same result as that to +— necessarily and demonstrably, though in one of the two applications there is the symbol of absurdity. In the same way the other cases may be proved, whence it follows that however many of these simple operations may be no result can arise except either that of the correct operation or one which may be brought to it by the operations on signs, already described.
We must here pause to remind the reader that errors, however palpable and admitted, are not necessarily productive of error. True reasoning, on true principles, mast lead to truth ; but if for true we write false, and for truth falsehood, wo have no longer any right to say must, but only most probably will. If then we can show of a particular class of errors that, used in a certain way, the results agree with those of true reasoning on true principles, %.e may demand the use of those errors as demonstrated means of finding truth. The mind of man would never stop at such a point ; but, for all that, we have the con clusion, as a logical consequence of the rules of arithmetic, that the mistake of the Impossible subtraction introduced in operations, and not having previously vitiated the interpretation by which the funda mental objects of operation (equations) were deduced from the condi tions of the problem, will produce no falsehood in the result..