2. Let us now examine the consequences of the error of interpreta tion. The effect of this is, that we write a—b instead of b—a, but at the same time we suppose the quantity of %%Well we are thinking to be of a diametrically opposite character to that which it ought to have. But also at the seine time we add this symbol where we should sub tract it, and rice rend ; so that where wo should take a—b, and add, giving c+ (a—b), we make one mistake in taking and another in subtracting, giving —(6—a). When mere rules come to be applied, we find the same result from both, namely, c+ a —b and e—b+ a. Wo might then so manage as to elude the actual press ntation of the noise tive quantity, as in the following problem :—Two persons are now aged 50 and 40 ; at what date is (was, or will be, as the case may be) the first twice as old as the second ? Let us suppose that we reach the date by going a years forward and afterwards b years back from the epoch to which we then come : here is a supposition which is per fectly competent to yield any result, before or after the present epoch, by properly assuming a and But we must now choose a supposi tion ; let it be that the ratio in question exists at some future time, that is, a is greater than b. In years then the thing happens; consequently, 50+(a—b)=2{40 +(a—b)} , . (1) 50 + a —b= 80 + 2a —2b . . . (2) 80 +2a-2b-50 —a+b=0 30+a—b=0 b=a+ 30.
Or any number of years forward and 30 more years back is all the answer the conditions of the problem will give, or the event took place 30 years ago. But the correctness of this reasoning is only a semblance, for the result contradicts the supposition on which it was obtained, namely, that a is greater than b. To increase 50 by the excess of a over 30 more than a is beyond the power of the arithmetician. If then it be taken that a is less than b, or that the event happened b—a years ago, we have 50— (b — a)= 2 (40 — (b— a) ) . . (3 50—b+a=80-2b+2a . . . . (4 and (4) is the same as (2); so that we arrive at the same result as before, and find our conclusion to justify the _supposition on which it was made. The steps (1) and (3) differ to the same effect as if an error of operation had been made on (4) or (2) in retracing the steps.
In the preceding, by the use of two symbols, a and b, we have enabled ourselves to obtain a correct and intelligible answer, even by the incorrect process, since we end with the determination of b—a ( =30), even where we reasoned on a—b. If however we had repre sented our unknown quantity by a single symbol, x, our first process would have stood as follows :— 50-1-x=2 (40 +x)=80+2x x = 50-80 And the answer is obviously impossible. Our second process is, 50—x=2 (40—x) =80-2x x=80-50=30.
From such instances as the preceding it may be collected that an error of interpretation, which causes us to write a—b instead of b—a, will, in finding the value of a—b, cause an impossible subtraction to appear ; and vice versa, that the appearance of an impossible subtrac tion in the result can arise from nothing but a primitive error of Interpretation in fixing the nature of that result. This point must be well ascertained by every beginner from repeated instances.
Such a result as 3 — 8 may be written 3-3-5, or 0-5 ; so that the error • of attempting to subtract 8 from 33 is reducible to that of attempting to subtract 5 from nothing. At our present point we can say that the occurrence of 0-5 shows us that the result which we supposed ourselves about to obtain was diametrically wrong in quality in our previous supposition : thus in the preceding problem we found 50-80, or 0-30, and the real answer is 30 in its magnitude, but instead of being, as we supposed, 30 years after the present time, it is 30 years before it.
Having arrived at this point, the earlier algebraists at once received such symbols as 0-5 and 0-30, which they wrote-5 and —30, into the list of algebraical objects of reasoning, calling them negative quantities, and treating them as diametrically opposite in meaning to 5 and 30, which should for comparison be written 0+ 5 and 0 + 30. These they called positive quantities. And, because, in all possible sub tractions the remainder is less than the minuend (a—b is less than a) they called 0-5 less than nothing. The fault committed by elementary writers, in beginning algebraical works by an exhibition of these definitions without the least warning of the manner in which arithmetical terms had been extended, converted the whole science into a mystery.
If we extend the notion of quantity so as to give different names to those of diametrically opposite kinds, we may call one set of quantities direct, and the others inverse. Thus property and debt, distance north and distance south, time before and time after, ascent and descent, loss and gain, progression and retrogression, &c. &c., are of different kinds ; either of any one pair may be called direct, but the other is then inverse. And in circumstances which require the addition of the direct quantity, the subtraction of the inverse is equally required : thus whatever an increase in A'e property will augment, a diminution of it will diminish ; whatever distance on a line of progression on that line will increase, retrogression will diminish. If then we have a+6 where we imagine both quantities were what we took them to be; but if it should turn out that b is of the contrary kind, we know that we should have had a—b. If we put for the quantity we thought we were using, and—b for its opposite, the ordinary rule of signs will be sufficient to make the conversions which the correction of the mistake requires. Thus if, attending only to the rule that like signs produce + and unlike signs —, we treat al-(1-b) and a + (—b) we find a+b and a —6; or, in this instance, the affixing of + or—to a quantity according as our initial supposition is correct or incorrect, leaves us with our result if we were correct and makes the necessary alteration if we were incorrect. The application of the same reasoning leads to the same conclusion in all the cases of addition and subtraction. Observe also that if any one, disputing the propriety of making the signs + and —take a new meaning, should prefer, say, to denote direct quantity by the prefix of and inverse quantity by that of §, the rule he would arrive at by induction is that like signs produce + for operation, and ¶ for interpretation, while unlike signs produce— for operation and § for interpretation ; here by like signs he would find he must mean + and + , or + and IT, —and—, or—and §, and all others unlike. His final rule then would be, use ¶ as if it were +, and § as if it were—, so that he would ultimately differ from the algebraist by the con tinual use of two new signs without any new uses or practical meanings.