t 



NEGATIVE QUANTITIES. 



NEGATIVE QUANTITIES. 



nothing more than arithmetic, instead of introducing those new 

 hsUsOiuua which are th basis of the separate science : to that 

 algebra. iuU*d of being nystemntically learnt, it collected by alow and 

 often dubious steps from arithmetical examples, in which the rule* of 

 operation of the former science an employed, preceded by the prin- 

 ciple* of the Utter. Few, therefore, acquire a real perception of the 

 mi-suing of the subject, except those who study mathematics to great 

 extent. It u matter of notoriety that difficulties attend the beginner 

 in algebra of a nature totally different from those which are found in 

 geometry ; <o that while a person who has read a few books of Euclid 

 may be imagined capable of writing an intelligent commentary on what 

 he know*, another who baa mattered a common elementary treatise on 

 algebra ia conscious only of a great increase of working power, with a 

 glimmering of principles which owe their reception more to the never- | 

 failing accuracy of their results than to native evidence or logical i 

 deduction from easily admitted truths. 



We must premise that, aa in all other cases where tin- lirst principles 

 of a science have been matter of dispute, it by no means follows that , 

 one riew of the subject is the most easy to every mind. Something 

 must depend on the intellectual constitution of the individual ; ami if 

 this be most probably true in geometry, the remark applies with still 

 greater force to algebra. 



The first abstraction which meets us in arithmetic follows the | 

 transition from actual magnitudes (concrete numbers, so called) to ' 

 their numerical representations. We then find general properties of ' 

 numbers, in which we learn to consider number independently of a 

 specific concrete unit. Thus weseein7+5 8 = 7 8 + 5a relation 

 equally true, whatever may be the nature or magnitude of the unit. 

 When we drop the concrete number and rise to the abstract, we gain 

 (something more by the transition than immediately appears ; and this 

 the student should particularly note, because some of the succeeding 

 difficulties which attend the passage into algebra are very similar in 

 character, though preceded by a stranger and harder process. The 

 operation of multiplication takes a power and a property which it had 

 not before : thus if we denote concrete number by Roman numerals, 

 and if we speak of yards, it is clear that 5 x VII = XXXV, or 7 yards 

 taken 5 times is 85 yards. But we may not, therefore, say that VII x 

 I X V, for VII x 5, the number 5 multiplied by 7 yards [MOLTI- 

 PLICATIOS], is an incongruous and unmeaning set of words, and it 

 would be equally improper to say that it it and it u not 35 yards. In 

 abstract numbers no such caution is necessary : 7 x 5 and 5x7 are both 

 the Fame. If men had never considered number independently of 

 magnitude measured or repeated by it, the arithmetician would have 

 confounded VII x 5 and 5 x VII, because he would soon have found 

 that no false results would have sprung therefrom; while VII x 5 

 would have been a sort of impossible quantity, useful in practice and 

 difficult in theory. 



We are now on the ground of abstract arithmetic, and on examining 

 the four fundamental operations, we see no difficulty in either addition. 

 multiplication, or division. So soon as we have mastered the subject 

 of frac- have clearly admitted the introduction of a part of a 



repetition [Mi'LTirucATrox], we say as follows : Let a and b be any 

 two numbers or fractions, and a + b, ab, and a : b must be real 

 numbers or fractions, assignable by demonstrated operations so soon 

 a* a and b are assigned. But there is still a restriction upon the 

 possibility of subtraction: a A has no imaginable existence, unless 

 a be greater than 6 ; when a-b, the magnitude of ttb vanishes 

 entirely, and when a is less than 6, the direction to perform o b a 

 just the same as asking for a part which shall be greater than the 

 whole of which it is a part. If we confined ourselves to particular 

 arithmetic, in which all numbers used have specific values, it would 

 most likely he thought of no use to carry the subject further, and in 

 one point of view correctly ; that is, it would be of little moment to 

 deduce methods by which an individual so careless as to write down 

 and operate upon such a symbol as 3 - 4 might be enabled to arrive at 

 a subsequent correction of the mistake which a glance at the symbol 

 should show him he has made. But when wo uxc general symbols of 

 number, we are liable to mistakes of two kinds, both dependent upon 

 our liability to invert the order of terms of which the less should be 

 subtracted from the greater. 



. we may mistake the nature of the -|u n.iu which results: 

 thiw if it be part of the conditions of a prohlnn that I pay a and 

 receive th, end if the application of the conditions requires that I 

 should state how much I gain or lose, the answer should be either a 

 lose of (a b) or a gain of (b a), according as a or b is the greater, 

 ive then the choice between adopting one of these with the 

 chance of being entirely wrong, or of working the problem in two 

 distinct ways. And if it should l,app-n that thu conditions of the 

 problem present this alternative in six distinct instances (and some- 

 time* it li would be no lew than 64 oases of 



oloti :illy (peaking, essentially different in th 



of obtaining the answer, whether the answers obtained be the same or 



Secondly, we may make an error of the same kind in the details of 

 operation. For instance, suppose we have i '< <, 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 + (-), we may 

 be in error ; for if 6 be less than c, the proper alteration is a - (c - b}. 



It is evident that both species of mistake ore 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 ft to be lost where 6 a ia really gained, and 

 if the problem, for instance, require the result of the proceeding to 

 be annexed to a lost *, we shall suppose there is altogether a lues of 

 x + (a i), whereas it should be a loos of only .r (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 + (ab) as a 

 loss, we shall evidently suppose it to be more of a loss than .r, or that 

 a 6 is lost besides; whereas, had we looked at ar (6 o), we should 

 have inferred that there is a less Ion than .r. 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 results. 

 We shall here only state the truths .it which he will linally arri\< 

 an example of each. The beginner cannot, as the proficient may do, 

 see a sufficient reason for these results in the common rules of alge- 

 braical operation; and we should <l-il.t that anything but a huge 

 number of examples would serve to give him 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 no more than those rules which apply in the rational 

 process. 



Thus if x-t-a- b, wrongly taken to be ,T+(a 6), 6 being greater 

 than a, be multiplied by .c + p q, wrongly taken as x + (j> q), q being 

 greater than p, we find as the (supposed) product 



i (a-b+p-q) x + (al) (p-q), 

 to which the application of the common rules gives 



a 3 + ax bx +#.c qx + ap bp ay + bq, 



precisely the same as the product of .r + ab and x -rfy. The reason 

 of this is as follows : In 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 



+ & (c d), or o + t (0 + c rf) 



=0+6 c + d 



(a-b) (c-rf) or (O + a-4) (0 + c-rf) 

 = + oc 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 + ab wrongly 



as # (& a), we see that when we come to add this, say to r, w> 



c +{*-<*-)} 



in which a, before it is disengaged, must come under the signs + , 



or, if the phrase be less objectionable, under the application of the 

 rules to these signs, successively. But the correct process would give 



e+ {* + (-)} 



in which a falls under the application of the rules to + + ; and such 



application to + + gives the same result as that to H , 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 performed, 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. 



\Vc must here pause to remind the reader that errors, however 

 palpable and admitted, are not necessarily productive of error. Tntc 

 reasoning, on Inn principle*, MH.II I, ad to irnlh ; but if for true we 

 ilse, and for truth falsehood, we have no longer any right to say 

 mutt, but only ,/".x? /.r..6uMy trill. If then we LUI .-how of a particular 

 class of errors that, used in a certain way, the result* agree with those 

 of true reasoning on true principles, we may demand the 

 errors as demonstrated means of finding truth. Tin- 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 oj.rnti"] 

 not having previously vitiated the i ''" funda- 



mental objects of operation (equal: 



li in, will produce no falsehood in the result. 



2. Let us now examine the consequences of the error oi 

 tion. The effect of this is, that we write a b in bead of bti, but at 

 the same time we suppose the quantity of which we are thinking to 

 be of a diametrically op]K>.<jtc thai, which it ought to have. 



But also at the same time we add tbi-t M-mhol where we should cub- 

 tract it, and riee rend ; sothv should take n--/i, and add, 



giving c + (a b), we make one mistake in taking ba, and another in 

 subtracting, giving e (60). \\ h< u mere rules come to be applied, 

 1 the same result from both, namely, c + ab and e b + n. \\ c 

 might then so manage as to elude the. n> tation of tli nega- 



