917 



NEGATIVE QUANTITIES. 



NEMESIS. 



913 



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 "6. 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 6. In a b years then the thing happens; 

 consequently, 



(1) 

 (2) 



80 + 2a-26-50-a + 6 = 

 30 + a 6=0 



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-(6-a) = 2 (40-(6-a)) (3> 

 50-6+0=80-26 + 2(1 . . . . (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 6 a 

 ( = 30), even where we reasoned on a 6. If however we had repre- 

 sented our unknown quantity by a single symbol, x, our first process 

 would have stood as follows : 



2;= 50 80 

 And the answer is obviously impossible. Our second process is, 



50-x=2 (40 *) = 80 2.r 

 a:=80-50 = 30. 



From such instances as the preceding it may be collected that an 

 error of interpretation, which causes us to write a 6 instead of 6 a, 

 will, in finding the value of a 6, cause an impossible subtraction to 

 appear ; and vice versd, 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 as3-8maybe written 3 3 5, or fi ; so that the 

 error of attempting to subtract 8 from B is reducible to that 

 of attempting to subtract 5 from nothing. At our present point 

 we can gay that the occurrence of 05 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 5080, or 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 6 and 30, which they wrote 5 and 30, into 

 the hat 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 + 5 and + 30. 

 These they called positive quantities. And, because, in all possible sub- 

 tractions the remainder is less than the minuend (a 6 is less than a) 

 they called 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's 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 + b 

 where we imagine both quantities were what we took them to be ; but 

 if it should turn out that 6 is of the contrary kind, we know that we 

 should have had a b. If we put + 6 for the quantity we thought we 

 were using, and 6 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 



a + ( + 5) and a + (5) 

 we find 



a + 6 and a-b; 



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 H, and inverse quantity by that of , the rule he would 

 arrive at by induction is that like signs produce + for operation, and If 

 for interpretation, while unlike signs produce for operation and for 

 interpretation ; here by like signs he would find he must mean + and + , 

 or + aud % and , or and g, and all others unlike. His final rule 

 then would be, use If 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. 



In the operations of multiplication and division the rule of signs is 

 thus shown : It is said that two negative quantities multiplied 

 together produce a positive quantity, which means that a mistake of 

 direct for inverse, or vice versd, made in both the terms of a pro- 

 duct, produces no mistake in the product, when the latter is formed 

 by the usual rules. Thus, if a, which should be a: y, has been taken 

 to be y x, and if b, which should be v w, has been taken to be 

 w v, the algebraical product 



(ro 1>) (y a;) or wy wx vy + vx 



at which we arrive in the mistaken process, is precisely the same as 

 (v w) (x y) or vxvytcx+wy 



at which we should have arrived in the correct process. 



The first step then from arithmetic to algebra is made by the follow- 

 ing definitions : 



1. Quantities are distinguished into positive and negative, which are 

 to be considered as of diametrically opposite kinds ; and common arith- 

 metical quantities ( abstract numbers without signs ) are to be con- 

 sidered as positive. 2. The rules of arithmetical algebra are to be 

 applied to the extended algebra, and in all cases in which the latter 

 presents a case unknown to the former, the rule of signs already known 

 in the former must be applied. The extension which takes place in 

 the terms less than nothing, &c., will be considered under the word 

 NOTHING. 



The preceding extension gives an extended meaning to all the terms 

 of operation ; thus addition is no longer the simple arithmetical pro- 

 cess, but a compound operation, first reducing a multiplicity of signs 

 to one alone, and then following the direction of that sign ; and the 

 same of subtraction. Thus a ( 6) is a + b. It may be asked then 

 how we are to trace our steps throvigh any problem so as to form its 

 equation out of symbols which seem to have various meanings ; for it 

 might appear as if the + of algebra were either the + or of arithmetic, 

 as the case may be. The answer is very simple : since the extended 

 algebra is no more than arithmetic in its actual operations, how- 

 ever the meaning of those operations may be extended, we may be 

 sure that if we assign a particular case of a problem, and treat it 

 entirely as in arithmetic, we are, though with one case only in view, 

 performing upon limited symbols (limited because we think at the 

 time only of a limited meaning) the same steps which we should have 

 to follow if we could, by one act of the mind, grasp the symbols in 

 their utmost generality. 



NEGATIVE RADICALS. [ORGANIC RADICALS.] 



NEGOTIABILITY. [BILL OF EXCHANGE.] 



NEMEAN GAMES, one of the four great national festivals of the 

 Greeks, derived their name from Nemea, a village in the north- 

 eastern part of Argolis, on the borders of the Corinthian territory. 

 They were celebrated under the presidency of the Corinthians, Argives, 

 and inhabitants of Cleonse (Scholiast, on Pindar) ; but in later times 

 they appear to have been entirely under the management of the 

 Argives. (Liv., xxxiv. 41.) They are said to have been celebrated 

 every third year ; and sometimes, as we learn from Pausanias, in the 

 winter (ii. 15, 2 ; vi. 16, 4). 



The Nemean games were (according to some legends) first established 

 by the Epigoni, in memory of Opheltes (Schol. on Find. ; Apollod., 

 iii. 6, 4; Pans., x. 25, 2) ; others attribute their foundation to Hercules, 

 in honour of Zeus, in consequence of his victory over the Nemeau lion. 



The games, at first of a warlike character, afterwards consisted of 

 horse and chariot racing, boxing, wrestling, archery, and casting the 

 spear. The crowns bestowed on victors were made of parsley [OLYMPIC 

 GAMES.] 



NEMESIS (Nejunns), a female Greek divinity, who appears to have 

 been regarded as the personification of the righteous anger of the gods. 

 She is represented as inflexibly severe to the proud and insolent (Pans., 

 i. 33, 2.) ; but she also sets right the inequalities of fortune, and 



