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XXV. On the Evles for Algehrmcal Muliiplicat ion. 

 By Mr. J. Dillon. 



To Mr. Tilloclu 



Sir, — iriAViNG seen in your Magazine for January some re- 

 marks by Sir H. Englefieki, tending to explain the algebraical 

 theorem, at once so necessary for the yoir.ig mathematician to 

 master, and yet so diiiicult for him fully to understand, that a 

 negative quantity, ujultiplicd into a negative quantity, gives a 

 positive result, I beg leave to add a few observations which have 

 occurred to ma upon the subject, and which may perhaps in some 

 degree tend to ;,>iace the niatter in a clearer point of view. 



The signs + and — appear too generally (at least in ele- 

 mentary works) construed to m.ean plus and minus; a sense 

 which, though perhaps always included in, does not appear to 

 constitute the whole of their definition. The sign + signifj-ing, 

 in fact, that the term to which it is prefixed is positive, and the" 

 sign — that such term is negative, that the one should be plus, 

 (or the Gbject of addition,) and the other minus, (or the object 

 of subtraction,) when addition or subtraction with other quan- 

 tities is in question ; these are rather consequences flowing from, 

 than essential parts of the nature of, such signs of + ar.d — . 



The fallacy of considering + and — as merely meaning phis 

 nnd minus, will plainly appear where multiplication or division is 



intended, as — a x l, or-^-r, v.here it is evident neither plu^ 



-r I.' 



nor minus can be meant by the signs + and — ; au^i it is in thi«! 

 fallacy, as it appears to me, that all the difriculties of tlie present 

 t]ue-.tion have their origin ; for, by always affixing the sense of 

 p.silive and negative to these signs, nearly all the obstacles which 

 fctnpede the progress of the learner on this subject will vanish. 



I scarcely need previously to observe, that the algebraist is as 

 conversant with the idea of a negative as of a positive ([uantity. 

 Considerable confusion appears, however, to have arisen from at- 

 tempts to render this idea familiar to minds not accustomed to 

 abstract reasoning. Thus, therefore, it is frecjueutly represented 

 that — a is not so much the negative quantity, a, as it is the 

 positive quantity a with a mark alhxed to it, signifying that it 

 is to be subtracted from some other quantity either actually 

 known, or to be discovered ; whereas, in fact, it should be con- 

 eidered as strictly a negative quantity, capable of destroying or 

 counteracting: a positive quantity of equal value, when it comes 

 n\ contact with such, and existing in the mind in a way perhaps 

 eomewhal tiniilar to the ideas of darkness, silence, or vacuum, 



which 



