13S On the. Rales for Algebraical Mulliplication. 



which may be considered as abstract negative ideas, expressive 

 of the al)sence of light, sound, or matter. 



To apply these premises to the subject under consideration, 

 namelv, the multiplication of algebraical quantities, under all 

 the varieties of the signs + and — : of this multiplication there 

 are three cases. 



1st. When the terms are both positive, + x +• 

 2d. Wlien only one of the terms is positive, — X + . 

 3d. When both the terms are negative, — x — . 

 For ihejirsl then, to take an instance, + o x + /' ; "'^ have 

 only to remember the well known principle that the multiplier is 

 merely an abstract quantity, expressive of the number of times 

 the multiplicand is to be added within itself, and we shall im- 

 mediately perceive that the result must be + , as it is merely pro- 

 posed that + a should be taken h number of times, without any 

 alteration of the signs, which are indeed expressly uffirmtd by the 

 iign -{- affiled to b. 



For the second case, —a x + b; it is equally evident that 

 — a taken /; number of times must on the other hand always 

 remain — , whatever may be the value of b. 



For the third case, —ax — b, where tlv difficulty is sup- 

 posed to rest, it may be previously remarked that if — a when 

 multiplied (as in the second case) by b or •{- b gives a nigative 

 result, then may it beforehand be expected that this same — a 

 when muitipHed by — b will give a contrary, that is a positive 

 result. 



It is a well known position in logic that two negatives make 

 an affirmative ; to say that a thing is vot vnt so, is in fact but a 

 more circuitous manner of saying that it is so, and exactly this 

 process appears to take place in the case before us. The result 

 of — G X — Ms + o^ for this reason ; a X b = ab, and the 

 negative sign of the a is (if I may so express it) itself jKgatedhy 

 the negative sign of the 0. The quantity a had, we must sup- 

 pose, become negative by some previous process ; the rei;ersal 

 therefore of this sentence of negation must be as necessarily 

 the consequence of its being multiplied into a rieg, iiire i\uantity, 

 as the continuing subject to that sentence would have resulted 

 from its being multiplied into an affirmative or positive quantity: 

 in other words, the sign — prefixed to the b is, in fact, the nega- 

 tion of the sign belonging to a, (the quantity to be multiplied,) 

 whether the sign of such quantity be positive or negative 



The origin of the error, and tlie consequent existence of the 

 difficulty in question, appear to be this : that the affirmation of 

 a positive quantity, (as a x + b,) and the negation of a negative 

 quantity (as — a x — /',) are supposed to be the contrast, or 

 antithesis of each other j whereas, in fact, so far from being op- 

 posed. 



