On the Rules for Algelraic Multiplication . 1 7 



the real lepresentation of that under consideration in the mind: 

 it is onlv a sort of assistance to the mental process. The line^ 

 though crooked, the angles though not right angles in the figure 

 drawn, represent straight lines and right angles, to the reasoner j 

 and still further, the figure uhich he thus i-onteniplates has no 

 determinate magnitude to his mind, and indeed cannot have 

 any; for, if it had, the deductions would not be universal, but 

 particular. It would, I believe, be of great advantage to the 

 young student to consider these matters attentively on his first 

 outset, as many false ideas would be thereby prevented from 

 entering his mind. 



Perhaps too much has been said on this even for a learner : — 

 to proceed to the common rules of what is called algebraic mul- 

 tij)lication. 



1st. + X -f is +plus, axl evidentlv means that a is to be taken 

 I times, a ^ b X c, that the sum of «^nd h is to be taken c 

 times: but as a and b cannot in this state be really added to- 

 gether, we can only vsay that v.hen they can, a is to be taken 

 c tunes, and b is to be taken c times, and the mode of writing 

 this direction shortlv is ca + cb. 



2d. That + x — is minus, admits of an equally clear de- 

 monstration or rather explanation. We must first remember 

 that no quantitv simply considered can be — minus. It must 

 be compared with some other quantity either greater than itself 

 or of an opposite direction. In the common operations of al- 

 gebra it is only used in the former sense, and a — b merely 

 means that b being less than a, it is when it can be done, to be 

 subtracted from a; a — h therefore really means t!ie difference 

 between a and b: if then a — b'i^ to be multiplied by c, having 

 first taken a, c times, we have evidently done too much, for that 

 would be the product of a only, taken c times ; what then is to 

 be subtracted from cu in order to have a true result ? evidently 

 b taken c tinres, cl, and the direction will be, multiply a hy c^ 

 and from the product take the multiple of b by c. That is in 

 the short-hand of algebra ca — cb: nor can this operation be 

 announced in tliis mode of directing our operations in any other 

 than this circuitous wav; but in practice the operator would 

 eertainlv go a shorter way to work, and having first subtracted 

 I) from u, would multiplv the remainder by c. 



3dly. — X — ]Jroduces -f . 



This most perplexing rule to every clear-headed student, and 

 which has been rendered onh- more obscure bv every attempt I 

 have yet met with to explain it, (for denuMistrations I cannot 

 t-ail them) may be equally explained by following one step 

 further the above train of reasoning. If a — bh to be inulti- 



Voi. 45. No.201. Jc«. 181.5. B plied 



