IS 



On llie Rules for Jlgeliaic Mulliplicat'ion. 



plied by c—d, or taken c — d times, the evident meaning is, that 

 the difference of a and /' is to he mnltiplied by the difference of 

 f'and (Z. Now we hiive before shown that if «-/■, be taken 

 C times, tlie direction for this operation is ca — tO. But at pre- 

 sent we are to have a less result, as the multiplier c is dimi- 

 nished by d. Something more therefore is to be taken from c a, 

 than in ease 2. Now as we can only operate on the symbols 

 separately, we will now take from ca, dn, and the expression will 

 be ca — ci — dn. But it is obvious that we have now taken 

 away too much from ca, for da is the whole quantity a taken 

 d times ; whereas the cpiantity we ought to have taken away 

 was only a lessened by l, d times. It is therpfore obvious that 

 having in the expression ca — cb—da, lessened ca too much by 

 dl, or be taken (/ times, we must add to that expression db, 

 and the true direction will be ca — ch—da + id: and as these 

 symbols are of no particular meaning the rule must be univer- 

 sal, and must extend tt> multinomials as well as binomials^ for in 

 all cases the reasoning is precisely the same. 



This may also be demonstrated geometrically with great 

 care. 



G 



D, 



H 



A C B 



Let the hue AB = fl, BC = /;, AD = c and DE = <f. Then will 

 AC be =:a — b, and EA = c— J. Complete the parallelogrwn 

 ABDF and draw CG and EH through C and E parallel with 

 DAandAB. 



Then the rectangle AF will represent ca, the rectangle CF, 

 cl, the rectangle EF, da, the rectangle GH db, and the rectangle 



CE, 



