270 Olservations on Equation of Payments; tending to prove 



treaties appertaining to scientific, that we at present possess on 

 the subject. — That the rule, however, is susceptible of further elu- 

 cidation, I shall endeavour to exemphfy in the sequel of consi- 

 deration. Witli a view of preparing to illustrate my position, 

 I shall abstract both the rules in their analytical form (which I 

 shall do from Bonnycastle's Arithmetic), and subsequently their 

 practical enunciation : 



1 /j , a+b\ /a.r.t + a + b 





Here a. the first debt. b. the second. /= time between the 

 payments, r. the rate ; and x. the equated lime from the first 

 payment. 



_ BT + dt 

 ^— D + d ' 



Here d= the first debt, and its term t. D= the second, and its 

 term T. x. the equated time. 



The first expression is Malcolm's rule, and the second is that 

 which I may properly term the old rule, as adopted by Cocker, 

 and nearly all the authors on practical arithmetic who have fol- 

 lowed in succession after him. It depends on the supposition, 

 " that the sum of the interests of the debts payable before the 

 equated time from their terms to that time, should equal the sum 

 of the interests of the debts due after the equated time from that 

 time to their terms." But which also includes this extension ; 

 viz. " that the interest of the sum of the debts from the equated 

 time to the end of the final term is equal to the sum of the in- 

 terests of the debts from their several terms to the end of the 

 final term*." 



And this I conceive to be the true intent and principle of the 

 rule ; namely, an equation of the party's separate right and in- 

 terest in the debts, so that neither shall sustain loss; — or where is 

 the utility of determining an e([uated time at all? and which also 

 I take for granted was the primary object contemplated by the 

 founders of it. But this object can never be accomplished in 

 Malcolm's rule ; although, from the plausibility of making a pro- 

 per allowance of discount, those who argue in favina- of the rule 

 maintain that it is: but it is an absolute fact, that one party will 

 be injured by it, md that party is alwavs the debtor. 



The main reason why Malcolm's rule fails in giving the true 



* Fw let there be two payments: a. the first, and b. the second; r.t.x., 

 denoting the same f|iiaDtitics as hefore ; tlicn (a.r.t) will he the entire in- 

 terest arisins from the debts at the eTuI of the last term ; and to equate tl)is 

 for the time(f— .r) we should have {a. r.ty— {a.r.t + b.t.7-—a.r.x+li.i-..v), and 

 by reduction ({a + b). r.r) = (6.r.i); or, (Jb.r.ti-,(a,r.x + brx))=0; from whicli 

 the condition is manifest. 



equated 



