I9S AS.ITHMETIC. 



2. Multiply twice the first payment by the rate, and call 

 this the second number. 



3. Divide 



The rule, given above, is the same as Mr. Malcolm's, cxcci^fe 

 that it is not encumbered with the time before any payment is 

 due, that being no necessary part of the operation. 



Demonstration of the Rule. Suppose a sum of money 

 to be due imraedrately, and another sum at the expiration of a 

 certain given time forward, and it is proposed to find a time to 

 pay the whole at once, so that neither party shall sustain loss. 



Nov/, it is plain, that the equated time must fall between those 

 of tlie two payments ; and that what is got by keeping the first 

 debt after 4t is due, should be equal to what is lost by paying the 

 second debt before it is due. 



But the gain, arising from the keeping of a sum of money after 

 It is 'due, is evidently equal to the interest of the debt for tliat 

 tune. 



And the loss, which is sustained by the paying of a sum of 

 money before it is due, is evidently equal to the discount of the 

 debt for that time. 



Therefore, it is obvious, tl^at the dcbtot rpust retain the sunit 

 immediately due, or, the first payment, till its interest shall be 

 equal to the discount of the second sum for the time it is paid be- 

 fore due ; because, in that case, the gain and loss will be equal> 

 and consequently neither party can be the loser. 



Now, to find such a time, let a = first payment, h •=. second,^ 

 and / rr time between the payments ; r =r rate, or interest of iL 

 for one year, and y rr equated time after the first payment. 

 Then arx = interest of a for x time, 



J htr — Irx J. i r 1 r I 



and =: discount of L for tiie time 



I -^-tr—rx 



Bat. 



