INTEREST. 



_ mp, 



and a will be = »» 



I VI — p 



p 



Dr. Mafkelyne, in his introdudion to Taylor's loga- 

 rithms, has, with the view of facilitating thefe operations, 

 had recourfe to the tables of logarithmic fi nes and tangents, 

 by putting r = /"- . A, 1 + *• = fee' A, I + /■'- = fee.'-, 

 B, &c. ; thus, in Cafe \, the value of r becomes = - x 



for - being 

 P 



— — X a. Inlikemanner, inCfl/fi2, (1= c=^-,,^— - 



. B 



becomes = 7 — - „ — 



fee. B — 1 » . 

 or m . «= . A X f - <= . B. But though the expreffions 

 appear more limple, I do not know that in general the opera- 

 tions will be found to be much facilitated by thefe means. 



In the preceding cafes money has been fuppofed to be 

 improved or difcounted yearly. But all thefe different 

 theorems may, however, be applied to the folution of cafes 

 which require money to be Improved or difcounted at 

 (horter intervals. Thus, fuppofing it were required to de- 

 termine the amount of 50/., when laid out half yearly, to 

 accumulate at 4/. per cent, for the term of 30 years : /> is 

 = 50, n = Z X 30 ^ 60, r th" int ereft of I/, for half 

 a year = .02, and m ~ p i -r r^' becomes = 50 x 

 i^ '> = 164.05. If improved quarterly it will be = 

 50 X I 01' " '« = l6j. 



A wain, if it were required to determine the amount of 

 5/. laTd out half yearly, to be improved at 4/. per cent, during 



a term of 10 years. In this cafe ; 



will 



proved quarterly, 



rill be 



[21.486, and if it be im- 



InUke manner, if money be difcounted at {horter intervals 

 thanayear, the prefent value will be obtained from theexpreffion 



. ' ; thusjlettheprefentvalueberequired of 20/. payable 

 I 4 '■ ' ' 



at the end of 40 years, fuppofing money to be difcounted every 

 half year at 61. per cent. I^re n becomes = 2 x 40 = 



80, r = --— = -OS' 3nd confequently v = -^^'^.i = 



-=-=-CT, = 1.8795. 

 1.03 ' 



Let h be any fraftion of a year in which money is to be im- 

 proved or difcounted, and the amount or prefent value may aU 

 •ways be found from the amount or prefent value when money 

 is improved or difcounted yearly, being the fame with fuch 



amount or value at - intereft for kn years. Thus, the 



amount of 50/. improved half yearly at 4/. percent, for 30 

 years is the fame with the amount of 50/. improved yearly 

 for 60 years at 2/. per cent. ; the amount of j/. per ami. im- 



proved half yearly for 10 years at 4/. prr rent., is the fame 

 with the amount of 5/. per ann. improv.d yearly for 20 

 years at 2/. prr cent. ; and 10/. difcounted half yearly for 

 40 years at 6/. />-r cent, is the fame with 10/. difcounted 

 yearly for 80 years at 3/ per cent. The different values of 

 annuities, when payable yearly, half yearlv, quarttrly, or 

 at fhorter intervals, may in the fame manner be deduced' from 

 the preceding theorems, but thefe are explained in the article 



AXNUITIEN,. 



M. De Moivre, M. D'Alembert, and fome others, inftead 

 aking theintercft of 1/. for the kh part of a year = — 



have chofen to make it = i + ,'* _ i, and hence the 

 amount of i/. in n yc^rs, or its prefent value at the end of 

 n years, will be the fame, whether money be improved or 

 difcounted ye.irly, or at fliorter intervals. But the amonnt 

 of i/./ifra«rt. will be to its amount when improved yearly 

 ill the conftant ratio of r to i x i +P' — i, and the 

 value of an annuity of it. will be to irs value, when 

 paid yearly, ir.verfely as r to jf , i -f- rM — i, whatever 

 the length of the term during which the money is to 

 accumulate, or the annuity is to continue. Now, it is 

 well known that the difference between the values of annui- 

 ties payable yearly, and their values, when pavable half- 



yearly, quarterly, or at fliorter intervals, is alwa 



:;n'ened 



the term is extend d, fo that if the annuity be perpetual, 

 the values will be the fame, whether the payments are made 

 yearly, orin any fractional part of the year ; which can never 

 be the cafe on the fuppofition alove menlioned. When n is 



= loa 



=^— -M02 



~l. 



-)■ r, an annuity pay. 



able every ith part of the year, will be equal to the perpetuity, 

 fo that an annuity payable haf -yearly, at 5/. per cent, for 

 90^ years, or quarterly for 80 years, will be equal to the 

 fame annuity payable yearly for ever. And at dl.per cent. 

 the annuities will be of equal value if the term be only 72.1 

 years in the one cafe, or 6j.; years in the other. But while 

 thefe rules, when the term is very long, give the values of 

 annuities payable at fhorter inter>-'als than a year too high, 

 they alwa; s, on the contrary, and efpecially w hen the term is 

 very long, give the amount of a fum much too low. Thus 

 fuppofing one penny to be laid out at 5/ percent, compound 

 intered: at the birth of our Saviour, or 16 10 years ago, it 

 will accumulate, when money is improved yearly, to a fum 

 which is equal to 381,860,000 globes of fohd'gold, each 

 equal to the earth in magnitude. When improved half-yearly, 

 to a fum which is equal to 1,121,470,000 of fuch globes, 

 and when improved quarterly, to a fum which is equal to 

 1,945,680,000 fuch globes ; fo that although in this long 

 time the accumulation is nearly three limes greater v^hen 

 money is improved half-yearly, and more than live time* 

 greater when improved quarterly than it is when money is 

 improved yearly, yet according to the fuppolition that 

 l+H* is the amount of i/. in the ith part of a year, or 



1 -(- r' ' = its amount in n years, its accumulation will be 

 the fame, whether money is improved yearly, hulf-yeariy, or 



quarterly. It may eafily be fhcwti that 1 + r I' — i is al- 

 ways lefs than - , or, in other words, that this exprcflion 



does not give tU- full intereft of i/. for the /th part of a 

 year. It is no wonder, therefore, that any theorcnia de ivcd 

 from a principle fo enooeous, fhould, like the preceding, 

 lead to concluficns which are not only incorrect but abfurd. 

 M m J Ti» 



