« Dr. Price and his Followers" 293 



—The series -^ + —L-g + -^=L^3 -=^» ^^ known 



1 T r \^r) l + r] 1 + r\ 



to express the value of an annuity of ll. for n years = 

 . For the same reason, if — th part of ll. be 



»■ r.l +r] 



1 J. ± 



paid a times in each year, the series — — r +^==.2 -f 7—^3 ... 



i I 1 



... -f- - = ^na will express the value of ll. 



a 



per annum payable every — th part of a year for n years. 



In the Preface to Taylor's Logarithms, Dr. Maskelyne as- 

 sumes r to be the interest of 1/. for one ti7ne, and supposes the 

 payment of the annuities to be made so many ti7nes. This is 

 in fact the same as Dr. Price, and I believe every other per- 

 son since his time have done, who have had a due knowledge 

 of the subject. It necessarily follows that temporary annuities 

 payable at shorter intervals than a year must be more valuable 

 than annuities payable yearly; and in consequence, Dr. Price 

 states the value of an annuity of ll. payable half-yearly for 

 five years at 4 per cent, to be 4-*4913, and its value payable 

 yearly to be 4*4518; or in other words, the two half-yearly 

 fractions in any year being greater than the single fraction in 



1 i 



the corresponding year, that is -p— 4- -^^=[2 being greater 



1 i i 1 



sum of the former must be greater than the sum of the latter. 

 But instead of comparing the sum of the two half-yearly terms 

 with the corresponding yearly term, Dr. Young compares the 

 second of each half-yearly term with half the corresponding 



i 1 1. _ 1 



vearlv term, or :r=?=r, widi -r^ ... ==:4 with :=:5 &c., and 

 '' ' To2\ 1-04 T02! 1-041 



by this means finds the discount taken half-yearly to be greater 

 than the discount taken yearly ; which if true, would make 

 the value of an annuity payable half-yearly to be less than its 

 value |>ayal)le yearly, which is self-evidently wrong*. — Dr. 

 Price is said to "have fallen into error by " adojjting tlie legal 



restraints 



• It may be easily proved that tiic value of the second half-ycarJy pay- 

 ment ill each year is always less than the value of half the payment at the 

 end of that year; and on the contrary that the value of the /(V.t/ half-yearly 



payment 



