'm^ 
 
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 THE UNIVERSITY 
 
 OF ILLINOIS 
 
 LIBRARY 
 
 "EST 5 
 
 
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 -,■■.''!,.. "t .V,-' ."■'):■,■ -".•.■'
 
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 y/^<-: {''-' 
 
 INVESTIGATION 
 
 OF THE 
 
 ERRORS 
 
 OF 
 
 ALL WRITERS ON ANNUITIES, 
 
 IN THEIR 
 
 Valuation of Half-yearly and Quarterly Payments^ 
 
 INCLUDING THOSE OP 
 
 Sir Isaac Newton, Demoivre, Dr. Price, Mr. Morgan, Dr. Hcttow, 
 
 §c. Isc. 
 
 WLiti) Cables, 
 
 SHOWING THE CORRECT VALUES WHEN PAYMENTS ARE 
 MADE IN LESS PERIODS THAN YEARLY, 
 
 AND A SPECIMEN OF 
 
 »^ Set of Tables on a new Principle, 
 
 (NOW IN THE PRESS) 
 FOR THE VALUATION OF 
 
 LEASES, ESTATES, ANNUITIES, CHURCH LIVINGS, 
 
 OR ANY INCOME WHATEVER. 
 
 By WILLIAM ROUSE, 
 
 Autlitr of the Doctrine of' Chancis ; and lUmarkM on Freehold and Copyhold Land, 
 AdiMisoiis, i^c. 
 
 LONDON: 
 
 Printed by Gye and Dalne, 39, Oracechurch-Strut, 
 
 ron THE AUTHOR, PUBLISHF.n BY LACKINGTON, ALLEN, AND CO. 
 
 FINSBIjKY SQUARK. 
 
 AND MAY BE HAD Of ALL BOOKSELLERS. 
 
 1816. 
 
 Price is. iSd.
 
 AN 
 
 INVESTIGATION, 
 
 XT may appear a bold assertion for an obscure individual 
 to make, that all the mathematicians, u'ho have written 
 OH the subject (even including Sir Isaac Newton and 
 Demoivre), have given erroneous rules and theoretns 
 for the valuation of half-yearly and quarterly pay- 
 ments of annuities or incomes, whctherybr life, for yearSy 
 or for ever. But truth is equally valuable, from whatever 
 quarter it may How, and mathematical truths seem to have 
 an advantage ovor all others ; for as they admit of demon- 
 stration, Ihoy (lisirm the sceptic, and must be equally 
 received without dispute, both by the enemy and the 
 friend. 
 
 When it is considered, that much the greater part of the 
 income of the whole country is received in \esa periods than . 
 yearly, it must sunly lit- of importance to have correct 
 notions of the dillercncc in the values betweoa ijuch pay- . 
 A 2 mcnts
 
 ments and yearly payments. If the case be applied to tli^ 
 national debt, by supposing the interest 32 millions of 
 pounds per annum, we shall find the following difference in 
 the amounts, whether the same be paid yearly^ and in- 
 creased at 4 per cent, per annum^ or paid in 4 quarterly 
 payments of 8 millions each, and increased at 1 per cent, 
 per quarter, being the usual mode of payment. 
 
 In 25 years the difference will be 31 millions of pounds. 
 In 50 years the difference will be 167 millions of pounds. 
 In 100 years the difference will be 2400 millions of 
 pounds ; — and this difference must increase ad injinitum . 
 
 But, by the theorems and tables now in use, we are 
 taught, that although there is a difference between the 
 values of two annuities, where one is paid yearly and the 
 other quarterly, if they are to continue 20 or 30 years, 
 yet if the same annuities are to continue for 100 years, or 
 for ever, there is no difference at all ! ! while, at the same 
 lime, it is admitted as self-evident, that the amount and the 
 value must be proportionate to each other. 
 
 In tracing these errors, it is curious to observe the incon- 
 sistency of each succeeding writer. Sir Isaac Newton and 
 Demoivre began with a mngle error, which increases in 
 every stage of computation, in the same proportion as the 
 difference between the yearly and less periods than yearly 
 amounts and values increases ; but at the value of the 
 
 perpetuity
 
 perpetuity it reaches its maximum, and then it will be in 
 the same proportion to the value of the perpetuity (if paid 
 yearly) that the first year's error was to the first year's 
 value (if paid yearly), as will be seen in Case I. 
 
 Dr. Price and Mr. Morgan, after asserting that this is 
 " sufficient to prove the fallacy/ of Mr. Demoivre's 
 " method of solution,'^ actually combine this very error 
 with another of much greater magnitude, and ground 
 theorems (in the same way as Dr. Hutton has done) on 
 such combination of errors, and which never can reach 
 A MAXIM u>r. Then ]Mr. Baily, who says, " he has been par- 
 " ticular in making these observations," having " no where 
 " seen the facts set in their proper point of view ;" after 
 animadverting on Dr. Price's rule, and declaring that 
 " the present value of an annuity, payable four times a 
 " year, must in all cases be greater than when paid only 
 " once a year, " gives an example from his own theorem, to 
 prove, that a perpetual income of four pounds per annum, 
 if the inlere^t be paid yearltj, is worth £100; but if the 
 interest be paid t/uarterlij, it is worth only £98: 10:3!!! 
 
 A like inconsistency appears in a work published under 
 the inviting title of " Sir Isaac Newton's Tables;" but 
 whoever the author of sucli part of the Tables might be, it 
 was cruel in him to make Sir Isaac give one rule in page 
 61, and forget it quite so soon as to begin a table in the 
 very next page, computed in some diflerent manner, but 
 
 how,
 
 6 
 
 how^ we are left to discover ; yet, as the calculator has 
 favoured us with the amount of £1 per annum, if improved 
 half-yearly at what he calls the same rate (4 per cent, per 
 annum) we are enabled, by comparing them with the 
 amounts, if improved 2/ear^y, to find the difference, and which 
 will always be proportionate to the difference of their values ; 
 for £200, to be received at the end of any number of years, 
 must always be of double the present value of £100, to be 
 then received (computing at the same rate of interest. ) Now 
 from these tables, in which the editors say, " iheij have not 
 ^^ been able to discover a single error^'' the following 
 Is the difference in amounts between yearly and half-yeasly 
 pa;^ents, at (what is called) 4 per cent, per annum, with 
 their difference of values; against which I have stated 
 what should be the true difierence of values. 
 
 niflSerence in 
 No. of amounts of yearly 
 
 Years. aud half-yearly 
 
 paymeuts. 
 
 £. s. d. £. «. d. £. s. d. 
 
 25 12 10 110 4 10 
 
 .50 3 9 I 4 9 9 
 
 100 49 10 4 19 7 
 
 So that, where the difference in the values should be greater, 
 they make it less ; but what is more singular, they con- 
 tinue their amounts to 150 years (when the difference 
 becomes £532 ; 15 : 0, and which requires a difierence of 
 £1:8:3 in the present value to meet it), yet they make 
 but one step ia their values from 100 years to eternity, 
 
 although 
 
 Difference in 
 
 True 
 
 \'aliics by tliLir 
 
 Difference <if 
 
 Tables. 
 
 Values.
 
 cdthough the adjoiiiiag column points out most clearly, that 
 the jiroportion of diflereoce between yearly aiid half- 
 yearly amuunts must increase ad injinitum j for this 
 differ e^ice.) wiiich iii 25 years is only 1-65*'' part of the 
 yearly amount, becomes iu 50 years the 44^ part ; in 100 
 years, the 25*'' part ; and in 150 years it becomes less thau 
 tlie 18*'' part ; and will, in time, exceed the yearly amount 
 by an unmeasurable sum. Now can any reasonable man 
 believe, that such part of the tables was the production of Sir' 
 Isaac Newton ? I admit that the rule given in page 61, 
 is the same as Demoivre's, embracing only a single error, and 
 probably was Sir Isaac's, as they were cotttemporary, and 
 rrct|ueutly submitted their calcidations to each other. 
 
 In Dr. Price's work, edited by Mr. Morgan, there is 
 an Essay which was read lo the Royal Society, in liie year 
 1775, containing "Short and easy Theorems for finding, 
 " in all cases, the (hflereuces between the Values of 
 *•' Annuities payable yearly, and of the same Annuities 
 " payable lialf-yeariy, quarterly, or momently ;" and it 
 stales, " In the foregoing theorems, it may be observed, 
 " that the ratio to one another of the values of annuities 
 " payable yearly, half-yearly, quarterly, and momently, is 
 " greatest when w" (the time) " is lea^st ; that it decreases 
 " conlitiualiy as /i" (the time) "increases, till at last it 
 " vaaishcs when ?>" (the lime) " becomes infinite, or the 
 " annuity is a perpetuity." And a note by the Editor 
 KJalos, " The difrcrcnce, however, between the vnlues of 
 
 " annuities
 
 8 
 
 " annuities payable yearly and at shorter intervals, is 
 " known to be continually lessening in proportion to the 
 '^ length of the term ; till at last, when the terra is extended 
 *' to a perpetuity, those values become the same, whether 
 '' the payments are made yearly or momently." And 
 Dr. Hutton, in his Mathematical Dictionary, where he 
 gives a set of theorems for finding the respective values, and 
 a formula for the maximum of difference^ writes thus, 
 " The difference in the value, by making periods of 
 '^ payments smaller, for any given term of years, is the 
 " more as the intervals are smaller, or the periods more 
 *' frequent. The same difference is also variable, both as 
 *' the rate of interest varies, and also as the whole term of 
 *' years n varies ; and, for any given rate of interest, it is 
 '' evident that the difference for any periods m of payments, 
 *' first increases from nothing as the term n increases, when 
 " n is 0, to some certain finite term, or value of w, when 
 '• the difference D is the greatest, or a maximum ; and 
 " that afterwards, as n increases more, that difference will 
 '' continugJly decrease to notliing again, and vanish when ii 
 " is infinite." 
 
 Now although the above remarks of Dr. Hutton are a 
 little at variance with those of Dr. Price and Mr. Morgan, 
 who assert, that the difference in the values is continually 
 lessening, yet all their theorems indicate a maximum of 
 difference in twenty-five years. 
 
 That
 
 That the generality of writers on annuities (who perhaps 
 seldom investigate rules) should copy theorems authorised 
 by such great name>, is not surprising, but it seems extraor- 
 dinary that a paper, from which such a palpable incon- 
 sistency flows, should have escaped the notice of the truly 
 learned and acute members of the Royal Society. 
 
 As Zeno's celebrated Problem of Achilles and the 
 Tortoise, with Dr. Hutton's reasoning upon it, is applicable, 
 I here "ivc it in his own words. 
 
 PROBLEM. 
 
 *■' If Achilles can walk ten times as fast as a tortoise^ 
 " V'hich is a furlong before him^ can crawl, will the 
 " former overtake the latter ^ and hoiv far must ho 
 " icalk hofire he does so ?" 
 
 " THIS Problem has been thought worthy of notice 
 *' merely because Zeno, the founder of tiie sect of the 
 *' Stoics, pretended !<» prove by a sophism, that Achilles 
 *' could never overlako the tortoise; for while Achilles 
 " said he, is walking a furlong, the tortoise will have 
 " advanced the tenth of a furlong ; and while the former is 
 " walking tliat tentii, the tortoise will have advanced the 
 *' hundredth part of a furlong, and so on in injinitum ; 
 
 " cousc(iuenlly
 
 10 
 
 ^ consequently an infinite number of instants must elapse 
 ** before the hero can come up with the reptile, and there" 
 ** fore he will never come up with it. 
 
 *' Any person however possessed of common sense, may 
 *' readily perceive that Achilles will soon come up with the 
 " tortoise, since he will get before it." 
 
 This is really a parallel case ; for the present value of an 
 income to continue any length of time, is just so much 
 money as will come up with, or amount to, the same sum 
 that the income itself will am aunt to (both being in- 
 creased by the same rate of interest) at the end of the 
 term. Now, if the amount of X'lOO per annum, to 
 continue twenty-five years (if received £25 quarterly, and 
 improved at the rate of one per cent, or one-fourth the 
 year's interest quarterly) exceeds the amount (when the 
 annuity is received and improved at four per cent yearly,) 
 by nearly 100 pounds, surely the present worth ought to 
 be more in one case than in the other, because it has not 
 only to come up with, or amount to, the same sum that the 
 yearly payments do, but must exceed such sum by nearly 
 100 pounds ; the difference given by their Theorems, (and 
 which is their maximum of difference in any period of 
 time) to reach this surplus amount is £13.. 10.. 5, but 
 this is at four per cent interest per annum (at which the 
 purchaser's value is limited, otherwise it becomes a new 
 question and a different rate) will not amount to more tlian 
 
 ^36..1..0.
 
 11 
 
 X'36. .1 . -0. In one hundred years, the difference of the 
 amounts (if lYccivcd and improved in the above manner) 
 is ujtwards of aovcn thousand^ five hundred poundft\ 
 Avhile the diflerence in the values given by the theorems^ 
 is only £2..1')..10, which, in the one hundred years, 
 only amounts to .4*141, and in three hundred years the 
 d}ffere7ice of such amounts is upwards of sixty-one 
 MILLIONS OF POUNDS ; vct according to their theorems, 
 ihe diffei'ence of the values now vanishes / 
 
 As to a maximum in the difference of values, (when a 
 greater part of an annuity is paid than ought to be, and 
 this 'm continually increased in a greater ratio than the value 
 is), such can never take place. Portions of time are 
 fractions of eternity, and however difl'erent men's jiotions 
 may be of that endless increase of time, yet 50, 100, or 
 1000 years can only be parts of such a term in all our 
 thoughts ; and " any person possessed of common sense, 
 **' may readily perceive," that a sum of money thus con- 
 tinually gaining upon another, must, like Achilles, not only 
 come up with llie tortoise who moves slower, but get before 
 
 Compound interest (or interest upon interest) is grounded 
 on a series of terms increasing in geometrical ])roportion'. 
 If £100 be lent at the rate of four per cent, per annum, 
 the lender at the end of thc^ tirst year w«hiI(I rrceive £104 ) 
 and if this be lent for another year, he would receive (fit 
 
 the
 
 12 
 
 the end of the year) a sum bearing the same proportion to 
 this £104, as the £ 104 did to the £100 first lent. Now , 
 as £100 is to £104, so is £104 to £108. .3. .21 ; and this 
 will be the amount of principal and interest at the end of 
 two years, and so on. Then if £100 be lent for only ha// 
 a year at the same rate of interest per annum, (and which 
 all the tables and theorems imply), such a sura should be 
 returned, at the end of the half year, as would (being in- 
 creased in the same proportion) amount to £104 at the end 
 of the year ; therefore, the lender ought to receive only 
 £101.. 19.. 8; for as £100 is to £101.. 19.. 8, so is 
 £101 . . 19. .8 to £104, and this sum it ought to be at the 
 end of the year (and no more) according to the agreed rate 
 of four per cent, per annum ; indeed, it seems quite as 
 reasonable to make an allowance for the payment of any 
 part of a year's interest before it becomes due, as to give 
 discount for the payment of a bill before it becomes due , 
 and although custom may be pleaded in giving 1-12**' part 
 of a year's interest for a month, it is by no means just ; but 
 as to questions of compound interest, it is inadmissible^ 
 for the principal thus becomes increased at a different rate 
 per annum. 
 
 When any sum of money is to be increased in a certain 
 ratio annually, as by 1-20*'' part, if at 5 per cent., or by 
 1-25*'' part, if at 4 per cent. ; such yearly ratio must be 
 compounded of, and equal to, the sums of the half-yearly, 
 quarterly, or monthly ratios, according as Ihe payments 
 
 may
 
 13 
 
 may be made in loss periods than yearly ; that is, if R 
 represents tlic yearly ratio, or £1 with its interest for a 
 year, then R- : R' : R^ - (being the square root, the fourth 
 root, and the twelfth root of R) will respectively represent 
 the half-yearly, quarterly, and monthly ratios, and R- : 
 R* : R'-' will each be equal to R it^self, and which is the 
 sum of their ratios. But this will appear plainer, by the 
 following ratios of £ I at 4 per cent, per annum, whether 
 paid annually or quarterly. 
 
 Amoimt l'«year = R 1-01— Amount 2* year R* 1-0816 
 
 1st Quarter. td Quarter. 3(1 Quarter. 4tb Quaiter. 
 
 R^ R- RA RA 
 
 1-009853 : 1-019804 : 1-02985 : 1-04 
 
 Jlh Quarter. 6ili Quarter. 7tli Quarter. 8th Quarter. 
 
 R^ Ri Rl Rf 
 
 1-05024 : 1-06059 : 1-07104 : 1-0816 
 
 It must be evident on inspection, that any terms com- 
 pared in each of these series, (being in proportion to 
 yearly and quarterly, or two in the first, and eight in the 
 second), will be the same in amount ; for the fourth root 
 of R raised to the cigi)th jjower, is the same as R raised 
 to the second power; therelJjre, whether interest be paid 
 yearly, or (if in a corrct^t manner) (juarlerly, or momently, 
 the results will be the same, however far the series may be 
 extended. 
 
 But,
 
 14 
 
 ■^I Ril, if interest be paid as the common theorems indicate, 
 Uiat is, a fourth of the year's interest quarterly, the series 
 will become 
 
 Amount if yearly 1'04. — Amount in two years 1*0816. 
 
 Isl Quarter. 2(1 Quarter. 3d Quarter. 4th Quarter. 
 
 1-01 : 10201 : 1 030301 : 1-040604 
 
 5th Quarter. 6tb Quarter, nh Quarter. 8tb Quarter. 
 
 1-05101 : 1-06152 : 1-07213 : 108285 
 
 It must appear manifest, that no terms in these series 
 will correspond ; and the farther they are carried, the 
 ^reafer will be the difference between the amounts oi 
 yearly and quarterly payments ; and as all present 
 values must be proportionate to their respective amounts, 
 if such differences continually increase, so will the 
 differences in the respective values continually increase. 
 
 An annuity is nothing more than a yearly interest on a 
 corresponding sum of money; for an annuity of j£], and 
 the interest of £25, if at 4 per cent., or of £20, if at 5 
 per cent., amount to one and the same sura ; therefore, if 
 an annuity is to be divided into quarterly payments, each 
 quarter's payment should be in the same ratio to the true 
 quarterns interest, that the annuity is to the true yearly 
 interest. Now the true yearly interest of £1 (the annuity) 
 at 4 per cent., is -04 ; therefore, as -04 is to 1 (liie annuity).
 
 u 
 
 so ife -0098533 (the true qiiarter's interest) to -2463325 
 (the true quarter's payment), being 4s. lid. instead of 
 f> shillings. 
 
 Wliether equal sums of money, paid periodically, are 
 called interest or annuities, their increase will be the same ; 
 and if correct portions of such sums are paid each tittiey 
 and these are increased according to the true ratio ; lljeir 
 amounts and their values will increase in the same pro- 
 portion, for any number of years, or for ever, whether 
 paid and increased yearly or momently ; but if the pay-* 
 ments, or the ratioy or doth, be wrong, the amounts, in 
 every period of time, must be dilTerent from the yearly 
 amounts^ as will be clearly seen by the following four 
 
 CASE I. 
 
 Where the part of the Yearly Sum, or Annuity, is 
 WRONG ; but the ratio of the Rate of Interest operating 
 
 upon it, is RIGHT. 
 
 THIS Case supposes '25, or \ of tlie annual sum to be 
 paid ywar/cr/^, but improved in the correct ratio 1 .001)8533 j 
 and as this diflerencu between the true payment -2463325, 
 
 and
 
 16 
 
 and '25, will be the same in each quarter, while the ratio of 
 the rate of interest operating upon it, is correct^ whatever 
 proportion the surjjlus interest or amount in one year bears 
 to the correct interest or amount for one year, if the like 
 proportion be added to any other correct yearlij value, in 
 any stage of time, it will give the correct present value, if 
 the annuity and interest be paid quarterly in this manner ; 
 for example, — 
 
 If each quarter's payment be '2463325, and improved in 
 the ratio 1*0098533, at the end of the year the amount 
 will be precisely £1, being the true amount of the first 
 year's payment ; but if each quarter's payment be "25, and 
 improved in the same ratio, the amount at the end of the 
 year will be 1*014888: and in the same proportion as this 
 exceeds 1, so will the true value for quarterly payments 
 (at 4 per cent, per annum) exceed the value for yearly 
 payments at the same rate of interest per annum. Now 
 the present value of £1, to be received at the end of the 
 year, is in the tables (at 4 per cent.) -961538 ; therefore, 
 as £1 is to jei-014888, so is £-961538 to £-975854, the 
 true value, if the £\ is to be made in four quarterly pay- 
 ments of five shillings each; and as £1 is to £1*014888, so 
 will £25, the value of the perpetuity, if paid yearly, be to 
 £25*37221, the true value to give the purchaser 4 per cent, 
 per annum, if the same be paid and improved in this man- 
 ner quarterly ; and a like reasoning will apply to the yearly 
 values for any intermediate stage of time. The proportion 
 
 of
 
 17 
 
 ol Ihi^ excess is about tiie 07"' purl of llie yearly value, 
 and always continues the same. 
 
 This is the error of both Sir Isaac Newton and 
 Demoivre ; their ratio of tiic rate of interest was righty 
 but the part of the annuity paid was wrong ; yet as this 
 excess of annuity is constantly improved by a correct ratio 
 of interest, such annual excess must require 25 years pur- 
 chase for the value of the perpetuity, equally with the 
 annuity itself; and this is precisely the result, for £1-014888 
 is rather more than £1..0..3^, and £25-37221, or 
 £25..7..5| is 25 times the annual payment, as it ought lo 
 be. Indeed it must be evident to common sense, that an 
 income of £1 : : 3", per annum, requires more purchase 
 mon?y than ill per annum, supposing the j?ame rate of 
 interest in both cases, which all the tables and theorems 
 imply. 
 
 CASE II. 
 
 Where the part of the Annuilg paidis right, as -2463325 
 quarterly ; but the Ratio of the Rate of Interest is 
 WRONG, being TOI quarterly, instead of 1-0098533. 
 
 xl ERE, the surplus interest, or amount, does not continue 
 
 each succeeding year in the same proportion to the respec- 
 
 B tive
 
 18 
 
 live yeatly amount that it bears the first year, but \v> 
 continually increasing ; for this difference^ which is not the 
 4000*'' part of the yearly amount, or £1 the first year, 
 becomes in 50 years the 52*^ part of the yearly amount in 
 that time ; in 100 years, it will become the 22* part ; 
 and in 500 years, this difference^ between paying and 
 improving only £\ per annum, yearly^ at 4 per cent, 
 or quarterly in the above manner, amounts to more than 
 two thousand six hundred millions of pounds!! which 
 IS nearly the third part of the whole yearly amount ; in- 
 deed, in 1200 years, the difference alone becomes equal to 
 (heiohole yearly amount in the time, and beyond this term 
 it exceeds the amount (improved yearly at 4 per cent.), and 
 will go on increasing, till such excess becomes an unmea- 
 surable sum. This case implies £1 per annum (or any 
 annuity), imjjroved in the ratio 1*040604, or about £*4..1..2| 
 per cent, per annum, which is as distinct from 4 per cent, 
 as from 14 ; and in proportion as such ratio increases any 
 annual sum more rapidly, so must the amount of any 
 present worth he increased (if improved at only 4 per cent. 
 per annum) to meet it. 
 
 CASE
 
 19 
 
 CASE III. 
 
 fVhere the pari of the Annuity and the Ratio of the Rate 
 of Interest are both wrong ; the Payment being One 
 Quarter of the yearly Sum, or '25 instead of 
 •2463325, and the Ratio being I'Ol, instead of 
 1.0098533. 
 
 -I. HE theorems of Dr. Price, Mr. Morgan, Dr. Hutton, 
 &c. are formed on this combination of errors, which will 
 account for the great and increasing differences between 
 amounts, when improved correctly, or according to their 
 theorems. 
 
 The differences arising in this case will be the sum of the 
 diflerences of tlic first and F?cond cases ; for, instead of JEI 
 per annum, increased in the ratio 1-04 (or 4 per cent, per 
 annum), it becomes £1..0..3l per annum, increased in the 
 ratio 1 040604, or JE4..1..2^ per cent, per annum. 
 
 The amount of £1 per annum, thus received and 
 improved quarterly for 100 years, will exceed the amount, 
 if correctly received and improved at the rale of 4 per cent, 
 per annum, by nearly 1-22' part of the yearly amount 
 (as in Case 11), added to nearly 1-67''' part (as in Case I) ; 
 b2 for
 
 20 
 
 Ibr the amount, if liius received and improved quarterly 
 for 100 years, is £ 1313 ; — but the amount of £ 1 per annum, 
 if improved at 4 per cent, per annum, for 100 years, is only 
 £1237 (leaving out decimals), and the difference, £76, will 
 be very nearly 1-22*^ part, added to 1-67*^ part of £1237, 
 and so is the dilTerence in the values ; for according to the 
 tables, the value of £1 per annum for 100 years, at 4 per 
 cent., is £24-5050, but this is as an equivalent for what 
 will amount to £1237, in the time : if the annuity be 
 received and improved quarterly, agreeably to this case, 
 the amount will be £1313, the true value of which, by the 
 tables now given, is £26*0003, and the purchaser will 
 make 4 per cent, per annum of his money, as much in 
 the one case as in the other. If an annuity be received and 
 improved in this manner quarterly for 1200 years, or for 
 4800 quarters, it will amount to more than double the 
 sum that the same annuity will amount to in the same 
 time, if paid and improved correctly at 4 per cent, per 
 annum, and of course will be worth twice the present 
 value of the other. It may appear extraordinary at first 
 view, that the perpetuity requires but 25 years purchase, 
 (howevei* often received and improved, if it be correctly 
 received and improved), and yet, apparently, this same 
 annuity for a finite term requires upwards of 50 years 
 purchase, which seems only to pay 2 per cent, instead of 
 4 per cent. 
 
 Probably such a first impression induced Mr. Morgan to 
 
 reject
 
 21 
 
 reject Demoivre's rule ; for INfr. M. says, '« According lo 
 " Demoivre's rule, if the term be exlcndeil, an annuity 
 " payable quarterly \vill be worth more than even the 
 " perpetuitij when the pnynienls are made yearly. Tiiis 
 " appears to be very erroneous, and sufficient to prove the 
 " fallacy of Demoivre's method of solution." But this 
 very circumstance is " sufficient to prove" the truth of 
 Demoivre's method of solution (as far as it goes) ; for it 
 must be evident to common sense, that more money is 
 required to purchase £\..l..'2j per annum for ever (or 
 even for a finite term, if extended), than to purchase £4 
 per annum for ever, supposing the same rate in both cases, 
 which is always presumed ; but if two equivalent sums are 
 improved by difiiprent rates of interest, the longer the term, 
 the more must they be separated ; and the circumstance of 
 the value being oO years purchase for an annuity (if paid 
 quarterly in this manner), and yet give the purchaser 4 per 
 rent, per annum interest, when duly considered, will 
 excite no more astonishment than for a man to employ his 
 capital at 4 per cent, interest per annum, who may pay 
 X*.')0() for only five .shillings a year for 30 or 40 years, witli 
 a valuable revertion at the end of the term. S<j it is in the 
 above case, for £1 per annum (as it is called), if paid at 
 five shillings per quarler, and improved in the ratio of 
 1 per cent, per quarter, will, in 1200 years, amount, not 
 only to the same sum tliat the L"l per annum \\ill, if paid 
 yearly and improved at 4 [ler cenl., but precisely to as 
 much more as the extra £2o will amount lo in the same 
 time, if iu)i»roved in the ratio of 4 per cent, per jiunum. 
 
 CASE
 
 22 
 CASE IV 
 
 Where the part of the Annuity, and the Ratio of the Ratn 
 of Interest, are both right ; the Quarterly Payment 
 being -2463325, awe? improved in the Ratio 1 '0098533. 
 
 tV hen a just proportion is paid of the annuity, and it 
 is increased in a correct ratio, according to the number ot 
 payments in a year, there will be no difference in any 
 corresponding period of time, between the yearly or 
 momently amounts and values. 
 
 If the value be required of £1 per annum for 50 years, 
 so as to pay the purchaser 4 per cent, per annum interest, 
 let the annuity, or £1, be equal to w, 1*04 (or £1 with a 
 years interest) = R, 50 years (or the time) = t, then 
 
 or — = value ; being £21 '4822 for 
 
 i2_l ' "^ -04 
 
 the yearly payments ; and if the payments are to be made, 
 
 1 
 m times in a year, the m'* root of 72, or R"" , will express 
 
 the ratio of the rate of interest for the timey and R '"• — ^1 
 will be the correct interest for the time. It has been shown 
 that the true part of the annuity for the time will be in 
 the same proportion to the true interest, that the annuity 
 is to the year's interest, and calling this part of the annuity 
 
 !7j
 
 y, it will be found thus: As /?— 1 : £i : : R'" -^i : 
 
 —jz — j— X i-'l ^- 9, this is merely dividing the trii4 
 
 interest for the time by the year's interest (as mul- 
 tiplying by £1 makes no alteration) ; for if the payment! 
 
 be quarterly, /?"^— 1, will be -0098533, and this divided by 
 
 R — Ij or '04, gives •2463325 for q, or the true quarter's 
 
 1 
 
 payment ; and if the payments be monthly, /? '" — I wiU 
 be -0032737, which, divided by -04, givea -0818425 for q, or 
 the true monthly payment. The value of the annuity for 
 / years, if the payments be made m times in a year, will 
 he thus found : 
 
 q 2- 
 
 li m 
 
 1 = value ; 
 
 /?^— 1 
 or if quarterly for 50 years, 
 •2463325 
 
 •2463325 — 
 
 -= i:21-4822, the same 
 
 -009S533 
 
 the yearly value. R raised to / years, is the same as raising 
 the m^^ root of R to m times t years ; therefore, "FoTi*" 
 may be used instead of l'04Y-7-. If the annuity is to be 
 perpetual, here ;, or the time, becomes infinite, and 
 whatever sum is supposed to be divided by an injinite 
 quantity, the quotient must be equal to nothing, oi ; 
 
 u q 
 
 therefore ^i in the yf^arhj payments, and -^ in the m 
 
 payments in a year, becoming equal to 0, may be left out ; 
 
 and
 
 24 
 
 and the other part ^ — r, being merely the annuity, or 
 yearly sum, divided by the interest for a year, will give 
 
 £1 
 the value of the perpetuity ; as ;t7-t- = £25 for the value 
 
 of the perpetuity, if in yearly payments ; and in the 
 
 q -2463325 
 
 theorem for m payments in a year — j , or ^3993533 
 
 = £25 (the same as the yearly value), being the correct 
 part of the annuity divided by the correct part of the 
 interest. 
 
 It may be proper to remark here, that the arithmetical 
 part of the annuity, or £-25, divided by a like arithmetical 
 part of the yearly interest, or £-01 (as by the common 
 theorems), will give the same quotient, or £25 ; and such 
 may have led to the opinion, that " when the term is 
 " extended to a perpetuity, those values become the same, 
 " whether the payments are made yearly or momently :" 
 but this by no means proves it ; for any part of any other 
 dividend, if divided by a like part of its divisor, the 
 quotient will be the same as the dividend itself, divided by 
 the divisor itself: 84 di\ ided by 12, gives 7 ; and so will 
 I of 84 (or 21) if divided by | of 12 (or 3). Indeed 
 the above proves (what is named in Case III.) that 
 the purchaser, instead of making 4 per cent, per annum, 
 makes £4. 1..2| per cent, per annum interest; for -01 
 is the correct quarterly payment of such a rate of 
 interest per annum ; and £*25 is the correct quarterly 
 part of £1..0..34 per annum; therefore he gives £25 
 
 for
 
 25 
 
 Ibr a perpclual income (as near as money can mark 
 it), of £*1..0..3y pt?r annum. The value of 6ft. per 
 quarter for ever (if improved as in Case I.), to give the 
 purchasrcr 4 per cent, interest per annum, is £2o..l..o\. 
 If the purchaser were allowed 5 per cent, per annum 
 interest, such a perpetual income would be worth only 
 JE20..7.. 1|. As the interest allowed is no more than 
 j£4..1..2{ per cent, per annum, the value comes out £25 
 precisely, as it ought to do; but this is as totally distinct from 
 any question of 4 per cent., as it is from 14 per cent, 
 interest per annum. 
 
 In all cases, for a finite term of years, the most simple 
 rule to ascertain the true value, will be — First to find the 
 AMOUNT of the annuity at the end of the term (whether 
 paid yearly, or divided into half-yearly, quarterly or momently 
 payments), according to the ratio by ichich it is increased 
 each TBfE of payment (without considering the per 
 centum jjer annum) : then, (whatever interest per annum 
 is to be allowed the purchaser) raise R (or £1 with the 
 interest to be allowed for a year), to the power denoted by 
 the given number of years, and divide the amount (first 
 found) b)) this sum^ the fjifofirnt irill be the true value, so 
 as to give the purchaser the rate per cent, per annum 
 agreed upon. 
 
 The amount at llie end of the term, however often the 
 payments are made, will he found by the following theorem. 
 
 Call
 
 Call the number of payments to be made, i. 
 Each payment u. 
 
 £1 with its interest for one time r. 
 
 y*— 1 
 
 then f- X w = amount. And if the amount of £25 
 
 per quarter, improved at 1 percent, per quarter, for 100 
 years, were required, we have t -ziz. 400 ; u ■=. £25 ; and 
 
 l-Oir'— 1 
 
 r=l-01. Therefore, X ^25 = amount, 
 
 •01 
 
 being £131313-5. 
 
 And to find the value, so as to give the purchaser 4 per 
 cent per annum interest, raise 7? =: 1*04 (being £1 with a 
 year's interest), to the hundredth power, for a divisor ; 
 then £131313-5, divided by 'VW\ or 50-5045, gives 
 £2G00, for the true value of £25 per quarter, supposing 
 it improved at 1 per cent, per quarter for 100 years ; 
 so as to give the purchaser 4 per cent, interest per annum. 
 
 By applying the foregoing theorem to the dividends on 
 the national debt, calling them 32 millions per annum, the 
 great difference between paying them in the usual manner, 
 of one-fourth part every quarter, and supposing them 
 increased in the ratio of one-fourlli the yearly interest quar- 
 terly, or paying them as the nature of the contract seems 
 to imply: viz. yearly (or correctly, if otherwise), will 
 appear as follows : 
 
 The amount in 25 years,. if paid in four quarterly sums, 
 of eight millions each, and improved each quarter by one- 
 
 foiu'th
 
 27 
 
 fourth of a year's interest, or £1 per cent, per quarter, 
 will be 
 
 •01 
 
 X ieSOOOOOO = X1363S64000. 
 
 But if paid and increased annually, or by a true part, 
 both of annuity and interest, (if paid quarterly), the 
 amount will be 
 
 1-O()98o33l"" ' 
 
 X 1*7882640 1= £1332001000, 
 
 •0098533 
 
 making a difl'erence in the amounts, in 25 years, of thirty- 
 one millions tao hundred thousand pounds. Proceeding 
 in the same manner, the difl'ercnce will be found in 50 
 years to be more than 107 rnillioiis ; and in 100 years, to 
 be upwards of TMO thousand four hundred shllions 
 OF pounds. 
 
 The following table shows, at one view, the precise parts 
 of £1, that ought to be paid, if the annuity be divided into 
 half-yearly, quarterly, or monthly payments, according to 
 the rates of 4 or 6 per cent, per annum interest. 
 
 Yearly payments £ 1 
 
 Half-yearly ditto 
 
 Quarterly ditto 
 
 Monthly ditto 
 
 4 per Cent. 
 ycr J-nnum. 
 
 6 jwr rent 
 
 !• 
 
 £1' 
 
 •4951000 
 
 •4939200 
 
 •2403325 
 
 •2451000 
 
 •0818425 
 
 •0814880 
 
 This Table is very readily formed, for an annuity of 
 ^1, being the same as one year*h interest on JC25 (if at 
 
 4 per
 
 28 
 
 4 per cent.), or on £20 (if at 5 per cent.), it is merely 
 multiplying the true interest on £1, for the timo^ by 25 or 
 by 20, as the case may be. The use of this table will 
 appear by the following examples: — 
 
 If a person enjoy an income of £100 per annum, and be 
 desirous of being paid quarterly; what ought he to receive 
 each quarter, computing at 4 per cent, interest per annum ? 
 The true part of £ I per annum being •2463325, it is mere!}^ 
 multiplying this decimal by the annuity or £100, which 
 gives £24-63325, or £24..12..8 is the true quarterly 
 payment. 
 
 In the same manner, the true quarterly payment of interest 
 on the national debt, supposing it 32 millions per annum, 
 is found by multiplying the above decimal by 32 millions, 
 which gives £7882640, instead of 8 millions, making a 
 difference of £117360 per quarter. 
 
 If a person enjoy an income of £600 per annum, and be 
 desirous of being paid monthly, or 12 times a year ; what 
 ought to be paid each month, computing at 5 per cent, 
 interest per annum ? Here, the true part of £1 per annum 
 for a month is -0814880, wliich multiplied by £600, gives 
 £488928000, or £48..17..10| for the true monthly 
 payment. And if any income, receivable in less 2JGriods 
 than yearly, is to be exchanged for an equivalent yearly 
 payment, it will be only reversing the operation; for if a 
 
 monthly
 
 29 
 
 monthly income of £48-8928000 is to be exchanged for an 
 equivalent yearly income, divide the monthly income by the 
 hue mondily part of £1 per annum, or by '0814880, and 
 the answer will be JL'600, as above. 
 
 The following Tables show at one view the amount of £1 
 per annum, for any number of years therein named, whe- 
 ther paid and improved annually, or paid in the usual 
 manner of 10s. half yearly, or5«. quarterly, and increased in 
 the ratio of one half', or o)ie quarter, the annual interest ; 
 also the present value of the same, so as to give the pur- 
 chaser 4 or 5 per cent, interest per annum, as named in its 
 respective column. 
 
 TABLE
 
 80 
 
 TABLE 1. 
 
 Amounts and Values of Half-Yearly Payments, at (what is called) 
 4 per Cent, per Annum. 
 
 Amount of £1 per annum, 
 if pairt and improved 
 yearly, at 4 per Cent. 
 
 Present value 
 of £1 per an- 
 num, if paid 
 and improved 
 yearlij, at -1 
 per cent. 
 
 NO of 
 Years. 
 
 Amonnt of ten shillings, If 
 paid and improved hnlj: 
 yearly, at 2 per Cent. 
 
 Present valne of ten 
 shillings, if paid and 
 improved half-yearly, 
 at 2 per Cent, so as to 
 give the purrhabcr 4 
 per Cent, per annum 
 Interest. 
 
 ^ Decimal 
 £ Parts. 
 
 „ Decimal 
 Jb Pans. 
 
 
 ^ Decimal 
 £ Parts. 
 
 Decimal 
 A Parts. 
 
 1 '0000 
 
 -9615 
 
 1 
 
 1 -0100 
 
 '97115 
 
 2 -0400 
 
 1 '8861 
 
 2 
 
 2 -0607 
 
 1 -9053 
 
 3 -1216 
 
 2 -7751 
 
 3 
 
 3 '1540 
 
 2 -8039 
 
 4 -2464 
 
 5 -4163 
 
 3 -6299 
 
 4 
 
 4 -2915 
 
 3 -6684 
 
 4 -4518 
 
 5 
 
 5 '4750 
 
 4 -5005 
 
 12 -0061 
 
 8 '1109 
 
 10 
 
 12 '1487 
 
 8 -2072 
 
 20 -0236 
 
 11 -1184 
 
 15 
 
 20 -2841 
 
 11 -2631 
 
 29 -7781 
 
 13 -5903 
 
 20 
 
 30 -2112 
 
 13 -7880 
 
 41 -6459 
 
 15 '6221 
 
 25 
 
 42 -2900 
 
 15 -8636 
 
 ! 56 -0849 
 1 Q5 '0255 
 
 17 -2920 
 
 30 
 
 57 -0260 
 
 17 -5822 
 
 19 -7927 
 
 40 
 
 96 -8870 
 
 20 -1805 
 
 : 152 -6671 
 
 21 -4822 
 
 50 
 
 1.56 -1175 
 
 21 -9678 
 
 237 -9907 
 
 22 -6235 
 
 60 
 
 244 -1309 
 
 23 -2072 
 
 364 -2905 
 
 23 -3945 
 
 70 
 
 374 "9150 
 
 24 -0770 
 
 I 551 -2450 
 
 23 -9154 
 
 80 
 
 569 -2537 
 
 24 -6968 
 
 827 '9833 
 
 24 -2673 
 
 90 
 
 808 -0308 
 
 25 -1481 
 
 1237 -6237 
 
 24 '5050 
 
 100 
 
 1287 -1407 
 
 25 -4856 
 
 63742 -5882 24 "9902 
 
 200 
 
 68843 -3809 
 
 26 -9899 
 
 82146679^4 -5283 24 "9999 
 
 500 
 
 9957263636 -3636 
 
 30 -3033
 
 31 
 
 TABLE 11. 
 
 Amounts and Values of Quarterly Payments^ at (what is called) 
 4 per Cent, per Annum, 
 
 Amoun'of ^1 per annum, 
 if paid ami improved 
 yearly, at 4 per cent. 
 
 I'rcsent value 
 of £1 per an- 
 num, if paid 
 and improved 
 yearly, at 4 
 per cent. 
 
 No. of 
 Years. 
 
 ,1 mount of nv« sliillinsg if 
 paid and improved quar- 
 terly, at 1 per Cent. 
 
 I'resent wlue of five 
 shillings, il paid and 
 improvt-tl quarterly, 
 at 1 per Cent, so as to 
 give tbe Purchaser 4 
 per Cent per annum 
 Interest. 
 
 „ Decimal 
 £ part*. 
 
 jP rteci.nal 
 A> Parts. 
 
 
 a Decimal 
 £ Parts. 
 
 J, Decimal 
 3L, Pans. 
 
 1 -0000 
 
 -9615 
 
 1 
 
 1 '0151 
 
 
 
 •97606 
 
 2 -0400 
 
 1 -886] 
 
 2 
 
 2 -0714 
 
 I 
 
 •9151 
 
 3 -1216 
 
 2 -7751 
 
 3 
 
 3 -1706 
 
 2 
 
 •8187 
 
 4 -2464 
 
 3 -6299 
 
 4 
 
 4 -3145 
 
 3 
 
 •6880 
 
 5 -4163 
 
 4 -4518 
 
 5 
 
 5 -5047 
 
 4 
 
 •5245 
 
 12 -006 1 
 
 8 -1109 
 
 10 
 
 12 -2217 
 
 8 
 
 •2565 
 
 20 -0236 
 
 11 -1184 
 
 15 
 
 20 -4176 
 
 11 
 
 •3372 
 
 29 -7781 
 
 13 -5903 
 
 20 
 
 30 -4181 
 
 13 
 
 •8824 
 
 41 -6459 
 
 15 -6221 
 
 25 
 
 42 -6207 
 
 15 
 
 •9878 
 
 56 -0849 
 
 17 -2920 
 
 30 
 
 57 -5102 
 
 17 
 
 •7315 
 
 95 -0255 
 
 19 -79-27 
 
 40 
 
 97 -8467 
 
 20 
 
 •3804 
 
 152 -6071 
 
 21 -4822 
 
 50 
 
 157 -9026 
 
 22 
 
 •2!90 
 
 237 -9907 
 
 22 -6235 
 
 60 
 
 247 -3174 
 
 23 
 
 •5101 
 
 364 -2905 
 
 23 -3945 
 
 70 
 
 380 -4449 
 
 24 
 
 •4321 
 
 551 -2450 
 
 23 -9154 
 
 80 
 
 578 -6536 
 
 25 
 
 •1047 
 
 827 -9833 
 
 24 -2673 
 
 90 
 
 873 -7604 
 
 25 
 
 •6091 
 
 1237 •6-237 
 
 24 -5050 
 
 100 
 
 1313 -1350 
 
 26 
 
 •0103 
 
 63742 -5882 
 
 24 -9902 
 
 200 
 
 71599 -2300 
 
 28 
 
 •0704 
 
 821466/924 -5283 
 
 24 -9999 
 
 500 
 
 10983475000 -0000 
 
 33 
 
 •4264
 
 
 
 32 
 
 
 TABLE III. 
 
 
 Amounts and Values of Half-yearli/ Payments, at fwh 
 
 at is called) 
 
 5 per 
 
 Cent. 
 
 per Annum. 
 
 
 Amount of ^1 per annum, if 
 
 paid aud irnpioved yvuily, at 
 
 five per cent. 
 
 Present value 
 of £1 per an- 
 num, if paid 
 and iniprov< d 
 yearly 5 per 
 cent. 
 
 No. of 
 Years. 
 
 Amount of ten shillings, if paid 
 and improved kalj'-iiearly , at 
 2 and a half per cent. 
 
 Present value of ten 
 shillings, if paid and 1 
 improved half-yearly, 
 at two and a half per 1 
 cent, so as to sjive the 
 purchaser 5 per cent, 
 per annnm inleiest. 
 
 _ Decimal 
 jt Parts. 
 
 jp Decimal 
 <i. Paris. 
 
 
 4? Decimal 
 <*' Parts. 
 
 J? Decimal 
 ^ Parts. 
 
 1 -0000 
 
 -9524 
 
 1 
 
 1-0125 
 
 -96428 
 
 2 -0500 
 
 1 -8594 
 
 2 
 
 2 •07Gi 
 
 1 -8832 
 
 3 -1525 
 
 2 -7232 
 
 3 
 
 3 -1^38 
 
 2 -7589 
 
 4 -3101 
 
 3 -5459 
 
 4 
 
 4 -3680 
 
 3 -5936 
 
 5 -5256 
 
 4 -3295 
 
 5 
 
 5 -6017 
 
 4 -3891 
 
 12 -5779 
 
 7 -7217 
 
 10 
 
 12 -7724 
 
 7 -8411 
 
 21 -5786 
 
 10 -3796 
 
 15 
 
 21 -9514 
 
 10 -5590 
 
 33 -OfiGO 
 
 12 -4632 
 
 20 
 
 33 -7014 
 
 12 -7017 
 
 47 -7271 
 
 14 -0939 
 
 25 
 
 48 -7424 
 
 14 -3938 
 
 GQ -4388 
 
 15 -3724 
 
 30 
 
 67 -9962 
 
 15 -7328 
 
 120 -7998 
 
 17 -1591 
 
 40 
 
 124 -1922 
 
 ] 7 -6409 
 
 209 -3480 
 
 18 -2559 
 
 50 
 
 216 -2761 
 
 18 -8601 
 
 353 -5837 
 
 18 -9293 
 
 60 
 
 367 -1666 
 
 19 -6564 
 
 588 -5285 
 
 19 3-427 
 
 70 
 
 614 -4183 
 
 20 -1936 
 
 971 '2288 
 
 19 '59Q5 
 
 80 
 
 1019 -57^5 
 
 20 -5718 
 
 1594 -6073 
 
 19 -7522 
 
 90 
 
 1683 -4600 
 
 20 -8528 
 
 2GIO -0252 
 
 19 -8479 
 
 100 
 
 2771 -3210 
 
 21 -0744 
 
 345831 7600 
 
 19 -9988 
 
 200 
 
 389554 -0000 
 
 22 -5272 
 
 7864f»6000000 -0000 
 
 19 -9999 
 
 500 
 
 1059082324455 -0000 
 
 26 -9327
 
 
 
 33 
 
 
 TABLE IV. 
 
 
 Amounts and Values of Q 
 
 uarterly Payments, at (what is calledj 
 
 5 per 
 
 Cent, per Annutn. 
 
 
 Amount of £1 per annum, if 
 paid and improved yearly, 
 at 5 per cent. 
 
 Present value 
 of £1 per an- 
 num, if paid 
 and improved 
 yearly, at 5 
 per cent. 
 
 NO of 
 Years. 
 
 Amountcfflvcshillings, if paid 
 and improved quarterly, at 
 1 and l-4tli per ceut. 
 
 Present value of five 
 Kliiiliues if paid and 
 imnrov^l qi^aKerly. at 
 1 & l-tlh per <:e,-t. to 
 asiogivetliepiirrhaser 
 6 pel ceni. per annum 
 iiiierest. 
 
 ^ Decimal 
 £ Farts. 
 
 p Decimal 
 c"t Farts. 
 
 
 ^ Decimal 
 ,±, Parts. 
 
 r, Decimal 
 £ I'arls. 
 
 1 -0000 
 
 -9524 
 
 I 
 
 1 '0189 
 
 -97028 
 
 2 -0500 
 
 1 -8594 
 
 2 
 
 2 -0897 
 
 1 -8954 
 
 3 '1525 
 
 2 -7232 
 
 3 
 
 3 -2150 
 
 2 -7773 
 
 4 -3101 
 
 3 '5459 
 
 4 
 
 4 -3978 
 
 3 -6180 
 
 5 -5256 
 
 4 -3295 
 
 5 
 
 5 -6407 
 
 4 -4190 
 
 12 -5779 
 
 7 •7217 
 
 10 
 
 12 -8723 
 
 7 -9025 
 
 21 -5780 
 
 10 -3796 
 
 15 
 
 22 -1434 
 
 10 -6513 
 
 33 -OCGO 
 
 12 -4622 
 
 20 
 
 34 -0294 
 
 12 -8253 
 
 47 -7271 
 
 14 -0939 
 
 25 
 
 49 -2676 
 
 14 -5488 
 
 CG -4388 
 
 15 -3724 
 
 30 
 
 G8 -8034 
 
 15 -9195 
 
 120 -7993 
 
 17 -1591 
 
 40 
 
 125 -9587 
 
 17 -8919 
 
 209 -3480 
 
 18 -2559 
 
 50 
 
 219 "9000 
 
 19 -1761 
 
 353 -5837 
 
 18 -9293 
 
 60 
 
 374 -3027 
 
 20 -0385 
 
 588 -5285 
 
 19 -3427 
 
 70 
 
 628 -0810 
 
 20 '6A2C) 
 
 971 -2288 
 
 19 •59G5 
 
 80 
 
 1045 •I9GG 
 
 21 -0889 
 
 1594 -0073 
 
 19 -7522 
 
 90 
 
 J 730 -7740 
 
 21 -4389 
 
 2GlO -02.52 
 
 19 -8479 
 
 100 
 
 2857 -GOOO 
 
 21 -7306 
 
 345631 -7000 
 
 19 -9988 
 
 200 
 
 414008 -1900 
 
 23 -9413 
 
 7Ht74GG000O0O -tJOOO 
 
 19 -9999 
 
 500 
 
 1233190571428 '0000 
 
 31 -3603
 
 34 
 
 By these tables, the amounts and values of any other 
 yearly^ half-yearly^ or quarterly payments^ or gross 
 sums ; also the value of reversions^ or renewal of leaseSy 
 may be readily found, as by the following examples: — ■ 
 
 What will be the amount and value of a yearly income 
 of £24, improved at 5 per cent, per annum, to continue for 
 80 years ? 
 
 In the first column of Table IV. the amount of £1 per 
 annum, if paid and improved yearly, is, for 80 years, 
 £971-2288 ; therefore, 24 times this sum, or £23309-4912, 
 equal to £23309 .. 9 ,. 9 will be the amount, and 24 times 
 £19-5965 (in the next column), or £470 .. 6 .. 3 is the 
 present value. 
 
 What will be the amount of a quarterly income of £6, 
 improved at 1| per cent, per quarter, to continue for 80 
 years ? and what is the present value, so as to give the 
 purchaser 5 per cent, per annum interest ? 
 
 This being 24 times the quarterly income, or five 
 shillings, at the head of the fourth column in the same table, 
 we must multiply £1045*1906 (against 80 years}, by 24, 
 producing £25084-7184, or £25084 .. 14 .. 4 for the 
 amount; and 24 times £21-0889, being £500-1336, or 
 £506 .. 2 .. 8, is the value ; so as to give the purchaser 
 5 per cent, per annum interest. 
 
 What
 
 35 
 
 What will £500 amount to in 80 years, at 5 per cent, 
 per annum interest ? 
 
 As each line in the first and fourth columns shows the 
 amount of its respective line in the adjoining or second and 
 fifth columns, either may be used. By the rule of propor- 
 tion, if £21-0889 amount to £1045-1966, £500 will (in 
 the same time, and at the same rate) amount to £24780918, 
 or £24780 .. 18 .. 4. And if the value of a reversion 
 were required, it would be only reversing the operation ; 
 for, if £1045-1966 (to be received at the end of 80 
 years) require (a present value of) ^^2 1-0889, the sum of 
 ^624780-9 18 (to be then received) will require ^500 in 
 present money. 
 
 The renewal of any lapsed term in a lease (whether for 
 years or for lives), is nothing more than paying the dif-' 
 fcrence between the value of a new lease for the original 
 term, and the value of the unexpired term in the old lease. 
 
 If the original term were 40 years, and 15 years are 
 unexpired, the dljjfrrencc bclween the value for 40 years, 
 and the ^alue for 25 years (the unexpired term) will be 
 the value for renewal ; the value of j^l per annum, for 40 
 years, at 5 percent, if paid yearly^ is ^£17-1591, and for 
 25 years, is 14-09.39; the dillerence, or 30652 years* 
 purchase, is I he value for renewal. Bid if Jho rent be 
 p-iid fjuarlcrhj, the (/*77f'/v»fo bcfwfcn »he values (or 40 
 r 2 and
 
 36 
 
 and 25 years, in the fifth column, will he too much; for 
 the amounts in the fourth column (and by which the fifth 
 is formed) increase in a greater ratio than 5 per cent, per 
 annum ; vet, as 10'6513, against 15 years, is the correct 
 value for 1 5 years (when the 25 years are expired), if 
 paid quarterly, we must find what present sum will, in 25 
 years, amount to ^10*6513, which is readily done by the 
 rule of proportion ; for if ^49*2676 (to be paid in 25 years) 
 require ,;£14-5488 in hand, ^10*6513 (to be then paid) 
 will require ^3-1453; or so many years' purchase is the 
 value for renewal, if the rent be paid in the usual manner, 
 quarterli/. 
 
 As to the tables published at rates of interest above 5 per 
 cent, per annum, when the principle on which they are 
 formed is considered, they will be found both impracticable 
 and illusive to purchasers. 
 
 The principle on which all the tables hitherto published, 
 for the valuation of terminable incomes, whether for years 
 or lives, have been formed, is, that the yearly income will 
 not only be equal to the interest per cent, named in the 
 tables, but as much more as will replace (at the end of the 
 term or life) the capital employed : for instance, if a person 
 pay i:802 for an income of ^100 per year, for 17 years, 
 he employs his money, or capital, at 10 per cent, interest 
 (according to the Tables), that is, he is supposed to receive 
 ^10 for every ^100 advanced, and as much more as will 
 
 replace
 
 37 
 
 replace the ,£802 at the end of the 17 years. Ten per 
 cent, on the capital employed is ^80.. 4 ..0, but as he will 
 receive i; 100 each year, the difference, or ^19..16..0per 
 year, is the sum to replace the ^802 at the end of the 
 term ; this it certainly will do, ifu. man can make 10 per 
 cent, interest on <^19,.1G..0 every time he receives it; but 
 at 5 per cent, the same sum will only amount to ^511 in 17 
 years. Now I appeal to the common sense of every man, 
 if it be practicable to improve small sums of money at a 
 greater interest than 5 per cent. ? — indeed, beyond this 
 rate it is illegal to lend money; and no leases, annuities, or 
 government securities, can be purchased with small sums 
 of a few pounds each, which in general form the excess of 
 interest, to replace the capital with, when the income 
 ceases. 
 
 Such being tlie principle on which all the tables of com- 
 pound interest, for the valuation of leases and annuilies for 
 years or lives are formed, they must be practically wrong, 
 where they exceed the rate of 6 per cent, which being the 
 legal interest of the country, all calculations to replace the 
 capital at the end of the term, ought to be made in this rate ; 
 and as these tuhles form the basis of all the calculations 
 for annuities on lices^ they must follow the same fate: 
 for the present value of an annuity certain for any number 
 of years, is the several present values of the several sums 
 to be received at the end of the first, second, third, &c. 
 years, to the end of the term, added together; and the 
 
 present
 
 38 
 
 present value of a life annuity is nolliing more than the 
 amount of the said several values, each diminished in pro- 
 portion as the respective probabilities of the person being 
 alive at the end of the several years to receive them, are 
 below certainty, and continued to the utmost probable 
 extent of life ; this, according to Demoivre's hypothesis, i« 
 eighty-six years, and, according to Price and others, is twice 
 the expectation found by the tables formed to show the 
 progression of mortality, which greatly varies in different 
 countries, and indeed in the same country at different 
 periods ; for the continual discoveries in medical philosophy 
 to lessen the mortality of diseases, seem to have rendered 
 the tables formed fifty or one hundred years ago too in- 
 correct for the calculations of the present time. A French 
 author has computed, that the mean duration of human 
 life is increased, at least three years.) by the vaccine inocu- 
 lation alone. Now, as the values of annuities for lives 
 depend on the combinations of these two sets of tables, if 
 one requires new modelling, and the other is practically 
 wrong, all the results at rates exceeding 5 per cent, must 
 be doubly incorrect. 
 
 The following is a specimen of a set of Pocket Tables, 
 (which are now in the press), showing at one view, what 
 interest any purchaser makes of his money in buying 
 leases, annuities, or any nett yearly income whatever ; and 
 so that the returning sums to replace the capital when the 
 income expires, are computed at 5 per cent, interest (which 
 
 may
 
 39 
 
 may be practicable) ; also showing what Freehold^ or 
 perpetual Income, is of equal value with the purchase in 
 question. 
 
 Vnmherof 
 ye:irsto 
 coiitimie. 
 
 Yearly in- 
 come fur 
 each £100 
 paid. 
 
 Interest per cent, for 
 the time aud capital 
 replaced. 
 
 A Freehold of equal 
 \alue. 
 
 
 £ 
 
 £ s. d. 
 
 £ s. d. 
 
 10 
 
 15 
 
 7....1....0 
 
 5.. .15.. .10 
 
 10 
 
 16 
 
 8....1....0 
 
 6..,. 3.... 6 
 
 15 
 
 11 
 
 6.... 7.. ..3 
 
 5...14....2 
 
 lo 
 
 13 
 
 8.. ..7. ...3 
 
 6.. .14... 11 
 
 20 
 
 9 
 
 5.. .19.... 6 
 
 5.. .12... .2 
 
 20 
 
 12 
 
 8...19....6 
 
 7.... 9... .7 
 
 25 
 
 8 
 
 5...18....1 
 
 5...12....9 
 
 25 
 
 11 
 
 8...18....1 
 
 7...15....0 
 
 30 
 
 7 
 
 5.... 9... 10 
 
 5.... 7.... 7 
 
 30 
 
 10 
 
 8.... 9. ..10 
 
 7...13....9 
 
 50 
 
 C 
 
 5.. .10.... 5 
 
 5.... 9.... 6 
 
 50 
 
 10 
 
 9.. .10.... 5 
 
 9.... 2.... 7 
 
 EXAMPLE. 
 
 If I^ive £100 for £10 per annum to continue for 10 
 years ; the first column shows the number of years (or 10); 
 the second, the yearly income for each i'lOO paid (or £16), 
 agaiast which, in the 3'" column, is £8..1..0, being the rate 
 of interest the money is vested at for tlio time ; and tlic 
 
 4.H
 
 40 
 
 ^'^ column shows that such an investment is equal to the 
 purchase of a freehold, or perpetual income of JE6..3..6 
 per annum, both of which are thus proved : the difference 
 between the interest £8..1..0, and the yearly income £16 
 is £7..19..0 per annum ; now this improved at 5 per cent, 
 per annum, will amount in 10 years (when the income 
 expires) to £100, being the purchase money; and the 
 difference between £6.. 3.. 6 in the 4**' column, and the £16 
 received, is £9.. 16.. 6 per annum, which, improved at 5 
 percent, will in 10 years amount to £123,.10..0, being 
 sufficient to purchase a perpetual income of the same amount 
 (£6.. 3.. 6) when the .10 years have expired. 
 
 On this simple principle the Tables have been formed ; 
 they are extended to 100 years, and will most readily 
 answer for any rate of interest required. They will be 
 accompanied with Tables of simple and compound interest ; 
 true and customary discounts, values of annuities on lives, 
 church livings, &c. with a short history of interest, and an 
 elucidation of the principles on which the valuation of life 
 annuities is grounded. 
 
 GYE and BALNE, Piiiifers, 38, Gracechurch-Street.

 
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 UNIVERSITY OF ILLINOIS-URBANA 
 
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