UNIVERSITY 
 
 OF CALIFORNIA 
 
 LOS ANGELES 
 
 SCHOOL OF LAW 
 LIBRARY 
 
 '^
 
 A N 
 
 ESSAY 
 
 To afcertain the Value of 
 
 Leases and Annuities 
 
 FOR 
 
 YEARS and LIVES, ^^r.
 
 B OO'kS 'printed for S, Bii^'t, in Ave-mary-Lane ; D. Browne, 
 nuithout Temple-Bar; and J. Shuckburgh, near the Inner- 
 Temple-Gate. 
 
 EVerv Man his o\vn Lawyer : Or, A Summary of the Laws of 
 Envland in a New and Inftruftive Method, under the follow- 
 ino- Heads t'iz. I. Of Aftions and Remedies, Wjjts, Procefs, 
 An-efts and Bail. II- Of Courts, Attornies and Solicitors therein, 
 Turies, Witnefies, Trials, Executions, iyc. III. Of Elbtes and 
 Property in Lands and Goods, and how acquired, Anceftors, Heirs, 
 E-xecutors and Admin iibators. IV. Of the Laws relating to Mar- 
 riac;e, Baftardv, Infants, Idiots, Lunaticks. V. Of the Liberty of 
 the Subject, Magna Ckarta, the Habeas Corpus Aft, and other Sta- 
 tutes. VI- Of the King and his Prerogative, the Queen, Peers, 
 Judees, Sheriffs, Coroners, Juftices of Peace, Conllables, &c. 
 VII. Of Publiclc Offences, Treafon, Murder, Felony, Burglary, 
 Robberv, Rape, Sodomy, Forgery, Perjury, Ufc. and their Pu- 
 nishment. All of them fo plainly treated of, that all Manner of 
 Perfons m;:y be particularly acquainted with our Laws and Statutes, 
 concerning Civil and Criminal Affairs, and know how to defend 
 themfelve^, and their Eilates and Fortunes, in all Cafes whatfoever. 
 The Second Edition corrected, with Additions. 
 
 The City and Country Purchaser's and Builder's Dic- 
 tionary : Or, The Compleit Builder's Guide. Containing an 
 ExpcANATioN of all the Tcrins of Art ufed by Workmen ; as 
 alio what is neccffar)- to be known in the Art of Buii-ding, as well 
 by Gentlemen as Artificers of every Detwminatiov, in the following 
 uifcful Particulars, ws. The Qualities, Quantities, Proportions 
 and Value, of all Materials us'd in Building; the beft Method of 
 preparing thofe Materials, the Methods of wf^/wm/^ molt Sorts of 
 Artificers Work, the City and Country ?;-/<-?; of Workmanfhip, 
 Jphorifvis or Rules neceffery to be obferved in Build jng, as to Situ- 
 ation, ' Contrl<va7ice, Comfutlnefs, Uniformity, Con-venieiicy, Firm- 
 7iefs, Form, &c. Eftimates of Building any -F^T/rAi, great or fmall. 
 Rules for the Valuation of Houfes. Rules to be obferv'd in Repairs, 
 Scz. The Piopagation and Culture of fuch Trees as are us'd in 
 Buildin-?, and planted, as well for Ornament, as Profit, with the 
 Soils, a'nd Management proper for cnch Species ; and Rules for 
 their McJuriiig," Felling, and Sawing, i^c. A brief and clear 
 Dcfcription of'lhe various Parts of Painting and Sculpture, 
 and the feveral Arts depending on them: Including briefly, the 
 Theory and Pradice of Architecture in \i% di\?icxcnt Branches. 
 Very ufeful to Gentlemen, Workmafters, and Workmen, in ma- 
 king of Bargains, Contrafts, or Computations, relating to any 
 part of Buildir.o-, (s'c. Originally written and compiled by Richard 
 Ne-Tje, Philoma%. The Third Edition, correfted and improv'd 
 throughout, with the Addition of 2700 new Articles, in Archi- 
 tcaure, Mechanicks, &;c. In one Volume Oftavo. Piice 5 -i. 
 
 The Englijhmani, Right : A Dialogue between a Barriiter at 
 Law, and a juryman. Plainly fetting forth, i. The Antiquity. 
 2. The excellent defigned Ufe. 3. The Oiftce and juft Privileges 
 of Juries, by the Law of England. By Sir John Haxdes, Knt. 
 Soliicitor-General to the late King WilVw.m. Price 1 s.
 
 E S SAY 
 
 To afcertain the Value of /r^,' ' ' 
 
 Leases and Annuities 
 
 FOR 
 
 YEARS and LIVES, 
 
 And to eftimate the 
 
 CHANCES 
 
 O F T H E 
 
 Duration of LIVES. 
 
 WHEREIN ARE 
 
 Many Observations od Bills of Mortality, on the 
 State of the Coin, the Intere{l of Money, and the Price 
 of ThinjJs, in different Ages: And Notice taken of 
 feveral Writers on this Suhje<^h 
 ALSO 
 
 R U L ES are therein given for reducing Leafes for Years or Li\-es 
 to clear Annuities ; fhewing what Dcduftions are necefTarv to be 
 made for that Purpofe, out of the improv'd Rents on fuch Eltates ; 
 without which tlie "\^alue of them cannot be rightly afcertain'd ; 
 and ftating at what Rate Interelf ought to be computed in pur- 
 chahng Annuities or Leafes, either for Years or Live^, if the Pur- 
 chafers expeft to have the Principal reimburfed , or an Equivalent. 
 
 A Method likevvife is Ihewn, of calculating to what Number of 
 Years any given Life or Live?, on an even Chance, may probablv 
 extend, whereby Ellates held by thofe Tenures are converted into 
 Terms certain, and the Value of them thereby readily fecn : And . 
 all other Schemes for adjufting fuch Value are fhewn to be falfeor 
 precarious. 
 
 To which are added, in the Appendix, 
 
 Tables adapted to thefe feveral Purpofes, and to moft Cafes which 
 
 may ariie on the Value of Eftates of this Kind. 
 
 By W E Y M A N Lee, Efq; 
 
 L O N D O N: 
 Printed for S. Bip. t, in Ave-mary-Lane ; D. Browne, at thi" 
 Black ^ic-*^^ without Tefnple-Bar', and J. Shuckburgh, at tbe 
 Sun ncur the Inncr'''Tefnfk-Gate, Fleet-Street. 1738. 
 
 U

 
 1 
 
 TO THE 
 
 R E A D E R 
 
 '7 V;'vn'^ 
 
 SHALL not need to give ajty 
 Reafoiis for writing the following 
 Treatife^ or for fubliping it now 
 'tts wrote y becmife the: SubjeSf^ 
 Matter is, of a very common and 
 general Concern: ^orjh all J trouble you with 
 a long T ref ace y to give an Account of mylln-- 
 dertakingy or to cummend my Performance. 
 If r/y Notions and Calculations are right y. and 
 of any Service^ their own Weight will and 
 Jhall bear them out ; and if they are wrongs 
 or are tfelefs^ I have not fo great an Opnion 
 of my own, or fo little an one of otherTeofle's 
 Under ft anding, as to expeB that the World 
 will have any Regard to what I may fay to 
 recommend them* 
 
 But as in writing on this SubjeEi I have 
 traveUd out ofmypoferTrofejfion, and tin- ■ 
 dertaken a Province which may feem not to 
 belong to me ; and info doing have encounter' d^ 
 not vulgar Ofinions only, but fame which have 
 been received on the Authority of great Men ; 
 andfince I have carried the Matter fo far, as 
 to attack Rules and Computations which are 
 grounded, orfuppofed to be grounded, on Ma- 
 thematical Operations-, I hope you will, and 
 Ldmibt not but yon will, do me the Favour, 
 
 and 
 
 c»( 1 
 
 >:» 
 
 Hi
 
 iv To the R E A D E R. 
 
 and yoitrfelf the Jufike^ to read^ examine^ and 
 weigh the Arguments which I bring for my 
 AJJertions^ before you fafs any Judgment on 
 them. When that is done^ and if on due Con- 
 federation^ they have not that Weight in them 
 which to me they feem to have ; you will be 
 at Liberty^ and to be fure you will ufe the 
 Liberty^ to witholdyoiir AJfent from any No- 
 tions which may have been taken up and en- 
 tertained^ by m£ or any Man elfe^ without 
 fufficient Reafon aiid Foundation. 
 
 For the reft : If you have any other and 
 better Thoughts^ on this SubjeSi^ than you meet 
 with here^ you will be fo good as to give me 
 ^art of them j but if not^ and until fitch do 
 afpear^ you are welcome to thefe^^ and to my 
 ^Pains herein. 
 
 Your's, 
 Oflober 14, 
 
 THE
 
 THE i 
 
 CONTE NTS. 
 
 DISCOURSE I. 
 
 rO afcertain the Value of Leafes for Years^ 
 &c. Page I. to 129* 
 
 That there is no fur e Rule extant for 'valuing 
 LcafeSy which is fuited to the prefent Ctrcum^ 
 fiances of Things I. 
 
 The Methods for valuing Leafes for Lives hy 'much 
 the mnjt erroneous j and more fpecially and 
 largely conjtdered 2* 
 
 So7ne Circumjlances the fame in Leafes of loth 
 Kinds ; and what is faid of one^ in thefe In-* 
 fiances^ meant to he applied to both ibid. 
 
 ^ great Variety in the Value of thefe Eflates arifes 
 from the Difference in the Land'Tax^ and in 
 the ^lantity and ^lality of the Buildings on 
 them 3. 
 
 To afcertain the Value of Leafes ; that they he re- 
 duced to Annuities free from all Reprizes \ and 
 that Money he computed at 6 1. per Cent., ot 
 one above the common Kate 4. 
 
 To do this ; the Renty and all payments certain, 
 to be deduced 5. 
 
 Jls alfo the Land-Tax y Repairs, and incidental 
 Charges ib. 
 
 The Reverend Author of a Treatife calfd. The 
 Value oi Church and College Leafes confi- 
 der'd, &c. charges thefe Articles to the Lejfee*5 
 Account 6* 
 
 His Reafonsfor this Afertion not valid, ib. 
 
 a His
 
 VI 
 
 CONTENTS. 
 
 Hfs Ifjfifjuatiotjs, that Leafeholders^ in purchafing 
 dvd renewing their EJiateSy make 6 1. per Cent. 
 of their Money, not true, though artfully es" 
 prcjs'd Page p. 
 
 His Injfance, to pow the Advantage of buying 
 Leafes rather than hands of Inheritance, con" 
 
 ■ fiderd li. 
 
 ^he Comparifon here not rightly injlituted 13. 
 
 ^he Jnjiance, being founded on wrong Suppofitions^ 
 proves nothing 14, 
 
 ^he Author of this Difceurfe proves from thence^ 
 that the Purchafor oj Lands of Inheritance muji 
 have by much the greater Advantage 15. 
 
 'that by the Decreafe of Intereji of Money, from 
 5I. to 4 1, per Cent., the Value of one RJiate is 
 raifed five Tears ^Pur chafe, and the other only 
 one Tear and a ^larter 16. 
 
 What Interefi may be made of Money, not laid out 
 in hand, by the Purchafor of a heafehold, iiO' 
 thing to the 'h^urpofe 17. 
 
 ji Computation oj the great Difference between the 
 Value of hands oj Inheritance and heafeholds, 
 in the Purchafe, on any Variation of the Inte^ 
 refl of Money 1 8. 
 
 j4 Rife, in the 'Purchafe of hands, of no Service to 
 Owners of heafes, or of Fee-Swiple hands, any 
 farther than they have ^Power to fell their 
 Eflates 10. 
 
 The Reverend Author, inflating his Cafe, kept a 
 Referve of fome Money, to raife his Tenant's 
 Fine for reneiving ', and therein a&ed judiciou/Iy, 
 but not very j airly 24. 
 
 This Fund will do very little, if any Thing, more 
 than raife the Fine, and make the Income of each 
 Purchafor the fame : But fuch Identity only will 
 
 • hardly fet them on a hevel 25. 
 
 If Interefi of Money finks, and Fines are raifed, 
 
 . . the very Income oj the heafeholder vmji fitik tn 
 ^Prcfortkn to loth of them 27, 
 
 The
 
 CONTENTS. vii 
 
 The true State of the Comparifon between the Pur- 
 chafors of one Eftate and the other ^ is^ What is 
 the natural 'Troduce oj eachy and hoiv it will 
 arife Page 2.8. 
 
 If the Land-Tax y Repairs ^ and incidental Charges ^ 
 are to come out oftheLeJJee's Ejlate'^ the Owner y 
 though the Inter eji oj his Money referved comes 
 in Jidy will have only 3 1, 10 s. per Cent, for 
 his Money 3 r 
 
 Mr, Richards, /;; his Treatife on this SuhjeB^ ad- 
 mits a Tenant is to he conjiderd jor the two lajt 
 Articles ^ but not for the Land-Tan ^ for a Kea^ 
 fon very particular ib. 
 
 The Author contends that he ought to be confiderd 
 on all the Jrttcles 3 '2. 
 
 This JJfertion proved ^ from the Cafe given by the 
 Reverend Author ^ for that otherwife fuch a 
 '^Pur chafer will not have 3 1. per Gent, for his 
 Money fo laid out ib. 
 
 The fame ■proved from the Reafon and Jufiice of 
 the Thing 36. 
 
 The Way to adjufl the '^lajitum of the Land^Tax 
 for a Term ; fiated at 10 1. per Cent, per An- 
 num ^^ ^ Medium 39. 
 
 Ihefime on Repairs j and fated at the fame Sum 
 in common Cafes 40. 
 
 On this AnicUy many Exceptions to he made out 
 cj the general R ule 4 1 . 
 
 On Houfes in particular ^ 10 1. per Cent, might be 
 little enough ib. 
 
 Ihe common Method^ in Cafe of HoufeSy viz. to 
 compute the Money laid out at a higher Rate of 
 Intereft^ not afaje one ; and compared' with the 
 Author's in an hiftance 42. 
 
 7 he Article for Charges and^ accidental Lojfes^ the 
 Author divides into two • one for managing the 
 Efiate, the other for afjuring it, and paying the 
 
 ■ P^ent in all Events : and (fates each at ^ 1. per 
 
 • <-cnt. 44, 
 
 a 2 Thefe
 
 viii CONTENTS. 
 
 I'hefe DedaBiotJs, in the wbole^ the fame asfiip^ 
 pofsd by the Reverend Author to be going out of 
 a Fee-Simple Efiate : viz. 30 1. per Cent. Pag. 46. 
 
 Injfead of thefe two Allowances of ^\. and 5 1. per 
 Cent. Mr. Richards, in fating the Value of 
 heafes^ computes the Tenant's Annuity at one 
 Rate of Interejtj and the Out-goings in Rent at 
 another 47. 
 
 This Method not right \ the Reafons why mt, and 
 the ah fur d Confequences pown ib. 
 
 The ObjeSfion^ that Tenants^ fpecially in ptihlick 
 LeafeSy are under Covenants to pay their Rents 
 free from Taxes and all Reprizes , of no Weight 
 
 53- 
 
 Money, laid out in thefe Efates, to be computed at 
 
 6 !. per Cent. 5^. 
 
 T'hat a Purchafor, fo valuing his Money, will make 
 6 1. per Cent., a vulgar Error j fiown by an In- 
 jiance what will be made ib. 
 
 Shown ; how Money laid out in fuch Pur chafes is 
 
 to be reimburfed, and computed what Part is to 
 
 arife from the Surplufage of the Rents, what 
 
 from the Simple, and what from the Compound 
 
 Intere/l of fuch Surplufage 59- 
 
 T'his Computation demonflrates that the Lefjee is not 
 to be charged with the Land-fas, Sec 61. 
 
 Notice taken, what Value others put upon thefe 
 Eftates', and how they compute the fame 62. 
 
 The Abfurdity and Faljity oj Jo me of the?n 6^. 
 
 Though thefe Efates are reduced to abfoJute Annui- 
 ties, yet notfo valuable as fuch', and the Rea- 
 fon 7^- 
 
 T'he Author's Method of computing the Fine to be 
 paid on Renewals 79' 
 
 T'his compared with the common one, and a new 
 one of Mr, Richards ; and the Difference pown 
 by lnjlanc.es ^83. 
 
 A particular Abfurdity following from Mr. Rich- 
 
 ards'i Method ^7' 
 
 When
 
 CONTENTS. ix 
 
 Where one Tear*s Rent only may he a jitji Fine for 
 renewing feven Page 89. 
 
 Mr. Richards Jtates the Fine on fuch Renewal 
 higher than his own Rule admits ibid. 
 
 He profejjes to have no Conjideration here of the 
 Land-'taa ; and yet infotne Cafes in FffeB he 
 does^ or miifl conjider it 91. 
 
 'fhe Jtithor doubts the 'Truth of the FaSi alledged^ 
 that the Laiety^ tn renewing feven Tears^ gene- 
 rally take 2 \ Tears Value 93. 
 
 The Reafon^jor raijing Fines above the old Price of 
 one Tear's Value^ conjider' d \ with fome Ohfer^ 
 nations on /Ecroid'j Tables 95. 
 
 The Cenfure of Tenants as impojing on their hand" 
 lords in their Fines^ and of Landlords asfub- 
 mitting to them, not fo well grounded as cofn- 
 monly imagined 98. 
 
 Where Churchmen and Colleges cannot get Fines to 
 the real Value of their Efiate ; the Remedy pro- 
 pofed for fuch '•Trat^ice, by letting the Leafs ex- 
 pirCy conjider d \ and the Succefs of it doubted 
 
 99- 
 
 The Author thinks, fuch a Procedure in them would 
 
 not be an AB of Prudence 10 1. 
 
 And that a Law to oblige them to it would be un- 
 reafonable 102. 
 
 That they are not obliged to do it^ either as a Point 
 of Duty or fujiice to their SucceJJ'ors, as the 
 Law now jiands 103. 
 
 For Churchmen and Colleges, now and then, to Jink 
 a Fine and augment the fmall Livings tn their 
 Gift ', a much better ProjeB for promoting the 
 Inter ejl of the Church of England and the Uni- 
 verjities, than taking their Ejiates into their own 
 Hands 106 
 
 What Influence the Decreafe in the Value of Moj/ey, 
 
 and the Increafe in the 'Tnce of the 'Necejfanes 
 
 of Life, have in this Matter-^ conjider d 107. 
 
 a 3 What
 
 X CONTENTS. 
 
 What have been the Variations in thefe 7hings^ 
 within thefe lafi fifty or fixty Tears, and the 
 EffeSf of them P^ge lo/- 
 
 T^he fame, as fir hack as 200 or 300 Tears ago 
 
 109. 
 
 The State of the current Coin of England in diffe- 
 rent Periods of Tims ibid. 
 
 'The ^iteration of the nominal or real Value of the 
 Silver Coin equally pernicious \' in Oppofitwn to 
 the Notion of Bipop Fleetwood, and others III. 
 
 j^n Alteration in either JVay would not make the 
 Species of Silver more plenttjul, fo as to anfwer 
 any good Purpofe 113. 
 
 Nor prevent its Exportation 1 14. 
 
 The Intcrc/i of Money, on Loan, at different '^e* 
 riods of Time, fated 1 1 5. 
 
 "From the Difference in the Intereji of Money, and 
 the intrinjic Value of it, at feveral Periods of 
 Time, may he colleBed the Difference in the 
 ^rice of Things ib. 
 
 This 'illnjirated, and fbown by a familiar Injiance 
 
 The real Value of the Neceffaries of Life always 
 the fame', but the Price or nominal Value varies 
 as the Meafure varies 'by which we coinpute 1 20. 
 
 Whence the Dfference in the Intersji of Money, at 
 dfferent Times, arifes 121. 
 
 Increafe in the '^rice of Things, how far a Reafon 
 for advancing Fines 1 2J. 
 
 The Chcapnefs of Money, or Dearnefs of the Ne- 
 ceffaries of Life, of no Service to the Owners of 
 Leafes or Money, but may to the Owners ofFee" 
 Simple Lands 124. 
 
 The ConcUifion : IVtth an Jc count of the Nature 
 and Ufe of the Table annexed, for purchafing 
 Annuities orLeafes for Tears, or in renewing 
 the.m 1 26. 
 
 DISCOURSE
 
 CONTENTS. xi 
 
 DISCOURSE II. 
 
 TO cfiimate the Chance of the Duration pf a 
 Ltfe ; and to a/certain the Value of Leafes 
 
 and ^Annuities for Life^ &c. Page 1 30 to 259. 
 Leafes for Life^ as well as Leafes for lears^ jnnji 
 
 be reduced to ahfolnte Annuities 130. 
 
 Leafes and Annuities for Life muft he refolved w- 
 
 to 'Terms for Tears ibid. 
 
 This to be done hy computing the Chance of the 
 
 Duration of the Life, on which ibid. 
 
 The Nece^ty of making fuch Computation, to put 
 
 a right Value on fuel) Interejis 131 
 
 ^he Author* s Table exhibits the Term to which 
 
 any /ingle Life, therein mentioned, is equal 134. 
 Dr, Halley'i Table, for the Value of Annuities for 
 
 a Life, which were confiruded by a different 
 
 Rule, not a right one 135. 
 
 The Author*s Exception to it, viz. that it does not 
 
 pre/erve a proper Decreafe in the Value of an 
 
 Annuity, or in the Term to which fuch Value is 
 
 equal, from one Period of Ltje to another ib. 
 What is meant by the Decreafe in the Value, and 
 
 in the Term, explain d l'j6. 
 
 The fame Exception lies agalnjl Mr. Hayes'j, Mr, 
 
 Richards'5, and Mr. Morris'j Tables, for the 
 
 like Annuities 1 37. 
 
 The Author confines himfelfto Dr. Halley'j Table, 
 
 andpows the Defe3 in two remarkable Injlances, 
 
 quoad the Decreafe in the Term 138. 
 
 The Age of 30 not the befl Age of Life whereon to 
 
 have aji Annuity, occajlonally fjown 141, 
 
 This Dejccf goes through the whole Table 143, 
 This floozvn to be the Cafe in the Decreafe of the 
 
 Value of the Annuity, as well as in the Term ; 
 
 in Jingle Inftances, and through the whole Table 
 
 146. 
 a 4 Thii
 
 xii CONTENTS. 
 
 T'hisjkiown in a fhorter and more eafy Way Pag. 15 1, 
 
 ^he "-['able^ in this 'Particulars in no Sort conform^ 
 able to the Breflr.w -/ahle^ though conjiruBed 
 from thence j oj ithich a remarkable Injiance 
 given 152. 
 
 Objection to Mr. Haves*j and Mr. Richards'^ 
 Tables^ that the Value of the Jnnuities there 
 given IS fitch^ that one and the fame Life is 
 equal to a different Term as the Kate of Inter eji 
 varies 154. 
 
 'This ObjeBion does not lie againfi Mr, Morris's 
 Table ib. 
 
 ^he Force of this ObjeBion 156. 
 
 Dr. Halley*j P.ule for 'valuing thefe Annuities con' 
 fder'd 159. 
 
 The Author a^voids the Ufe of the Jford Odds, be- 
 caufe an equivocal Term 161. 
 
 Jin Arithmetic al^ ar adox^ given by Mr. Roberts, 
 a meer ^libble on the Word Odds ; the 'very 
 Odds mifcomputed^ and the Whole a low Perfor- 
 mance ib. 
 
 Dr. HalleyV Ruky for valuing thefe Annuities^ 
 fet forth 164. 
 
 Th^ Value of an Annuity for a Life of ten Tears 
 Age, given in his Table^ not the true Value^ as 
 it arifes on a Calculation made by his Rule j and 
 the Calculus fet forth 1 65. 
 
 That the Value of fuch an Anpuity, where rightly 
 computed by the Rule ^ is not the true one \ proved 
 hy a ^ofition of the DoBor's 168. 
 
 The fame proved hy another Po/ltion of the DoBor's 
 
 171. 
 
 T^e Fa/fty of the Rule Jhown from hence ; for that 
 
 it produces Ciich Sums as cannot he the Value of 
 
 fuch Annuities 174. 
 
 For hifiance ; that the Term^ to which the Value 
 produced is equals in younger Life is muchjkorter 
 ^han the Terniy to which the Life is equal on an 
 
 even
 
 CONTENTS. xiii 
 
 even Chance ; hut in older Life the T'etms the 
 fame Page 174, 
 
 Another Inftance \ that the Term in the Annuity ^ 
 and the Chance of the Diiratton of the fame 
 Life J differs as the Kate of 1/jteref} differs iy6 
 
 That this Difference is a neceffary Refult of the 
 Rule lyy. 
 
 Where a certain Term and Value are given^ and a 
 DeduBion he made of a proportionate ^art ; 
 great Difference whether the Deduction he made 
 out of the Value y or out of the Term ib. 
 
 That a Term and a Value given will reciprocally 
 C6rrefpond\ but if we add to^ fubjlra3 from^ or 
 multiply that Value and that Term in a certain 
 Proportions the new Term and Value will not 
 correfpond 1 83, 
 
 Shoivn^ wherein the Error of the Rule lies^ viz. 
 in fippojing that in each Tear of Life a Value 
 anfwermg to the Chances of Mortality arifng in 
 that Tear of Life are then to he deduBed \ 80. 
 
 //" on the Chance of Mortality fome Value were to 
 he deduBed in each Tear of Life^ the Sum prO' 
 duced by the Rule not the right one 182. 
 
 Thefe Pojitions dejiroy the Foundations of Dr, Hal- 
 ley 'j and Mr. Moivre'i Rule ^ applied in an 
 Injlance 1 86*. 
 
 The Author s T'oftion^ that the Value correfponding 
 to the Chancer of Mortality arifing in each Tear 
 of Life is not to he deduSisd in fuch Tear ; 
 proved 1 88. 
 
 The DoBors Method of computing founded on fome^ 
 thing which can't pojjtbly happen ip3. 
 
 The Author s ^ofition proved in another Way i ^5. 
 
 The Difference between the Value of an Annuity 
 
 for one Tear^ if a T'erfon fo long lives ^ and the 
 
 Value of it for the fir(i Tear of the Life of the 
 
 fame 1?erfony where the Annuity is for his Life 
 
 ahfolutely 1 97, 
 
 Tkc
 
 xiv CONTENTS. 
 
 ^he DoHors Rule conjider'd in another Light Pag. 200. 
 ^he Sum given in his I'able^ as the Value of an An- 
 nuity for a Lifsy not the Value of an Annuity for a 
 Lije, but of feveral Annuities put together ; and 
 fhown what, and how they differ 202. 
 
 ^he Rule faulty in making Suppofitions not confiftenty 
 
 • or not ju/i 209. 
 
 I'he DoBors Maxim^ that the Purchafor of an An- 
 nutty for a Life is to pay for fuch Parts only of the 
 Value of it as he has Chances that his Nominee is 
 livings is true,, but not well applied 2 1 3 ' 
 
 This fhown by an Injiance^ by which the Faljtty of the 
 Rule is evidently feen 214. 
 
 The Senfe of the Maxim, that the Chances to be paid 
 for muft be equal in Value as well as in Number^ 
 to the Chances to receive 219, 
 
 'J^hii Cafe vefembled to a Lottery 221. 
 
 ^his Maxim does not /how the Chance of Vitality of a 
 Life^ or the Parts of the Value of the Annuity to be 
 received; or indeed prove any 'Thing 225. 
 
 The Conclufion of this Head^ with a flmt Recapitula- 
 tion 227. 
 
 The Chance of the Duration of a Life the fole Meafure 
 of the Value of the Annuity depending thereon 228. 
 
 The Author^s Method of forming his Table of the 
 Chances of Vitality 229. 
 
 This Table drawn chiefly from the Bills of Mortality 
 /ok London, and fome Obfervations on them ib. 
 
 A Life of ten Tears Age the befl, and equal to a Term 
 (f 2S Tears 230. 
 
 A Life of that Age^ in London, equal to a Term ra- 
 ther longer 231. 
 
 Obfervations on a Treatife caWd^ A true Eftimate 
 of the Value of Leafehold Eftates ; in which a 
 Life of ^o Tears Age is flated as the befiy and equal 
 only to a Term of i/\ Tears ^ and other Lives in Pro- 
 portion 233. 
 
 The
 
 CONTENTS. 
 
 XV 
 
 'The Rule to compute the Chance of Vitality of a Life 
 
 Page 24:?. 
 
 Shown in an Inflame oj the Bifhops of the Kingdom^ 
 and the Alderme}2 of London 245. 
 
 ^he fame^ in an In fiance of the Inhabitants of thePa- 
 rjftj mentioned by the Author lafi cited 247. 
 
 Obfervations on the Bills of Mortality for London 250- 
 
 The fame, on a certain Society kncvon to the Author 251. 
 
 7he jame, on the Nomineei in the Government Annui- 
 ties on Survivo;fJ:)ip ' 252. 
 
 An ObjeElion to the Bills of Mortality for London, 
 viz. that they are not a proper Rule to determine the 
 Chances oJ Vitality of Lives, confJer'd; and how 
 far they may be proper for this Pinpoje 254. 
 
 Sotne Calculations on thejn^ to fhoiv the Chance of tlje 
 Duration of the Lives of the Inhabitants there', and 
 their Longevity 
 
 DISCOURSE III. 
 
 rO e/limate the Chance of the Duration of two or 
 more Lives^ and to afcertain the Value of Annu- 
 ities and Leafes for fuch Lives^ dec. Page 260. 
 This the mo/i difficult Part of the PVork, and mofi 
 weakly performed by all M^riters on this Subjecij not 
 excepting Mr. Richards, whojolhws Mr. Moivre'i 
 Rules z6o. 
 
 The Alethod of computing the Value of thefe Annuities^ 
 prefcribed by the Author, to tvhofe Tabks Sir Ifaac 
 Nevvton'j Name is prefix' dy a very bad one 262. 
 Air. Hayes*i Method worfe than the jormer 265. 
 
 Air. Morris'i the fame with Air. Moivre'^, and con- 
 ftde/d with it 266, 
 
 Mr. Richards'j Tables, formed by Air. Moivre'.v 
 Rule, do not give the true Value of theje Annuities 
 
 166. 
 
 Thefe his '^T'ables wrong, becaufe his Principles, viz. his 
 
 T'ables for the Value of Annuities for a ftngle Life 
 
 are
 
 xvi CONTENTS. 
 
 are wrong : But not chiefly for this Reafon Pag. 2^7. 
 If thofefirfi T'ahles were right ^ yet thefe here are wrong ; 
 
 for that the Rule, by which they were conftruSiedy 
 
 is wrong 268. 
 
 An Inftance of an Error in thefe Tables, and Jbown 
 
 that it arifesfrom a Defeci in the Rule 268. 
 
 A fecond Li/lance to the fame Purpofe 270. 
 
 A Cafe flat ed by the Author, which Jhows thefe 'Tables 
 
 to be jalfe throughout 273. 
 
 Jn this Cafe, an obvious Error in the Rule obferved 
 
 2j6. 
 
 From this Cafe, a little varied, fhown to what fort of 
 Annuities on Lives the Value given in the 'Tables 
 may fometimes be equal 278 
 
 Mr. Moivre'j Rule for valuing thefe Annuities fet out, 
 and confide/ d 280, 
 
 Shown to be falfe, as grounded on a wrong Way of 
 Reajoning 281, 
 
 Shown more clearly by an Inflame 287. 
 
 The Rule confide/ d in another Light ; and/hown, in 
 an Inflame, that the Annuitant cannot pojflbly have 
 thofe Parts of the Value of the Annuity which the 
 Rule allots to him 292. 
 
 The Rule put into another Form; and the fame Thing 
 fhown from thence 301. 
 
 Ihe Rule, if applied to adjufl the Term to which two 
 Lives are equal, may give the true Term ; though, 
 when 'tis applied to adjufl the Value, it does not 
 give the true Value 305. 
 
 The Reafon of this Difference fhown 307. 
 
 That the Rule carries to the Annuitant* s Account either 
 more or greater Values than belong to him, fhown 
 by an eafy and obvious Argument 308, 
 
 Dr. Halley'j Hypothefis, for adjufling the Chance of 
 the Duration of two Lives, confide/ d 3 10. 
 
 The DoElo/s and Mr. Moivre's Met hods, for afcevtain- 
 ing the Value of Annuities for two or for three Lives 
 0nd the Survivor, not the fame 311. 
 
 The
 
 CONTENTS, xvii 
 
 The DoEio/i HypothepSj for adjufling the Chance of 
 the Duration of two Lives, not right ; fiown in 
 Numbers 313. 
 
 7'hii Hypothe/tfj and Mr. Moivre'x Method for ajcer- 
 taining the Value of Annuities on two Lives, in fome 
 Points do agree 322. 
 
 I'hisfame Hypothefis not right, Jhown by Lines and a 
 Scheme 324. 
 
 'The Error of this Hypothejis, and of Mr. Moivre'f 
 Method^ Jhown by Mr. Richards'j Tables conftruc- 
 ted from thence 3 3 (5. 
 
 The Author admits he has not reprefented the DoEio/s 
 Reafoning on Lines and Scheme juji in the Manner 
 in which the DoSior reprefents it ; but afjerts^ that 
 his Reprefentation of the two is the more right one 
 
 341. 
 
 The Error of the Hypothefis fhown by a Calculation of 
 the Chance of the Duration of two Lives made by 
 the Breflaw Tabk^ which was framed by the Doclor 
 for computing fuch Chance 3 /\6. 
 
 An Obfervation of the extravagant Eflimate of fuch 
 Chance by that Table 349. 
 
 The Author doubts the Truth of the DoSiors Pofjtion, 
 viz. that the Chances of Vitality and Mortality on 
 one Life, multiplied by the Chances of a fecond Life, 
 give us the Chances on both Lives 350. 
 
 The Error of the Hypot he/is fhown by a more familiar 
 Argu?nent, and an eafy Demonfiration in Numbers 
 
 359- 
 Mr. Richards^ Tables^ for the Value of Annuities for 
 
 three Lives, examined 353. 
 
 A grofs Error in them, fhown in an Inflame ib. 
 
 Another fI:>oivn, which goes through the whole Table 355 . 
 A Cafe jiated by the Author demonjirates the Falfuy of 
 
 this Table ^67. 
 
 The Errors in them arife chiefiy from the Rule by which 
 
 conftruEied 371. 
 
 Another Error in them, viz. that they do not obferve a 
 
 Decreafe
 
 xviii CONTENTS. 
 
 Decreafe in the Value of Annuities in any fort of 
 
 Proportion to the Jncreaje of Age; foown in fever al 
 
 bifiances Pag. 375. 
 
 'This Fault arifes from a Defect in the original Hypo- 
 
 thejis 384- 
 
 Ihat the Term^ to 'xhich the Value of Annuities for 
 
 one or more Lives^ is equal, (Joould vary as the Rate 
 
 of Inlerefl varies, which is a nectffary Confequence of 
 
 the Hypothefis; greatly ab fur d 385; 
 
 "that the Term to which the Value of an Annuity Jor 
 
 three Lives of iz Tears Age given in the Tables, is 
 
 equal, (hould be longer than the Term to which any 
 
 cm of the Lives have a Chance of Duration, where 
 
 fuch Chance is computed by the H\pothefs \ greatly 
 
 ahfurd 3^9- 
 
 A Computatimmade of fuch Chance by the Hypothecs 390. 
 
 Mr. Moivre'i Rule jor valuing thtfe Annuities fet out 
 
 391. 
 Shown to be falfe, becaufe it allots to fuch Annuitant 
 more or greater Values than do belonf^ to him 393. 
 Shoxm, that the Parts allotted by the Rule cannot pof- 
 ftbly belong to the Anmiitant 1)91' 
 
 Shown, in an Inflame, what are the very Parts of the 
 Value of the Annuity too much, which by the Rule 
 are brought to the Annuitant's Account 199- 
 
 This confirmed by a Cafe (tated by Mr. Richards 402. 
 Shown, that Dr. Halley'j Hypothefis makes more 
 Chances of Vitality on three Lives, than there really 
 are, viz. feven inflead oj fix 4^5. 
 
 That this mufi be the Cafe, fhown by a fecond Argu^ 
 ment 4°7- 
 
 Mr. Moivre'j Rtile, if confonant to the Rule, mvfl be 
 therefore falfe 4° 9* 
 
 His Rule not confonant to the Hypothefis in all Inflames 
 
 ibid. 
 His Rule does not agree with the DoSio/s Rule for cal- 
 culating the Value oj Annuities either for two or 
 three Lives : Shown by an Infiance 410. 
 
 The
 
 CONTENTS. xix 
 
 I'he Rule agrees with the Hypothejis in fuppofmg /even 
 Chances on three Lives, inflead of fix Page 41 3. 
 
 An additional Patilt in the Rule, by giving too great 
 a Value jor fuch fupernumerary Chance ibid. 
 
 "This Fault in Mr. Moivre'^ Rule caufes a moji extra- 
 vagant Eftimate of the Value of fuch Annuities 417. 
 
 The Author s Rule for valuing Annuities for two or 
 more Lives, by computing the Chance of the Dura- 
 tion of the given Lives 41^' 
 
 "Ihe Author's Method of computing the Chance of the 
 Duration of two or more Lives and the Survivor 420. 
 
 I'be Method illuftrated by an Inflame on Lives of equal 
 Ages 421. 
 
 Compared with Air. Moivre'j Rule 427. 
 
 7'he Method the fame, and illuftrated by an Inftance^ 
 on Lives of unequal Ages 43 1- 
 
 ^he Ju/iice of the Method fhown, and proved from 
 Dr. Halley'j Way of Reafoning 434, 
 
 The Contingency of Survivorfjjip between Lives of un- 
 equal Ages, how adjufted 43 8. 
 
 A Comparifon of the Chance of the Duration of two or 
 three Lives, where the Computation is made in Dr, 
 Halley*j or Mr. Moivre'x Method, and when in 
 the Author's 441. 
 
 An algebraical Calculation, to adjufl the Duration of 
 the Credibility of human T'efiimony, grounded on ar- 
 bitrary and wild Suppo/itions ; and if thofe were 
 right, the Conclufion abfolutely falfe 443. 
 
 tables 455, &c. 
 
 Adde?ida
 
 Addenda & Emendanda. 
 
 PAG E 14, Line s, for or read /or. Pag. 6l, 1. 23, for Valnatien 
 read rjfef. After Pag. 62, for 61 read 6j. Pag. 64, 1. 26, for 
 200/. read 2000.'. Pag. 77, 1. 20, for they read we. Pag. 92, 1. 21, 
 for Annuity, foleiy read Annuity ftldy. Pag. 125', 1. zj, for «/, r/;j; read 
 of that. Pag. i?7, 1. 25', for Merris's read Morris's. Pag. 159, 1. 17, 
 for aw^ in the read ^«^ f/je. Pag 219, 1. 6, for that there read f/;ere. 
 I'ag. 2J7, 1. 6, forfoMwry read Coimtry. Pag. 246, 1 29, add, but not out 
 tf the Indhiduals who voere fuch at his Mccejjlon. Pag. 266, 1. 21, for theft 
 read the. Pag. ib. 1. 22, for the read tkefe. Pag. joy, 1. j, for the read 
 /inrf the. Pag. 305, 1. 10, fox Lift makfs read L:/e, wJd/;*;. Pag. Jjj, 
 1. 20, for ^s read ff. Pag. ?J7, I. 6, for coi«y;iie read coincide. Pag. 
 364, crofs Lines, for ten read wc/T/e. Pag. J75, 1. j2, for mj^ read 
 tak(s, Pag. 379, I. 17, for aiovethe read ^ioi/e, tie. After Page j8i, 
 for J73 and 372 read 382, ^83. Pag. 373, 1. 9, for Regard c/ read 
 'Regard. Pag 414, 1. 10, ioT three Lives Kzi. three 'joint- Lives. Pag. 426, 
 1. 1 1, for at read of. Pag. 428, 1. 33, for 47 read 44. Pag. 452, 1. 24, 
 for 10 read 11. Pag 449, 1. 30, for commmate read cemmy.nicate. 
 
 AN
 
 A N 
 
 ESSAY 
 
 To Afcertain the VA L U K of 
 
 I E A S E S, ^c. 
 
 PROPOSE to confidcr here, 
 whether there be any, and 
 what i^, the Rule to compute 
 the Value of Lcafcs for Terms 
 of Years, and for one or more 
 Lives ; fo far that Perfons who 
 have Occafion to tranfadl thofe 
 Affairs, either to buy, cr to fell, or to renew, 
 (and one or other of thefe Tranfadions almoft 
 daily occurs,) may know whether the Bargain 
 they make is an even and a fliir one. That there 
 Ihould be a Want of fuch a Rule at this Time of 
 Day, after fo many Affairs of this Kind have 
 been tranflidied, may fcem a little flrange ; and 
 yet none has been produced, or at leaft none has 
 fallen under my Obfervation, to which fome ma- 
 terial Qbjedion, as I apprehend, dees not lie in 
 A the
 
 [ ^ ] 
 
 the prefent State of Things : And I am not fo 
 fond of my own Produdions as to imagine no 
 Objedion can or will be made to the Method I 
 mean to offer : Thus far only I fhall go ; I will 
 propofe it vvith my Reafons to fupport it, and 
 hope it will appear to come nearer the Mark 
 than any now extant, or commonly made Ufe 
 of- and that, with fome lew Additions and fuch 
 Alterations as may make it fuit the particular 
 Eftate in View, it will anfwer the Purpofes in- 
 tended. 
 
 The common Methods for computing the 
 "Value of Leafes for Lives are, in my Judgment, 
 much the moft erroneous, and therefore I mean 
 to conlider them more fpecially • and the Me- 
 thod I propofe for that Purpofe, is by reducing 
 them to Terms for Years, and by eflimating them 
 as fuch. Now if this can be done within any 
 tolerable Degree of Probability, 'tis certain the' 
 Value of them may be fo far known ; for the 
 Value of Terms for Years m.ay be computed to 
 the Exadlnefs of a fmgle Shilling, if necelfary, 
 at any given Rate of Interefl : And the Thing, 
 I think, may be done as near the Truth as in a 
 Matter of fo much Uncertainty can be cxpedlcd ; 
 and is to be done, by fuppoling fuch Leafes to 
 have a Duration equal to the Term to which the 
 given Life or Lives on an even Chance are efti- 
 mated to be equal : And then the Value of them 
 likewife may be adjufted, and known to a reafo- 
 jiable Degree of Probability. 
 
 But when Leafes for Lives are converted into 
 Terms for Years, or may be confider'd as fuch, 
 
 there
 
 C 3 ] 
 
 there fccm to me fome other CIrcumftances to be 
 taken into the Account, before we can determine 
 the V^alue of. them : And Jmce thefe Gircum- 
 Itanccs are fuch as do or may attend all Leafes, 
 whether for Lives or Years, I fhall not need to 
 make any Diftindion between fuch Leafes with 
 Regard to thefe ; but vvhat I fay upon Leafes 
 for Years under this Head, I think is applicable, 
 and I mean fhould be applied, to Leafes for 
 Lives likewife. 
 
 It has been often, and I believe very truly, 
 faid, that there is hardly one Caufe comes into 
 a Court of Equity, but has fome Circumftances 
 attending it, peculiar to itfelf, which make it 
 vary from any other : And, I think, it may as 
 juftly be faid that fcarcely two Leafehold Ellates 
 in the Kingdom, whether for Lives or Years, 
 are in all Inllances juft fo much alike, that any 
 one common Rule for cllimating the Value, ei- 
 ther in Purchafes or Renewals, will quadrate 
 with them : But that fome other Particulars, be- 
 fides thofe of the Referv'd and Improved Rent, 
 are to be taken into Conlideration, and yet fuch 
 others arc rarely or very flightly regarded. 
 
 In the North and Weft of Evgland the Tax 
 on Lands is in general much eafier in Proportion 
 than in the South and Eaft Parts • and even in 
 the fame Countries Eftates of the fame Value in 
 Rent are rated in thofe Aifelfmcnts in an un- 
 equal Manner. In fome Countries the Buildings 
 in general are of Stone or Brick, in fome of 
 Timber and Lime, and in others of fomethmg 
 iefs durable ; and Eftates have feme more iome 
 A 2, fewer
 
 ■ [ 4 ] 
 
 fewer Buildings on them, and thofe in a better 
 or worfe Condition. Now where Ellates differ 
 in any of thefe Gircumftances, if the Difference 
 be to any Degree, I confefs I cannot fee any 
 Reafon or Equity to make ufe of a general Rule 
 of fo many Years Value of the improved Rent, 
 after deduding the referved Rent only, either in 
 taking, buying, or renewing fuch Kftates. And 
 where thefe Things, as undoubtedly in fome 
 Cafes they muft, and in Fad are, taken into 
 Confideration ; I very much doubt whether it 
 be done in a regular and even Way. 
 
 I have before obierved that the Value of 
 Leafes for Years may be computed to a 
 great Exadnefs, and molt certainly it may ; 
 but then thofe Leafes mufl be firil reduced 
 to abfolute Annuities, by which I mean Sums 
 certain, clear of all Deductions and Out- 
 goings whatfoever ; for 'tis on fuch Suppolition 
 only, that their Value can be cxadtly afcertain'd, 
 Befides this ; I think an Allowance lliould be 
 made to the Tenant for alfuring fuch Eftate and 
 the Rent from all cafual Lodes and Damages, 
 and alfb for his Trouble in managing and look- 
 ing after the Eftate, and receiving and paying 
 the Rent, and that thefe ought to be made in 
 Proportion to the improv'd Value of the Eftate. 
 And then, that the Surplufage or Neat-Money 
 ariling, after fuch Reduction and Allowances 
 made, be looked upon as the LelTee^s Eftate and 
 Intercft ; for 'tis well known that the Proprietors 
 of abfolute Annuities arc free from all Engage- 
 ments which can poffibly make them fubject to 
 
 any
 
 ^ [5 3 
 
 anv Dcdudions, or to any Loflcs or Clwfrge>^ 
 ■whatever. This Neat Produce then being Hated 
 as the Lefiee's Intereft, I propofe that the Mo- 
 ney laid out on a Purchafe, or Renewal of it be 
 reckon 'd at an Interell of 6 /. per Cent, or one 
 per Cent, above common Intereft : By which I 
 would not be undcrilood to mean that fuch Te* 
 nant is to make or will make 6 /. per Cent, of his 
 Money ;. for though it fliould be fo reckon'd in 
 Computation, yet in Fadt there will not be fo 
 much coming to the Tenant for his own Ufe. j 
 as I fhall take Occaiion to fliow under the pros- 
 per Head. 
 
 I am very fenfible that mofl Perfons, in Tranf-^ 
 aftions of this Kind, conlider and allow only 
 the Rent referved to the Landlord \ but furely 
 all other Payments made by the Tenant, whe- 
 ther Fee-Farm, Chief, or Quit-Rent, or what- 
 ever other Denomination fuch Payments como 
 under, if they arc certain and perpetual, and 
 are chargeable on and iffuing out of the Eftate, 
 and are not ufually and commonly difcharged by 
 an ordinary Tenant at Will, muft be and will be 
 admitted to come into the fame Clafs j and aii 
 they are determinate Sums, and are or may be 
 certainly known, I apprehend there can't well 
 be any Difputc about them, if they are, and 
 ■where they are, of fuch a Value as to delerve 
 Notice.^—-* 
 
 As for thofe three Articles, 'viz, the Land- 
 
 Tax, Repairs, and an Allowance for Charges 
 
 and accidental Lofles, all which I have propofed 
 
 and contend ought to be difcountcd b) the Lef- 
 
 A 3 ioVf
 
 [ 6 ] 
 
 for, before we can truly and properly fay the 
 LeiTcc's Eflate is an abfolute Annuity; about 
 thefe there is great Difagrecment, as wcU in 
 Pradicc, I believe, as in Theory. The Author 
 of a Trcatife call'd, The Value of Chnrcb and 
 College Leafes coiijiderd^ who has been f lid in the 
 common Prints to be a Dignitary or Prelate of 
 the Church, urges^ that all thefe Burdens ai'e tp 
 lie on the Tenant : And Mr. Richards^ in aTradt 
 publifhed fomc few Years fince, which' he calls, 
 The Gentlemaj/s Steivard atid 'Tenant of Mduors 
 inJirnBed^ admits that a Tenant is to have Com- 
 penfatidnfor the two laft, but not for the firfl 
 Article; and I ain of Opinion, and find I am 
 not lingular in Opinion, that the Landlord ought 
 to make ah Allowance for every one of thefe 
 Article's." Before I give my Reafons for my 
 Opinion, I intend to*conlider what is alledged 
 for the contrary one. The Pvevereud Author 
 tells us. then, That Taxes, Repairs, and Acci- 
 dents,; are not peculiar to Church or College 
 Eftates ; but Eitates of Inheritance are equally 
 liable to the fimc Inconveniencies : I add, nei- 
 ther are the Eftates of the Laiety, where let at 
 a fmall Rent with a Fine taken, exempted from 
 thefe Burdens ; for I mean to make no Difference 
 between them, unlcfs any particular Argument 
 neceilarily leads me into it. Now I make no 
 Doubt but that forty Years E:^perience, and 
 more, has made moft People fenlible of this great 
 Truth, that Lands of Inheritance are fubjed to 
 the Land-Tax ; and I Ihould be much pleafed 
 if I could have the Opportunity of concurring 
 
 with
 
 [ 7 ] 
 
 with this learned Writer, in ranking it among 
 the Inconvcniencics. ^attending Eftates in Land 
 which arc accidental only, for at prcfent it mult 
 be number'd among the certain ones : But lc£ 
 the Taxes and Rcjyairs be certain or accidental 
 Inconveniencies, I confefs I don't fee the Gon- 
 fequence. Owners of Lands of Inheritance, if 
 they let thofeLands-at the rmprove4 Rent, muil 
 pay the 'Taxes, and do Repairs, and not their 
 Tenants; therefore Owners oi Lands of Inhe- 
 ritance) where they let their Lands at a fnvall 
 annual Rent, and take a Fine inlkad of and 
 equivalent to the reft, mult not pny the Taxes, 
 or do Repairs, but their Tenants mull : Where- 
 as, if any Inference can be drawn from this Way 
 of Reafoning, the Rcverfe is the mofl natural 
 and moll obvious one. ; ;■ 
 
 If we fnould ask the Reafon why Owners of 
 Lands oi Inheritance, if they let thofe Lands at 
 the improved Rent, muft pay the Taxes ; we 
 fhall undoubtedly be told, becaufe fjch Owners 
 receive the Rents, or may take the Profits, of 
 fuch Eltates. For the very fame Reafon, and 
 on the fame Grounds^ if Landlords letting at a 
 fmall Rent do, and fo far forth as they do, re-^ 
 ceive the Rents and take the Profits of their 
 Eftates, though in a Manner and at a Time dif- 
 fering from that in. which the Owners of Eil^tes 
 who let at the improved Rent, receive and take 
 theirs, then fo fir and for fo much of the Rents 
 and Profits of their Eftates as comes to their 
 Hands, they muft like wife pay the Taxes ; they» 
 as well i^s the reft of the Kingdoip, being obliged 
 A 4 tQ
 
 [ 8 ] 
 
 to take and hold their Eftates with fdch Burdens 
 as the Laws of the Land lay on them. 
 
 I obfcrve that this Author walks over this 
 Head very tenderly, and contents himfelf with 
 telling us he believes, where the Rent referved 
 is a third or a fourth Part of the improved Value, 
 that there a Confideration and Allowance is had 
 by fuch Landlords ; which is a tacit ConfeiTion 
 that there is fome Equity and Reafon in the Ex- 
 pe(flation of Tenants in fome Cafes. I fuppofe 
 therefore, when he tells us in general Terms, 
 that Tenants are unrealbnable and partial to 
 themfelves, if they expe6t to hold their Leafe- 
 hold Efiates free from thefe Inconveniencies, he 
 docs not mean that there is an Argument to 
 prove that all Tenants ought to be charged with 
 them, but fuch only as hold at a fmall Rent : 
 But then he is fo good to Landlords as to leave 
 them a great Latitude in judging, or rather makes 
 them fole Judges, what they fhall call a fmall 
 Kent, or what fliall be iook'd upon as a reafo- 
 nable Confideration on a great Rent. -But if 
 Tenants, whether holding at a great Rent or at 
 a fraall Rent, fliould give the fame civil Appel- 
 lations to their Landlords, and call them unrea- 
 fonable and partial Men, if they expedt to be 
 exempted from thefe Burdens ; would not the 
 Argument on their Side have juft the fame 
 Weight, and full as much Manners, as on the 
 Other Side? 
 
 We are told farther, that thefe Charges on 
 Lands of Inheritance, in fome Parts of England 
 zt leaft, make a ]3raw-back of ^o /. per Cent.
 
 [ 9 3 
 
 ^hich we admit may be true; but then 'tis equally 
 true with Regard to Leafe-hold Eltates, at leaft 
 jn the fame Parts of England : The ncccfrary 
 Confequence of which is^ that the Owners of 
 fuch Eflates, and in thofe Parts of the Kingdom, 
 can receive for their own Ufes no more than 70 /. 
 per Jnmim out of an Ellate which is call'd, and 
 at the improved Rent is worth, ipo/. per An" 
 num. Now though fuch Tenants, or the Perfon 
 whom he drelles up and introduces in a Fool's 
 Coat, that he may have an Opportunity of Ihcw- 
 ing him away, do not underlland, or will not 
 attend to, nice Calculations ; yet, I fuppofe, 
 they may have Senfe enough to find out that yoL 
 is not quite io much as 100/. and that the one 
 is not fo good an Interell: for the Sum of Money 
 fuppofed to be laid out in the Purchafe of this 
 Kflate, as the other is ; though, perhaps, they 
 may not be able to difcover exadlly what Inte- 
 refl they have for fuch Money, for want of a 
 little of that fame which fome others are fp great 
 Mailers ofl 
 
 And if this Author would have paid a proper 
 Regard to his own Aflertions, or had attended 
 here to Calculations, I don't fay any nice ones, 
 but one of his own, I think he would not, I am 
 fure he fhould not have complain'd of the Gene- 
 rality of the Worldj as unrcafonable and partial 
 to themfelves, in exped:ing to make full 6 /. per 
 Cent, of their Money, above all Deductions, 
 when they purchafe Leafe-hold Eftates : Or if 
 thefc Expectations in fach Purchalers are really 
 pfeafonable, he, of all Men, Ihould not have 
 
 givea
 
 [ ^o 3 
 
 given them Grounds whereon to ra'ife f^ich Ex- 
 pedtations, and yet he has done it in the ftrong- 
 eil Terms. 
 
 In Page the fixth of his Treatife, having in- 
 formed the Tenant, to whom he addrcifes his 
 Letter, that on the nicell Calculations, the Rule 
 "for renewing feven Years lapfed in a Leafe of 
 'twenty-one Years, fuppoling the Intereft of Mp- 
 ney at 6 /. per Cent, is near two Years and an 
 half Value ; with an Air of Infult, he asks hin^, 
 if 2it this Time he can put out his Money oji 
 Land Security, and' have 6 /. per Cent, for it : 
 And in a Line or two after tells hini, if hi.s 
 Landlord, on renewing his Leafe,. had treate^ 
 him on that Foot, that he ought to be thankful 
 that his Money was laid out fecurcly upon Land, 
 after' the B-ate of 6 /, per Cent, The Inilnuati- 
 oris here, that a Perfon renewing his Leafe at the 
 Hate here mentioned, would have 6/. pet Cent. 
 for his Money, are exprefs'd in Terms fo itrong, 
 or fo artful, th.it from thence, and thence only, 
 every Reader mull have concluded that he meant 
 i^a^yfuch "a' Tenant .would really make 6 I. per 
 'Cent, o£ his Moi^ey/ In this Senfe he has been 
 Mnderftbod by fohic, and mufi and will be under- 
 wood by every Man, who docs not attend to his 
 fubfcquciit declarations ; which plainly enough 
 ihow, that fofnething elfe was his Meaning. 
 ■ Itiimediatcly after, he gives us what he calls a 
 ^"miliar Inftance, to ihovv the great Advantages 
 which a Purchafer will make in laying out his 
 Money at the Rate of 6 /. per Cent, in buying 
 Lcafe-hold Eft'ates, He does not indeed aflert, 
 
 that
 
 C " ] 
 
 jthat a Purchafer of fuch Eftates, in the Ihftance 
 and on the Terms there mentioned, makes 6 /. 
 per Cent, of his Money ; for he pfcs this wary 
 Expreffion, that the Purchafer values his Money 
 laid out at 6h per Cent, but does not fay that he 
 makes -6 /. per Cent, of his Mopey fo laid out. 
 But tliough he does not aflcrt, he ought t<i havb 
 afferted and proved, that fuch Purchafer makes 
 , 61. per Cent, of his Money, if he would have 
 proved any Thing pertinent to the Purpole ; 
 for let the Valuation be what it will, if the ef- 
 fe(5i:i?al Produce does not anfwer the Computa^ 
 tipn, the Manner of computing the Interef^ 
 where two Perfons difpofe of Money in two dif- 
 ferent Ways, will never fhow which of the two 
 has the greater Interefl: for his Money, or the 
 better Bargain. We may compute the Intercii: 
 at 6 /. per Cent, and in Fad make but 3 /. as 1 
 Ihall take Occafion, in another Place, to fhow, 
 that a Purchafer, in the Inftance here given, will 
 make no more, if thefe inconvenient Draw-backs, 
 the Land-Tax, Repairs, arid cafual Lofles and 
 Charges, are to come out of his Eftate ; and wc 
 may compute the Interell at 5 /. and in Fa6t 
 make 4/. or 3 /. 10 s. and then there can be no 
 Queftion who has the larger Interefl for his Mo- 
 ney, and the moft advantagious Bargain. 
 
 And in this Cafe, if it were not neceflary to 
 zffcn and prove that fuch a Purchafer would 
 make 6 /. per Ce.nt. of his Money ; yet I think 
 J:he Author, out of Regard to the fuppofed Ig- 
 norance of his Leafe-hold Tenant, to whom he 
 js giving Advice, ought to have been more ex- 
 plicit.:
 
 [ 12] 
 
 plicit : For we know all Men will not attend to 
 nice Calculations, and perhaps all Men do not 
 undcrlland Expreffions fo well guarded. In 
 Fa6t, fo it comes out ; for there is a learned 
 Land Surveyor, who fays, that a Purchafer of 
 Leafe-hpld Eltates on . the Terms mentioned by 
 this Reverend Author makes 7 /. per Cent, of his 
 Money, and if he makes 7 /. to be furc he makes 
 6 /, and cites this Inllance as a Proof and Dc- 
 monftration of it. 
 
 Since we hiive mention'd this Tnftance, and it 
 is brought to induce Perfons to become Pur- 
 chafers of Leafes, and to prevail on Tenants of 
 fuch Eftates to pay their Fines with a free and a 
 liberal Hand ; and if an advanced Fine is paid^ 
 polTibly it mav be no great Matter whether 
 Taxes, Repairs, and other Charges on the 
 Eflate, are or are not taken into the Confidera- 
 tion : And lince- the Author tells us ^tis a plain 
 and familiar Inflance and a Demonftration fuited 
 to every Man's Capacity, and the Writer juft 
 mention'd, I fuppofe, that he might not be 
 deem'd to want common Capacity, very readily 
 gives his Affent to it, as he docs to whatever 
 the other advances, though never fo extravagant; 
 ^nd another, and a third Writer, quote it as a 
 ruled Cafe that is to govern every Thing to be 
 offcr*d on this Subjecb; let us lee what this In- 
 ftance is. and what mighty Matters it proves. 
 
 The Reverend Author alTerts, that 'tis better 
 Husbandry to buy Leafes at twelve Years Value, 
 and to renew from feven Years to feven Years, 
 jftfr the Hate of two Years and an half Value, 
 
 or
 
 [ 13 ] 
 
 or thereabouts, than to give twenty or twenty 
 one Years Value for Lands of Inheritance j and 
 to prove the Aflertion gives this Cafe. 
 
 The Lcale-hold to be purchafed, he fuppofes 
 to be 100 /. per Juntim, befidcs the referved 
 Rent : That if a Man gives iiool. for this 
 Eftate, he gives 800 /. or poo /. lefs than he mufl 
 give for an Eftate in Fee, and confequently that 
 he has 8co /. at leaft to improve ; though he has 
 the fame Income as if he had laid out 2000 /. in 
 Lands of Inheritance. And as for the 8co /. 
 Surplus Money, this, at the End of feven Years 
 produces aSo /. at 5 /. per Cent. Simple Intereft ; 
 that if the Landlord Hiould take 250 /. which 
 is two Years Value and an half, for a Renewal, 
 fuch Tenant would be a Gainer 30 /. befides the 
 Improvement he might make of his Intereft; 
 and that if the Landlord take only one Years 
 Value, as Churches and Colleges now do, the 
 Fine would be lefs by 150 /. (it fhould be faid 
 180 /.) than the Simple Intereft of the 800 /. ; 
 the Tenant confequently is a Gainer to that 
 Value. 
 
 This Infi:ance, though it may fcem a familiar 
 one, will never be admitted to be a fair one, be- 
 caule the Comparifon ought not to have been 
 framed between 2000/. laid out in Lands of In- 
 heritance of one Side, and on the other Side, 
 1200 /. laid out on Leafe-hold Lands, and 800/. 
 referved at Intereft, but the whole 2000/. on both 
 Sides Ihould have been fuppofcd to have been 
 vefted in Lands ; and if *tis good Husbandry to 
 lay out J^loney on Leafes, the more Money is 
 
 laid
 
 [ ^4] 
 
 kid out in thofe EflateSj Co much the greater 
 Advantage will be rcap'd from thence. But the 
 Author, I fuppofe, ftated his Cafe in this Man- 
 ner, that his Leafe-hold Tenant might be fure 
 of a Fundfor Payment of his Fine at the End 
 of feven Years j and that being a provident 
 Care for the Landlord as well as the Tenant j 
 the Cafe fhall e'en fland as here put. 
 
 But this Cafe, even as 'tis Hated, is fo far 
 from being a Demonftration, that the Leafe-hold 
 Purchafer has the Advantage of the Fee Simple 
 Purchafcr, that it proves juft nothing at all; be- 
 caufe it afferts, or takes for granted, that the 
 Owner of the Leafe-hold Eftate has the fame 
 Income out of his purchafed Lands, as the 
 Owner of the Fee-Simple has out of his ; which, 
 in Fad, is not true, and the very Cafe fuppofes 
 it not to be true. The Cafe fuppofes, and if it 
 did not fuppofe it, there ncceffarily mufl be a 
 Rent referved ; now for this referved Rent the 
 Land-Tax muft be paid ; for this a proportion- 
 able Allowance for Repairs, and other Out- 
 goings, mull be made ; and confequently the 
 Income of one mull be lefs than that of the 
 other, jufl io much as thefe Charges on the re- 
 ferved Rent amount to. I obferve, the Author 
 is fo wary as not to mention what the referved 
 Rent might be : And if we fuppofe it to be 
 equal to a third or a fourth Part of the improved 
 Rent, the Deductions on thofe Heads will con- 
 iiderably abate fuch Tenant's Income : But be 
 it more, or be it lefs, fomething it neceifarily 
 mull be, and the very Cafe isfoput, and there- 
 fore
 
 [ ^5 ] 
 
 fore the Demonftration, let k be fulted to whofe 
 Capacity it will, fails, and comes to nothing. 
 
 This Reverend Author having affured usj 
 that this is a plain and familiar Inftance • tho' 
 as plain as it is, we fee fomc Perfons have made 
 a Shift not to underlland it, or to underiland it 
 to prove fomething which it does not, nor I 
 fuppofe, was ever intended to prove : I beo- 
 Leave to make Ufe of it to ihovv, as I think I 
 can do, I will not iay to a Demonllration, but 
 to fomc Degree of Probability, that a Purchafer 
 of a Leafc-hold will have a much worfe Bargain 
 than he that purchafcs a Fee-Simple, on the 
 Terms here propoled. We will fuppofe then, 
 that one Pcrfon purchafed a Fee-Simple of the 
 Value of ICO /. per Jfnium^ about the Year i ^i 8, 
 the Time when this Treatife was firft publillied ; 
 and becaufe I am deiirpus that he Ihould have a 
 wcll-condition'd Eflate, I will fuppofe that he 
 gave for it twenty one Years Value, and that 
 another Pcrfon about the fame Time, being wclj 
 inflruded from this Inflance how to buy Bar- 
 gains, purchafed a Leafe-hold of the fame an- 
 nual Value, over and above the refcrved Rent, 
 and gave for it twelve Years VaJue ; and I will 
 fuppofe this Eftate too to be a well-condition'd 
 one, and fo I am fure I ought, and I will Hate 
 the referved Rent annually payable out of it at 
 no more than ten Groats, if the Author will af- 
 fure me any fuch is to be found in the King- 
 dom ; and that this laft Perfon difpofed of 900/. 
 which is the Sum that the Leafe-hold Eitate 
 eoit lefs than the Fee-Simple did, in Public, or 
 
 any
 
 [ i6] 
 
 firiy other Securities, juft as he fhall think fit to 
 diredl : Let us fee then how the prefent Cafe of 
 thefe two Purchafers Hands, and examine whe- 
 ther now, in I y^^i-i they are the fame ; or whe-» 
 ther, and how far, they differ, and on which 
 Side the Advantage, if any, lies. 
 
 I have ilated the Rent payable out of this 
 Lcafe-hold Eftate fo low, that the Income of 
 both Eftates, arifing from the meer Rents, muft 
 be admitted to have been the fame, within a 
 Trifle, ever fince the fuppofed Purchafe ; but 
 'tis not neceffary to admit that the Owner of the 
 Fee-Simple Land has all this Time made no Ad- 
 vantage by breaking up Meadow, or Paflure- 
 Land, by cutting Timber, or by fome cafual or 
 accidental Profits arifing from his Lands, fuch 
 Advantage as the Owner of the Leafe-hold has 
 no Right or Power to meddle with. But not 
 to infill on this ; though I muft obferve, if no 
 fuch cafual Profit has been made, the Eftate is 
 become fo much the more valuable to the pre- 
 fent, or any future Owner : Here I put the whole 
 Difference on that fingle Point, the Rife of the 
 Value of Lands, in the Purchafe, within this 
 Compafs of Time. 
 
 On this Head, I do not think I make an over 
 Eftlmate, when I compute the Value of tl>e 
 Fee-Simple Lands, I mean the Inheritance, at 
 500 /. that is at five Years Purchafe, more than 
 feventeen or eighteen Years ago ; and I am fure 
 I make a fair Allowance if I compute the Value 
 of the Leafe-hold Lands, I mean to be fold, at 
 120 /. that is, at one Year and near a Quarter'3 
 "\ ■ Purchafe,
 
 [ ^7] 
 
 Purchafe, more than in 1718 : For the Fall of 
 the Intcreft of Money from 5/. to 4 /. does not 
 make a Rife of full one Year and a Quarter's 
 Value in a Term for twenty one Years, though 
 it makes a Rife of five Years Value in Lands of 
 Inheritance. This Computation, which is a 
 fair and a plain one, and, I believe, fuited to 
 every Man's Capacity, leaves a manifcft Diffe- 
 rence of 380 /. clear Gain, on the Side of the 
 Owner of Fee-Simple Land ; and when, and 
 where, and out of what the Owner of Leafe- 
 hold Land can make up this Difference, I be- 
 lieve, mull be left to his learned Advocate, for 
 I am furc the Tenant can't make it up out of 
 this his Eftate. 
 
 As for the 900 /. fuppofed to be retain'd in 
 fuch Purchafer's Hands, or at lead not invefted 
 in Land, and the Interell arifing upon it ; the 
 Author, or his Tenant, may difpofe of it juft 
 as they in their great Wifdom fhall think conve- 
 nient. They are at Liberty to employ it in re- 
 newing the Leafe at fevcn Years End, at any 
 Rate they fhall fix, or let it alone ; they ihall 
 put out the whole, or any Part of it, at Interell 
 at 5/. or at 10/. per Cefit. and accumulate Inte- 
 rell upon Interell, if Aey can; and they fhall 
 fuppofe it, at feven Years End, to be 2000 /. or 
 at twenty one Years, to be 20,000/. juftasthey 
 pleafe : And yet, if all thefe fine Things were 
 really fomething more than meer Speculation, 
 they are nothing to the Purpofe ; for the Quelti- 
 on is only about the Produce of Money laid out 
 in the Purchafe of Lands of Inheritance, or 
 B Leafe-
 
 [ i8 ] 
 
 Leafe-liold Lands, and not about the Produce 
 of Money which is not invefted in Lands, but 
 put out at Intereft : And this, I think, plainly 
 juflifies my former Obfervation, that the Au- 
 thor's Comparifon in the firft Inftance was not 
 rightly inftitutcd. 
 
 And lince this Author fo frequently puts us 
 in Mind of the Probability of the Decreafe in 
 the Intereft of Money, and fo ftrongly inculcates 
 the Neceffity of having Regard to the Price 
 paid on the Purchafe of Fee-Simple Lands ; 
 after fo much Provocation, I mull do him the 
 Juftice to take fome farther Notice of it. I 
 have already obferved what Influence fuch De- 
 creafe has on the Value of a Fee-Simple, and 
 on a Leafe-hold Eftate in a Purchafe; viz. that 
 a Fall from 5 /. to 4 /. per Cent, inhances the 
 Price of one full five Years, and the Price of 
 the other only one Year and near a Quarter, that 
 is, it advances one from twenty to twenty five 
 Years, and the other from twelve and three 
 Quarters, and fomething over, to fourteen Years 
 Value. To which I add farther, that if the 
 Fall of Intereft be from 4/. to 3 /. the Rife of 
 Lands of Inheritance is to thirty three Years 
 and eight Months Value, and the Rife of a 
 Term for twenty one Years to fifteen Years one 
 Quarter ; and if the Fall be to two and a Half^ 
 the Rife, in one Cafe, is to forty Years Value, 
 and in the other to lixteen Years and one Eighth. 
 If this were purfued farther, and Money fup- 
 pofed to be fb low as one per Cent. I believe, 
 though I have not computed it as I have done 
 
 ia
 
 [ 19] 
 
 in the other Inftanccs, that it would come out, 
 that a Leafe for twenty one Years would not be 
 worth twenty Years Value, and confequently 
 the fuppofed Leafe-hold Eftate not worth 2000/. 
 when 'tis plain and obvious to every Man's Un- 
 derllanding, that the fuppofed Fee-Simple 
 Eftate would be worth an Hundred Years 
 Value, that is, 10,000/. I am furprized there- 
 fore, that any Perfcn, who had the Decreafe of 
 Interell profefledly in his View, and fuppofed 
 the Thing might probably foon take Effed:, and 
 the Event has Ihown the Jullnefs of the Thought, 
 could conceive it to be better Husbandry to lay 
 out Money in Leafe-hold Eftatcs at twelve Years 
 Value, than in Fee-Simple at twenty or twenty 
 one : And to urge this View and fuch Suppofi- 
 tion, which he frequently does, as an Argument 
 to prove it, is abfurd and ridiculous. 
 
 I have taken the more Notice of the different 
 Influence which the Variation of Intercll has on 
 the Value in the Purchafe of the one Kind of 
 Eflates, from what it has in the other ; becaufc 
 I believe many Perfons are apt to imagine, that 
 on a Decreafe of Interelt the Value rifes in the 
 fame Proportion in the one as in the other 
 Eftate ; which the Inftances I have already given 
 fhow manifeftly to be a falfe Notion. And it 
 we look back and examine what Influence the 
 Decreafe already incurr'd has had on thefe dif^ 
 ferent Eftates, we fhall fee the P^allity of the 
 Notion, perhaps, ftill more clearly. When 
 Money was at 10 /. per Cent. Lands of Inheri- 
 tance were v/crth ten Years Value;, and now that 
 B 2 it
 
 [ 20] 
 
 it is at ^I, per Cent, the Worth is twenty Years 
 Value J at the one Time the Value of a Term 
 for twenty one Years was about eight Years and 
 an Half, and now 'tis about twelve and three 
 Quarters, that is, in one Cafe the Increafe in 
 Value is fully double, and in the other one Half^ 
 or thereabouts. 
 
 Since then the Decreafe of Intercft has an In- 
 fluence on the Value of both thefe Sorts of 
 Eftatcs, and where the Decreafe is from 5 /. to 
 4 /. the Increafe in Value will be in fuch a Pro- 
 portion, that in an Eftate of 100/. per j^nnumy 
 thePvife, if it be a Fee-Simple Ellate, will be 
 500/. and if it be a Leafe-hold lao/. only; 
 and by the common Tables for Annuities this 
 appears plainly to be the Cafe : It may be 
 asked, What is become of, and who has the 
 Advantage of, the remaining Sum, 'viz. 3S0 /. ? 
 This is certain ; if an Ellate in Fee-Simple, 
 ■which is let at the improved Rent, is advanced 
 in Value 500/. by fuch Variation of Interells ; 
 an Eflate in Fee-Simple, which is let on Leafe 
 at a fmall Rent, muft be advanced in Value the 
 like Sum one Way or other, where we fuppofc, 
 as we do here, that the real Value and other 
 Circumftances of both Eftates are the fame, the 
 Rent refcrved only excepted. In the Cafe then 
 where a Term for Years is exifling at a fmall 
 Rent, the Term and Reverfion may be and are 
 confidered as fcparate Ellates, and as fuch may 
 have a diHindt Valuation fet upon them ; and 
 the Value of the Term and Reverfion put to- 
 gether, is one and the fame as the Value of the 
 
 Fee-
 
 [ 21 ] 
 
 Fee-Simple entire. To the Queftion then, 
 What rs become of this 380/. and who has the 
 Advantage of it ? The Anfwer is, That it ac- 
 crues to the Reveriionary Part of the Eftate, 
 and belongs to the Landlord. But then with 
 Regard to Church-Men and Colleges, whom 
 this Author had chiefly in View, this Circum- 
 Itance is of no Service ; for the Advance in Va- 
 lue of this Part of the Eilate will create no Ad- 
 vantage cither to Landlord or Tenant ; for it 
 cannot be fold, mortgaged, or run into, any 
 farther than by filling up the ufual and ancient 
 Term, or the cuftomary Lives ; nor, if the 
 Leafes were fufter*d to expire, can the annual 
 Rent be raifed mecrly upon this Account : So 
 that the whole Advantage coming to the one or 
 the other, or both, from this Advance in the 
 Value of the whole Eftate, mult arife folely 
 from, and is confined to the Term, for the one 
 cannot fell, or the other buy, more than the ufual 
 and accuftomed Term. 
 
 It may, perhaps, be more material to have an 
 Anfwer to another Qucftion which may be ask'd, 
 viz. Who is to have and will have the Benefit 
 of the advanced Value in the Term here fup- 
 pofed, I mean the 110 1.? I doubt the Reve- 
 rend Author will find a Difficulty in anfvvering 
 this Quellion in fuch a Manner as fliall be con- 
 iiftent both with the Advice to his Tenant, and 
 •with his Reafoning on Behalf of Landlords. If 
 he admits that the Tenant is to have the Ad- 
 vantage of this Rife, he dire6lly contradids his 
 pwn Principles ; for this greater Value of the 
 P 3 Term^
 
 [ 22 ] 
 
 Term, unlefs there be a greater Rent, which is out 
 of the Qucfticn here, is the lolc P'oundation on 
 which Landlords can raile any Right or Pretence to 
 advance their Fines, and his Arguments tend all to 
 this Purpofe. Nay, this is not only fo far true, but 
 even this lower Interell, which produces this pro- 
 portionably greater Value, is the very Rule which 
 he gives and urges to be the Rule by which the 
 Quantity of fuch advanced Fine mull be adjuftcd. 
 
 If the Author leaves his Tenant to make the 
 beft of his Bargain, and infifts, as I doubt not 
 but he will, that this Advance in the Value of 
 the Term belongs to the Landlord ; and it muft 
 belong to him, or he can receive none from this 
 Change of Cireumftances ; and if he claims and 
 takes this as a Pcrquifite appertaining to his In- 
 terell, as he has Power to do on fubfequent Re- 
 newals, then the Tenant can have none at all : 
 For we are fpeaking here folely of the Advan- 
 tage ariling on the Term, and 'tis impoifible 
 both the one and the other Ihould have one and 
 the fame Thing ; fo that we ihall be at a Lofs 
 to find out the Wholefomenefs or Dilinterefled-? 
 nefs of the Advice here given. 
 
 A Tenant indeed, who purchafed when the 
 higher Rate of Intcreft prevail'd, and confe- 
 qucntly bought at the lower Price, if he takes 
 Time by the Fore-lock, and meets with an un- 
 wary Purchafer, may fell as foon as the lower 
 Rate of Intcreft takes Place, and by that Means 
 will have the Advantage of the Advance in the 
 Value of the Term ; but if he miffes his Oppor- 
 tunity, and waits a Renewal, the Landlord, 
 
 according
 
 [ ^3 ] 
 
 according to this Dodrine, has a Right, and 
 poflibly may advance his Fine, and then the 
 imaginary great Bargain amounts to juft nothing. 
 
 This then being the Cafe, that the whole Ad- 
 vantage ariling by the Increafe of the Value on 
 Purchafes accrues to the Purchafcr of Lands of In- 
 heritance 'j and httle more than one f.fth Part of 
 fuch Increafe comes to the Purchafer of Leafe- 
 hold Lands, and even this lictle will be coming to 
 him only on a Suppofition that he makes the beft 
 of his Bargain that he can poffibly make; that is, if 
 he difpofes of his Eftate, and has the good For- 
 tune to do it at the right Juncture of Time, or, 
 upon keeping his Eftatje, if his Landlord is fo 
 good to him as to renew at the old Fine : If this 
 be the true State of the Cafe, 1 liippofe Perfons 
 will not be much at a Lofs to find out where to 
 difpofe of their Money with the befl Husband- 
 ry, if they are minded to vefl it in Lands. 
 
 I intend to conlider elfcwbere, whether a 
 Landlord has, and how far he has, a jufl Pre- 
 tence to raife his Fine on the Decreafe of Inte- 
 reft of Money ; or, which is the fame Thing, 
 on the Increafe of the Value of a Term of 
 Years. Here I have been Hating only the Dif- 
 ference between purchafing the one Eftate and 
 the other, on a Suppofition of fuch a Decreafe : 
 And on the Comparifon I think it appears very 
 plainly, that in the one Cafe there is a very 
 great Advance ; and that the Right to it, the 
 whole of it, is indifputably with one Party ; 
 and in the other Cafe, that the Advance, com- 
 paratively, is very inconfiderablc, and that the 
 B 4 Right
 
 [ 24 ] 
 
 Right to it is at leaft very precarious with the 
 other Party, fince it depends on the good For- 
 tune of the Proprietor, or the Good-nature of 
 his Superior, of whom he holds it. 
 
 Hitherto we have been only difcufTing that 
 Point, which of two Purchafers of an Eftate of 
 thefc different Tenures will have the better Bar- 
 gain, on a Suppofition that both of them buy 
 with a View to part with, or in Fadt do part 
 with, their Eftates foon after their Purchafc ; 
 and upon the whole, I think it abundantly ap- 
 parent on which Side the Advantage lies, and 
 that it lies on the Side of a Purchafer of Fee- 
 Simple Lands to a great Degree, if look'd on 
 in this View. Let us go on now to examine 
 how this Matter will ftand, if we fuppofe both 
 Parties to keep their refpcdive Eftates, and to 
 employ the Rents and Profits of them to fuch 
 Ufes in Life as they have Occafion for. 
 
 This Author, in ilating his Cafe, kept aRe- 
 ferve of Soo/. or poo/, in Hand, in Order to 
 provide, as I fnppofed, for the Payment of the 
 Fine for renewing at the End of feven Years, 
 In that Refpc6t he flatcs his Cafe very judici- 
 ouily, becaufe he might think, pofTibly he might 
 know, that fuch Tenants frequently delay Pay- 
 ment of their Fine, or wrangle and quarrel about 
 ■ the Quantity of it, more for that they have not 
 the Money ready to pay the Fine, than that 
 they have any juft and reafonablc Obje6tion to 
 make to it; and that when Payment can be no 
 longer fhuffled off, the Debate ends in a Mort- 
 gage of the Eftate, this Mortgage continues 
 
 'till
 
 [ 25 ] 
 
 'till another Renewal comes, and then an addi- 
 tional Mortgage is made, and lb, totics qnoties^ 
 'till feveral Renewals have cat out the Principal. 
 To prevent this Mifchief, which but too fre- 
 quently happens upon thefe Sorts of Eftates, I 
 mufl agree that the Provifion here made is a 
 very proper and a prudent one : But then I 
 doubt it will be enumerated among the Incon- 
 veniencies which attend thefe Eftatcs ; and we 
 are to feek the Remedy for it out of fomething 
 which does not arife from the Eft ate itfelf. 
 
 In another P^cfpcd, and which is the only 
 one that fhould have been here regarded, that 
 is, whether it would be Prudence to lay out 
 Money on fuch Eftates, it muft have been better 
 to have fuppofed the whole Sum laid out in a 
 Purchafe ; becaufe, in Point of Computation at 
 Icall, fuch a Purchafer muft have had the Ad- 
 vantage, juft as much as Money at 6 /. per Cait. 
 if it were really to be had, will produce a bet- 
 ter Income than the fame Sum at 5/. per Cent. 
 and as a larger Eftate, when fold, will yield a 
 greater Gain in Proportion than the lefs Eilate, 
 on a Suppofition of a Rife in the Value. 
 
 But what if this 8co/. or 900/. which is fo 
 providently retain'd in Hand as a Fund to raife 
 the Fines on Renewals, and out of which fuch 
 vaft Mountains of Gold are to be raifed, fhould 
 be ftiort enough to anfwer the firil Purpofe, 'uiz, 
 to raife the Fine ? The Interell of Money, 
 fince the Year 1718, has fallen, though not by 
 Law, yet in Fad: and in Pradice, fo that this 
 I'rincipal referved and put out at Intereft, fup- 
 
 pofing
 
 [ 25 ] 
 
 poiiiig it to be 900 /. in fevcn Years Time, at 
 4 /. per Cent, will produce no more than 252 /. 
 at Simple Intcrell ; and if the Tenant was fo 
 wary and fo happy as to accumulate Inte- 
 reft, and he muft have a great Share of Sagacity 
 and good Fortune too that can do it, yet the 
 Produce would not be 300/. Now if 'twere 
 right, in 17 18, for a Landlord to take, and 
 prudent in a Tenant to give, two Years and an 
 half Value, that is 250 /. for a Renewal of this 
 Eftate ; then, in 1736, it would be Jullice in 
 the Landlord to demand, and Prudence in the 
 Tenant to pay, near three Years Value, that is, 
 near 300 /. for fach Renewal. To which it we 
 add the Charges of Journeys, Attendances, 
 Fees, 6^^. to buy this Bargain at firll ; and the 
 Pleafurc of repeating it once in feven Years for 
 ever ; the Upfhot is, that this mighty Fund, 
 out of which fuch vail Savings were to be made, 
 out of which fuch Heaps of Riches were to be 
 amafs'd, and a Fortune and Family to be raifed, 
 is barely fuiUcient to bear its own Charges. 
 
 But if this Fund were more than fufEcient to 
 anfwer the Purpofe of railing the Fines, and 
 Ihould even produce a Surplus, as it will do on 
 a Suppofition that an Interell of 5/. per Can. is 
 made of it, yet this Author does not fee, or if 
 he does, fome of his auxiliary Writers do not 
 fee, that the Intercft of this Money, or fo much 
 of it as is applied for Payment of the Fines, is 
 funk, and entirely loft to the Owner ; fo that 
 the Proprietor of this Leafe-hold Eflate, who 
 has laid out i2co/. in the Purchafc of it^ and 
 
 has
 
 C 27] 
 
 has placed out 800 /. at Intereft, has only 70 /. 
 per Jnmim in the whole, as the Income of his 
 2000 /. that is no more thun 3 /. lo s. per Cent, 
 for his Money. I fay, the whole of the Income 
 is no more than "jo I. per Junum^ for the Inte- 
 rcft arillng from the Money, or the greatefl 
 Part, mull be applied to the Payment of the 
 Fine, and out of the 100 1, per Juimm in Land 
 are to be dedud:ed the Taxes on it, the Repairs, 
 and accidental Loffcs, which this Author ftates 
 at 30 /. per Cent, and when thcfe are deducted, 
 as he contends they ought to be, the Neat Mo- 
 ney left is only 70 /. The Income ariling from 
 the Lands of Inheritance being fuppofed to be 
 the fame, and in the prefent State of Things, 
 the Surplufage coming from the Money at Inte- 
 reft being an inconfiderable Sum, and if Things 
 were to remain on the fame P'oot that they new 
 do ; yet I do not think, nor do I believe the 
 moft fond Admirer of thcfe Eftates can think, 
 that this Circumftance only, 'viz. the Identity of 
 Income ariling from both Kilates fets the two 
 Owners in a Condition equally good. 
 
 But admitting it does fo, if Intereftof Money 
 fliould fall, which this Author fuppofed very 
 probable, and fince the Publication of his Trea- 
 tife in Practice it has fallen ; the neceflary Gon- 
 fequence of fuch Fall is, that the Income arifing 
 from the Money will fink, and at the fame 
 Time, and for the fame Reafon, the Fine will 
 be raifed on Renewals ; and all the while the 
 Rents of the Lands of Inheritance and of the 
 fiCale-hold will continue the fame ^ *fo that the 
 
 Income
 
 [ 28 ] 
 
 Income of one probably may, is fubjed at leafl 
 to., be diminifliM, when the other is liable to no 
 fuch Contingency. And if we Hate this Cafe 
 fairly, and as it ought to have been flated, and 
 put both Perfons on an equal Foot, we muft 
 fuppofe the Purchafer of Lands of Inheritance 
 to lay out only 1200/. in the Purchafc of fuch 
 Lands, and to place out 800 /. at Interefl-. If 
 we do fo, the Lands purchafed will be about 60/, 
 per Annum • and if we allow for the Land-Tax, 
 Repairs, and accidental LoiTes, after the Rate 
 of 30 /. per Cent, the Rent Neat will be 4a /. 
 thelnterefl of the Soo/. will be 40/. and both to- 
 gether will be 82 /. which is a better Income 
 than the LefTce-Tenant will fee from his Land 
 and Money, unlefs his Landlord be more kind 
 to him in his Fine than fome Writers are willing 
 to allow he fhould. 
 
 After all, when the Debate is, What Advan- 
 tages are to be made by purchafing Leafe-hold 
 Eftate6, 'tis nothing to the Purpofc to examine 
 what another and a different Eftate will produce, 
 or whether a certain Sum of Money, not laid 
 out in Land, but put out at Interell, will make 
 fuch a Return as fiiall be fuflicient to fatisfy this 
 or that, or any otlicr Demand ; fmce the fole 
 Queflion here is, What is the natural and genu- 
 ine Produdl of this Eftate in particular ? If the 
 Tenant has an Opportunity, by artificial Aids 
 drawn from another Eftate, to help himfelf fo 
 far as that the Bargain fliall not be an ill one, 
 and, farther than that, this Contrivance will not 
 carry him,' yet this is only an accidental Relief, 
 
 and
 
 [ 29] 
 
 and does not arlfe from the Eflate itfelf. And 
 when we conlider the Eilate naked as it is, and 
 llripp'd of this foreign AITiftance ; if we fhould 
 admit that it might be purchafed at a Rate where 
 Interefl is computed at 6 /. per Cent, and a Fee- 
 Simple where 'tis computed at 5 /. only ; and 
 though we Ihould fuppofe that the Land-Tax, 
 Repairs, and accidental Lofles, were not to be 
 placed to the Termer's Account ; yet even up- 
 on thefe Terms, I am not of Opinion that the 
 Prudence and good Husbandry is on the Side 
 on which this Author has put it. In Theory 
 and meer Computation it mufl be with him, be^ 
 caufe, pall all Peradvcnture, 'tis better to have 
 61. per Cent, if 'tis to be had, than to have 5 /. 
 •whether it be for Expence or for Increafc, if 
 there were nothing to be regarded but meerly 
 the Income ; nay, and if the meer Income is 
 confider'd, I doubt, in Pradice and in the Event, 
 even this will not, and in ¥^St rarely does, an- 
 -fwer fuch Expedations. 
 
 The Owner of a Fee-Simple has a flx'd per- 
 manent Eflate ; his Property is an entire Thing, 
 lies together, and is ready to his Hand ; and 
 being veiled in Terra Firma, is fubjed to no Ha- 
 zards, Accidents, or Contingencies, none I 
 mean but what a Leafe-hold Property is equally 
 liable to : For I have no Regard to the Infinu- 
 ation, that a Tenant by Leafe has an indifputable 
 Title to his Eflate, fince there are many more 
 Suits between Landlord and Tenant, cither 
 about the Eflate granted, or the Terms on 
 which 'tis granted, than there are between the 
 
 Buyers
 
 [ 30 ] 
 
 Buyers and Sellers of Lands of Inheritance. ' 
 But where and whence is the Tenant by Leate 
 to have his Principal, or his Fine, if he has the 
 Favour to renew ? He is to have it by faving 
 theSurplufles of the Rents of his Ellate as they 
 arife Yearly or Half-Yearly, by putting fuch 
 Surplulfes out at Intereft, and accumulating In- 
 tereft. But if his Under-Tenants, or any of 
 them, omit making regular Payments of their 
 Rents at the Day ; if he himfelf mifles an Op- 
 portunity once only during the whole Term in- 
 flantly to difpofe of his great SurplufTes, and 
 the Interell of fuch Surplulfes ; if in any Part 
 of the Time he takes but one bad Security, nay, 
 if a Security be only fo far bad as not to make 
 pundtual Payment of his Interell ^ or if the In- 
 terell of Money fhould decreafe within the 
 Time : If any of thefe Contingencies happen, 
 the Principal, or the annual Income, mull fo far 
 fall Ihort. 
 
 Now if this Author will (how me one Lcafe- 
 hold Tenant who ever efcapcd all thefe Incon- 
 veniencies, which are not only pofTible but pro- 
 bable ; I may fifely engage to produce him an 
 Hundred, who, inltead of accumulating Inte- 
 rell, putting out their Surplulfes at Simple In- 
 terell, or even of faving thofe Surplulfes, have 
 mortgaged their Ellates to raile Money for their 
 Fines on a Renewal. Therefore, however ex- 
 act and regular this Scheme may be in Numbers 
 and Figures, however fine and well it may look 
 in Notion \ yet we arc to conlider whether it 
 can, or how lar it can, be reduced into Pradice, 
 
 and
 
 [ 31 ] 
 
 and what ufually and commonly is the Event of 
 fuch Undertakings : And we are likewife to take 
 Mankind to be fuch as by Experience and in 
 common Life we find them to be, and not that 
 all Leafe-holdcrs are as provident, and as dex- 
 trous, Rebus agundis^ as this Gentleman himfelf 
 may be, or as he would have us fuppofe all fuch 
 Tenants to be. 
 
 But then, in dating the Produce of this 
 Leafe-hold, if we charge the Land-Tax, Re- 
 pairs, and accidental Lolies on the Tenant, we 
 have already feen that the clear Income from the 
 Land and Money together is no more, or very 
 little more, than 70/. per Anniim^ or about 3/. 
 loj-. pr Cent, which I can hardly perfwade my- 
 felf any one will think to be a fufficient Produce 
 for Money fo laid out. Nay, even this 3/. loj-. 
 per Cent, arifes by calling in the Affiilance of the 
 Money not laid out in Land ; and if that be left 
 out of the Cafe, as in all Reafon it ought to be, 
 it being an Aid entirely foreign to the Eftate, 
 we ftiall fee prefcntly, that the Neat Income 
 arifing from the Eltate folely will not be fb 
 much as after the Rate of 3 /. per Cent, for the 
 •Money laid out in the Purchafe. 
 
 I would firft take Notice, that Mr. Richards^ 
 whom I have mention'd before, admits that 
 fuch a Tenant is entitled to a Gompenfation, in 
 fome Way or other, for Repairs, and other 
 Out-goings, whether certain or cafual ; but he 
 excepts the Land-Tax, for a Reafon peculiar to 
 himfeli, 'uiz. becaufe, as he fays. Money is 
 chargeable to the iKfeffments on Land. That 
 
 Money
 
 [ 32] 
 
 Money is by A6t of Parliament made liable to 
 the Aid on Land is undoubtedly true ; but 'tis 
 as notorioufly true that it very feldom or ever is 
 fo charged : And when it becomes the common 
 Pradtice to aflefs it, that there may be Time 
 enough to debate that Point. Hov/ever, I con- 
 cur in Opinion with this Gentleman, that thefe 
 Eflates are to be look'd upon and treated rather 
 as Money than Land ; but it will be very hard 
 on one Side to tax them becaufe they are Land, 
 and on the other Side becaufe they are Money, 
 for that will be placing the poor Tenant between 
 the Hammer and the Anvil, and then he is fure 
 to be cruih'd to nothing. 
 
 But I fay, in whichibever Light we look up- 
 on the Leffee's Intereft, it ought to be exem- 
 pted from all thefe Burdens ; and to prove it, ar- 
 gue thus. Whenever a Pcrfon takes a Leafe for 
 a Term of Years, or purchafes one already 
 granted, he mult have this in his View, viz. out 
 of the Rents and Profits of this Eftate, to take 
 fo much yearly for his own Ufe as amounts to 
 the Intereit of the Money he lays out, com- 
 puting it at fuch a Rate as he is contented to 
 take, or might clfcwhere have had for fuch 
 Money ; and with the Overplus of fuch Rents 
 and Profits to re-imburfe the Principal during 
 the Term, or to raife Money for a Fine to re- 
 new. Suppofing then, that the Rate of Inte- 
 reft on which he buys be Hated at 6 /. and I 
 think in ail thefe Cafes it ought to be io Hated, 
 that is at one per Cent, higher than common In- 
 tcreli:, and under another Head fnall give my 
 
 Reafons
 
 [33 i 
 
 Reafons for it, the Perfon may give for a Term 
 of twenty one Years is 1 1 Years J Value, as 
 may be fcen in the common Tables, Now tho' 
 fuch Purchafer makes his Computation at 6 /. per 
 Cent, and though he fhould receive the whole of 
 the ftated Income in neat Money ; yet, in Fad, 
 he will have little more than five and a Half for 
 his own Ufe, as I fhall take Occalion to Ihow 
 elfewhere. But if he mull go further, and out 
 of the Rents and Profits of this Eftate mull dis- 
 count Taxes, Repairs, and accidental LolTes, 
 which this Author has aflerted, and I have ad- 
 mitted, may amount to 30 /. per Cejit» on the full 
 or improved Rent ; fure I am, that fuch a Pur- 
 chafer mull fit down contented with lefs than 
 3 /. per Cent, as the Interell of his Money to be 
 applied to his ©wn Ufe, or his Principal at the 
 End of the Term, or his Fine to renew, will 
 come fhort. 
 
 The Calculations, in common Ufe, for the 
 Value of fuch Eftates on the original Purchafe, 
 or on the Renewal of them, let them be framed 
 or approved by Sir Ifaac Newton^ or the moll 
 exad: Calculator in the Univerfe, are not appli- 
 cable to thefe Eflates, unlefs or until they arc 
 reduced to abfolute Annuities ; for the Truth 
 and Jullnefs of thefe Computations depend on 
 that Point, that the Ellates arc clear of all De- 
 dud:ions : And to make Ufe of one and the fame 
 Rule to adjull the Value of Interefts fubjedl to 
 Draw-backs, and of Interefts fubje<5l to no Draw- 
 backs, is abfurd, to fay no worfe. And then to 
 be told, as we arc by one with a grave Face, 
 C how
 
 •*:.S» 
 
 [ 34] 
 
 how we may difpoie of Money where Vis valued at 
 6 h perCetJt. Intercll, that is, in Purchafes of Leafes • 
 and afterwards to be told by him, as we are almoft 
 in the fame Breath, that the Owners are unreafon- 
 able and partial Wretches if they expect to make 
 6 h per Cent, of their Money, over and above Tuxes, 
 B.epairs, and other accidental Loffes ^ that is, if 
 they expedl to make 3 /. per Cent, of their Money ^ 
 for, in Fadt, they will not make fo much, if thefe 
 Deductions are made out of their Eftates: This 
 is to fuppofe that the Generality of Mankind, at 
 leaft one largePart of 'em,want common Senfe * and 
 that the Author had a Right to impofe on them. 
 I have aflertcd, if thefe Incumbrances are to 
 be charged on a Tenant, even fuppoJing him to 
 have purchafed where Interefl is computed at 
 6 /. per Cent, that he will not be able to make fo 
 much as 3 /. per Cent, of his Purchafe-Money for 
 his own Ufe ; and to prove it, fhall make Ufe 
 of the Inftance before mention'd, and, when ap- 
 plied to my Purpofe, I hope it will appear as plain 
 and as familiar as when applied to fome others. 
 We will fuppofe then, the Eftate to be taken, 
 or to be purchafed, is 100/. per Annum., over 
 and above the Rent referved, and that it is to 
 continue for twenty one Years ; and then, if 
 Interefl be computed at 6 /. per Cent, the Price 
 to be paid for it, if we ftate it at the exa<^Sum, 
 is 1 176/. Now if this 100/. per Annum^ fup- 
 pofed to be the Tenant's Annuity, were a neat 
 Income, free from all Reprizes, I agree that 
 fuch Tenant will make about five and a Half 
 .^frG/;^ of his Money, and fome thing over; but 
 
 ■what
 
 [35] 
 
 what Interell will he make, if the Dedudions in 
 Controvcrfy are made out of his Ellate ? For 
 there is but fmall Comfort in valuing his Money 
 at 6 /. per Cent, in the Purchafe, if in the Event 
 it does not produce half the Sum. 
 
 We have fuppofed thefe Dedudions, in com- 
 mon Cafes, and in common Years, may amount 
 to 30 /. per Cent. ; and fince we have grounded 
 our Suppofition on a very great Authority, and 
 find it confirmed by much Experience, I appre- 
 hend this will not be controverted. If the Rent 
 referved on the Leafe of this Eftate be only 10/. 
 per Jntitim^ then the Value at the improved 
 Rent will be no/, per Anntim^ and the De- 
 duftions out of this no/, if made after the 
 Rate of 30 /. per Cent, will be ^"i^ /. per Annum, 
 In this Cafe a Fund muft be provided out of the 
 Eftate for re-imburfing the Principal by the Ex- 
 piration of the Term for twenty one Years ; and 
 this Fund, I fay, mult be 33 I. per Annum^ for a 
 lefs annual Sum, though computed at Compound 
 Intereft, and at 5 /. per Cent, will not produce 
 1 176/. in twenty one Years Time.^ Now when 
 thefe three Sums, 10 /. o^"^ I. and 0^^ /. making 
 all together y6 /. are dedudbed out of the im- 
 proved Rent no/, per Annum y there remains to 
 the Tenant for his Intereft no more than ths 
 Sum of 34 /. per Annum., which is not 3 /. per 
 Cent, for his Principal Money of 1 176 /. 
 
 This Inftance, I think, is fuited to the Ca- 
 pacity of every Man, at leaft to the Capacity of 
 every one who has common Underftanding in 
 Numbers and Figures \ and evidently fhows, 
 C 2 that
 
 [ 36} 
 
 that in the Purchafe and Renewal of Leafe-hold 
 Kflates, the Articles of Taxes, Repairs, and 
 accidental LofTes, are not to be placed to the 
 Tetiant's Account, lince no Man will pretend 
 that 3 /. per Cent, is a reafonable Interell for Mo- 
 ney laid out in that Manner : And even this 3 /. 
 fer Cent, arifes on a Suppofition that Compound 
 Intereft is in Fad made ; which is almolT: as cer- 
 tainly not true in Pradice, as in Computation 
 'tis true. 
 
 If we vary this Cafe, and put it on a Foot of 
 an higher Rent referved, we fhall fee flill more 
 plainly the Juftice and NecelTity of placing thefe 
 Reprizes to the Landlord's Account, and not 
 the Tenant's. If the Eflate at the improved 
 Rent be aoo /. per Jmium^ and the Rent re- 
 ferved be 100/. the Annuity left to the Tenant 
 will be ioo /. ; and if no Regard is to be had to 
 any Out-goings but that of the Rent referved, 
 the Value of this Eftate in the Purchafe will be 
 the fame as of the former \ the annual Sum to 
 re-imburfe the Principal muft ,be the fame here 
 as before, "u'lz. ^0, 1. ; the Dedudions, if com- 
 puted at the former Rate, will be 60/. : And if 
 thefe three Sums, 100/. ^^L and 60/. making 
 in all 193 A are deducted out of aoo/. the Re- 
 mainder is no more than 7 /. as the Annuity or 
 Income for the Principal Sum of 1 176 /. 
 
 I might very well reft the Matter here, fincc 
 this Inftance demonftrates in Numbers and Fi- 
 gures, which are very llubborn Things, that 
 thefe Reprizes are not to be made out of the 
 Tenant's Eftate : But fince this Reafoning is 
 ' founded
 
 [ 37] 
 
 founded on Suppofitions, and thofe may be un- 
 fair or precarious ones, tho* this cannot well be 
 obje6ted to me, fince I made not the Suppoliti- 
 ons here, but aflented to thofe ready made to 
 my Hands ; I will proceed to fhow, that the 
 Argument drawn from the Reafon and Juftice of 
 the Thing is with me, as well as that in Num- 
 bers and Figures. 
 
 If an Eftate were out of Leafe, and let to a 
 Tenant at Will at the improved Rent ; I would 
 ask. Whether the Landlord then, let him be 
 Church or Lay-Man, muft not pay or difcount 
 the Land-Tax for the whole Eftate ? The Col- 
 Icdor of this Tax, or His Majelly's Receiver- 
 General, to be fure will not be fatjsfy'd with 
 •feme little Part, with a fmall Allowance, or 
 with fome equitable Confiderations had of the 
 Taxes, but niufl and will have the whole. And 
 if it fliould fo happen that the Dwelling-Houft", 
 «a Barn or two, or the like, belonging to this 
 Eftate, fhould tumble down, or want Repairs 5 
 and fuch Things may happen in this the molt 
 profperous State of the Nation ; I fuppofe the 
 Owner would think convenient to rebuild or re* 
 pair at his own Gofts and Charges, rather than 
 iuffer the Eftate to be untenanted. And if it 
 be the conftant and daily Practice for the Land- 
 lord to pay or allow the Taxes and Repairs to 
 his Tenant at Will, or to make him a Satisfadi- 
 on for them by an Eafement in the Rent, and 
 the Right and Juftice of the Thing is that he 
 Ihould do fo, which no Man can controvert ; 
 i:hen it muft be right and juft for him to pay or 
 G 3 al!o>Y
 
 [38] 
 
 allow the fame to his Tenant by Leafe : For 'tis 
 impoffible to lliow the Difference between the 
 Cafes where a Perfon receives the whole Produce 
 of his Eftate by Yearly or Half Yearly Rents, 
 and at feveral Inftalments ; and where he re- 
 ceives fuch whole Produce, Part by Way of An- 
 ticipation, and as a fore-hand Rent, and the re- 
 maining Part by Yearly or Half- Yearly Rents^ 
 and at feveral Inftalments : Therefore, where 
 the Tenant cannot have an Allowance for the 
 Land-Tax and Repairs out of the Rents paid 
 by Inltdments, as on a Leafe at a fmall Rent 
 he cannot have, he mull, and in Equity and 
 J'lftice ought to have a Gompenfation for them 
 out of the anticipated or fore-hand Rent, that 
 is, out of the Fine, whether paid on an origi- 
 nal Grant, or on a Renewal. 
 
 To purfue this Matter a little farther : If one 
 of thcfe Eilates, fuppofed to be out of Leafe, at 
 the improved Rent were worth an lOp/. pr AtH 
 mim^ and the PoflefTor were delirous of a certaia 
 Rent, the bell he could get, without taking any 
 Fine or Premium ; would any Man of common 
 Honefty and Integrity exped: to have, or any 
 one of common Underllanding agree to give, an 
 abfolute Rent of lOO/. per Annum for it j that 
 is, charge himlelf with the Land-Tax and Re- 
 pairs, without having a Confideration for them, 
 either by Abatement of the Rent to be referved, 
 or by Difcount on it when to be paid ? And if 
 a Fine were to be paid on finking Part of the 
 Rent, could any Proprietor in Confcience ask, 
 or would any Tenant of common Senfe pay, i 
 
 Fine
 
 C 39 ] 
 
 Fine for fuch Part of the improved Rent as was 
 to be abated or difcounted for the Land-Tax 
 and Repairs ? Or if fo much Rent only were 
 funk as was jull equal to thefe Out-goings, could 
 the one ask, or would the other pay, any Fins 
 at all ? It is ridiculous in Notion, and in Prac- 
 tice would be lomething worfe ; and if this be 
 the Cafe in a Grant on an original Leafe, as un=> 
 doubtcdly It is, it will be impoffible to make 
 out that a Grant on an old Leafe renewed dif- 
 fers in the Reafon and Juftice of the Thing. 
 
 For thefe Reafons I am clear of Opinion, 
 that the Land-Owner is to pay or allow thf 
 Taxes and Repairs ; fo let us now confider how 
 thefe Articles are to be adjulted ; and thpfe, I 
 think, depend on, and muft be determined by 
 the particular Circumflances in this Kind, at« 
 tending every individual Eftate to be dealt in. 
 Though there feem to be fome DiiFiculties in thp 
 Way, yetlobferve they are no other than what 
 occur in the Purchafe of Lands of Inheritance j 
 and which, I think, with fome Enquiries made^ 
 a little Experience and Difcretion ufed, apd 
 fome Share of Honefly obferved, which in all 
 thefe Cafes is neceflary, may be conquered. 
 
 As for the Land-Tax, though the Rate of it 
 in the different Parts of the Kingdom varies very 
 much, to a Degree beyond what could well be 
 imagined, if Matter of Fad: did not prove it, 
 and in the fame County in different Diflrids 
 there is great Difproportion, yet the Rate at 
 •which any particular Eltatc is affefs'd, by propec 
 Enquiries;^ may be known ; and when a Yearly 
 C ^ Paymei^f
 
 [ 40 ] 
 
 Payment is afcertain'd, the Payments for a Num- 
 ber ^of Years may be fettled by a proper Medi- 
 um : And if we take the Mean between the 
 higheft and loweft, and for a Term, I fuppofc 
 10 /. per Cent, will be a moderate Computation. 
 The Matter of Repairs has Hill more Difficul- 
 ties in it ; for, belides that every individual 
 Eltate has its particular Circumftances of this 
 Kind, it may not be right, perhaps, to Hate the 
 Charges for a Number of Years from that of 
 any one Year : And yet even this may be got 
 over ; for by looking back fome Years, and 
 computing what has been the Charge, and by 
 looking forwards and eftimating what the future 
 Charge may be, and upon a View or Enquiry 
 of the Quantity and Quality of the Buildings 
 on the Premifes, fome Sort of reafonable Mea- 
 fure might be framed by Perfons of Skill in thefe 
 Affairs ; and if lo /. per Cent, were fix'd for 
 common and ufual Cafes, with an Exception to 
 fuch as have any Particularities in them, I believe 
 we fhould not be far off the Mark. 
 
 The Reverend Author, whom I have fo fre- 
 quently before mention'd, fuppofes Taxes and 
 Repairs, throwing in accidental LolTes, to be 
 2,oJ. per Cent. '^ and I would ftate them at the 
 fame Rate, if he would give me Leave to add 
 an Article of Charge, for managing the Eftate 
 and receiving and paying the Rent ; but, in 
 Truth, unlefs this Article be included, which 
 there does not feem to be the fame Reafon for 
 in Lands of Inheritance as in Leafe-holds, I 
 tthiok the Author here flretch'd the Point a little, 
 
 to
 
 [41 ] 
 
 to ferve a Turn ; for his Argument led him to 
 depreciate the Value of Lands of Inheritance, 
 and he therefore ftates thefe Deducflions at the 
 moll advanced Rate ; and never confiders, that 
 having laid the Load of thefe Dedudions on his 
 Leafe-hold Tenants, he does thereby link the 
 Value of their Eltates in the fame Proportion. 
 
 Though I have here ftated lo/. per Cent, on 
 the improved Rent, as a reafonable Allowance 
 for Repairs in common Cafes, yet there are Ex- 
 ceptions to be made out of this general Rule : 
 For where an Ellate conlifts in the whole, or the 
 greatcft Part, in Meadow or Pallure Land, no 
 Repairs, or very little, can be call'd for, even 
 during a long Term : On the other Side, if the 
 whole, or greateft Part, of the Eftate confilts ia 
 Houfes, though they be in the belt of Repair 
 and Condition, yet from the natural Decay of 
 Buildings, from Change of Tenants, and other 
 Accidents, which in the Courfe of fome Years 
 mull happen, it will become necelTary to lay out 
 Money in Repairs, in Alterations, or in Orna- 
 ments, or it may be impolTible to keep up the 
 Rents of fuch Houfes : To this if we add the 
 Hazard of Want of Tenants during fome Part 
 of a Term, for which Time, in the Cafe of 
 Houfes, there is a Lofs of the total Profit : If I 
 ihould Hate thefe Charges and Contingencies at 
 S.O I. per Cent. I believe it will not be thought 
 extravagant ; and Cafes undoubtedly may be 
 put, where, from the ruinous State of the 
 Houfes, their unfortunate Situation, or fome 
 pther bad Qualities accompanying them, it would 
 
 not
 
 [42 ] 
 
 not appear too much to allow one half of the 
 real Rent on this Article. 
 
 I know, in this and fuch like Cafes, the com- 
 mon Method and Pradice is, to allow a Tenant 
 pr a Purchafer an higher Rate of Intereft, as a 
 Compenfation ; but this, in Truth, is a fallacious 
 pne ; and though it may not lie open to the Eye 
 of a common Obferver, for all the Writers on 
 this Subjedl that ever I met with ufe it and re- 
 commend it, yet 'tis greatly to the Prejudice of 
 any Perfon's Purchafing, and will not anfwer the 
 Purpofe delign'd, unlefs the Rate of Intereft be 
 fet ib high as would appear extravagant to all 
 Mankind. 
 
 To fhow how thcfe two Methods differ from 
 one another, and what are the feveral Refults of 
 each, I will take Leave to put a Cafe, which, 
 perhaps, may better explain my Meaning, and 
 at the fame Time fhow which of the two is more 
 eligible. We will fuppofe that an Eftate in 
 Houfes is to be let, or fold, for a Term of 
 twenty one Years ; and that the Rent of it, clear 
 of all Dedudions, excepting this one Article of 
 Repairs, is an lool.per Jnnum ; and that it will 
 coft 50 /. per Jnnum^ communihus Jnnis^ for the 
 Term, to keep it in Repair : In my Method I 
 reckon the Money laid out by the Tenant in the 
 Purchafe at 6 /. per Cent, and then the whole 
 Rent being fuppofed to be an 100 /. pet Amtiniy 
 and the annual Sum to be paid, and which I fay 
 is to be put to the Landlord's Account for Re- 
 pairs, being 50 /. pr Annum^ if this be dedu6leci 
 out of the former Sum, the Neat Rent remain-
 
 [ 43 ] 
 
 ing to the Tenant will be 50 /. per Ammm ; and 
 this being the Intercfl which the Purchafcr is to 
 enjoy and to pay for, the Price to be paid, In- 
 tcreft being reckoned at 6 /. pr Gent* will be 
 588/. 
 
 In this Way of putting it, the Purchafer 
 plainly fees what he buys, and the Price he pays 
 for it, and *tis evident that he has an adequate 
 Compenfation for taking upon himfelf the Re- 
 pairs \ for they are fuppofed to be at the Rate 
 of 50 /. pr Annum y and for doing this he has 
 50 /. per Annum allotted to him, by Way of 
 Difcount ; and with the other 50 /. the whole 
 being fuppofed to be 100/. he pays himfelf In- 
 tercfl for his Money laid out, and re-imburfes 
 the Principal during the Term. 
 
 In the other Method, which is to allow a 
 Purchafer an higher Rate of Interefl, Who Ihall 
 fay what is fuch an higher Rate of Intereft as 15 
 juft an Equivalent for thefe Out-goings in Re- 
 pairs, and how Hiall we determine the Price to 
 be given for the Eftate ? I'll take upon me to 
 fay, and no great Undertaking neither, that it 
 cannot be adjufled but by one of thefe two Ways \ 
 cither by deducing this Charge as an annual 
 Charge out of the fuppofed annual Rent, and {o 
 valuing the Eftate at 50 /. per Annum only, which 
 is the Method I propofe ; or by putting a Value 
 on this Incumbrance in a grofs Sum, and then 
 deduding fuch Sum out of the Purchafe- Money 
 to be paid, valuing the Eflate at an 100 /. per An- 
 mum : But then the Value of this Incumbrance, 
 jn the Qrofs and for the whole Term, cannot be 
 
 known
 
 [44] 
 
 known but by ftating firfl what will be the annual 
 Charge; and this, inEfFed:, is my Method Itill, 
 only we take a longer Courfe to come at the 
 f$me End. 
 
 If a Perfon, without Computation in one or 
 other of thefe Ways, will lit down to Guefs- 
 Work, and determine what is the higher Rate 
 of Intereft to be reckonM in this Cafe, I doubt 
 he would run himfelf into an Error; for, I be- 
 lieve, molt Men would be apt to conclude, that 
 an Allowance of 12. /. per Cent, for the Purchafe- 
 Money, upon the whole Rent of an 100 1, per Jn' 
 mim^ would be equal to the Allowance of 6 /. 
 fer Cent, upon 50 /. per Ammm ; whereas if the 
 common Tables are confulted, it will appear, 
 that for an 1 00 /. per Jnniim at an Intereft of ill. 
 per Ceht. 75a /. and above, muft be given, when, 
 in my Way of computing, no more mull be 
 given for this Eftate than 588 /. ; and 'tis worth 
 no more. But if he will call the Eftate here an 
 100/. per Jnntimy and will ftate his Intereft at 
 fuch a Rate as will repay him his Principal, In- 
 tereft, and the fuppofed Charges of Repairs, he 
 muft then reckon it at ly I per Cent, as near as 
 can be ; which, as I obferved before, is fo great 
 a Rate of Intereft that a Man could hardly 
 think of, or have the Face to demand, and yet 
 at fo much he muft ftate it, or he will be a Lofer 
 by the Bargain. 
 
 Having confider'd, under the Head of Out- 
 goings, what Articles of Deductions are neceir 
 fary to be made to reduce thefe Eftates to abfos- 
 lute Annuities, and in what Manner, and at wh^l: 
 
 Rate
 
 [ 45 ] 
 
 Rate thefe Reducements are, in a general Way, 
 to be made, leaving a Latitude for peculiar Cir- 
 cumftances that may attend particular Eftates j 
 \ye have now to fettle the Articles of Allowance 
 to be made to the Tenants, for their Trouble 
 and Charge in managing their Landlord's Ellate, 
 and for afluring it and the Rents from accidental 
 Lofles ; which I choofe to divide into two 
 Heads, and to fet each at 5 /. per Cent, oa the 
 improved Value. 
 
 I fuppofe a Landlord, whether Ecclefiaftical 
 or Lay, does -not expert that his Tenant will be 
 his Bailiff, and fet and let his Eftate for him, 
 and infpecSb the State of it from Time to Time ; 
 that he will receive and pay his Rent, and that, 
 perhaps, at a Place many Miles dillant from the 
 Place where the Ellate lies, at a Time, to be 
 fure, many Months before he can receive it from 
 the Under-Tenants, and that he will be out of 
 Pocket, at leaft, to the Amount of half a Year's 
 Rent during the whole Term ; all thefe Things 
 no one can imagine will be coming to a Land- 
 lord from a Tenant, out of ftark Love and 
 Kindnefs : No ; the Tenant has, in this Cafe, 
 undoubtedly, a Right to be conlider'd for his 
 Trouble as well, and as much, as any other 
 Perfon would be who fhould take thefe Matters 
 upon himfelf ; and for the Management of this 
 Ellate, where there's added an Engagement to 
 pay at a Day and Place certain, which is more 
 than Stewards or common Agents undertake, 
 the Tenant, to be fure, may well deferve 5 /. 
 per Cem, at leall, on the improved Rent, when 
 
 'tis
 
 [ 46 ] 
 
 'tis no more than the cuflomary Allowance, in 
 many the like Cafes, to meer Stewards. 
 
 As to the Premium to the Tenant, for afTuring 
 the Eftate againft the Cafualties of Fire, Water, 
 and accidental Loffes ; I Itate that likewife at 
 5 /. per CetJt» in the like Manner : And thefe two 
 Allowances, put together, amounting to lo/. 
 per Cent* this Sum is the Medium, or near the 
 Medium, at which the Author, fo often cited, 
 values thefe Undertakings and Charges in Eftates 
 of Inheritance ; for out of fuch Eftates he makes 
 a Draw-back of 30 /. per Cent, in one grofs Sum, 
 for Taxes, Repairs, and accidental Loifes. In 
 my Way I have divided thefe Reprizes into 10 /. 
 per Cent* for Taxes, 10/. for Repairs, and 10/. 
 more for accidental LolTes, or rather for ma- 
 naging and affuring the Eftate and Rent. If the 
 Partition of thefe Deductions, thus made by me, 
 will not fuit with all Eftates alike, as undoubt- 
 edly they will not,a Deduction in a grofs Sum will 
 ftill lefs fuit ; but in this Method the Parties in- 
 terefted, both of one Side and the other, will 
 have a more clear View of the State of their 
 Cafe, and be able to form a better Judgment, 
 whether on any, and on which Articles an Ad- 
 dition or a Subftradtion may be made, than when 
 all are thrown into one Article. Upon the 
 •whole, when thefe Out- goings and Allowances 
 are deduded, the Rent remaining to the Lefiee, 
 ■which, in common Cafes, may be about yol.per 
 Cent, on the improved Rent, is the neat Rent, 
 or the abfolute Annuity, which the Tenant is 
 to pay for in his Purchafe. 
 
 Inftcad
 
 [47] 
 
 Inflead of this Allowance of lo/. per Cent, to 
 the Tenant, for his Trouble and Charge in ma- 
 naging and afluring the Eftate of his Landlord, 
 or to make him a Compenfation for the Hazards 
 which he runs on his Side, that he fhall be a 
 Lofer by fome Contingencies or other, during 
 his Terra, and that his Superior, on the other 
 Side, is made fecure in all Events; Mr. Richardsy 
 in ftating the Value of Leafes, makes Ufe of 
 another Method of Computation to adjufl this 
 Difference. He reckons the Out-goings, that 
 is, whatever is paid to, or on Account of the 
 Landlord, as an Annuity certain, as it really is, 
 becaufe in all Events it mufl be paid, and firll 
 paid, and values it at a Rate of 4 /. per Cent, In- 
 tereil ; and the Annuity belonging to the Te- 
 nant, either on Purchafe or Renewal, he reckons 
 as an uncertain one, as it really is ; becaufe this 
 Annuity is to arife out of the Surpluflage of the 
 Rents of the Eilate, and thofe Rents may noC 
 be fufficient to anfwer both Annuities, and values 
 it accordingly at 6 /. per Cent. ; and this different 
 Valuation of the Annuities he judges to be an 
 Equivalent to the Tenant for his Undertaking, 
 and then fubftrading one Value from the other, 
 he ftates the Remainder as the Value of the 
 Leafe. 
 
 That a Tenant, or a Purchafer, is entitled to 
 fome Compenfation for his Undertakings of this 
 Kind, is agreed between us ; and On my firft 
 Thoughts upon this Scheme for that Purpofe, I 
 confefs I came into this Notion : But upon a 
 Calculation of the Equivalent which would arife 
 
 to
 
 [48 ] 
 
 to a Tenant on fuch Scheme, upon Suppofitlon 
 of greater or lefs Out-goings, I have alter'd my 
 Opinion ^ for this Reafon, becaufe where the 
 Out-goings are large, this Method produces too 
 great a Satisfadion, and where they are fmall, 
 too little a Satisfadion for this Purpofc. My 
 Notion here is, that the Provilion for fuch Pur- 
 pofes ought in all Inftances to be equal ; I mean 
 that it ought to be one and the fame for an 
 Eftate of one and the fame Value, be the Out- 
 goings more or lefs ; for that fome very odd 
 Confequences will follow from a different Provi- 
 'iion. 
 
 This Author obferves, that Cafes may be Co 
 put, that the Value of a Leafe of a Ihorter Con- 
 tinuance may be greater than the Value of a 
 Leafe of a longer Continuance on the fame 
 Eftate ; and this, he fays, will be the Cafe, 
 where the Sum paid out is large in Refped: to 
 the yearly improved Rent, and at the fame 
 Time the Rate of Intercft to be allowed a Pur- 
 chafer is very great, on Account of the ruinous 
 Condition of Houfes, or the like ; the Confe- 
 quence of which muft be, that it may be worth 
 the while to take a Leafe of fuch Houfes for 
 twenty one Years ; but when feven of them are 
 expired, it may not be fo to renew the Leafe, 
 and to make it up twenty one again. 
 
 From this Obfervation, which is a juft one, 
 
 he raifes an Objection againft his own Method 
 
 of allowing a Purchafer a greater or lefs Rate of 
 
 Intereft for his Money laid out, as the Eftate is 
 
 .more or kfs liable to Cafualties, or the Charge 
 
 of
 
 [49] 
 
 of Repairs are greater or lefs. To this Objec- 
 tion, urged thus far and no farther, and to that 
 Purpofe he mentions, and no other, viz. where 
 the Out-goings for Repairs are very great, his 
 Anfwer may be right enough, 'viz. that Houfes 
 may cafily be fuppofed in fuch a State as to ftand 
 and be kept up at a reafonable Charge for a fmaU 
 Term, but in the Compafs of a longer mud be 
 rebuilt in Part, or in the whole, fo as the fhorter 
 Term Ihall be of greater Value than the longer. 
 But this Obfervation may be carried farther, 
 and has more in it than this Author faw, or at 
 leaft has taken Notice of, viz. that this will be 
 the Cafe, that a Leafe for a Ihorter Term will 
 be of greater Value than a Leafe for a longer 
 Term, fuppofing his Manner of calculating be 
 a right one, not only where the Out-goings for 
 Cafualties or Repairs are great, but even in all 
 Cafes where the Out-goings are to a certain 
 Value, let thefe Out-goings be of what Kind 
 foever you will: For Inftance • if the real Value 
 or improved Rent of an Eftate be an lOo/. peir 
 Jinmimy and the Rent refervcd to the Landlord be 
 80 /. per Anniim^ or four Fifths of the whole ; 
 though there be no Out-goings for Taxes or Re- 
 pairs, or if the Landlord takes them upon him- 
 felf ; yet if the Method here propofcd for ad- 
 jufting the Equivalent to a Tenant for under- 
 taking to pay this Rent in all Events, is a right 
 one, it muft follow, that a Leafe of fuch an 
 Eftate^ though it fhould confift wholly of Mea- 
 dow or Failure Ground, for feven or fourteen 
 Years, is of fome Value ; for twenty one Years, 
 D of
 
 [ 50] 
 
 of iefs Vatuc ; for thirty one Years, at Par, oi? 
 thereabouts ; and for fifty two Years, worth Iefs 
 than nothing confiderably : And this, I think, 
 has fo much of the Abfurd in it as to be an Ob- 
 ie6tion to his Method, when applied to pur- 
 chaling, and has more of it, if ufed, in renewing* 
 I know few or no fuch Church or College- 
 Leafes can well be fuppofed ; but it may be fup- 
 pofed, and in Fadt is true, that Gentlemen of 
 great Ellates frequently let them in fuch a Man- 
 ner ; taking upon themfelves the Land-Tax, 
 allowing Timber and Materials for Repairs, and 
 putting Contingencies and Cafualties on their 
 Tenants, and then referving an abfolutc Rent 
 •without a Fine. Now to fay, that an Ellate of 
 an ICO I. per Jnnumy let at 80/. per Annum Kent 
 on fuch Conditions for a Term of thirty or fifty 
 Years, is of no Value, or worth, iefs than no- 
 thing, is abfurd ; and yet 'tis a neceffary Con- 
 fcquence of this Method of adjufting this Equi- 
 valent. But in my Method I ftate an annual Pre- 
 mium of 10 /. per Cent, to the Tenant for taking 
 upon himfelf thefe Contingencies and Trouble, 
 and dedud it out of the improved Rent, and 
 then there remains 10/. per Annum neat Annu- 
 ity to the Tenant; and in this Way of computing 
 'tis plain, that a Leafc of fuch an Eftatc, and 
 upon fuch Terms, is of fome Value, and in- 
 creaies in Value in Proportion to the Number of 
 Years for which 'tis to continue ; and I believe 
 all Mankind will judge that it muft do fo. 
 
 I have beiore aflerted, that this Method pro- 
 vides too great an Equivalent where the Out- 
 goings
 
 [si] 
 
 goings arc large, and too fmall an one where 
 they are little : And, upon a Computation, it 
 comes out, where thefe Out-goings are 80 /. in 
 the 100 /. or four Fifths of the whole Rent, the 
 Deduction to be made under this Head is 20/, 
 per Cefjt, on the improved Rent, which furely is 
 too favourable to Leflees : On the other Side, 
 where the Out-goings are low in Proportion to 
 the whole Rent, fuppofe 20 L in the 100/. or 
 one Fifth of the Whole, then the Dedudion to 
 be made on this Head will not amount to fo 
 much as 5 /. per Cent, on the improved Rent ; 
 which feems to me to under-do the Matter as 
 much as in the other Cafe it over-does it. 
 
 I mufl confefs I don't fee how the ^aafitum of 
 the Out-goings can in any wife be made a Rule 
 to govern thefe Allowances ; for whether they 
 be great or fmall, the Charge and Trouble of 
 managing the Eftate is the fame, the cafual 
 LofTes incident to it are the fame, and both de- 
 pend on the ^antum of the improved Rent, 
 and nothing ellc : But to make the Out-goings 
 a Rule to regulate the Value of this Charge and 
 thefe Lofles, is to make fomething a Meafure of 
 the Value of the Charge and Lofies, which is 
 plainly no Meafure, cither of the Charge or of 
 the Lolfes. The Charge and Trouble of ^ct- 
 ting and letting the Eftate, of receiving and 
 paying the Rents, and the like, arife from and 
 mufl be governed by the Quantity of the whole 
 Eftate, and not by the '^lota Pars coming to 
 the Landlord and Tenant refpedively ; and the 
 Cafuakies and LofTes which may happen muit 
 D 2 and
 
 L 52 J 
 
 and will happen in, or in Proportion to, the 
 whole, and not to any particular Part that may 
 be going out. 
 
 . It is, or perhapvS "may be, urged, that where a 
 Tenant undertakes for fb large Out-goings as four 
 Fifths of the full Value of the Eilate, he runs a 
 •greater Hazard than if he undertakes for one Fifth 
 jonly : And in one Cafe, that it may come out, by 
 LoiTcs which may happen, that fome Years the 
 Eilate will not be fufficient to anfwer the Demands 
 to be made lapon it ; fince the Out-goings certain 
 are fuppofed to be four Fifths of the whole, and 
 confequently that the Tenant muft make good 
 fuch-Ueficiency out of his own proper Eftate : But, 
 on the other Side, that this cannot be fuppofed 
 ever to happen in an Ellate where the Out-goings 
 are only a fifth Part. Admitting that this may 
 happen where the Out-goings are large, as it 
 may, and fornetimes does ; and admitting it can 
 never happen, where thofe Out-goings are 
 fmall, yet the Lofs to the Tenant is the fame, 
 whether the Payments on thefe Articles are one, 
 or four Fifths of the whole Rent. In the firft 
 Cafef if the Leafe Ellate falls fhort of anfwcring 
 the Demands upon it, the Tenant muft make 
 them good diredly and immediately out of his 
 own feparate Ellate ; and in the latter Cafe, 
 though the Leafe Ellate does not fall fhort of 
 anfwering thefe Demands where they are the 
 largeft that well can happen, yet fuch Ellate is 
 the Fund out of which the Tenant is to receive 
 his Annuity or Interell, call it which you will, 
 and ©ut of which he is4ikewife to re-imburfe his 
 
 Priacipal ;
 
 [ 53 3 
 
 Principal ; and upon fuch Contingencies hap- 
 pening, as wc are now fpeaking oi, this Fund 
 muft be deficient with Refped: either to the An- 
 nuity or the Principal, and fuch Deficiency muft 
 be made good out of fuch Tenant's feparate 
 Eftate : So that diredlly or indirediy, and in 
 the Event, this Lofs falls upon, and. muft be 
 fupply'd out of, the Tenant's proper Eftate, in 
 the one as well as in the other Cafe. 
 
 All than I fhall obferve farther on this HcacJ 
 js, that if thefc Out-goings are about two Fifth 
 parts, or fomething between that and a Moiety 
 ,of the whole Rent, then the V^lue of an Eftate 
 for a Term of Years, either in the Purchafe or 
 Renewal, will be the fame both in this Author's 
 and my Method of making and computing thefe 
 Allowances : And thofe vvho think his to be a 
 j'eafonable and equitable Provilion between Land- 
 lord and Tenant, and are {o far cotiverfant in 
 Numbers and Figures as to go through his Ope- 
 ration, now they fee the different Refults of botI> 
 Methods, are at Liberty to be furc to make Ufe 
 of his ; though mine feems to me the moft equal, 
 and in Pratflice, I beHeve, will be found the 
 more eafy and more fafe one. In the main, we 
 agree that fuch an Allowance is to be made, and 
 thaffome Method of obtaining it muft be ob-» 
 ferved, or the Price of fuch an Eftate will not 
 te fairly and evenly adjufted. 
 
 There is an Objedion remains behind, which 
 goes to all thefe Articles, the Taxes, the Re- 
 pairs, and accidental LoHes, or to the Allow- 
 ances for managing and affuring the Eftate ; for 
 
 p 3 \^^9\^
 
 [54] 
 
 whioh Rcafon I referved the Gonlideration of it 
 to this Place ; and it arifes from the ufual Cove- 
 nants in fuch Leafes. The Allegation then is, 
 that the Tenants, by the Covenants in fuch 
 Leafes, have obliged themfelves to pay the 
 Land-Tax, to keep their Eltates in Repair, and 
 to anfwer all Lofles and Out-goings, whether 
 certain or cafual ; and have alfo thereby under- 
 taken to pay the Rents at the Places and Times 
 fix*d, though it Ihould prove never fo expcniive 
 or inconvenient. All this is certainly true, but 
 is fond Talk to amufe themfelves, or to impofe 
 on ignorant or unwary People ; for where fuch 
 Engagements arc entered into, as in moft Leafes 
 belonging to corporate Bodies they are, it niuft 
 be taken for granted, in Regard to that Julticc 
 and Equity which is fuppofed to be between 
 Landlord and Tenant, and that common Senfe 
 which a Tenant, imlefs carved out on Purpofe 
 to ferve a Turn, may have, that a Compenfati- 
 on, adequate to thefe Covenants, is, fome Way 
 or other, provided for fuch Undertakings. The 
 Queftion then is not, What are the Effects of 
 fuch Covenants, and what will be the Confe- 
 quence of them ; for every Man knows that 
 they mail be perform 'd, if the Party is able to 
 perform them : But, Whether any, and what 
 Compenfation, in Reafon and Jullice, a Man is 
 intitled to, upon or for entring into fuch Cove- 
 nants, and taking upon himfelf thefe Charges 
 and Incumbrances. 
 
 Since this Objedion has but little Weight ia 
 it, ^nd whatever it h^Sy the fame has been, in a 
 
 great
 
 [ 55 3 
 
 great Meafure, obviated by what I have already 
 alledged under another Head, where I have 
 proved that the Articles of Taxes and Repairs 
 are not to be placed to the Tenant's Account ; 
 I Ihall only add here : The Cafe being fo, that 
 thefe Covenants mult of Neceffity continue to 
 be infer ted in Church and College-Leafes, or 
 other wife they would not be valid j if, by Vir«> 
 tue of fuch Covenants, the Owners of thole 
 Eflates are now become burdened with Incum* 
 brances which were either not at all in Being, 
 or were of fmall Significance, when thofe Cove* 
 nants were firft inferted ; and if it be reafonablc, 
 as the State of fuch Incumbrances now Hands, 
 to make fuch Tenants a Compenfation for fuch 
 new Burdens, which were then unforefeen, and 
 therefore not poffible to be provided for, the 
 Jullice of which, I think, no Man can deny; and 
 lince no Equivalent can be contrived for fucb 
 Leflees, but from the Fine to be paid on Re^ 
 newals, out of the Fine it ought and mufl come, 
 and be difcounted. 
 
 Having offer'd to Confideration the feveral 
 Articles of Dedu6lions and Allowances, whichi 
 I apprehend the Owners of Leafe-hold Ellatea 
 may fairly infill on in Order to reduce fuch 
 their Eftates to abfolute Annuities, and in what 
 Manner, and at what Rate fuch Reducements iti 
 a general Way may be made and computed ; I 
 fhall proceed now to confider and ftate what 
 may be a rcafonable Interell to allow them, for 
 the Money they lay out in purchafing or renew-* 
 ing fuch Leafcs. 
 
 D 4 I think
 
 C 56] 
 
 X think then, as I propofed at firit, that for 
 the Principal Money laid out in the original Pur- 
 chafe, or in the Renewal of thofe Eftates, In- 
 tereft ftiould be computed at 6 /. per Cent. ; but I 
 do not mean here, that on fuch Computadon the 
 Intercft, or Annuity, coming to the Purchafer 
 for his own Ufe, Ihould be, or will be 6 /. per 
 Cent, for his Money fo laid out ; for I know of 
 NecefTity it mufl be lefs. To fee how this Mat- 
 ter flands, and what Intereft will in Fa6l be 
 coming to one purchafing on fuch Terms, wc 
 will flate a plain and obvious Cafe ; and from 
 thence every Man may draw his own Conclufi- 
 ons, and form a Rule for his own Pradice, juft 
 as he likes bell. We will fuppofe then the 
 Eftate to be purchafed or renewed, at the im- 
 proved Value, is worth aoo/. per Annum ; the 
 Out-goings and Allowances, which under the 
 former Heads we have flated at 30 /. per Cent. 
 mull here be 60 /. annually, and if the Rent re- 
 ferved to the Landlord be 40 /. per Annum, when 
 thefe two Sums are dedu6led, the neat Income re- 
 maining to the Tenant will be an 100/. per Annum. 
 For a Term of twenty one Years in this clear 
 100/. per Annum, I fay, a Perfon may give 
 II ^ Value, or in Money 1 175/.; and for re- 
 newing it at feven Years End may give two Years 
 Value, and near an Half, or in Money 246 /.; 
 and this is the Price to be paid for the one and 
 the other, Intcrell computed at 6 /. per Cent. 
 
 I don't doubt .but moll Perfons have a Notion, 
 we have feen in Fad fome have, that a Purchafer 
 on thefe Terms makes 6 /. per Cent, of his Mo- 
 ney,
 
 [ 57 ] 
 
 jiey, but this is a great Miflake ; for if we cn-f 
 quire what it is that fuch a Perfon propofes to 
 himfelf, and what he is to do, and mull do, on 
 fuch a Purchafe ; we fee moll plainly, that his 
 iExpedlations are, and the £fFe6l of his Under- 
 taking mull be, that the Rents and Profits of the 
 Ellate purchafed ihall yield him a reafonable In- 
 terell for his Money, to be applied to fuch Ufes 
 of Life as he fhall fee convenient ; and a farther 
 iSum, to be applied to re-imburfe the Principal 
 by the End of the Term, or to pay the Fine 
 from Time to Time, as he ftiall renew the 
 Leafe. 
 
 The Interell of 1 175/- the Sum fuppofed to 
 be laid out in this Purchafe, at 6 /. per Cent, is 
 yoL 10 J. ; and if this Sum be dedu6led out of 
 the 100 /. per Annum^ which is the Neat Income 
 left to the Tenant, the Surplufage will be 2p/. 
 \os. Now this Surplufage will not produce a 
 Principal of 11 75/. within the Term, but upon 
 a Suppofition that you can, and in Fa6l do, 
 make 6 /. pr Cent, of it, and that you can and 
 do put it out at Compound Interell j and all 
 this at regular and lifted Times. But 'tis no- 
 torious, that by Law you cannot let it at more 
 than 5 /. per Cent, and almoll as notorious, that 
 in Pradlice you cannot get fo much ; the necef- 
 lary Confequence of which is, that the Pur- 
 chafer muH fink his Annuity, or lofe Part of his 
 Principal, in Proportion to fuch Decreafe in 
 common Interell. 
 
 The Fqnd, out of which the Return of the 
 principal is to be made, and out of which the 
 
 Interell
 
 [ 58 3 
 
 Intereft or Annuity is to be paid, being one and 
 the fame, viz, the neat Rent ; in Point of mcer 
 Computation, 'tis one and the fame Thing whe- 
 ther the Principal or Intereft falls Ihort ; but 
 ilnce, in Point of Prudence, the Proprietor would 
 chufe to link his Annuity rather than his Princi- 
 pal, I will fuppofe this Deficiency in the Fund, 
 which is an unavoidable one, to fall on the An- 
 nuity folely. If then the Purchafer, out of the 
 Rents of this Eftate, will referve fuch a Sum as 
 will be fufficient, at a Rate of 5 /. per Cent, to 
 make a Return of the Principal, he muft apply 
 annually 33 /. to that Purpofe, and he nmft make 
 both Simple and Compound Intereft of it like- 
 wife ; and then there remains to the Owner for 
 his Intereft, and as the ^antum which he may 
 employ for his own Ufe, no more than 6y L per 
 Annum ; which comes out to be 5 ^ per Cent, 
 for his Money, or fomething lefs : So that the 
 Law has cut him Ihort \ per Cent, at leaft, if 
 there were nothing elfe in the Cafe. 
 
 But then we are to confider farther, that for 
 large Sums the common Pradice now is to ac- 
 cept 4/. or 4 i per Cent.'j and for fmall Sums, 
 fuch as this Surplufage is, very great Trouble 
 and Difficulty will attend the placing them out 
 at all, and to do it in the ftill fmaller Sums of 
 the growing Intereft, to be fure, is impradicable* 
 And if both thefe were, and where they are, 
 practicable, yet the Proprietor of the Eftatc 
 muft wait for doing it 'till his Tenants pay their 
 Rents, and his Securities pay their Intereft ; 
 and though none of them Should prove fo bad 
 
 a?
 
 t 59 ] 
 
 as to be totally infolvent, yet the Delays in fach 
 Cafes are fo notorious, that no prudent Man will 
 or does reckon on thofe Payments at the Itated 
 and limited Times to an Exadncfs, and yet the 
 Juftice of the Computations, both on Simple 
 and Compound Intcrelt, depends on the Pund:u- 
 ality of Payment, even to the Nicety of a Day, 
 Thefe Contingencies, though there are others 
 attending the Affair of Intereft, I think, may 
 modellly be computed to link the Tenant's An- 
 nuity ^ per Cent, more, and reduce it to 5 /. ; 
 ipd I believe the World will judge that he is 
 li-ot only a provident Tenant, but an happy one 
 too, if he can make it hold up to this Rate 
 through all thefe Difficulties. 
 
 If we examine this Matter more clofely, and 
 compute what Proportion of this 11 75/. Prin- 
 cipal Money is feverally to arife from thefe Sur- 
 plufages, what Part from the Simple Intereft, and 
 what from the Compound Intereft, of fuch Sur- 
 plufages, we fhall be able to form a better Judg- 
 ment what Part of his Income of an 100/. peif 
 Aumim he may apply to his own Ufe, falely and 
 confiftently with re-imburfing the Principal. 
 We have already feen that he mufl make a Re- 
 ferve of the annual Sum of 33 /. as a Fund, fbi: 
 the Purpofe of re-imburfing the Principal, for 
 that no lefs annual Sum, even computed at Com- 
 pound Intereft, will produce 1 175/. in twenty 
 one Years ; and, upon a Computation, it ap-» 
 pears, that from the meer Surplufage, '■oiz» the 
 2,"^ /. per Jnnuniy there arifes in twenty one 
 Years no more than 6^o^ /. fo that the remaining 
 
 SuiJ3
 
 [ 6o] 
 
 Bum of 4S2 /. muft arife from the Interefl: ; and 
 it appears likewife, that from the Simple Inte- 
 rell only of thofe Surplufages there arifes in the 
 fame Time no more than 346/. loj. fo that 
 there is a Deficiency of 136/. 10 j. which is left 
 to arife and be gained folely by accumulating In- 
 tereft. If Compound Interefl: is not taken into 
 the Gonlideration, and in Truth it hardly de- 
 ferves it, and the Surplufage of Rents, and 
 Simple Interefl of fuch Rents, be look'd on as 
 the Fund to raife this Principal, the annual Sum 
 to be refer ved for this Purpofe mull be 38 /. aq|^ 
 above ; and then the Annuity left to the Pro-* 
 prietor of the Leafe, for his own Ufe, will be 
 near 62 /. and about 5 /. 5: per Cent, for his Mo- 
 ney : But even this is grounded on fomething 
 which undoubtedly in Pradiice is never the Cafc, 
 that the whole of the Rents is pundlually paid 
 to a Day at the Year's End, and the Overplus 
 as regularly placed out. If Simple Interefl be 
 likewife left out of the Confideration, as it ge- 
 nerally may be, if Confideration is then only to 
 be had of it when and where it is actually made, 
 and the meer Surplufage of the Rents be the 
 only Fund for railing this Money, then the Sum 
 referved for that Purpofe mull be the exa6l Sum 
 of S^ /. pe'i" JiiJJum ; and then 44/. being left as 
 the clear Annuity, does not leave full 4/. p^ 
 Cent, to the Tenant, as Interefl for his Principal 
 Money, Having thus Ihown the Manner of 
 making Computations of this Kind, and fairly 
 ftated how and from whence the Money is to 
 strife to make good fuch Computations, I leave 
 
 it
 
 L 6i J 
 
 it with the Reader to determine what Part of 
 his Rents a Purchafer of fuch an Efiate, and on 
 fuch Terms, may fafely apply to his own Ufe, 
 without Danger of diminifhing his Principal. 
 
 If any Leafe-hold Tenant Ihould be fo un- 
 reafonabie and partial to himfelf, as to think 
 that the Overplus Rent only fhould be look'd 
 on as the Fund for raifing this Principal, or the 
 Fine to renew, and that no Regard ought to be 
 had to the Intereft of fuch Overplus, either 
 Compound or Simple ; or if any fuch Tenant 
 lhoul(i be fo unwary or fo unhappy as to con- 
 tent himfelf meerly with faving fuch Surplufles 
 to anfwer thefe Purpofes ; and were it not for 
 Fear of Incurring feme Gentlemen's DIfpleafure 
 and Cenfure, I fhould readily fuppofc, that fomc 
 fuch there were j then fuch Tenants at leafl will 
 have no mpre than 44 /. per Annum out of an 
 100/. which is lefs than 4/. per Cent, for the 
 Money laid out in the Purehafe of thefe flftates* 
 and this is the Cafe, though Intereil be computed 
 at 6 /. per Cent, in the Purehafe, and the Ellate 
 be a clear neat Income, or an abfolute Annuity. 
 But if fuch Tenant is likewife to be faddled with 
 the Land-Tax, Repairs, and other Charges in- 
 cident to Eftates in Land, and to havenoGom- 
 p'enfation for fuch Burdens, which being ftated 
 at 30/. per Cent, will, in this Cafe, amount to 
 60/. per Annum \ if this Load is likewife to be 
 laid on fuch Tenant, there will be nothing re- 
 maining for his own Ufe, or nothing to re-imburfe 
 the Principal. 
 
 On
 
 [ 62 ] 
 
 On a View of thele Calculations I cannot help 
 being of Opinion, that 5 /. per Ce»t. is as much 
 as can reafonably be expeded to arife to a Te- " 
 nant purchafing on thefe Terms ; and tho' this 
 be a greater Interell than will be coming to one 
 ■who purchafes Lands of Inheritance, and fome- 
 thing more than in fome Cafes may be coming 
 to one who difpofes of his Money in Mortgages, 
 yet *tis no more than legal Intereft ; and fo 
 much, in Confideration had of the Hazards and 
 Undertakings he engages in, is, I think, no 
 more than what he is well entitled to. What is 
 nfually pradtifed in thefe Cafes, I don't pretend 
 to know, nor can any one know without apply- 
 ing it to Particulars ; and in different Cafes the 
 Circumftances may be fo various as to be a Se- 
 cret to any one but thofe who are immediately 
 concerned in buying and felling ; and if we con- 
 fult them, we fhall only learn from thence, that 
 the Seller fold dear and the Buyer bought cheap, 
 no Body being willing to be thought to have 
 been over-witted in a Bargain. 
 
 I am led to take Notice here what Valuali**' 
 other Writers on this Subject put upon thefe 
 Sorts of Eftates ; and I obferve, that the Reve- 
 rend Author, whom I have fo often mention'd 
 before, fuppofes a Purchafer to buy at twelve 
 Years Value, or where Money is computed at 
 6 /. per Cent, j but then he does not fay, that a 
 Purchafer on thefe Terms will make 6 /. per Cent, 
 of his Money ; and to be fure he did not think 
 he makes fo much, becaufe he rails molt plenti- 
 fully at the Unreafonablcnefs and Partiality of 
 
 fuch
 
 [ 6i3] 
 
 fuch who cxped it : But then this Opinion is 
 not grounded on the Reafons I have given, viz, 
 the Decreafe of Common Intereft, and the Con- 
 tingencies of not making Compound or even 
 Single Intereft on fuch fmall Sums, and in a 
 Manner fo regular as the Computations fuppofe; 
 but becaufe he thinks that the Load of Taxes, 
 Repairs, and accidental Charges, are to lie on 
 fuch Purchafer's Ellatc. And yet, whatever he 
 thought, or would be underftood to have thought, 
 a fecondary Writer, who implicitly follows him, 
 or means to follow him, underftands him in that 
 Senfe ; that the Purchafer will make 6 /. or even 
 7 /. per Cent, of his Money, and quotes his Au- 
 thority, and his Inllance, as a Demonllration of 
 it. Now whoever will read, and does attend 
 to what he reads, will certainly fee, that this 
 Reverend Author does not prove, or undertake 
 to prove, any Thing more by his Inftance, than 
 that one who difpofes of his Money in fuch a 
 Manner as he there directs, mz, partly in a 
 Lcafe, and partly in Securities at Intereft, will 
 have the fame Income out of his Land, befides 
 fome Savings over out of his Intereft, as ano- 
 ther Ihall have who lays out the like Sum on 
 Lands of Inheritance. And if we enquire how 
 much his Income will be from 2000/. laid out 
 in Lands of Inheritance, his Argument necefla- 
 rily leads him to fay, and he fays it in fo many 
 Words, that he will have about yohperjfjfjum, 
 that is, 3 /. 10 J. per Cent, for his Money fo laid 
 out, and confequently the Purchafer of Leafe- 
 fcolda is to expedt no more from his Land mcer-
 
 [ 64] 
 
 ly; and indeed, in the Cafe as dated, 'tis fup- 
 pofed that the total Income of both Eftatcs in 
 Land is one and the fame, and both liable to the 
 fame Dedudtions. But then the Interefl of the 
 Aloney on Securities being more than fufficient 
 in all Events to pay the Fine from Time to 
 Time, on the Account of fuch Overplus he 
 gives the Preference to the Leafe-hold Tenant. 
 But whatever the firft Writer aflerts, or what- 
 ever he meant or has been underllood to have 
 aflerted, 'tis demonflrable that the Purchafcr of 
 the Leafe-hold does not make fo much as 5/. 
 fer Cent, of his Money. The neat Produce of 
 the Land which colt 1 200 /. is 70 /. per Jnntim ; 
 the Interefl of his 800 /. on Securities can be no 
 more than 40 /. ; both together are an 1 10 /. the 
 Fine to be paid at feven Years End, if the Su- 
 perior takes only one Year's Value, will be 100 /. 
 and to raife that Sum, the Tenant muft refervc 
 out of his Income more than the annual Sum of 
 10/. for \ol. per j4nnu7?i for feven Years, at 
 Compound Interefl, will not produce an 100 /. 
 and confequently the Neat Income to the Te- 
 nant, both from his Land and his Money, mufl 
 be lefs than an 100 /. that is, Icfs than 5/. per 
 Cent, for his lood. 
 
 It feems to me, as if this Secondary Writer 
 -was determined in all Events, whenever the firft 
 advances any Thing extravagant, to affert the 
 feme Thing with fome Additions ; whether he 
 -did underfland or did not underfland the firft, 
 or whether he faw any Reafon for it, or faw 
 none : And feeing the Eitimate which they have 
 
 both
 
 [ 65 ] 
 
 both given of the Value of thefe Eftates is a very 
 high one, and greatly exceeds the Value whidh 
 I have here put upon them, we will examine 
 them both, though the latter is only an ill Copy 
 of an ill Original. 
 
 The Firft then affirms, that a Perfon might 
 well afford to give fifteen or fixteen Years Pur- 
 chafe for a Leafe of twenty one Years, if he 
 could be alTured that his Superior would always 
 take one Year's Value for the Renewal of feven : 
 And his Second fays, that if a Perfon gives fix- 
 teen Years Pnrchafe for fuch an Eftate, with a 
 View of renewing at two Years Purchale and an 
 half, he will have near 5 /. per Cent, for his Mo- 
 ney, Thefe two Eftimates, we fee, differ very 
 widely from one another, as well as both do 
 from mine ; for one Year's Purchafe on a Re- 
 newal is very fhort of two and an half^ and a 
 View only is not fo good as an Affu ranee ; tho' 
 I confefs, a View of renewing at two Years Pur- 
 chafe and an half may amount almoll to an Al^ 
 furance. 
 
 If we would know what Tnterefl a Purchafer 
 will have for his Money, where he buys at fix- 
 teen Years Value, as this Secondary Writer flates 
 it ; let us take the fame Inftance we ufed before, 
 and fuppofe the Eftate at the full Rent to be an 
 110/. per Amium.^ the Rent relcrved to the 
 Landlord to be 10/. and then the Sum to be 
 given for this Eftate, at this Computation, will 
 be 1600/, The Deductions for the Out-goings, 
 which he admits are ufually borne by Tenants, 
 and means that of Right they ought to be ^Oy 
 E ftating
 
 [ 66 ] 
 
 noting them at 30 /. per Cent, by a Medium for 
 ' Common Cafes, amount to o^'^L per Anuiim j the 
 jinnual Sum to be referved to re-^imburfe the 
 Principal Sum of l<5ob /. muil be 45/. or very 
 near it, and this mull be computed at 5/. per 
 Cent* and with Compound Intercft : And when 
 thefe two Sums, 33 /. and 45 /. making together 
 78 /. are dcdudcd out of the 100/. the cleat 
 Annuity left to the Ovvner^ as the Intereil of hiS 
 Money, will be only 11 /. which is lefs than i \ 
 per Cef/t. for his Principal Money laid out. 
 
 If we take into our View, that this Leafe may 
 he fenew'd at feven Years End, and fo from fe- 
 teii Years to feven Years, at 2 i Years ValuCj 
 the Fine on Renewals will be 250 /. ; and to 
 faife that Sam in feven Years, computing Money 
 at 5 /. per Cent, and Compound Interell:, there 
 mull be referved 31 /. annually, or very near it j 
 and when this Sum, and the Sum of ^^ I. to be 
 fet oif for Out-goings, making together 64 /. 
 are deduced out of this Ellatc, the Remainder 
 to the Tenant will be no more than "^(^ I. per 
 Anniim^ which is jull 2 \ per Cent, for his 1600/. 
 
 There is a Way, indeed, of making the Com- 
 putation here, which may feem to produce an 
 Intereft almoll of 4 -^, though, by the Bye, that 
 is not near 5/. and is thus made. The Rent re- 
 ferved to the Landlord is 10/. and the Out- 
 goings in Taxes, and other Charges, are ^2> ^* 
 per Jnnnm'j which two Sums being deduded cut 
 of the no/, per Jnnur,i^ the Neat Income re- 
 maining to the Proprietor is 6y I. : Now if the 
 Whole of this 67 A be put to the Account of 
 
 Intereft,
 
 [ 67 ] 
 
 Jntercft, the Tenant then will have 4 ^ per Cent,' 
 for his Money, or near it. But whoever com- 
 putes in this Manner, and I cannot conceive by 
 what other Method this Writer can produce 
 fuch an Intereft as he talks of^ dreams fb much 
 and fo long upon the Intereil, that he forgets 
 that the Principal is by this Way funk, and the 
 Efcate entirely loll. If the Purchafcr here takes 
 near 5 /. per Cent, as the Interell: of his Money 
 laid out, he muft take near 80 /. per Jnnum of- 
 the 100/.; nay, if he takes only 4 -^ he muft 
 take 68/. and if the Tenant mult bear the Bur- 
 den of the Land-Tax, and other Out-goings, 
 as this Writer afferts ; and if thefe. Charges 
 amount to ^^"2^ I. per Anniim^ and they are gut and 
 carved out according to the Model framed by 
 his Friend from whom he copies; 'tis obvious 
 there is nothing left to rc-imburfe the Principal, 
 dr to raife the Fine for renewing, and conle- 
 quently the Principal funk, and the Eitate totally 
 loft. 
 
 But to leave this Author to his own Under- 
 ftanding, in which he greatly abounds, let us 
 examine the AlTertion, or rather Concellion, of 
 the original Author, which is not quite fo extra- 
 vagant ; who only fays it is a Matter confefs'd, 
 that a Perfon may well afford to give fifteen or 
 lixteen Years Purchafe for fuch an Eftate, on an 
 Affurance of renewing always at one Year's Pur- 
 chafe for fevcn Years. The total Rent, at the 
 full Extent, being an no/, and the Rent refcrved 
 to the Landlord 10/. the Price to be paid for 
 the 100/. per Annum retting, at fifteen Years 
 E 2 Valucj
 
 [ 68 ] 
 
 Value, will be 1500/. The Dedudidns for the 
 Land-Tax, and other Out-goings, will be, as 
 before, 33 /. per Jtmum^ and the Sum to be an- 
 nually refervcd to raife the Fine of an 100/. will 
 be about 13 /. if we compute at 5 /. per Cent, 
 Compound Intereft. And if thcfe three Sums, 
 ^^''* 33^' and 13/. making together 56 /. are 
 dedu6i:ed out of an 1 10 /. the Remainder will be 
 54/. which is exad:Iy 3 f per Cent, for the Pur- 
 chafe Money of 1500/. ; and I doubt not but 
 this Author did himfelf fee that this, or fome 
 fuch like Sum would be the Intereft coming to a 
 Perfon purchafing on thofe Terms, whatever his 
 blundering Second faw, or thought he faw. 
 
 If indeed the Alfurance of renewing once in 
 feven Years for ever, at one Year's Value, were 
 a legal and binding one, fuch an one I mean as 
 could be juftified in WeJimhifter^Hall'^ or if the 
 Leafe were a perpetual one on thofe Terms, 7;i;3. 
 that the Tenant once in feven Years fhould pay 
 the whole Rent of his Eftate at the extended 
 Value, and no more, I fliould agree that fuch a 
 Leafe might be worth fifteen or lixteen Years 
 Value ; and the Reafon of it is obvious. By 
 thefc Tv'Icans the Intereft is become a certain and 
 permanent one ; and though in the Terms of the 
 Law it cannot be call'd a Fee-Simple, yet in 
 cominon Senfe, and common Underftanding, 'tis 
 equal to one, it being a Perpetuity ; and the 
 Fine or Premium to be paid once in feven Years 
 is afcertain'd and fix\i, and the Value of it may 
 be adjufted with great Exaftnefs. But that a 
 Leafe, where the Tenure is fo far precarious 
 
 that
 
 [69 ] 
 
 that the Superior may or may not prolong it, as 
 he judges convenient, and from Time to Time 
 may advance the Fine at his Will and Pleafure ; 
 that fuch an Ellate fhould be worth fifteen or 
 lixteen Years Value, I can by no Means admit : 
 And I believe this Author, with all his Learning 
 and Abilities, will never be able to make out, to 
 the Satisfaction of Perfons who are unprejudiced, 
 and fee vvhat is the Produce of it. 
 
 The Truth is, that a Purchafcr of fuch an 
 Eflate, if it were a neat Income, at fixteen 
 Years Value, would make little more than 2 /, 
 10 5. per Cent, of his Money ; and a Purchafer at 
 filteen Years Value, little more than 3 /. per Cent, 
 of it : For the Value of an abfblute Annuity, 
 or of a Leafe reduced to an abfolute Annuity, 
 for a Term of twenty one Years, is very little 
 more than lixteen Years Purchafe, if Money* 
 were at 2 j per Cent, and very little more than 
 fifteen Years Purchafe, if it were at 3 /. per Cent. : 
 Or, which is the fame Thing, and the PropoJi- 
 tion is convertible, that a i, or thereabouts, is 
 the Intereft which is made of fuch an Annuity, 
 if valued at lixteen Years Purchafe, and 3 /. per 
 Cent, if valued at fifteen Years Purchafe. And 
 fince thefe Writers lay the Burden of the Land- 
 Tax, and other Out-goings, on the Tenants, 
 'tis certain they will have no more than 1 \ in 
 one Cafe, and 3 f per Cent, in the other, as the 
 Interefl for their Money laid out at thofe Rates ; 
 and even thus much will be coming to them, on- 
 ly on a Suppofition that the Term is continued 
 by repeated Renewals, the former at the P^ate 
 E3 of
 
 [ 7o] 
 
 of 2 i Years Value, and the latter at one Year's 
 Value only ; therefore I cannot but concur with 
 this Reverend Author in his Advice, that Per- 
 fons who buy thefc Ertatcs Ihould be upon their 
 Guard, how and at what Rate they purchafe 
 upon fuch Expectation's. 
 
 However, though thefe Authors exalt the 
 Value of a Term for twenty one Years to iix- 
 teen Years Purchafe, when they are in their Al- 
 titudes ; yet they fcem to fet it no higher than 
 twelve Years, where they are more immediately 
 confidering how and in what Manner fuch Value 
 is to be afcertain'd. And I cannot forbear ob- 
 ferving the pleafant Method which one of them 
 takes of doing this, where he tells us, that the 
 inoft agreeable Proportion between Leafe-holds 
 for three Lives, and thofe for twenty one Years, 
 is as fixtccn to twelve ; which is neither better 
 nor worfe than it would be to tell us, from the 
 Contents of a Circle, which never were known, 
 and perhaps never will be known, you may know 
 the Contents of a Square, which are known, or 
 may be known, to a mathematical Point. 
 
 But in the Gomparifon made of thefe two 
 Ellates, 'tis not the only P'ault that 'tis trifling 
 and limple ; for there is another worfe Fault, 
 and which indeed induced me to take Notice of 
 it, viz. that it is grounded on fbmething which, 
 I doubt, in Fad is not true, viz. that Eflates 
 lor three Lives are worth, and commonly fold 
 for, fixt^en Years Value, and that Covenants arc 
 frequently cnter'd into to give and take at that 
 Price. That fuch Eltates are worth h'xteen 
 
 Years
 
 [ 71 ] 
 
 Years Value, when Eftates for twenty one Years 
 are worth twelve Years, depends on a Suppoli- 
 tion that an Eftate for three Lives is equal to a 
 Term for lixty Years ; and whether there be 
 any Senfe or Reafon for making fuch a Suppofi^ 
 tion fliall be difcufs^d elfewbere ; apd it will be 
 there demonftratcd that there is no Foundation 
 for making any fuch extravagant Eftimatc, That 
 Eftates for three Lives, where nothing more or 
 farther is granted, are comirjonly fold for fix- 
 teen Years Value, furc enough qiuft be falfe iri 
 Fadl ; bccaufe this is the Market-Price of Eltatea 
 for three Lives, to which is annexed a Right of 
 Free-Bench, which is a Right to a Widow's 
 Life, that is, a fourth Life to commence aftef 
 the Determination of the other three. This I 
 take to be the common Price of thefe Sorts of 
 Eftates, bccaufe I have been fo inform'd by thofe 
 who ufually do, and have themfclves grante4 
 rnany fuch ; and this Author himfclf does, in 
 EfFed:, inform us the fame Thing, when he tells 
 us, that Copyholds for Lives, to which are an-? 
 nexed the abfurd Cuftom, as he calls it, of the 
 Widow's Free-Bench, are of equal Value with 
 Leafes for three Lives, Now if we look \^pon 
 fuch fourth Life, not as a Life concurrent with 
 the other three, but as a Life in Reverfion after 
 the other three, as in Fad: it is, we may, at 
 ^ Medium, fuppofe it to be of the Age betwcea 
 the oldeft and youngeft, and then it may very 
 well be eftimatcd equal to a Term of twenty 
 Years, or thereabouts ; and I believe thofe who. 
 have dealt in granting fuch Eijatcs have foun4 
 E 4 E^^'.foa
 
 [ 72 ] 
 
 Reafon to make the Eftimate at this Rate : And 
 fuch an Eilate indeed, in my Way of Compu- 
 tation, may be equal to a Term of fixty Years, 
 and commonly fell at the Price mentioned. But 
 to talk of Ellates for three Lives, meerly and 
 limply fuch, as the fame Things as Eftates for 
 three Lives, and a fourth in Reverfion after the 
 End of the other three, and to put both on the 
 fame Foot in Point of Value, has fomething i^ 
 it which I cannot account for, fo I mufl: leave it 
 with the Author. 
 
 I have propofed, that in the Purchafe of thefe 
 Ellates Intereft fhould be computed at 6 /. per 
 Cent, which, under a good Management, may 
 produce about 5 /. for the proper \Jic of the 
 Owner, thiiiking for mvfelf, that fuch an Inte- 
 reft is not an unreafonable one on thefe Eftates : 
 But my Intention here is not to prefcribe what 
 Intereft ftiall be made by a Purchafer ; let every 
 Man judge of that for hinifelf. My Aim is 
 chiefly to point out, if fuch or fuch an Intereft 
 is expelled to be made, fuch or fuch a Manner 
 of computing muft be ufcd in the Purchafe ; 
 which fome People either do not themfelves fee, 
 or are unwilling others fiiould fee. But where 
 Perfons plainly fee and kriovv what Intereft they 
 ihail' h-ive on any particular Computation, if they 
 will accept of 2 ^, or 3 f per Cent, for theif 
 Money, I have nothing farther to fay ; if they 
 are contented, fo am I. 
 
 It will be all edged, perhaps, iince I look up- 
 on the Tenant's Intereft here as an Annuity, and 
 Jiave guarded it fo ftrongly againft all pofhble 
 
 Hazards
 
 [73] 
 
 Hazards and Contingencies, as indeed T mean it 
 fhould, as far as the Nature of the Thing will 
 admit, that fuch an Annuity is, upon the Mat- 
 ter, as certain and as good as any other what- 
 ever, and equally valuable ; and therefore that 
 'tis unreafonable for fuch a Tenant to expe6b 5 /. 
 per Cent, for his Money, when a common Annu- 
 itant would be well contented with 4 /. Upon 
 this I obferve ; where Perfons buy thefe Eftates 
 meerly for their Convenience, fuppofe an Houfc 
 in Lofidofiy or elfewhcre, for their own Dwelling, 
 a Farm or Land for their own Occupation ; or 
 becaufe they are contiguous to, or lie intermix*d 
 Vk'ith, other of the Purchafer's Eftates, or in any 
 other Refpe(3:s are greatly commodious to him: 
 With fuch Views Perfons may be induced, and 
 there may be Reafon to be contented with a 
 lower Rate of Intereft, or not to ftand fo ftrid:- 
 ly to the Allowances for accidental Lofles and 
 Charges, becaufe by occupying the Premifes they 
 pay themfelves their Intereft, and the Contin- 
 gencies of Lolles and Charges are by that Means 
 Icflen'd. On fuch Occalions I have only this to 
 fay ; if in the Main they keep thefe general 
 Rules in their Eye, they cannot be much amifs 
 in their Purchafes. 
 
 But in common and ordinary Gales, fuch 
 where Perfons buy thefe Eftates meerly for the 
 Sake of the Rents, and folely as a Manner of 
 difpofing their Money on Land Security to re- 
 ceive Intereft for it, there I can by no Means 
 admit that fuch a Tenant's Annuity is as good 
 as any other, or ought to be ib valued. Sure 
 
 enough,
 
 [ 7+] 
 
 enough, an Annuity certain, one where the 
 Fund is iiifc, and the Day and Place of Payment 
 is fix'd, mult be better than one where the Fun4 
 may prove deficient, and where the l>ay of Pay- 
 ment is uncertain, at the bell, is always a long 
 Day, and fometimes may never happen to come 
 at all. 
 
 Now fince wc may conlider thefe Eftates as, 
 having feveral Annuities ifTuing out of them to 
 different Perfons, or for different Purpofes, and 
 fomc of them are niore and fpme are Icfs certain, 
 it cannot, with any Reafon, be expedted, that 
 the Proprietors of them iliould incumber theit 
 Eftatcs with the Payment of the certain Annuir 
 ties, and themfelves be content with a lels cer- 
 tain one, unlefs fuch lefs certain one have the 
 Advantage of being larger. And fince the 
 Rent payable to the Landlord, the Out-goings 
 for the Land-Tax, and other Charges incident 
 to the Eftate; as alfo the annual Sum to be re- 
 ferved for re-imburfmg the Principal, or railing 
 the Fine to renew, may all be look'd upon as^ 
 certain and abfolutc Annuities ; and they are 
 Teally and truly luch, and irjull be cilimated as 
 fuch ; for in all Events they mult be paid or 
 provided for, whether the Owners can or can- 
 not raife them out of the Leafe-hold Eltate ; for, 
 not only that their Eflate, but all other their 
 Eltate whatever, is charged or chargeable with 
 them ; if thefe Annuities or Payments were 
 valued at a Rate of 4 /. per Cent, the Ellimatc, 
 perhaps, would not be thought too high. 
 
 But
 
 [ 75 ] 
 
 But the Annuity coming to the Tenants, 
 which is the Intereft they are to receive for their 
 Money, cannot, to be fure, be Hated at a Rate 
 Co high • becaufe the Fund for Payment of their 
 Annuity is one and the fame as is to pay the 
 former Annuities, and thofe other Annuities, in 
 Point of Charge and Payment, are all precedent 
 to this ; fo that if the Fund ever falls fhort of 
 paying all the Annuities, and in the Compafs of 
 fome Years, 'tis almoll impoffible but that ic 
 Ihould fometimes happen, fuch Deficiency mull 
 all fall upon the Tenant's Annuity. And if the 
 Fund fhould prove fo good as in no one Year to 
 fiil, yet the very PofTibility on that Side againft 
 the Non-PofTibility of the other Side, is of 
 fome Confideration ; and in all Events, fuch 
 Tenants, as they ftand lafl in Courle of Pay- 
 ment, cannot be paid their Annuity 'till all 
 the others are fatisfied, and mult therefore of 
 Neceffity take up with the Remnants of Rent, 
 and fuch as come lall to Hand. On thefe Ac- 
 counts I think fuch Tenants may very well 
 expedt and deferve an Annuity of 5 /. per Cejit^ 
 as the Intereft of their Money thus laid out j 
 and is no greater an Annuity than every Man, 
 I believe, will expect, who buys an Annuity 
 with his Kyes open, where the Fund, out of 
 which 'tis to iffue, may with any Poflibility be 
 ever deficient, and fuch Deficiency muft be his 
 Lofs, and where the. Times and Manner of 
 Payment are fuch as of Courfe they wilt be ia 
 Cafes of this Nature. 
 
 Thefe
 
 [ 76] 
 
 Thefe Eliates, if purchafcd at the Rate and 
 upon the Terms I have niention'd, or there- 
 abouts, may be delirable Things ; but furel/ 
 they over-a6t a Part who fct them out fo pom- 
 poufly, as if there were hardly any other in the 
 Kingdom wbrth enquiring after, in Comparifon 
 of thefe ; and who magnify the Advantages of 
 laying out Money in the Purchafe of them, as if 
 no Way of difpoling of Money in Land could 
 come into any Sort of Competition with this* 
 If we look upon them as Annuities, they are 
 determinable ; if as Mortgages, or Land Secu- 
 rities attended with Interelt, they are redeem- 
 able ; if the Landlord thinks fit to fuffer the 
 Term to expire, or the Lives to fall in. They 
 do indeed differ from common Mortgages, be- 
 caufe they cannot be redeem'd abfolutely at the 
 Will of the Lelfor, but mull continue for a Time 
 certain, and in that Refpe6l may have fome Ad- 
 vantage of ordinary Mortgages • yet flill they 
 are determinable by EfHuxion of the Time : 
 And if in Fait they are not fo determined, but 
 are fuffer'd to continue by Renewals, there is 
 Trouble, fometimes very unneceffary Trouble, 
 and Difputes, and always fome Charge, attend- 
 ing fuch Renewals. 
 
 To this we may add ; that there is much more 
 Care and Caution to be ufed in managing thefe 
 Eftates, than in abfolute Annuities, or ordinary 
 Mortgages, with which fome People cannot, and 
 others will not, incumber themfelves ; and that 
 they are not fo eafily and fo readily fold and 
 transferred as thofe Securities are. On thefe Ac- 
 counts
 
 [77] 
 
 counts thefe latter, though attended with an In- 
 ter ell of :\L per Cent, only, (hall be generally 
 preferr'd to the former, though the Intereft be 
 5 /. per Cem. or more : For the Certainty of the 
 Income, and the punctual Payment of it at the 
 Times flated, is to fome, and the Facility in 
 felling and transferring their Intereft is to others, 
 of fo great Value and Conlideration, that we 
 fee in daily Experience it. makes a Difference of 
 one per Cent, at leaft. 
 
 We are told indeed, that this Manner of dif- 
 poling of Money is laying it out in the Purchafc 
 of Lands ; and on that Account frequent Cau- 
 tion is given to the Gentlemen of the Church 
 and Colleges, that in making and renewing their 
 Leafcs,they are fo far felling their Eftates; which 
 may be very proper Advice, but comes from 
 this Hand with an ill Grace ; yet if we call it 
 buying and felling Lands, for I Ihall have no 
 Quarrel about the Terms if -4^6^ agree in the 
 Thing ; What do we learn, or what can be in- 
 ferred from thence ? Why truly, nothing that 
 I know of, but that the one buys and the other 
 fells the growing Rents of the Eftate, and them 
 only, for a Term of Years, or for one or more 
 Lives ; and if their Tenants renew their Leafes, 
 then there is buying and felling from feven Years 
 to feven Years, and from Life to Life, and fo on 
 to the End of the Chapter ; and by that Means 
 the Owners oi Leafes have frequent Opportuni- 
 ties to repeat the Pleafure of laying out their 
 Money in the Purchafe of Lands. 
 
 Now
 
 [ 78 ] 
 
 Now admitting that the Purchafe of a Leafe 
 is a Purchaie of Land, yet that does not make 
 the Ellate in it to be a perpetual or a permanent 
 Interelt, nor is it fuch; but mull be efteem'd and 
 is treated only as a Mortgage redeemable, or an 
 Annuity determinable, at the Pleafurc of the 
 Leffor under fome Limitations. And that ^tis 
 no other, or better, is plain not only from thence, 
 bccaufe 'tis entirely in the Leflbr*s Ereaft whe- 
 ther he will renew or not, and if he does not re- 
 new, there is an End put to the Intcrcll in 
 whatever View we look upon it; but becaufe if 
 he does renew he may raife the Fine, and in 
 thefe our Days of Profperity, that is no fuch 
 wild or imaginary Suppofition ; and every Ad- 
 vance in the Fine, though never fo fmall, is fo 
 far forth a Determination of the old Annuity, 
 and a Subftitution of one lefs in its Room ; for 
 every Rife in the Fine does of NccelTity make 
 an Abatement of the Annuity. 
 . Nay, we may go farther ; for tho' a Church 
 or College, or other Landlord, has taken one 
 and the fame Fine, fuppofe at one Year's V'alue 
 on the neat Rent, for thefe one Hundred or two 
 Hundred Years lafl pall, or Hiould continue to 
 do fo for as many Years to come, yet the very 
 Change of Times, and the Decreafe in the In- 
 terefl of Money, does neceffarily itfelf determine 
 or diminifh thefe Annuities or annual Income ; 
 for every Decreafe in the Interefl of Money 
 makes it neccffiry to enlarge the Fund for rai- 
 ling the Principal, or the Fine to renew, and this 
 cannot be made greater, but by making the an- 
 nual
 
 [ 79] 
 
 bual Income lefs/ And Imce a farther Decreafc 
 of Intereit may polHbly, and probably will come 
 to pafs in the Courfe of fome Years, though 
 perhaps not fo foon as may be commonly ima- 
 gined, and this will, and unavoidably mult, draw 
 along with it fome Diminution of the Annuity; 
 and lince the Advancement of Fines is labour'd 
 fo ftrongly, and fo warmly purfued, and this 
 muft make another and a firthcr Diminution of 
 fuch Annuity : Theie Gonfiderations, added to 
 thofe already allcdged, have induced me to think, 
 that, upon Purchafes of theie Eltates, Interell, 
 at prefent, lliould be computed at 6/. perCetit,'^ 
 upon which Computation a Purchafer may ex- 
 pect to get about 5/. per CetJt. for his own Ufe. 
 If thefe Calculations of mine, and the Reafon- 
 ings grounded on them, do not appear to others 
 to have that Weight in them which I have given 
 to them, fure enough, they are at Liberty to 
 think, and will ad:, for themfclves as they fee 
 convenient : And there I reft this Matter. 
 
 We fhall go on now to confider what is the 
 Method of adjufting the Fine to be paid for re- 
 newing, or filling up the Years lapfed in an old 
 Term ; or for adding any Number of given 
 Years to the Years in Being in any Leafe. In 
 general, to be fure, the Rule for renewing a 
 Leafe muft be the fame as for purchafing ; for it 
 can never be confonant to Juftice or Reafon, to 
 aflign one Rule for purchaling the Whole, and 
 another for purchafing a Part of the fame Whole. 
 To make a different Rule for thefe Purpofes, let 
 the Deviation be on which Side we will, is, in 
 
 Effca,
 
 [ 80 ] 
 
 £fFed, to fuppofe that feven Years (for Tnllance) 
 to be added on a Renewal, and which, in Time, 
 tvill come to be feven Years in PolTefTion, will 
 not be the fame Thing when they are become 
 feven Years in PolTefTion, as the like Number of 
 Years now in PofTeffion is. 
 
 A Renewal then being only a new, or another 
 Purchafe, and to be govern'd by the fame RuleSj 
 we muft reduce the Eftate to an abfolute Annu- 
 ity, in the Manner before direded • we muft 
 make our Computation at the Rate of 6 /. per 
 Cent, ; and then, if we fubftra<5t the Value of the 
 Leafe in Being from the Value of the whole 
 Leafe, or fubllrad: the Years Purchafe of the 
 Term in Being from the Years Purchafe of the 
 whole Term, the Remainder is the Value or the 
 Years Purchafe to be given for renewing. If 
 the Term be not a certain one, or not known, 
 as in the Cafe of a Life or Lives to be added it 
 will not,, there this previous Operation will be 
 neceflary, viz. we mufl calculate to what Term 
 the Life or Lives in Being, in Conjun6tion with 
 the Life or Lives to be added, when all put to- 
 gether, are equal ; or we mufl compute the 
 Value of them, when all put together j and the 
 Term to which they are equal, or the Value of 
 them, when put together, is the Term or the 
 Value out of which the Subflradion is to be 
 made. Out of this Term or Value a Subflrac- 
 tion muft be made of the Term or Value of the 
 Life or Lives in Being, and the Remainder will 
 be the Term and Value refpedively of the Life 
 or Lives to be added, as in the former Cafe : 
 
 And
 
 [ Si ] 
 
 And in the fame Manner we are to proceed, 
 whenever the new Term is to vary from the old 
 one, by fixing firft the Number of Years of fuch 
 new intended Term. 
 
 To this Method I fee no Objedion, except 
 any fhould arife from that Part of it which di- 
 reds that thefe Eftates lliould be firft reduced to 
 abfolute Annuities j and therefore, tho'the Aro-u- 
 mcnts which I have already offered under another 
 Head, where I treated of the Rate of purchafing 
 iuch Eftates, have, in great Meafure, obviated 
 any fuch Objedion here ; yet, it being a Point 
 of fome Moment in the Cafe, I will take the 
 . Liberty to repeat, in Part at leaft, what was 
 there alledged to remove this Difficulty. 
 
 We will fuppofe then that an Eftate Is out of 
 Leafe, and to be now let to a Tenant at Will, 
 or from three Years to three Years, at a Rent 
 certain, the beft that could be got without take- 
 ing any Fine or Premium ; and let us fee what 
 Rent a Perfon would give, where he undertakes 
 to make Payment at a Day and Place certain, 
 where he difcharges the Land-Tax, and all 
 other Out-goings whatever incident to the 
 Eftate, out of his own Pocket, and engages to 
 leave the Eftate In fuch Repair and Condition 
 as a future Tenant fliall approve. To imagine 
 that any Man, for an Eftate of an lOo/. per Jri' 
 finm at the improved Rent, would give an lOO /. 
 per Annum neat Rent, is ridiculous, and to prac- 
 tife it, or attempt it, would be fomething worfe. 
 In my Way of Thinking, and according to my 
 little Experience, I ftiould very much queftion 
 F whether
 
 [ 82] 
 
 whether any one, I mean any one of Serife 
 enough to know what he engages to do, and ot 
 Subftance and Honefty enough to make good 
 his Engagements, would give in clear Rent any 
 more than two Thirds, or thereabouts, of the 
 improved Rent, on the Terms I mentioned. 
 
 I don't fuppofe here, that the Taxes, Repairs, 
 and other incidental Charges would amount to 
 the other third Part, and yet in Truth they will 
 net fall very much fhort of it, becaufe they 
 have been already Hated at 30 /. per Cent, in com- 
 mon and ordinary Cafes where Lands are let at 
 "Will on a Rack-Rent : But then 'tis to be con- 
 fider'd, that it very rarely happens that any Per- 
 fons will undertake thefe Things on any Terms, 
 fpecially the Payment of the Land-Tax \ and many 
 are not to be trufted with it ; and thofe who will 
 undertake them, and can be confided in, will net 
 doit, on an uncertain and precarious Tenure, un- 
 lefs they have a conllderable Allowance, fuch an 
 one as, perhaps, upon a long Term, and a more 
 certain Intereft, they would not infift on. 
 
 If we carry this Matter farther, and fuppofe 
 this Eftate to be let out in future Times by a 
 Leafe for feven, fourteen, or twenty one Years, 
 vet thefe or the like mult be the Terms and the 
 Rent ; and then 'tis evident that the one Tenant 
 and the other has an Equivalent for taking thofe 
 Charges and Incumbrances on himfelf, as in 
 Tuftice he ought. If we proceed, and fuppofe 
 fuch Landlord to fell, and fuch Tenant to buy 
 off Part of this Rent, 'tis weak, or wicked, to 
 think that one may fell, or the other ought to 
 
 buy,
 
 [ 83 ] 
 
 buy, any Thing but Part of the Neat Rent ; for 
 the other Part is fomething that does not, and is 
 fuppofcd, not to accrue either to the Landlord 
 or Tenant. If then a Renewal is only a Pur- 
 chafe repeated, or only a new Purchafe call*d by 
 another Name, as in Truth 'tis no other, 'tis 
 impoffible, in the Right and Juftice of the Thing, 
 to diftinguifh between letting an Eliate at Will, 
 on a Rent certain, granting a Term in it with- 
 out Fine on the like Neat Rent, granting a 
 Term in it on a Fine proportionate to the 
 Abatement of fuch Rent, and renewing fuch 
 Term on a Fine anfwerable to the Number of 
 Years lapfed : And therefore, in all the Cafes, 
 Conlideration muft be had of thofe Out-goings, 
 in the laft Cafe, equally with any of the former. 
 The Juftice and Realbnablenefs of the Method 
 propofed by me for adjulling Fines-, and wherein, 
 and how far it differs from, or may be preferable 
 to any others ^ either that in commonUfe, or a new- 
 one recommended by Mr. Richards ; cannot better 
 be difcover'd than by computing the Fine in an 
 Inftance or two in each Method : And I will take 
 the Liberty to do it on two different Cafes; in one 
 where the Rent referved to the Landlord lliall be 
 fuppofed a great one, and in the other a fmall one, 
 in Proportion to the whole Value of the Eftate. 
 The common Method is to fubftra(5l the Lord's 
 Rent only out of the whole Rent, mine is to fub- 
 ftradt the Lord's Rent, and all other Out-goings 
 and Allowances, out of the whole Rent ; and in 
 both Cafes to take the Rent remaining as the Sum 
 on which to compute the Fine : And Mr. Rtch- 
 F 2 ardi^Q
 
 C 84] 
 
 {jrds's Method is ; he reckons the whole Rent for 
 the Years left in Being in the Term as an Annu- 
 ity at 6 /. per Cent, and the Out-goings for the 
 fame Number of Yea;rs as an Annuity at 4 /. 
 per Cent. \ he fubflradls the fecond Annuity out 
 of the firll, and takes the Remainder as the 
 Value of the Annuity for the Term left in Be- 
 ing ; and then, if the Value of the Annuity for 
 the Term ftill in Being is fubftraded out of the 
 Value of the Term when it was full, fuch 
 Value of the full Term being firft adjulled in 
 the fame Manner as the Value of the Term reft- 
 ing is adjufted ; the Sum remaining is the Fine 
 to be given for renewing or filling up the Term. 
 We will ftate Interefl at 6 /. per Cent, at which 
 Computation the Fine will be 2 i Years Value, 
 or near it ; and the Eftate we will fuppofc to be 
 200 /. per Jnimm at the improved Rent ; and 
 the Rent to the Lord in one Cafe to be 20 l.per 
 Jmnim^ and in the other lao/.; and let us com-' 
 nute what the Fine will be in thefc two different 
 Cafes, according to thcfe fevcral Ways of ad- 
 jufting it. On the firft Cafe ; in the common 
 Method, where the Lord's Rent only is de- 
 duced, the Rent remaining to the Tenant will 
 be 180/. and the Fine will be 450/. for renew- 
 ing feven Years lapfed in a Term of twenty one 
 Years : In my Method, where the Lord's Rent 
 of 2C /. and the Out-goings in Repairs, in the 
 Land-Tax, and for Contingencies, come to 60/. 
 more, the Rent remaining to the Tenant will 
 be X 20 /. and the Fine 300 /. for a like Renew- 
 al : And in Mr, Richards's Method, where the 
 Lord's Rent only, and the Out-goings for Re- 
 pairs
 
 [ 85 ] 
 
 pairs are dcdufled, and which will come only to 
 40/. the Rent remaining to the Tenant or his 
 Annuity will be 160 /. and the Fine will bs 
 354 /. ^ and his Computation is thus made ; 
 
 The Value of 200 /. per Ammm ^ L 
 for twenty one Years, at 6 /. > 2352* 
 per Cent, is j 
 
 The Value of 40 /. per Jmitm^ -n 
 being the Out-goings in Rent f 
 and Repairs, at 4 /. per Cent. C * 
 
 is -^ 
 
 If this Sum be dedu6led out of -n 
 
 the former, the Remainder / jyQ2»-, 1702 
 
 will be the Value of the Leafe T — — 
 
 for twenty one Years, and is J 
 
 The Value of aoo /. per Jmium \ 
 
 for fourteen Years, at 6 hper > i860. 
 
 Cent, is 3 
 
 The Value of 40 /. per Anniim^ ) 
 
 being the Out-goings for four- ^ 422. 
 
 teen Years, at 4 /. per Cent, is 3 
 If this Sum be dedu(5ted out of ->. 
 
 the former, the Remainder / 
 
 will be the Value of the ^' 1438 — 1438. 
 
 Leafe for fourteen Years, V. 
 
 and is ^ 
 
 If the Value of the Leafe for "^ 
 
 fourteen Years be fubflrac- 
 
 ted out of the Value of the ' ^___^ 
 
 Leafe for twenty one, the 
 
 Remainder will be the Fine, 
 
 ^nd is y
 
 [ 86 ] 
 
 On the other Cafe, in which the refer ved 
 Rent is fuppofed to be 120/. in the common 
 Method, where the Lord's Rent only is de- 
 dufled, the Rent remaining to the Tenant will 
 be 80/. and the Fine aco /. for fevcn Years; in 
 my Method, where the Rent and Out-goings 
 will come to 180/. all together, the neat Rent 
 to the Tenant will be 20 /. only, and the Fine 
 50/.; and in Mr. Rtchards's Method, who rec- 
 kons for Lord's Rent and Repairs 140/. only, 
 the neat Rent, or Annuity to the Tenant, will 
 be 60/. and the Fine only 8 /. for feven Years 
 additional Term. 
 
 I fhall leave thefe feveral Computations to the 
 Reader's Refletlions, without any of my pwn^ 
 except by remarking only, that if we vary this 
 laft Ciife io far us to Hate the Repairs on it at 
 Q.^1. inftead of 20/. as in Mr. Richards'sj and 
 in my Way of Thinking we may do, fince we 
 agree that the Repairs are to be allow'd lor, at 
 fuch a Rate as, in Fad and Reality, they come 
 out to be, as near as can be computed ; then the 
 Fine, in my Method, would be 40 /. and in his 
 nothing at all. The Rent and Out-goings, in 
 my Way of reckoning, would then be 184 A 
 the Rent left to the Tenant no more than 16/. 
 and the Fine for that 40/. ; And in Mr. Richards's 
 Way, the Rent and P^epairs only being Out- 
 goings, and which come to 144/. the Compu- 
 tation will iland thus : 
 The Value of 200 /. ^er Annum ^ I. 
 
 for twenty one YcarSj at 6 A > ^35-' 
 per Cent, is ^ 
 
 The
 
 [ 87] 
 
 The Value of 1 44 /. per Junam, -v /, 
 
 being the Out- goings in / ^^^^ 
 
 Rent and Repairs, at 4 /. per T 
 
 Cent, is 3 
 
 If this Sum be deduded out of ->. 
 
 the former, the Remainder /^ 
 
 will be the Value of the > 2>35 "^ ?t'^S^ 
 
 Leafe tor twenty one Years, V. 
 
 and is -^ 
 
 The Value of 200 /. per Annum ^ 
 
 for fourteen Years, at 6 /. ^1 86o» 
 
 per Cent, is 3 
 
 The Value of 144/. per Annum ^ ^ 
 
 being the Out-goings for 14 ^ liip* 
 
 Years, at 4/. per Cent, is J 
 If this Sum be deduced out of 
 
 the former, the Remainder 
 
 will be the Value of the Leafe 
 
 for fourteen Years, and is 
 
 341 — 34i» 
 
 This laft Remainder, which is the Value of 
 the Leafe for fourteen Years, being greater than 
 the firlt Remainder, which is the Value of the 
 Leafe for twenty one Years, there is rnanifeftly 
 nothing left for a Fine ; on the contrary, the 
 Value for fourteen Years being greater than the 
 Value for twenty one Years, by 6 /. the Land- 
 lord ought to pay the Tenant that 6/. to in-» 
 duce him to take an additional feven Years, 
 
 I have before obferved, that this Gentleraai^ 
 
 makes Ufe of the fame Method in fettling the 
 
 Price to be paid on taking or purchafing fuch an 
 
 pftatc, as he does in renewing it j and under 
 
 F 4 that
 
 [ 88 ] 
 
 that Head I found Fault with it, as producing 
 a very abfurd EfFe<5i:, viz. that a Leafe lor twenty 
 one Years would be of fome Value, but a Lcale 
 for fifty Years would be worth nothing. The 
 very fame Abfurdity, or rather one more grpfs, 
 recurs in renewing ; for if this Rule of com- 
 puting the Fine is a right one, the Tenant ought 
 not to renew, even though no Fine was de- 
 manded, in the Cafe laft ftated ; which is, in 
 Effed, to fay, that a Leafc of twenty one Years 
 was of Value, if made in 1720, but that a like 
 Leafc of the fame Eftate, if made in 1 727, was 
 "Worth nothing. This indeed may come out to 
 be the Cafe in fome Inftanccs ; that is, where 
 the Eftate conlifts moftly in Houfes, which .in 
 fcvcn Years Time may be much impair'd, and 
 become ruinous ; but then the Rule cannot be 
 a general one, and applied in all Cafes alike ; 
 and to be fure is not applicable to the Cafe jufl 
 mention'd, where the Repairs are not of much 
 Confequence. Nay, the- Rule will not do in any 
 Cafe .where the Out-goings amount fo high as 
 144 /. in 200 /. let thofc Out-goings be in the 
 Lord's Rent only, as in Cafe of Meadow or 
 Pall; u re Land, where no Repairs at all may be 
 neccflary • for if this Rule be obferved, and no 
 Fine is to be paid on renev.^ing the Leafc of fuch 
 an Eftate, the neceflary Confequence will be, 
 that a Leafe for twenty one Years of fuch an 
 Eftate would be a valuable one, but a Leafe for 
 fourteen Years and feven Years in Reverfion im- 
 mediately following, that is, a Leafe for twenty 
 one Years of one aiid the fame Eftate, would 
 
 he
 
 [ 89] 
 
 be of no Value, but worth lefs than nothing j 
 which is as much as to fay, that one and the 
 fame Thing, for one and the fame Time, is of 
 fome Value, and of no Value, 
 
 I aifert farther, and if any one will give them- 
 felves the Trouble to maice the Calculation as I 
 have done, they will find it to be true, that a 
 Fine of one Year's Value, as now frequently 
 taken, does then become a juft Fine for a feven 
 Years Renewal, according to this Rule, when 
 the Out-goings in Rent or othcrwife are about 
 if Parts, or about a Moiety of the whole Value. 
 For which Reafon I am a good deal furprized 
 that the Framer of this Rule Ihould on one Side 
 lay aLoad on the Clergy and Fellows of Colleges, 
 for accepting one Year's Rent, as a Fine for 
 adding feven Years to a Leafe of fourteen ia 
 Being, and on the other Side Ihould charge their 
 Tenants with Extortion and Opprcffion, in 
 forcing their Landlords to fubmit to fjch Terms; 
 fincc undoubtedly there are fome, very probably 
 there are many of thefe public Eftates, where 
 the Lord's Rent and the Out-goings, efpecially 
 if the Land-Tax is brought to that Account, 
 as molt certainly it is, or ought to be, mult 
 amount to more than a Moiety of the full Value • 
 in which Cafe, fuch a Fine cannot be taxed of 
 Partiality of the one Side or the other. 
 
 For the fame Reafon I do as much wonder, 
 tliat the fame Perfon fhould affert in general 
 Terms, that feven Years in Reverfion after four- 
 teen is worth and fold for 2 i Years Value, if 
 belonging to the Laiety ^ for wherever Ellates 
 
 arc
 
 [ 90] 
 
 »re under the Circumftances which have been 
 juft mentioned, let the Owners be who they will 
 or what they will, if they take more than one 
 Years Value for adding feven, they who take, 
 and not they who pay fuch a Fine, will be 
 chargeable with Extortion and OpprcfTion. 
 
 When the Rent and Out-goings are lefs, that 
 is, more properly fpeaking, where the Neat 
 Rent, or Annuity coming to the Tenant, is 
 greater, the Fine undoubtedly fhould be larger ^ 
 for moil certainly it mufl bear a Proportion to 
 that which he is to receive, whatever that be, 
 and not to fomething which he never can re-* 
 ceive. And yet, notwithftanding this Writer 
 rnagiflerially pronounces that a revcrfionary 
 Term of feven Years is worth 2 ~ Years Value, 
 I very much queflion whether he can produce a 
 fingle Inflance of a Leafe-hold Eftate in the 
 Kingdom, where fuch feven Years are worth 
 two Years Value, computing by his own Rule : 
 For if the Our?goings in the Rent to the Lord, 
 and in Repairs, are 15 /. per Cent, the Fine will 
 be Ihort of two Years Value. To make the 
 Out-goings come to 15/. per Cent. I have in- 
 ferted the Repairs, becaufc he agrees they are 
 to be allowed, but have" omitted the Land-Tax; 
 ^ld if that i3 likcwife to be inferted, as I muft 
 infill it ought to be, it will be impoifible to find 
 <put an Eftate where the Circumltances are fuch, 
 that the Out-goings, under all thcfe Heads, 
 ihall be fo little as 15/, per Cent. 
 
 That the Repairs belong to the Landlord he 
 admits, $nd in his Computations always re<;kon9
 
 [ 91 ] 
 
 on them as fuch ; and furely he mufl be flupid 
 himfelf, or mufl think others to be To who main- 
 tains the contrary, if thofe who are Owners of 
 the Eftate are to bear this Burden ; and if we 
 may not efleem and call them Owners of the 
 Eftate who take the whole Profits and Produce 
 of the Eflate, partly in the annual Rents, and 
 partly by a fore-hand Rent ; I don't know in 
 what Light we mufl look on, or under what De- 
 nomination we mufl put, them who take the 
 Whole in annual Rents, 
 
 For this Reafon, which I have mentioned 
 under another Head, and fhall not need to re- 
 peat any farther, I aflert that the Land-Tax is 
 to be put to the Landlord's Account, as well in 
 Hating the Price to be paid on renewing, as on 
 purchafing thefe Eftates. Nay, this Writer 
 himfelf, though he charges this Tax to the Te- 
 nant in all Cafes, and without Diflindlion, yet 
 in fome Cafes he docs, in Effedl, and in the 
 Event, make the Tenant an Allowance for them; 
 not indeed by the Name of the Land-Tax, but 
 under the Notion of Contingencies and Cafual- 
 ties : And the Allowance he makes for this Pur- 
 pofe is a very ample one, for it exceeds not 
 only what I crave Ibr fuch Purpofe, but it even 
 exceeds any Rate at which that Tax ever yet 
 was, or, I hope, ever will be affefs'd ; it being, 
 in fome Inftances, above 2% 1. per Cent, where a 
 Renewal, and the Fine to be paid upon it, is 
 under Confideration. This appears plainly in 
 the particular Inftance laft mentioned, where I 
 
 computed
 
 iC 92 ] 
 
 computed the Fine, or rather no Fine to be 
 paid, on fuppofing the Lord's Rent and Re- 
 pairs only to be 144/. fer Jlnnum : Where we 
 Ice that the reft of the Rent, w^. j6 /. pcT Au- 
 mim^ is funk and loft to the Landlord under the 
 Article of Contingencies or Cafualties ; but 
 what thofe fliould be, which happen alike in all 
 Eftates which are of a like Value, unlefs the 
 Land-Tax be accounted as one and put under 
 that Cover, is paft rriy Comprehenlion. 
 
 If we make Ufe of the other Inftancc, on 
 which I calculated the Fine to be paid, where 
 the Lord's Rent and Repairs arc comparatively 
 fmall, there indeed the Allowance to the Te- 
 nant for the Land-Tax is very fmall, or rather 
 none at all, on a fuppofed Renewal : But where's 
 the Reafon or the Juftice oi this great Diftindi- 
 on? The neat Rent or Annuity to the Tenant 
 in one Cafe is ftatcd at ^d l. per Anntim^ and in 
 the other at 160 7. ; bqc I fqppofe the UifFerence 
 in the Tenant's Annuity folely^will hardly pafs 
 for a juft Caufc of exempting one and charging 
 the other with this Tax. *Tis true indeed, that 
 the Engagements and Incumbrances lying on 
 one Tenant, if compared with thofe lying on 
 the other, are very fmall \ but then 'tis true 
 likewife, that he who has the lefs or the fewer 
 Incumbrances on his Eftate has paid a Fine to 
 his Landlord in Proportion to his Exemption, 
 and for fuch Excmptiono 
 
 In ihort, let us take it in which Light wc 
 pleafc, whether this Article of Contingencies, 
 was or was not intended to make a Provifion fo? 
 
 the
 
 [93 ] 
 
 the Land- Tax, the Method of computing will 
 Itill come out wrong. If it does include the 
 Land-Tax, then there is no Provilion at all made 
 for it in Behalf of a Tenant who has a large 
 Annuity, though the Provilion for him who has 
 a fmali Annuity is io very liberal an one • and 
 yet the one has all the Eftate he has paid for, 
 and the other has paid for all he has. If the 
 Land-Tax was not intended, nor is included in 
 this Provifion, it will be very difficult to fay 
 what thofe Contingencies or Cafualties are, which 
 in fuch an Ellate can amount to fo great an an- 
 nual Sum ; and much more difficult to give a to- 
 lerable Reafon for making fo great a Diftindion 
 on one and the lame Eftate, where *tis fuppofcd 
 to differ in one Circumftance only, that of a 
 greater or a lefs Rent to the fuperior Lord. 
 
 As to that Matter, what Fine the Laiety or 
 private Perfons do in Fad take on fuch Renew- 
 als, I fliall not pretend to fay, not knowing all 
 the Perfons in the Kingdom, much lefs their 
 Eflates ; and therefore it would be idle in mc to 
 contradid thofe who affirm that they take 2 -, 
 Years Value, for fomc People may know what 
 others do not know ; but if thofe Perfons mean 
 to fay that this is univcrfally the Practice among 
 the Laiety, as they Itrongly infmuate, if they 
 don't plainly fpeak out ; in this I will take up- 
 on me to contradid them, becauie fome Gentle- 
 men, to my Knowledge, never take more than 
 one Year's Value. I will venture to affirm far- 
 ther, that thofe who regulate their Fines by de- 
 duding the Lord's Rent only out of the tota\ 
 
 Rent,
 
 [ 94] 
 
 Rent, and take i \ Years Value after fuch De- 
 dudion, will neceflarily be guilty, even in this 
 Writer's Eftimation, of extorting more than is 
 ju% due to them. Nay, I may add, and I 
 think I have already proved, that if they take 
 two Years Value in the Manner he computes, 
 they will go to the very Extremity which 
 even his Rule, in any Inftance whatever, will 
 well admit. 
 
 According to my Method of adjufting a Fine 
 for feven Years, the Fine, if computed and re-' 
 gulated by dedu6ling the referved Rent only, 
 will come out to be one Year s Value, where the 
 referved Rent is a Moiety of the improved Value, 
 or near it ; where the referved Rent is three 
 Parts in four of the Whole, or thereabouts, no 
 Fine will be due ; and where the referved Rent 
 is a tenth Part only of the Whole, there the 
 Fine will be one Year's Value and an half, and 
 fcmething over ; in common and ordinary Cafes. 
 By common Cafes I mean fuch where the Land- 
 Tax and Repairs may be ftated, each of them at 
 lo/. per Cent,: But as thefe, in particular 
 Eftates, will be more or lefs ealy, there is Room 
 left to vary the Fine ; and where thefe, and all 
 other Out-goings, are ftated at the loweft Rate 
 that can well be fuppofed to happen in any 
 Eftate v/hatever, a Fine of two Years Value is 
 the higheft that can be given, even by the moft 
 willing Purchafcr. 
 
 Here I make no Difiindion between Lay and 
 Ecclcfiaftical Eflates; and whether Churchmen 
 and Colleges will think it proper and prudent 
 
 to
 
 [95] 
 
 to be more eafy in their Fines than Lay-Leflbrs 
 generally are or fliould be, and how tar they 
 ought to carry this Point, that there I fhall leave 
 to their Confideration : For I do not intend 
 here to obtrude my Advice on thofe who arc 
 wife enough to think, and able enough to judge, 
 for themfelves ; nor fhall I make any Suggeftions, 
 which they will readily enter into and make a 
 proper Ufe of, without any Interpolition of 
 mine. 
 
 It will be alledged, perhaps, by thofe who 
 take two Years or two Years and an half Value 
 for a Fine, if any fuch there be, and by thole 
 who are defirous of getting fuch a Fine, and 
 poffibly there may be many of them ; that a 
 Term for twenty one Years, and confequently a 
 Term for feven Years being a Part of it, increafes 
 in Value and Propordon to the Decreafe of 
 Intereft of Money ; that, when feven Years, Part 
 of a Leafe for twenty one Years, are elapfcd, 
 thofe feven Years in Reverfion are become the 
 Property of the Landlord, and therefore that 
 his Intereft in fuch feven Years mufl: be propor- 
 . tionably increafed in Value; and if he fells them 
 • to a Stranger, or adds them to the Term in Be- 
 ing by renewing to the old Tenant, that the 
 Price on Sale or Renewal ought to be advanced 
 in Proportion. 
 
 This Allegation I have already taken fome 
 Notice of; and having firft premifcd, that as 
 Intereft has decreafcd on one Side, fo as to give 
 a greater Value to fuch reverfionary Years, fo 
 on the other Side there may pofTibly be fome 
 
 new 
 
 xj
 
 [96] 
 
 new Charges on Leafe-hold filiates, which may 
 have prevented fuch Increafe in Value, in Part,' 
 or in the whole ; and whether that be the Cafe, 
 or how far it is the Cafe, I referve to be dif- 
 cufs'd hereafter : I fay, under fuch pofTible Li-r 
 mitations I admit the Allegation to be true, and 
 Hiall coniidcr it as fuch. 
 
 I cannot fay, with any Certainty, when Mr. 
 Ceroid lived, and framed his Tables for Fines 
 on Renewals ; but I conjecture he might make 
 them about two Hundred Years ago, or towards 
 the latter End of the Reign oi Henry the Vllltb; 
 and pofTibly for the Ufe of the Gentlemen into 
 whofe Hands the Church-Lands were difpcrfed, 
 on the Difiblution of the Monafteries. Be that 
 as it will, it appears pretty plainly from the 
 Conllrudion of thefe Tables, that the Intereft 
 of Money was then about lo/. p^r Cent. ; and 
 at that Rate, a Term for twenty one Years was 
 worth eight Years and half, 51 Days Value, and 
 the Fine for renewing fcven Years lapfed in fuch 
 aLeafe he Hates at one Year's Value. Amongft 
 the many Writers on this Subjed, fome of whom 
 are very fujl of Complaint, or rather Clamour ; 
 for in fome of them 'tis meer Clamour without 
 Knowledge, of the prefent fuppofed Minutenefs 
 of Fines taken by Churchmen and Colleges ; I 
 meet with none who finds Fault with this Gentle- 
 man for fixing them at the Rate he has done for 
 his own Time, and for that Rate of Intereft. 
 At the prefent Time, the Intereft of Money is 
 at 5 /. pr Cevt. and at that Rate, a Term for 
 twenty one Years is in Value twelve Years, three 
 
 Quarters
 
 [97] 
 
 Quarters^ 25 Days; and lince the Value of fuch 
 a Term at the former Rate of Intereft was 8-2-51, 
 by this Decreafe of Intereft the Increafe in Value 
 on the whole Term is one half, that being the 
 ■ Proportion as near as can well be fix*d : There- 
 fore the Increafe of the Value on the feven Years 
 lapfed mull likewifc be one half^ that is, half a 
 Year ; confequently, the Fine to be paid for fuch 
 revcrlionary feven Years, if Regard is to be had 
 to thefe Tables of Mr. jTlcmd^ and a Fine at 
 prefent is to bear a Proportion to a Fine as for- 
 merly, will be one Year and an halPs Value, 
 the Landlord's Interell here being manifeflly no 
 other or more than the feven Years lapfed. 
 
 To this if we add another Conlideration, 
 which I refer vcd to myfelf a Liberty to add, 
 w;3. that there may be fome Charges now im- 
 pofcd on thefe Eflatcs to which in former Times 
 •they were almoll Strangers, and if thofe Charges 
 are to come entirely out of the Landlord's In- 
 tereft, which they muft do, unlefs a Difference 
 can be fliown between a Payment of one and the 
 fame Value where made by annual Inftalments 
 and where made by feptennial ones ; and if this 
 cannot be efteded but by an Abatement in the 
 Fine, here will be a farther Reducement of fuch 
 Fine. The new Impolition I mean will ealily 
 be underftood to be the Tax on Land, which in 
 j^croid's'Timz was never laid, and if any other 
 were laid of the fame Nature, it was only occa- 
 lionally and rarely done, whereas it is now annu- 
 ally done ; and whoever knows any Thing of 
 the prefent Crow^n Revenue, and the Monies 
 G ufually
 
 [ 98 J 
 
 ufuallv faifcd for the current Services pf each 
 Year, muft acknowledge that it will be a conftant 
 and certain Tax for twenty one Years at leaft, 
 if not for ever. 
 
 If this be the Cafe on Mr. j^croid's Tables 
 for Pvenewals, I fuppofe we fhall hear no more 
 Arguments drawn from thence, or any farther 
 Appeals to them \ but then I muft infift that it 
 •will not be right neither to apply to this Pur- 
 pofe feme modern Tables, which are plainly and 
 avowedly calculated on a Suppolition that a Leafe 
 of Houfcs or Lands, deducting only the referved 
 Rent, are the fame Thing as abfolute Annuities; 
 tvhich no Body will admit but the moft ftrcnu- 
 ous Advocates for advancing Fines, and even 
 thefe are not all of them willing to go the whole 
 Length. And yet thefe Gentlemen, as if the 
 Juftice of thele Tables when applied to thefe 
 JPurpofcs, was as clear as when applied to abfo- 
 lute Annuities, run Riot againft Tenants, charge 
 them with Tyranny, Extortion, Oppreffion, 
 and other foul Practices ; and give their Land- 
 lords fome Language not much more civil, the 
 one for paying, and the other for accepting, 
 Fines at one Year's Value only. 
 
 But pray, who has given them Authority to 
 call me Fool, or my Tenant a Knave, becaufc 
 V7e can and do agree about a Fine, or I fell and 
 he buys fbmething, without confulting them, or 
 abiding by their Tables ? Am not I Mailer of 
 my own, and may not I do with it what I think 
 proper, without asking their Leave ? If I fet 
 an higher Value on my Eafe, or on my Conve- 
 nience,
 
 [99] 
 
 nience, or a lower Value on my Eftare, thati^ 
 fome others may do, fureiy I may exchange the 
 one for the other at what Price Ipleafe without 
 incurring their Cenfure ; where, to be fure, I 
 opprefs no Man. And I would not have them 
 to be fo certain and Co pofitive as fome of them 
 feem to be, that none of the Laiety do now 
 grant or renew Leafes on thefe eafy Terms, be- 
 caufe this is to aflert a Negative, and it happens 
 that I can and do affirm it to be falfe on myowa 
 Knowledge. 
 
 But to let this pafs ; for a Comparifon of what 
 is done by the Laiety with what is done by the 
 Clergy, is very little to the Purpofe, unlefs the 
 Pradice of one would juftiiy or excufe the Prac- 
 tice of another, without Regard had to the Equi- 
 ty or Iniquity of the Adion : I will fuppofe 
 that fuch Pradice of taking low Fines prevails 
 only or moflly amongft Eccleliaflical and Colle-^ 
 giate Perfons, and that 'tis really a Grievance ; 
 but how is it to be removed, if Tenants are ob- 
 llinate, and will not advance their Fines ? Why, 
 their Landlords muft fuffcr their Leafes to ex- 
 pire, and take their Eftates into their own 
 Hands ; and this is urged upon them not only 
 as a Matter of Prudence and Advantage, but as 
 a Point of Duty neceffarily incumbent on them, 
 in whatever Circumftances they may be. 
 
 Whether fuch a Proje<5i: as this, if it could be 
 and were carried into Execution, would produce 
 any fuch great Increafe in the annual Income of 
 thefe Bodies of Men as may he commonly ex- 
 pected, is a Thing which fome People, perhaps, 
 G 2 will
 
 [ 
 
 100 
 
 tvill make a Doubt of. Thofe Pcrfons will then 
 be fully convinced, if it will be any Satisfadiob 
 to them to be convinced, that Landlords mull 
 ; bear the Burden of the Land-Tax ; and they 
 ' will find that they mull fubmit to many other 
 . Inconveniencies likewife, of which, in their pre- 
 fent Situation^ they are not fcnfible, as being 
 wholly exempt from them. I have before made 
 an Eilimatc, if thefe Ellates were in Hand and 
 let out to Tenants at Will on a Rack-Rent 
 without Fine, that they would hardly yield fo 
 the Owners in neat Money more than two Thirds 
 of .the improved Value ; and I cannot imagine 
 but that, in the prefent Way of managing them, 
 they mull produce more than a Moiety of fuch 
 improved Value, the annual Rents and Fine be- 
 ing put together ; and where they do in Reality 
 produce lefs, I think the one may reafonably ej^- 
 pe6t a larger Fine, and the other may well com- 
 ply with it, and I will not fuppofe any fo foo- 
 lilhly obftinate as not to comply with it; and 
 -therefore I do not fee the Neceffity or the great 
 Ufc bf. this Device, the Event of which no Man 
 - can -forcfee. 
 
 I know very well that this Courfe has been 
 •frequently purfued of late Years amonglt Ibme 
 of the Laiety, but with what Succefs it has been 
 attended, or what Service it has done the Own- 
 ers, I do not pretend to know: But if a Man 
 were to make a Judgment of it from the Multi- 
 tude of Farms which are now in the Owners 
 Hands for want of Tenants, where to be fure 
 they do not fee half Rent, he would not be very 
 
 hally
 
 [ lOI ] 
 
 hafly' to run into this Projedl ; but, perhaps, 
 would be apt to fufpec^l rather, that at the Bot- 
 tom this Advice had no more in it, than that 
 one Fox had loft his Tail amongft the Briars or 
 in a Gin, and would willingly draw in his Brc;- 
 thren to cut off theirs. 
 
 But whatever may have been the Succefs of 
 thefe. Undertakings by private Perfons, I car^ 
 in no Sort agree that this is a good Reafon foi 
 Perforis in a publick Capacity to engage in fuch 
 an Affair. It may be fuppofed, and in this Cafe 
 it muft be fuppofed, where a private Pcrfon can 
 and docs forbear taking a Fine^ that his ftated 
 annual Income will fupply his common and or-? 
 dinary Occafions ;. and 'tis certain that the Be*? 
 neflt of fuch Forbearance in Time will come 
 to him or his ; polfibly to himfelf, in all Events 
 to fomq of his F'amily. With Regard to Per-?^ 
 foos in a public Capacity, the Cafe is quite otherr* 
 wife ; for' if thefe forego their Fines in their 
 Courfc, fome of them will be almoft deftitute of 
 a Subfiftence, others will be greatly ll:raiten'4 
 by it; and even thofe whofe Circurnflances are 
 fuch a5 leave Room to forego thefe accidental 
 Advantages can have no iV^'^fance, perhaps no 
 Probability, that they fhall have any Benefit them-* 
 felves of fuch Forbearance : And therefore, ta> 
 deny themfelves their Fines, and the Opportu-? 
 nities of providing thereby for their own or their 
 Families Subfiflence, would be an A(5t of down- 
 right Folly in fome of thcrn, and in none of then; 
 •would it be an Act of Prudence, if the pruden-? 
 ^ial Part only be regarded.
 
 [ 102 ] 
 
 The Advifers of this Scheme forget furcly 
 that, when thefe Leafes expire, the then Propri- 
 etor of the Ellate, if he be a fingle Perfon, or 
 the Community, if it be a Corporation aggregate, 
 as the Law ftands now will have Power, and 
 therefore may very legally and rightly grant new 
 Leafes at the ancient Rent, and take a Fine pro- 
 portionate to the reft ; fo that this admirable 
 Project will produce no more than this, that it 
 will flop the Hands of the prefent Proprietor, 
 or of the Individuals who now com pofc a Com- 
 munity, from taking a Fine, to give an Oppor- 
 tunity to a SuccciTor, or to the Individuals who 
 may hereafter be the Community, to take a 
 larger Fine. 
 
 A Law indeed may be made, as one of them 
 feems to hint, which fhall oblige thefe public 
 Perfons to permit their old Leafes to expire, 
 and which fhall reftrain them from granting any 
 new ones afterwards, on any other Terms but 
 at the moft improved Rent which can be got 
 without taking any Fine; and I muft acknow- 
 ledge this would deilroy the whole Iniquity of 
 Fine-taking, let it be of which Side we will : 
 But I very much doubt the Parliament will not 
 come into fuch an Act, unlefs the Houfes fhoul4 
 be very thin indeed, becaufe this is to deprive 
 one Man of his Property, in fome Cafes polTibly 
 of his whole Subfiftence, meerly to enlarge the 
 Property of another ; and this is to be done 
 without making any Compenfation to the Party 
 who is (b deprived, and in a Cafe where the 
 Intereft of the Perfons lofing and of the 
 
 Perfons
 
 [ 103 i 
 
 '■ 'i r ■ -...,■■■ 
 
 Pcrfons gajnlng is of equal Conc€r,a to r. the 
 
 Public. 
 
 If the Cafe be fuch, that the Revenue arifing 
 from thefe Eftates, unlefs you alter the Manner 
 in which they arc now managed, will not main- 
 tain the Succeflbrs anfwerable to their Fundions, 
 or their Dignities ; that there may be a good Rea- 
 fon to make fome farther Provilion for them ; 
 but then that Reafon is equally good to make a 
 farther Provifion for the now PoffeiTors, fince 
 the prefent Generation is as much concern'd that 
 the now Proprietors of thefe Offices or Prefer* 
 ments Ihould have an adequate Maintenance, as 
 the fucceeding Generation can be that the future 
 Proprietors ihould have fuch : And 'tis a very 
 forry Reafon to diminifli the Income of the im- 
 mediate Owners, which the Argument fuppofes 
 to be full low enough already, only becaufe 'tis 
 proper to augment the Income of their Succef^ 
 fors. 
 
 As for what is further urged, that 'tis the 
 Duty of the prefent Owners of fuch Eftates, 
 whatever their Gircumftances may be to forego 
 the Advantage of their Fines, for the Benefit, 
 poflibly of themfelves, and more certainly of 
 their Succeflbrs; I fee fome unfair and not very 
 decent Infinuations in the Allegation, but for 
 Reafon or Senfe in it I fee none. The Laws of 
 the Land have provided that thefe public Perfons 
 may let Leafes for twenty one Years, or three 
 Lives, in Pofleffion, and farther than that thofe 
 Laws have prohibited them to go ; a private 
 PerfoHj by Settlement, has refer ved to himfclf a 
 G 4 Power
 
 [ 104 ] 
 
 Power to let Leafes under the fame Limitatioms, 
 and farther he cannot go : But will any Man 
 fay, that the one or the other is guilty of In- 
 juftice if he does go fo far ? Surely the Laws of 
 the Land in one Cafe, and a Law of his own 
 making in the other Cafe, is the Meafure of 
 Juftice, or there's none. Or fhall we fay that a 
 Man in private Life and Concerns may jullly go 
 as far as he has referved to himfelf a Po^yer to 
 go, and that one in public Station is guilty of 
 Breach of Duty if he goes as far as the Law of 
 the Land gives him Liberty to go ? This is to 
 make a Diftindtion where there is really no Dif* 
 ference, or if there is any, the Grounds of Com- 
 plaint from the other Party are much better 
 founded. 
 
 Every one who keeps his Leafes always full 
 by renewing, whether he be in public or private 
 Life, though he does no more than what he le- 
 gally may do, and in taking a Fine takes no 
 more than what is rightfully his own, moft cer- 
 tainly floes fo far depreciate the Eft^te belonging 
 'to him ^ but the one does a Detriment to his 
 SuccefTorj, that is to fome Body, he knows not 
 whom, one who is a perfed: Stranger to him • 
 and the other takes to himfelf fomething, which 
 would otherwife come, poffibly to an eldeft Son 
 and Heir • but in all Events to fome Body of his 
 Family to whom he Hands nearly related. Not* 
 ' withflanding which, as much Clamour as is raifed 
 gigainfl the one Clafs of Men for renewing their 
 "Leafes and taking Fines, we hear of none againfl 
 the other for doing the fame Thing.
 
 [ i°S ] 
 
 Ay, but Perfons in a publick Station of Life, 
 Clergymen fpecially, arc to take particular Care 
 of their SuccefTors, and to manage their Eiiates 
 for their Benefit in a peculiar Manner. If ,by 
 this no more is meant, than that they ought to 
 make Ufe of their Eftates in fuch a Way as' to 
 do no Injury to their SuccefTors, this moH cer- 
 tainly is true ; but then 'tis equally true with 
 Refped to all Mankind as well as them, and in 
 all other Inftances as well as this ; for every Man 
 is obliged fo to ufe his own as not to injure the 
 Property of another. If more be meant here, 
 and that they are in Duty obliged to let their 
 Leafes expire, and leave them in that State for 
 the Benefit of their SuccefTors, this Notion I 
 can by no Means come into. 
 
 What they ought, or mull in Juflice do, we 
 have already feen j for the Laws are the Meafurc 
 pf that, or 'tis pafl my Underltanding to find 
 out what is the Meafure of it. What they fhould 
 do as an A61 of Charity or Munificence, muft be 
 determined by the Circumflances and Abilities of 
 the Proprietors ; and even where thefe are fup- 
 pofed to be fufHcIent for fuch Purpofes, yet fure- 
 ly we mull leave it to their Judgment and Dif^ 
 f retion : Or we fhall be forced to hy in one 
 Cafe, that one Man mud debar himfelf of the 
 Conveniencies, poffibly of the very Neccflaries 
 of Life, in order that another may enjoy them 
 in a greater Plenty than himfelf : And in the 
 other Cafe, if thofe whofe Condition is fuch that 
 they are in a Capacity to do this are under Obli- 
 gation to do it^ whether their own Judgment 
 
 concurs
 
 [ io6 ] 
 
 ^ioncurs in it or not, that very Obligation makes 
 the A6t ceafe to be, what it is intended for, an 
 Adt of Charity or Munificence. 
 
 Where thefe Bodies of Men, or any private 
 ones amongil them, are in fuch Circumftances 
 that they can well fpare taking a Fine, as fome 
 of them to be fure occafionally may do ; if they 
 are difpofed to Ads of Charity or Munificence 
 to thofe of their own Order, I cannot but think, 
 if they will look into their own Eftates, and ob- 
 ferve what they frequently confifl of, that the 
 Vicars and Curates of the Parifhes where their 
 Eftates lie, and out of which their Eftates arc 
 derived, will appear more proper Qbjeds of their 
 Benevolence, than their Succeflbrs can do. It 
 fo happens, not perhaps from the original Do- 
 nation, but by fomc fubfequent artful Exchanges, 
 that many of the Eftates belonging to thefe Per- 
 fons are Redories and impropriate Tythes, and 
 where they are fo, the Provifion left for the Mi- 
 nister, if any be left, is fome wretched Pittance ; 
 ^nd where this is the Cafe, and 'tis much tod 
 common a Cafe, furely the Pradice of thofe 
 Clergymen and Colleges is greatly to be com- 
 mended, who, on renewing their Leafes, fink 
 the Fine in the Whole, or in Part, and inftead 
 of it, put their Tenants under an Obligation to 
 imake a proportionable Augmentation of the 
 Minifter's Stipend : And as this may be done 
 %ifh Eafe and Convenience, and to the Satis-* 
 fadion of all Parties interelted, fure enough it 
 will promote the Intereft of the Church of 
 "EngUfid more effedually than any Scheme for 
 
 letting
 
 [ 107 ] 
 
 letting Church Lcafes run out,- and for Clergy- 
 rnens taking their Eftates into' their own Handsi 
 And fince the Bounty of Qnccn J/im gives thefc 
 Gentlemen, and iadeed all others, an Opportu- 
 nity to double their Benefaction ; if fuch a Me- 
 thod for augmenting poor Livings continues to 
 be encouraged by the Clergy of fuperior For- 
 tunes, and if the Laiety are ready to concur in 
 profecuting fo good a Delign, the inferior 
 Clergy may hope, in fome reafonable Time, td 
 have a competent Maintenance, and the People 
 an able and difcreet Miniiler as a Guide for 
 their Souls. 
 
 Before I conclude this Head, it will be ex- 
 peded, perhaps, I fliould, and therefore I fliall, 
 take Notice of another Thing, which is fre- 
 quently mentioned, and infifted on as a Matter 
 of great Moment, to determine this Gontroverfy 
 about Fines ; I mean the low Value of Money, 
 and the high Price of the Neceflaries of Life, 
 comparing the prefent with former Times* If 
 by former Times we underltand here only forty 
 or fifty Years ago, or the Times of King Charles 
 and King James the Second, which one of thefc 
 Writers has his Eye upon, the Difference of the 
 Price of Money, or of the NecefTaries of Life, 
 will make nothing for advancing Fines. The 
 Decreafe in the Price or Value of Money has 
 been one per Cent, fince thofe Times, but fuch 
 Decreafe is over-balanced by the Incrcafe of the 
 Charges which lince then have been laid on 
 Lands; the Article of the Land-Tax alone, 
 not to mention others^ being equal at leaft to the 
 
 Gain
 
 [ io8 ] 
 
 Gain on the other Side ; fo that fuch Decreafe 
 operates nothing towards this Purpofe. n 
 
 As to the NecefTaries of Life, it: will be ad* 
 rnitted that many of them arc advanced in their 
 Price within this Period ; but every Body knows, 
 who is not wilfully blind, that fuch additional 
 Price is to be attributed to the modern Duties 
 of Exdfe and Cuitoms; but fuch of thofs 
 yhings which are the dired and immediate Pro- 
 duce of Land, and which are exempt from thofe 
 Duties, *tis not true that their Price is generally 
 inhanced j and if 'tvvere true., the neceifary Con- 
 fequcnce of fuch Advance fnould bt^ and would 
 be, that the Lands which pro^^lice them would 
 yield a better annual Rent, and ye£. this certain- 
 ly is not the Cafe, unlefs the Lands have received 
 ^rae Improvement? ; and if that were the Cafe^ 
 ^d where it is the Cafe the Lelfors have the Be-? 
 ncfit of fuch Addition in the Rent by ^n Additi^ 
 on to the Fine in Proportion, even on a Suppoliti-f 
 on that they take no more than one Year's Value, 
 But for a Landlord to raife his Fine, which is 
 but of the Nature, of a fore-hand Rent, mecrly*/ 
 becaufe many Things in common Life are grown 
 dearer by new Excifes or new Cuftoms, by 
 which Means the Expences of him or his Fami- 
 ly are enlarged, has no more Senfe in it thanr 
 there would be if he fhould pretend to raife the 
 annual Rent of his Eftate becaufe there is an an- 
 nual Tax on it, and his Income thereby lefFen'd r 
 For the Diminution of the Income in one Cafe 
 is juft as good a Reafon for raifing one Kind of 
 Rentj as the Enlargement of the Expence in 
 
 the
 
 [ I09 ] 
 
 the other Gafe is for raifing the other Kind* But 
 .*J doubt few Landlords will find their Tenants 
 Tuch Afles, notwithflanding the low Opinion 
 which fome of them entertain of their Under- 
 Handing, as to fubmit to this Burden, unlefe 
 they can put them into a Way of throwing it 
 off from their own Shoulders, and laying it on 
 the Confuraer of the Produce of the Land, as 
 common Tradermen in excifeable Goods do, by 
 railing the Price of the Commodities they vend : 
 And yet, in the Event, even this would bring 
 it back on themfelves in great Meafure. The 
 Truth is, if Land Owners, in this and other In- 
 fiances, can and do prevent the Load of a Tax 
 from falling diredly and immediately on them- 
 felves, yet in the lalt Refort there it will fall, 
 let them Ihift it feemingly as far off as they will 
 in the firfl Impofition ; and, perhaps, juft fo 
 much farther off from them as 'tis laid in the 
 firll Inflantj and according to common View and 
 Eftimation, juft fo much the more heavily it 
 comes upon them at the laft. 
 
 If by former Times we mean thofe of two 
 Hundred or three Hundred Years ago, this may 
 require a diltind Confideration. Every one, who 
 has at all look'd into Things of thisNature, knows 
 very well^ that in the Times ofHemy the Fourth, 
 or about three Hundred Years ago, a Pound of 
 Silver Money in Tale was an actual and efFedive 
 Pound of Silver, wanting a few Shillings, and 
 allowing a fmall Matter for Allay ; a Solid or 
 Shilling, or what was Co denominated in Ac- 
 counts, was a twentieth Part of a Pound, and a 
 :.. . Penny
 
 [ no ] 
 
 3^enny a twelfth Part of a Shilling. From hence 
 Ivithout Doubt, was introduced the Method of 
 Reckoning by Pounds, Shillings, and Pence ; mean- 
 ing thereby fuch Quantities of Silver : And we 
 •continue Hill to make our Accounts by Pounds, 
 ■Shillings, and Pence, tho* it is now.mecrly a Fic- 
 tion, and fubfifts no where but in Imagination ; 
 ■for a Pound in Money, as now call'd, does certain- 
 ly contain no more than a third Part of the ancient 
 •Poiwl, which was a Pound in Weight as well as 
 in Tale. 
 
 It is known too, equally well, that fifteen 
 "l^ounds in Silver, whether in Cpin or in Bullion, 
 they being both the fame or near it, v/ere equal 
 to one Pound Weight in Gold : And a Pound of 
 ^Silver in Weight being now cut into 3 /. 2 s, or 
 fixty two Shillings, fifteen fuch Pounds make 
 now 46 /. 10 5. in Silver Money in Tale ; and a 
 Pound of Gold is now cut into forty four Gui- 
 neas and an Half, which makes in Gold Money 
 in Tale 46/. 14 s. 6 d. each Guinea reckoned 
 at I /. I s. : So that Gold and Silver in our Days 
 bears the fame, or very near the fame. Proporti- 
 on to one another, as they did in former Days." 
 
 How the State of the Silver Coin flood about 
 two Hundred Years ago, or in the Times of 
 -'Hei2ry the Eighth, is not fo eafy to fix without 
 diftinguilhing the Times; however, it may be 
 fufficient here to fay in general, that in the Be- 
 ginning of his Reign a Pound of Silver made 
 "a/. 5 J. in Tale, and at the latter End 7 /. 41.; 
 fo great was the Debafement of the Coin then 
 by the Mixture of Allay : And yet once, iii 
 
 the
 
 the Time of his Son and Succellbr, Edward 
 the Sixth, this Coin was made much worfe, 
 a Pound of Silver fine making 14/. 8 s, in Talc. 
 
 I cannot forbear obferving here, that many 
 Authors, ^mongft others Bifhop Fleetwood in 
 his Chronifott Preciofum^ take Notice of this De- 
 bafement of the Coin, and complain of it as a 
 great Grievance to the Nation, as undoubtedly 
 it was; but aflert, that an Advancement of Mo- 
 ney in the Denomination, where the Standard of 
 which *tis made remains the fame, is no Preju- 
 dice to thePublick. Nay, fom$ carry it fo far 
 as to. contend it would be of Advantage ; where- 
 as an Advancement and a Debafement mofl cer- 
 tainly are in equal Mifchief^ where made in an 
 equal Degree. If the Silver Species now in Be- 
 ing was all caird in and re-coin'd half of Silver 
 and half of Allay, the Grievance of this is readi- 
 ly feen and admitted ; but if a Crown Piece were 
 caird an Angel, and made to go for ten Shil- 
 lings, and the reft of the Silver Pieces in Pro- 
 portion ; in this, fay they, there's no Harm, 
 the Change is meerly nominal. 
 
 As to Sales or Bargains to be made in future, 
 I agree fuch Change in general is meerly nomi- 
 nal ; becaule the Parties felling or contracting 
 will have Regard, not to the Denomination 
 folely, but to the real Quantity of Silver or 
 Gold which is known, or fuppofed, to be con- 
 tained in the Money in which fuch Contracts are 
 prefumcd to be paid and performed ; and there- 
 fore the Owners of Goods would then expe(5t to 
 have double the Quantity of Money in Pounds, 
 
 Shillings,
 
 t 112] 
 
 Shillings, and Pence, for the fame Commodities 
 which they would now fell for the fingle Quan- 
 • tity • So that in thefe Inftances there would be 
 neither Prejudice nor Benefit accruing to either 
 : Party. But would not the Cafe be the fame, 
 •and the Change be meerly nominal, if the Mo- 
 ney were debafed in an equal Proportion ? Ex- 
 iidMy the very fame, and for the fame Reafon : 
 For there vvould be the like Right, the like 
 Opportunity, and the like Method, of Pcrforls 
 helping themfclves, viz. by taking then a double 
 Quantity of Money, in Monies number'd, in-- 
 -ftea'd of a fingle one which they now take. 
 
 But, belides thofe Tranfadions of buying and 
 •felling, and Contfads, in future, there arc other 
 Money Matters which are to be regarded, and 
 where this Exchange will produce an Effect 
 which fliall be more than Names and meer 
 Words. There are fuch Things as old and un- 
 alterable Rents, Annuities perpetual and certain, 
 Debts fubfifting, and Contracts in Being, both 
 of a public and private Nature, where the Sums 
 are fix'd and determinate, to the Value of many 
 Millions, perhaps not much Ihort of a Moiety 
 of the Value of the EUate of the whole King- 
 dom ; the Proprietors of all which neceflarily 
 mull lofe juft as much as the Money is nominally 
 advanced, for that they will receive juft fo much 
 • in Silver lefs, and in Gold proportionably, as is 
 equal to the fuppofcd Increafe in the Denomina- 
 - tion of the Money. With Regard therefore to 
 thefe Perfons, an Increafe in the nominal Value 
 t of Money, and a Dccreafe in the intrinfic Value, 
 ^s^ul'l:. .. fuppofing
 
 [ "3 ] 
 
 fuppofing them to be in one and the fame Pro- 
 portion, will have one and the fame mifchievous 
 Effect ; that is, the one or the other will fink a 
 Moiety of their Subltance. Whether fuch a Pro- 
 ject as this might not pafs in a neighbouring 
 Kingdom for a reafonably good Way of dif- 
 charging public Debts, I can't tell ; but I be- 
 lieve he who fhould advife it, and put it in Prac- 
 tice in Evglandj in moil People's Opinion, would 
 deferve an Ax or a Halter. 
 
 And for the Advantages which are furmifed 
 would arife from hence to the Public, viz. that 
 fuch a Scheme would make the Species of Silver 
 more plentiful, and prevent its being melted 
 down and exported ; they are both of them 
 very vain and very groundlefs Expedlations. If 
 the Silver Coin, fuppoling there are now fix 
 Millions of that Species in the Kingdom, were 
 advanced to double the prefent nominal Value, 
 we might indeed fay then, that we had twelve 
 Millions of that Species ; but this fame Silver 
 Species, when call'd twelve Millions, will ope- 
 rate in Trade, or any Bufmefs of the Nation, 
 juft fo far, and no firther, than the fame Species 
 now does, which is call'd only fix Millions : For 
 that, after fuch Alteration, there can be no 
 Money Tranfa(5i:ions in which Silver is made ufe 
 of, let the Tranfadion be in a greater or a lefs 
 Sum, but we mull necefl^arily then make Ufe of 
 double the Quantity of that Money in Tale, or 
 the fame Quantity in Weight, which we now 
 make Ufe of, for any Purpofe whatever, ex- 
 cepting that of paying old Debts, and making 
 H good
 
 [ "+ 3 
 
 good previous Contrads. Tofayotherwifc, of 
 that the fame Quantity of Silver in Coin, when 
 cali'd twelve .Millions, would be of more Ufe 
 than when cali'd fix Millions, has no more Truth 
 or Senfe in it, than there would be in faying ; 
 if a Piece of Cloth of five Yards long were cut 
 into ten Pieces, and thofe Pieces were called 
 Yards, that then it vyould go farther in making 
 a Suit of Cloaths, than it would do if thofe 
 Pieces were cali'd, as they really are, half Yards 
 only. Jufl in the fame Manner, if a Pound of 
 Silver, which is cut into fixty two Pieces, and 
 ivhich arc now cali'd Shillings, fhould hereafter 
 be cali'd Solids, or by any other new Name, 
 and made to pafs for two Shillings • yet a Pound 
 of Silver, v/hether the fixty two Pieces, of which 
 *ti3 made, are cali'd Shillings or Solids, is one 
 and the fame Thing, and of one aad the fame 
 Value ; and an old Shilling will go as far at the 
 Market as a new Solid. 
 
 On this Account, and for the fame Reafon, 
 any fuch Scheme as this, whether it be by in- 
 creafing the nominal Value of Money by giving 
 it a new Name, or by decreafing its real Value 
 by debafing it, will have no Sort of EfFed: to- 
 wards preventing the Exportation of our Coin, 
 cither of Gold or Silver. Traders and Mer- 
 chants, efpecially thofe who deal in foreign 
 Commodities, will not have Regard to the 
 Pounds, Shillings, and Pence in Tale, to which 
 fo many Pieces of Silver or Gold Coin may a- 
 mount; but to the Pounds, Ounces, and Grains, 
 which thofe Pieces will make in Weight at the 
 
 Scale ;
 
 [ 1^5 ] 
 
 Scale; and therefore, for a Parcel of Goods, the 
 fame in Quantity and Quality, they will then ex- 
 ped, and in Fad will receive. Silver or Gold in 
 Exchange, the fame in Quantity and Quality 
 which they now do : And confcquently m^y ex- 
 port Silver or Gold with the fame Eafc and equal 
 Profit, whatever the State of the Coin Ihall be : 
 And if they can do it to Advantage, I fuppofe 
 they will do it ; and I confefs I do not fee any 
 Reafon why they fhould be prohibited. 
 
 To return from this Digrellion. "What was 
 the Interell of Money, or of Silver and Gold 
 of which it conliHed, or the Price given for it 
 on Loan, in the two feveral Periods I have been 
 fpeaking of, t;/;s. two Hundred and three Hun- 
 dred Years ago, I confefs I have not been able 
 to difcover, fo as to fix it with any Certainty. 
 Thus far, however, we may fafely go, and per- 
 haps that may be fufficient for our Parpofc, as 
 to affirm, that about two Hundred Years ago it 
 was at leall as high as lo/. per Cent, per Anntun^ 
 there being an Ad: of Parliament made at the 
 latter End of the Reign of Henry the Vlllth, 
 that it Ihould not exceed that Rate ; io that it 
 is a very fair Conjecture, that it had been before 
 that Time higher, and in the Times of HetJry 
 the I\'th, poifibly as high as 15/. per Q fit.: And 
 every Body knows, that the Interell or Price of 
 it now is no more than 5 /. per CeiJt. fctting it at 
 the highefl Rate it can be let at. 
 
 From thefe Premifesj I mean froni -the diffe- 
 rent State of the Silver Coin, which generally 
 governs that of the Gold Coin, and from the 
 H 2 different
 
 [ "6] 
 
 different Rates of Intereft, in the two former ot 
 in any other Periods, and in the prefent Period 
 of Time, I make thefe Dedu(5lions : That when 
 ^ Pound of Money in Tale was an effecftive 
 Pound of Silver in Weight, and the Intereft or 
 Price of Money was at lo per Cent, if we com- 
 pare the Price of Commodities in Evgland now, 
 with the Price of them at thofe Times, the Price 
 now fhould be in the Proportion of fix to one \ 
 and if the Price or Intereft of Money was at 15 
 ^er Cent, and the like Comparifon were made, 
 the Proportion now fhould be as nine to one. 
 My Notion in this Matter is, that the Price of 
 Things at one Time will bear that Proportion to 
 the Price of them at another Time, which the ef- 
 fective Silver in the nominal Pound at one Time 
 bears to the effective Silver in the nominal Pound 
 at another Time, and the Intereft or Price of fuch 
 Pounds on Loan at the different given Times ; 
 Computation being made upon both thofe Heads. 
 From hence I infer, if in the Times oiHenty 
 the Vlllth a Pound of Silver was cut into forty 
 or forty five Shillings, and now into fixty or 
 fixty two, that the Price of any Commodity 
 •which then was two muft now be three Pounds 
 on the Account of that Difference only ; and if 
 the Intereft or Price of Money was then 10 
 fer Cent, which is now only five, the Price on 
 that Account muft be farther doubled, and, in 
 all, be fix Pounds ; that is, the prefent Price 
 muft be three Times as much as in that Reign, 
 In like Manner, if in the Times of tienry the 
 Vlth a Pound of Silver was cut into thirty, and 
 
 now
 
 [ "7] 
 
 now into fixty two Shillings, from thence only the 
 Price of Things now mull be double the Price of 
 them then • and if the Intereft of Money then was 
 at 15 pe/" Cent, as very probably it might, that 
 Difference from the Intercft now will make an. 
 Addition of a treble Price, and the Whole be as 
 fix to one. So in the Times oi Henry the IVth, 
 when a Pound of Money in Tale was a Pound 
 of Silver in Weight, and the Intereft was 15 pev 
 Cent, which is treble the Value and treble the Price 
 of each now, we muft treble the Price on each Ac- 
 count, and the Price now will be as nine to one. 
 To Ihow this, I will make Ufe of a familiar 
 Inftance. A Farmer or Merchant, in the Times 
 when a Pound of Money in Tale was an effective 
 Pound of Silver in Weight, and the Price of it 
 on Lean was 10 per Cent, is poflefs'd of a Par- 
 cel of Sheep, Oxen, Wool, Corn, or other Ne- 
 ceflaries of Life, to the Value of an 100/. at the 
 Market Price ; when he fells them at that Rate, 
 will receive an too /. in Silver, in Weight as well 
 as in Tale, or in Gold proportionably. A Far- 
 mer or Merchant in thefe Times therefore mull 
 have 300 /. in Money for a like Parcel of Goods, 
 or elfe he will not have the fame Quanfity of 
 Silver or Gold in Weight, though the Goods 
 fold are in Quantity and Quality the fame ; that 
 is, the prefent Owner muft have three Times the 
 Price, in Money as no^v told, as the ancient 
 Owner had, for the fame Things. That this is 
 the Fa(5t in the Inftance of Gold, as a Commo- 
 dity, is evident ; for that a Pound of Gold in 
 Eenry the IVth's Time, was fold for 15 I. or 
 H J there«
 
 [ 1^8 ] 
 
 thereabouts, ^in Money in Tale, and now it klls 
 for 45 /. and fomething over, in the like Money 
 in Tale, that is, at three Times the Price. 
 
 To 2:0 on to the Difference in the Rate of In- 
 tereft now and formerly. If thefe Merchants 
 have not prefcnt Occaiion for their fcveral Sums 
 in their Way of Trade and Bulinefs, or are de- 
 firous to let them out at Interelt : The Merchant 
 of old Times, on the Loan of his ico/. at the 
 End of the Year will receive 10/. in Silver, in 
 Weight as well as in Tale, fuppoling Interefl 
 to have been then at 10 per Ceiit.'^ but the 
 Merchant of thefe Times, for the Intcrcll of his 
 300/. vkfill .have no more than 15/. in Tale, 
 which is equal only to 5 /. in Weight : There- 
 fore, that the one and the other may have equal 
 Advantage from the fuiie Parcel of Goods, the 
 modern Merchant mull fell his Goods at 600/. 
 for no lefs Sum than that will yield him 10/. of 
 Silver in Weight for Intcrcll by the End of the 
 Year, that is, he muft fell them at jix Times the 
 Price: or'othcrwife thefe two Perfons, at the 
 End of fuch Year, will not be in equal Circum- 
 llances, though they were fo at the Beginning 
 when they were poflefs'd of their Goods. 
 
 If thefe Merchants fnould invcfl their Money 
 in Lands of Inheritance, inflead of putting it 
 out at Interefl, the Cafe will fiiil come out the 
 fame. Vv^hen Money is at fo per Cent, the 
 Price of Eilates in Fee-Simple is ten Years 
 Value : So that an 100 /. in old Times would 
 have purchafed an Eftate of 10 /. per Amnim^ 
 and a Rent of 10 /. per Annum was a Rent of 
 
 fo
 
 [ "9 ] 
 
 fo many Pounds Weight in Silver, or near ft, 
 and of Gold in Proportion. If a Man were novv 
 to purchafe an Eftate of equal Goodnefs, Ijc 
 muft pay for it 300 /. even though Intereft were 
 fuppofed to be the fame novv as formerly, and 
 there were no Advance in the Year's Purehafe ; 
 for no Rent lefs than 30/. per Jnnmn will pro- 
 duce ten Pounds in Silver in Weight, or propor-- 
 tionably in Gold. And when we take into the 
 Account the Difference of Intercft, 'tis plain, 
 that the Fall of Intereft to one Half makes a 
 Rife of Land in the Purchafe to a double Price : 
 and therefore, now Intereft is no more than 5 
 per Cent, he muft and does pay Ctoo I. for an 
 Eftate of 30 /. per Anmim\ that is, for an Eltate 
 of equal Goodnefs, and which fliall yield an 
 equal Quantity of real Silver or Gold, which 
 might have been bought 300 Years ago for one 
 100 /. he muft now pay juft fix Times as much. 
 
 Or take the Matter in this Light. The Sum 
 of an 100 /. in former Ages, if laid out in Lands, 
 or let out at Intereft, would, in a Year's Time, 
 produce to the Owner ten Pounds Weight of 
 Silver, or a proportionable Quantity of Gold ; 
 the Sum of aoo/. now, if laid out in Lands, or 
 let out at Intereft, would, in a Year's Time, 
 produce to the Owner ten Pounds in Money, as 
 now counted : But ten Pounds in Money, as 
 formerly counted, had three Times as much 
 Silver or Gold in it, as ten Pounds in Money, as 
 novv counted : Therefore, to produce as much 
 Silver or Gold within the Year as formerly, 
 three Times 500 /, or 600 /. as Money is now 
 H 4 counted
 
 [ I20 ] 
 
 counted, mull be laid out in Land, or put out 
 at Intereft ; that is, fijf Times the Money mull 
 now be employ'd to produce the fame Quantity 
 of Silver or Gold as was produced formerly by 
 one I do/, only. The Confequence of this is, 
 that the NecefTaries and Conveniencies of Life, 
 •which are the Things out of which Money is to 
 arife, when they come to a Market, muft now 
 be fold at fix Times the Price, or lix Times the 
 Pounds, Shillings, and Pepce, which they were 
 formerly fold at ; or the very fame Things will 
 not' anfwer the fame Purpofes of Life npw, 
 which they would have done three Hundred 
 Years ago. 
 
 It will be needlefs to repeat here this Rea- 
 foning, and fliovv that 'tis applicable to the Cafe, 
 which I have fappofed poiTible, that the Intereft 
 of Money has been in fomc Times at 15 A per 
 Cent, and that the Price of Goods between thofe 
 Times and thefe fhould be in the Proportion of 
 nine to one. It may fuffice to fay in general, 
 that it will hold good in that Inflance, and in 
 any other which may be fuppolcd ; fo that where- 
 ever the Quantity or real Silver in a nominal 
 Pound, and the current Intereft of Mpney can 
 be known and determined, there the Price of 
 Things may be known and determined likewife. 
 My Notion is here, that the real and intrinfic 
 Value of the abfolute Necelfaries of Life, fuch 
 as Food and Raymcnt, were always, and always 
 will be, much one and the fame, except where 
 an accidental Plenty or Scarcity makes a tempo- 
 rary Variation j but if we compute the Value of 
 
 thofe
 
 [ ^21 3 
 
 thofe NecefTaries by a third Thing as a common 
 Mcafure between them, the Price, or nominal 
 Value fo meafured, muft vary as fuch Meafurc 
 itfelf varies. If Silver be made that Medium, 
 as for many Ages it has been in thefe Parts of the 
 "World ; and if in former Times a real Pound 
 of Silver was call'd a Pound, a twentieth Part of 
 a Pound was call'd a Shilling, and the twelfth 
 Part of a Shilling a Penny ; if in Times fubfc- 
 quent the Meafure itfelf be varied, and a third 
 Part of a Pound of Silver be denominated a 
 Pound, and Shillings and Pence in the like Pro- 
 portion, the Price or nominal Value of the Ne- 
 cefTaries of Life will and mull vary with it. A 
 Quarter of Wheat or Malt, a Pound of Beef^ 
 or a Yard of Cloth, are now of the lame real 
 Value as heretofore ; for they vy^ill go as far to- 
 wards the Support of human Life now as they 
 did five Hundred or a Thoufand Years ago, and 
 no farther ; but their Price or nominal Value, 
 as meafured by the current Coin of the King- 
 dom, muft vary as the Coin itfelf varies ; and 
 this we muft admit to be the Cafe in Silver itfelf^ 
 or we muft be forced to fay, that one third Part 
 of a Pound of Silver, fuppoling no more than a 
 third Part of a Pound of Silver to be in a pre- 
 fent nominal Pound, is of equal Value with a 
 whole Pound. For if we confider Silver, not as 
 a Meafure of Traffick only, but as a Neceflary 
 or Convenience of Life, and as a Merchandizablc 
 Commodity, as we certainly may and do ; one 
 Pound of Silver is of the fame Value as another, 
 and of the fame Value at one Time ^9 at ano- 
 ther.
 
 C 122 ] 
 
 ther, greater or lefs Plenty excepted, but the 
 Price or nominal Value was formerly one Pound 
 only, but now we fee it is three Pounds and 
 above in Silver, as a Mcafure : And we may 
 rightly fay, that it ceafes to be one and the fame 
 Meafure, when it c:afes to be one and the fame 
 Thing in its Weight and Value. 
 
 From hence we plainly difcern one Reafon 
 why Money now is not of the fame Value as it 
 •was two Hundred or three Hundred Years ago, 
 W3. becaufe the Silver of which it confifts in any 
 given Sum in Tale is now only a Moiety, or a 
 third Part of the Quantity which was formerly 
 in the like Sum in Tale; and it being a Conve-^ 
 nience of Life and a faleable Commodity, the 
 real Quantity of Silver in the Money is the true 
 Ivleafure of its Value. But bcfides this Decreafe 
 in the Value of Money ariling from the Variati-^ 
 on in the Species ; we find another in the Inte- 
 Tci\, or annual Premium for it. This, to be fure, 
 has proceeded from another Caufe ; and has 
 Hrifen, as I apprehend, from the great Incrcafe 
 of the moveable and cafily transferable Eltates 
 in the Kingdom, which for fome Time has been 
 growing upon us, and oi" late has been fo vallly 
 inlarged by the public Funds, and not from the 
 Incrcafe of Gold and Silver in the Kingdom, 
 either in Coin or Bullion, any otherwife than as 
 they conllitute a Part of fuch moveable Eflates, 
 How, and in what Manner, thefe Decreafes of 
 the Value of Money in both Kinds have an In- 
 fluence on the Price of the NecelTaries of Life, 
 Y(C have already taken Notice '^ we Ihall proceed, 
 
 thcfclbrcj^
 
 [ 123 ] 
 
 therefore, to confider how fuch Decreafes, and 
 the Increafe in the Value, or rather Price, of 
 thefe NecelTaries, are or may be applied to the 
 ■Purpofe of advancing Fines lince thofe Days. 
 
 What Influence the Dccreafe in the Intereil 
 of Money fliould have in this Affair, we have 
 already conlider*d ; and I lliall not need to rer 
 peat what has been ofFcr'd to that Purpofe under 
 another Head. As to the Increafe in the Price 
 of Things, it can be nothing to the Purpofe, 
 unlefs we could fuppofe that the very numerical 
 Fine was now taken for the fame individual 
 Eilatc, which was taken two Hundred Years 
 ago, which every Body knows to be falle. If a 
 Landlord, in his Catalogue of the Neceflkries of 
 Life, fhould infert Wine, Brandy, Coffee, Tea, 
 and Chocolate, and alledge that the Price of 
 thefe Things is greatly inhanfed of late Years ; 
 I doubt his Tenant, who holds by the Year^ 
 will hardly be brought to admit, that an Advance, 
 in the Price of thofe Commodities, or in the Ufe 
 of them, is a fufficient Reafon to raife his Rent 
 which is paid annually, nor his Tenant, who 
 holds by Leafe, to raife his fore-hand Rent or 
 his Fine, which is paid at once : becaufe the 
 Produce of his Land, out of which fuch Rents 
 are to arife, to be fure, is neither made better, 
 nor of greater Value, by any fuch Advance; 'tis 
 well if 'tis not made worfe, and of lefs Value. 
 
 If the Price of Beef and Mutton, of Wool 
 and Corn, or the like, is advanced threefold fup- 
 pofe, fince the Times of Henry the Vlllth, or 
 in any other Proportion lince any other deter- 
 minate
 
 [ 124 ] 
 
 minate Period of Time ; whether fuch Rife haa 
 proceeded from lowering the Intereft of Money, 
 or from diminiihing its intrinfic Value, or from 
 both together ; fuch Advancement in one Kind 
 has given Occafion for, and produced, an Ad- 
 vancement in another Kind, viz. in the annual 
 Rent : For the Rent of Land and the Price of 
 the Things which that Land produces, always 
 do, and neceflarily muft rife or fall together; 
 And fince the annual Rent is the Meafure of the 
 fore-hand Rent, an Enlargement of one has 
 made an Enlargement of the other likewife in 
 Proportion ; fo that a LefTor, in the prefent 
 Way of Computation, has already raifed his 
 Fine anfwcrable to the Rife which has been in 
 the Price of thofe NecefTaries of Life, in which 
 he can any wife be concern 'd, or his Ellate be 
 affeded by : And where the Price of thofe 
 Things which the Land produces has not been 
 raifed, there's no Senfe or Reafon in railing the 
 annual or fore-hand Rent. 
 
 I ihall add here, that the Cheapnefs of Money 
 and Dearnefs of the Necellaries of Life, which 
 already have or hereafter may come to pafs, have 
 no Influence on Leafe-hold Eltates which in any 
 Rcfped are beneficial to the Owners, but di-? 
 rc6tly the contrary : though with Regard to 
 Fee-Simple Lands, the Proprietors of them have 
 a vifiblc and immediate Advantage if they part 
 with their Intereft, and if they keep it, in all 
 Probability, in the Courfe of fome Years, muft 
 have one in another Kind, The Confequence 
 of this Dccreafe in Interelt on one Side is, th^t 
 
 the-'
 
 [ 125 ] 
 
 the Annuity coming to the Owner of the Leafe 
 muft fink, as certainly, though not altogether in 
 the fame Proportion, as if his Money were in- 
 vefted in South-Sea^ or other public Annuities, 
 and the Government Ihould lower thofe Annui- 
 ties ; or if his Money were out on a Mortgage, 
 and the Interefl of it were decreafed by Law ; 
 and the Principal, in either of thofe Cafes, can 
 in no wife be enlarged : And, on the other Side, 
 the Proprietor of inheritable Lands has a great 
 Advance in his Principal, by the Rife in the 
 Value on Sale, and his Rent or Annuity out of 
 all Danger of finking. 
 
 If this Decreafe in the Value of Money, and 
 Increafe in the Price of the Necefl^aries of Life, 
 which always go together, or rather are one 
 and the fame Thing, fiiould occafion a Rife in 
 the Rent of Lands, as in the Compafs of fome 
 Time it certainly will, tho' it proceed by De- 
 grees not eafily perceptible ; the Advantage of 
 fuch Rife will redound wholly to the Proprie- 
 tor of Fee-Simple Eltates, and the Leafeholder 
 can have no Share in it, for his Landlord has un- 
 doubtedly the Right to, and will undoubtedly take 
 the wholeBenefit of. that,in his fubfcquent Fines. 
 Now I fuppofe a lefs Income arifing to a Te- 
 nant, and at the fame Time a greater Expence 
 becoming neceflary, as they always accompany 
 one another, is no very agreeable Thing in En- 
 joyment, or comfortable in Profped: : So that I 
 humbly apprehend thofe many Harangues upon 
 the prefent profperous State of the Nation, and 
 the Intimations of the Probability of a farther 
 
 Decrcalc
 
 [ 126 ] 
 
 Decreafe in the Intereft of Money, which fo fre* 
 quently occur in fome of the Writers on this 
 Subjed, however true they may be, and of Ser- 
 vice to Tenants of Lands in Fee-Simple, if ap- 
 plied, and when applied to Tenants by Leafe, 
 and Monied Men, arc not exceedingly much to 
 their Satisfadion, or to the prefent Purpofc. 
 
 Since then, upon the whole, Eflates for a 
 Term of twenty one Years, or any other deter- 
 minate Number of Years, may be afcertain'd to 
 the Exadnefs of a fingle Shilling, if there were 
 Occafion for it, provided they be, and when 
 they are, reduced to abfolute Annuities : And 
 having offer'd to Confideration the feveral Ar- 
 ticles of Deductions and Allowances to be made, 
 which the Purchafers of fuch Eitates may furly 
 infill on, in order to reduce them to fuch Annu- 
 ities : And having Hated what I apprehend to 
 be a reafcnable Intereft for the Money laid out 
 in purchafing or renewing them, which I think, 
 in a general Way, fhould be one per Cent, higher 
 than the common current Rate : I fliall only 
 add, that I have annexed a Table, N^ I., for 
 valuing Annuities for any Term for fifty one 
 Years, at the feveral Rates of 3, 4, 5, 6, 7, and 
 8 per Cent, Intereft ; in which, inftead of Deci- 
 mals for Parts of a Year or Parts of a Month, 
 I have inferted Quarters of a Year and Days, 
 and in the following Difcourfe exprefs my felf in 
 that Manner; apprehending fuch a Form would 
 be more ufeful and more ready for other People, 
 as I found it to be for myfelf. 
 
 Int
 
 [ 127 ] 
 
 In one. Cafe indeed, where the annual Rent or 
 Annuity is an even loo/. or is eafily reduced to 
 fuch, there a Table by Yc;ars and Decimals, or 
 rather Gentelimals, is more ready than by Quar- 
 ters and Days ; for there the very Figures, which 
 cxprefs the Years and centefimal Parts, exprels 
 the Value in Money : For Initance, if Enquiry 
 was made, what is the Value of an Annuity, or 
 a Leafe reduced to an Annuity, for twenty one 
 Years, Interell computed at 4/. per Cent. ; the 
 Anfwer is, 14-01 in Years and Centefimals, 
 which in Money is 1401 7. For which Reafon, 
 and becaufe fome Perfbns may have a better 
 Tafte, or have been more accuflomed, to thefe 
 Accounts in the Way of Decimals ; and fince in 
 the following Difcourfe I often make Ufe of, and 
 have Reference to, a Table drawn in that Form ; 
 to my Table for the Value of Annuities I have 
 added a third Column, which gives the Decimal, 
 or rather Centefimal, Parts of a Year, corre- 
 fponding to the Quarters and Days. I mult ob- 
 serve here, that Mr. R/V^^r^j-, in his Table, and 
 throughout his Difcourfe, calls thofe Parts of a 
 Year Decimals ; for what. Reafon I cannot un- 
 derftand • but not intending to have any Difpute 
 about Terms, where the Meaning of thcn^ is 
 well enough underllood, I have complied with 
 the Ufe of his Expreffion ; though the Parts in- 
 tended are really Centefimal Parts of a Year, and 
 ought to be fo call'd. 
 
 The Ufe of this Table in purchafing thefe 
 Eilates, whether they are originally and pro- 
 perly Terms for Years, or are Eltates for a Life 
 
 or
 
 C t28 ] 
 
 6r Lives, and are reduced to Terms for Years, 
 is fo obvious and fo well known, that I need to 
 add nothing to explain it. In renewing fuch 
 Eflates of either Kind, there is Occafion only 
 for an eafy Operation: that is, in Cafe of Leafes 
 for Years, to put together the Number of Years 
 remaining in the Leafeand the Number of Years 
 to be added : in Cafe of Leafes for Lives, to put 
 together the Number of Years to which the Life 
 or Lives remaining are equal, and the Number of 
 Years to which the Life or Lives to be added are 
 equal ; to Hate the Value of the whole Term : 
 and then out of fuch Value to dedud the Value 
 of the Term of Years in Being in the Leaie ; 
 and the Remainder is the Value of the Number 
 of Years, or of the Life or Lives, to be added ; 
 that is to fay, is the Fine to be paid for renew- 
 ing. For this Reafon I thought it unneceflary 
 to infert any Table for renewing Leafes ; but it 
 may, perhaps, be of Service to fuch who may 
 be concern'd in thefe Sort of Tranfadions to 
 give them a Caution, that they firft reduce the 
 Eftate in View to an abfolute Annuity; for that 
 thofe fame Tables, either for purchafing or re- 
 newing, are very improperly, I may fay very 
 unjuftly, applied to Eftates held by Leafs, un- 
 lefs and until they are fo reduced : For the Com- 
 putations, by which both the one and the other 
 are conftrufted, are made on this Foot, that the 
 Annuities are abfolute. And this Caution ap- 
 pears to me the more neceffary, becaufe the com^ 
 mon Tables of Renewals may tend, and per- 
 haps were calculated and publifhed with a View , 
 
 to
 
 [ 129 ] 
 
 to take off Peoples Minds from attending to this 
 Point, viz, what Parts of the annual Value of his 
 Kftate a Tenant is to pay for in the Fine on a 
 Renewal ; whereas, in my Judgment, this is the 
 only Matter that deferves or requires any Atten- 
 tion, and which I have been here attempting to 
 adjuft and eftablifh ; and I hope have put into 
 fuch a Method as will appear to be a fair and 
 equitable one between Landlord and Tenant. 
 
 End of the Firji Book* 
 
 BOOK
 
 130 
 
 •^\ 
 
 BOOK 11. 
 
 'Jn 'E SS AY to eftmate the Chance of 
 the T>uration of a Life, and to a/certain 
 the Value of Leafes and Jnmiities for 
 Life-i &c. 
 
 E proceed now to confider of, and 
 propofc, a Method to afcertain the 
 Value of Leafes for one or more 
 
 Lives, and thofe, I think, as well as 
 
 Leafes for Years, mufl firft be reduced to abfo- 
 lute Annuities; and this we mud do by the fame 
 Jlules, and in the fame Manner, as has been prc- 
 fcribed in common Leafes for Years. When 
 this is done, or provided for, Eftates, whether 
 held by Leafe, or by any other Tenure for a 
 Life, will have one and the fame Gonfideration ♦, 
 and to adjtjft the Value of fuch Intereits, I ap- 
 prehend it to be the bell:, and, indeed, the only 
 Method, to refolve them into Terms for Years 
 certain. To do this, we muft compute to what 
 Age there is a Chance or Probability that any 
 Perfon or Pcrfons named fhall live ; or, which ia 
 the fame Thing, to what Term of Years any 
 given Life or Lives are equal ; fo far, I mean, as 
 the Chance of any Perfons living can be efti- 
 jnated ; And when this is dpne, and from hence, 
 as 1 have""befoie obferved, we may fee, and rea- 
 dily
 
 [ 131 ] 
 
 dily determine what is the Value of fuch an 
 Ellate, to fuch a Degree of Exadnefs and Cer- 
 tainty as the Chance of the Life can be afcer- 
 tain'd. 
 
 We want to know, for Inftancc, what is thd 
 Value or the Sum of Money to be paid for a 
 ilatcd Annuity for a given Life ^ or, fuch a Sum 
 of Money we have to difpofe of, fuch is the 
 Annuity we would have during a Life, and arc 
 deiirous to know what Soft of Life, or what 
 Aged Life, muft be nominated, i^o as to buy an 
 equal Bargain. If thofe Cafes were put upon a 
 Terrti of Years, if we are not contented to abide 
 by the Tables in common Uiej the Method to 
 Calculate this, and by which the common Tables 
 were framed, is this. In the firfl Cafe, where 
 the Term or Number of Years is agreed on, wc 
 add together the Sum of the Reverlions, or the 
 prefent Value of the Sum to be paid at the End 
 of each Year, at the ftated Intereft, for the whole 
 Term of Years : and the Total arifing from 
 thence is the Value of the Annuity, or the Sum 
 to be paid for it, at fuch a R^te of Intereft. In 
 the other Cafe, where the Sum of Money to be 
 paid and received is agreed on, the Method is 
 this : Out of the ftated annual Income we de*- 
 du6t the Interelt which \^e expcd for our Mo- 
 ney, and compute in what Number of Years 
 the Surplufage of the Annuity, at Compound 
 Interell, at the current Rate, will produce the 
 Principal 5 and the Number of Years which 
 produces the Principal is the Term for which 
 the Purchafer .mull enjoy his Annuity* 
 
 I 2, Now
 
 [ 132 ] 
 
 Kow let this be applied to an Annuity to fee 
 purchafed on a given Life ; to adjuft the Valttc 
 of which we muft add together the Sum of the 
 Reverfions, at the Itated Intercft, for the given 
 Xife-; but before we can efFe6t this we muft de- 
 termine, by a Computation made, in the beft 
 Manner we can make one, for what Term or 
 Number of Years there is a Chance or Probabi- 
 lity that the Perfon whofe Life is given may 
 live ; and then, indeed, we may proceed as in a 
 Term of Years, it being now reduced to fuch. 
 But unlefs this be done, and until it is done, I 
 fee no Rule or Reafon why in adding together 
 Reverlions we may not Hop at five Years End, 
 or go on to fifty, every one according to his 
 own particular Conceit : And in Truth fome 
 of the Computations of this Kind are fuch, as 
 would tempt a Man to think that they were made 
 without either Rule or Rlafon. ..^ 
 
 In the Second Enquiry upon a Life ; that is, 
 if we would know what Sort of Life we muft 
 have, or of what Age the Perlbn to be named 
 Ihould be, for a determinate Sum, and at altatcd 
 Intereft : How are we to proceed there ? By the 
 fame Method as is obferved in a Term for Years : 
 We are to dedudt Intereft out of the annual Pay- 
 ments at a ftated Rate ; to compute in what 
 Time the Surplufage after fuch Deduction, at 
 Compound Intereft at the common Rate, will 
 produce the Principal ; and the Time which 
 produces the Principal is the Time for which 
 the Purchafcr muft enjoy the Annuity. But, by 
 knowing the Time for which the Annuitant is to 
 
 enjoy
 
 [ 133 3 
 
 enjoy his Eftate, we are not come to the Know^ 
 ledge of the Sort of Life to which fuch Time is 
 c<^ual, or what Aged Life muft be nominated to 
 anfwer fuch Purpofe : Nor can we ever, as I ap- 
 prehend, come to fuch Knowledge, without 
 computing firll and determining, by an even 
 Chance or on a reafonable Probability, what 
 Sort of Life, or what Aged Life, may be equal 
 to, or be likely to continue to, fuch a Time. 
 
 Now if the Chance of Life can be adjufled 
 upon any good and probable Grounds ; and we 
 muft fuppofe that this is capable of being done 
 to fome tolerable Degree, for if the Chan.ce of 
 the Duration of a Life cannot be adjufted with 
 fome fort of Probability, all the reft lignifies jufl: 
 nothing: I fay, if this can be done, and when 
 'tis done ; that is, if we can make, and have 
 inade, a reafonable Eftimate how long any Per- 
 fon named may live ; fince the Leafe or Annui- 
 ty is fuppofed to continue as long as the Life 
 continues, all farther Calculations, whether built 
 pn imaginary or on real F'pundations, are, in my 
 Judgment, perfectly ufelefs : for they can only 
 difcover fomething which is already fufficiently 
 difcoyer'd, fince the only Thing which remains 
 to be known is the Value of the Term to which 
 we have eftimated fuch or fuch ^ Perfon is likely 
 to live, which the common Tables |br the Value 
 of Annuities for Terms of Years Ihow us at 
 once, and that with Certainty, and to the Ex- 
 adnefs of a fiqgle Shilling, and at every Rate of 
 Jntereft : And to fearch farther is neither better 
 por worfe, if I may have Leave to ufe a famili3,r 
 I 3 Simile
 
 [ »34 ] 
 
 Simile, than it would be to feek for a Needlq \t\ 
 a Trufs of Hay when we have it all the whilq 
 between our Finger and Thumb. 
 
 In Purfuance of, and in Conformity to, this 
 my Way of thinking as to the Method of com- 
 jDuting the Value of Annuities for a Life, I have 
 drawn the Scheme in Table N? II., in the two 
 firfl Columns of which is exhibited the Term of 
 Years to which any fingle Life is equal, com- 
 puting for every Eve Years of Age of Lite, from 
 ten Years to eighty Years of Age, inclufive of 
 both : And fuch a Scheme for a fingle Life mult 
 be, as I apprehend, the Foundation of any that 
 can be framed for two or more Lives. This I 
 have formed upon the beft Obfervations and Cal- 
 culations that I could think of, and thofe not a 
 few, fome of which I fhall hereafter take Notice 
 of: And yet I do not expert it will be agreeable 
 to every Man's Tafte, and that no Objedlion can 
 be raifed againft it; for in Things of thisNaturc, 
 "which are fubjed to a vague Eftimate, every one 
 has his particular Way of Thinking : All that 
 I {hall fay of it is, that to me it feems liable to 
 fewer Objections than any that has yet appeared, 
 to me at leaft, not excepting that which was 
 4 rawn by the belt Hand. On which Account, 
 though there are many Tables already extant 
 for the Valuation of thefe Eltates, yet if the 
 iN^ethod of making fuch Valuation was meerly 
 imaginary, and not known before Dr. Halk/s 
 Rules for that Purpofe came out, as has been 
 confidently affirmed, and I cannot contradict ; 
 And if the Tables, which fince that Time have 
 
 been
 
 [^35] 
 
 been publiih'd were conllruded by thofe Rulcg, 
 I will venture to affirm, that there are none now 
 extant free from Errors, and, if I may have 
 Leave to fay fo, Errors apparent and amend- 
 able ; this will be a fufficient Excufc for offering; 
 a new one : fpecially if it avoids the Errors of 
 former Tables, and keeps clear of new ones ia 
 any tolerable Degree : But whether it does, and 
 how far it does this, or exceeds any other Tables, 
 is fubmitted to the Reader's Judgment. 
 
 I am very fenfible, that a Man ought to be 
 wary, and go on good Grounds, when he pre- 
 fumes to attack any Calculations made by fo 
 great a Mafter of thofe Things as Pr. Halky^ 
 and yet if his Calculations appear to me liable 
 to Exceptions, and thofe Exceptions can be 
 maintain'd, I doubt not but the World and the 
 Dodtor himfelf will excufe my Attempt : And I 
 verily perfwade myfelf, when the Animadverfi* 
 ons I Ihall make on his Tables and his Rule arc 
 thoroughly confider'd, it will appear to others, 
 as it does to me, that they are not framed with 
 that Accuracy which ufually attends this Gentle* 
 man's Performances of this Kind. 
 
 The chief Exception which I take to the 
 Doctor's Table is, that where the Value of an 
 Annuity for a Life in one Period of Life is com* 
 pared with the Value of an Annuity for a Life in 
 another Period of Life, the Term to which fuch 
 Value in one Period is equal does not differ or va-^ 
 ry, by a juft and proportional Degree of Decreafe^ 
 from the Term to which fuch Value in ano- 
 ther Perio4 is ecjual : By Means, and in Con-f 
 I 4 fe(^uenc5
 
 fequence whereof, the Chance of the Dura- 
 tion of the Annuity does not keep equal Pace 
 with, and fink by the fame Degrees as the 
 Chance of Vitality on the Life decreafes : Nor 
 does the Value of the Annuity itfelf, if it 
 were confider'd, and could it be confider*d 
 fingly and abftra(5tedly from the Life on which 
 it depends, fink and decreafe from one Peri- 
 od of Life to another, in a jull and proper 
 Proportion. 
 
 What I mean by the Decreafe in the Value of 
 the Annuity, and by the Difference of the Term 
 to which fuch Value is equal, from, one Period 
 of Life to another, will appear beft by an im- 
 rnediate Infpedion into the Tables ; but to give 
 an Inftance of it here, it (lands thus. An An- 
 nuity for the Life of a Perfon between the Age 
 of 15 and 20 is in Value 13-33, which is equal 
 to a Term of 27-2-61 ; and an Annuity for the 
 Life of a Perfon between the Age of 20 and 25 
 is in Value 1^-78, which is equal to a Term of 
 ij-o-oo, according to the Doctor's Table : And 
 if we dedud 12-78 out of 1 3-33, the Remain- 
 der will be 00-55, ^"^ ^^^s ^s the Decreafe in the 
 Value ; and if we dedud 25-0-00 out of 27-2-6 1 , 
 the Remainder will be 2-2-61, and this is the 
 Difference in the Term, in thofe two Periods. 
 
 But this, perhaps, will be flill more plain and 
 obvious if exprefs'd in Terms more common 
 and familiar ; and fo exprefs'd ftands thus. A 
 Perfon of the Age between 15 and 20 has a 
 Chance to live 27 Years, 2 Quarters, and 61 
 Days; and one of the Age between 20 and 25
 
 [ 137 ] 
 
 has a Chance to live 25 Years ; that Is, Icfs than 
 a Perfon of the precedent Age 1 Years, 1 Quar- 
 ters, and 61 Days : And this is what I call the 
 Difference of the Term of Years from one Pe- 
 riod of Life to another : And iince the Decreafc 
 in Value will always correfpond to the Difference 
 in the Terrn, the Value of the Annuity in the 
 fecond Period is lefs than the Value of the An- 
 nuity in the firft Period by QO-55 in ]])ecimals, 
 which is equal to a Term of 2-2-6 1 in Revcrlion 
 after a Term of 25-0-00. 
 
 I have been the more exa6l in explaining this 
 Difference in the Term or Dccreafe in the Value, 
 which are much one and the fame Thing or al- 
 ways co-incide, becaufe the Objcdlion I make to 
 the Dodor's Table is, that it does not prefervc 
 a juft and proper Proportion in this Inllance: 
 fuch a Proportion I mean as can be juflificd by 
 any Bills of Mortality, or can be accounted for 
 by any Fa6ts that ever happen in Nature. To 
 Ihow this Defed in the molt clear and dillin(^ 
 Manner I can, I have inferred the Doctor's Table 
 in N^ III., in the Appendix; and fince the fame 
 Exception lies againft Mr. Hayes's, Mr. Richr 
 ^n/j's, and Mr. Marris's Tables for the Value of 
 Annuities on a Life, I have inferted them like- 
 wife, or Part of them, in the Tables N^ IV, V, 
 and Vr. and to each of them I have added three 
 Calculations of my own : The firll Column of 
 which contains an Account of the Term of Years 
 and Parts of a Year, to which every Life is 
 equal as it correfponds to the Value in the Tables 
 rclpef^ively : the fecond gives the feveral Diffe- 
 rences
 
 [ 138 ] 
 
 rences in the Term of Years, and the third 
 the Decreafes in the Value of the Annuity, 
 from one Period of Life to another, as they 
 come out on the feveral Tables relpedivcly. 
 In my own Table, N^* II. I have inferted 
 folely the Term to which a Life is equal, be- 
 caule the Value of the Annuity will on Courfe 
 attend on and be equal to the Term ; and to this 
 I have added 4 thjrd Column, which gives this 
 Difference in the Terni according to fuch Table, 
 and the Decreafe in the Value will be in the very 
 fame Proportion : from whence, and by an In-? 
 ipedlion into the feveral Tables, and a Compa- 
 yifon of them together, we may fee which of 
 them befl preferves that proportional Difference 
 in t\]e Term and Pecreafe in the Value, which I 
 fay, and undoubtedly^ ought to be observed in 
 fhis Cafe. 
 
 Since then thefe Tables are all of them de- 
 fective in thi§ Point, fome in a greater fome in a 
 lefs Degree, and this appears by a meer Inlpec- 
 tion into the Tables thernfelycs ; I fhall confine 
 my Reflexions here folely to Dr. Halky^s Table, 
 but intend that my Reafoning in general on this 
 Head fhould be applied, as it certainly is appli-? 
 cable, to all of them equally. 
 
 The particular Periods of Life which I lay 
 my Finger on, and fay are Faulty in this Table 
 in this Rclpedl, are 20 and 40 : in the firfi of 
 ■which the Difference in the Term is much too 
 ^reat, and in the latter too fmall. In the ibft 
 Column of this Table, which contains the Age§ 
 ©f Pcrfons^ we have 20^ which is the Period in- 
 cluding
 
 [ 139 ] 
 
 cli^ding the Lives from 2q to 25 ; and againfl k^ 
 in the fourth Golumnj which gives the Difference 
 betw^een the Term in that Period and the Tern; 
 in the preceeding Period, we find the Difference 
 to be 2-2-6 1 : And in the fame fourth Column 
 we fee the Difference for the preceeding Period 
 to be 00-2-30^ and that of the fubfequent one 
 ^o be 2-0-46 : and this Difference 2-2-61, com- 
 pared with fuch the precedent and fubfequent 
 Difference, is fuch a Proportion, I fay, as can- 
 not be maintain'd by any Bills of Mortality, at 
 Icaft any that ever I faw, or by any Reafon ia 
 the Nature of the Thing. 
 
 If we confult the Bills of Mortality for Lo;;- 
 do»y I mean the modern ones, where the Ages 
 of Perfons dying are diftinguiih'd in Periods 
 from ten Years to ten Years, and . the Extract 
 which I have added in the Appendix, it appears 
 |:hat the Number of Perfons dying in the Period 
 between 20 and 30 is lefs than the Number of 
 thofe dying hi the Period between 30 and 40 y 
 and if we Ihould fuppofc of thofe dying between 
 20 and 30, one half were of the Age between 
 20 and 25, and the other half between 25 and 
 30, which is plainly a more favourable Suppofi- 
 tjon than there are juft Grounds to make; yet 
 fmce the Difference in the intermediate Period, 
 ^hat is, in the Period between 20 and 25, ought 
 to bear a Proportion to the precedent and fub- 
 fequent one, it ihould therefore be fomething 
 greater than the precedent and fomething lefs 
 than the fubfequent one : Whereas in this inter- 
 fiiediate Period the Difference is 2-0-31 more 
 
 than
 
 [ X40 ] 
 
 than In the precedent one, which is an cxtrava" 
 gant Decreafe j and 'tis alfo more than the fub- 
 fequent one by 00-2-15, when the fubfequent 
 Difference, if there were any Variation, un-^ 
 doubted! y ought to be the greater ^ for that few- 
 er Perfons out of an equal Number die in the 
 Period between 20 and 25, than in the Period 
 between 2.5 and 30. And if we fhould fuppofc, 
 as we have before fuppofed, that the Number of 
 Perfons dying of the Age between ao and 25 
 and betvveen 25 and 30 was the fame • yet the 
 Number of Perfons co-exifting in tfie firll of 
 thefe two Periods mull be greater th?,n in the 
 fecond, and confequently the Chance of Morta- 
 lity in the latter Period muft be greater than in 
 the former, the fame Number dying out of a 
 lefs Number living : and therefore the Diffe- 
 rence in the latter ought to be greater. 
 
 I lliould take Notice, however, that, at the 
 Time when the Dodtor com,pofed this Table, 
 the Bills of Mortality for London were not pub- 
 h'fhed in the Manner they have been of late Years, 
 that is, with the Dillindion of the Ages of the 
 feveral Perfons dying j fo that thefe Bills could 
 be no Guide to him, nor his Table be foun^ 
 Fault with meerly becaufe it docs not quadrati? 
 with a Rule which was not then in Being. But 
 then the Bre/Jaw Table, which, if I underiland 
 liim rightly, was in great Iv|eafure of his owp 
 Formation, was in Being and before him when 
 he wrote ; and if this be the Standard, and ip 
 Truth 'tis the only proper Standard, whereby 
 to try this Table for the Value, o^ Annuities fox 
 
 a Life,
 
 [ 141 ] 
 
 a Lifcj my Objedtion of the Want of a propor- 
 tionable Difference in the Term and Decreafe in 
 the Value is ftronger upon an Infpediion into chat 
 Tabic than in any other Light whatever. 
 
 If then we confult the Bre/Jaw Table, which 
 I have inferted in the Appendix, it will appear 
 that the Number of Perfons dying in a Year in 
 the Period- between 15 and 10 is in the whole 
 30j the Number dying in the Period between 
 20 and 25 is 31, and between 25 and 30 is 36. 
 Upon this I obferve, tho* the Number of Perfons 
 co-exilling in the firft of thefe Periods of Life 
 is greater than the Number co-exifting in the 
 fecond ; yet that the Number of Perfons dying 
 in the firft is lefs than the Number of thofe dy- 
 ing in the fecond Period ; and the fame Obfcr- 
 vation holds good in comparing the fecond with 
 the third Period : From whence it neceffarily 
 follows that the Difference in the Term ought 
 to be in a Proportion correfpondcnt, or, in other 
 Words, fuch Difference in the Term ought to 
 bear a Proportion anfwering to the Increafe in 
 the Chance of Mortality : And if the Breflaw 
 Table be a Rule for the Increafe of the Chance 
 of Mortality, it mufl be a Rule for the Difference 
 in the Term, and yet for this Difference in the 
 Term we fee that the proper Proportion is di- 
 redly contradided, or at leaft apparently un- 
 preferved, in this Table of the Value of Annu- 
 ities. 
 
 Many Perfons, I know, have a Notion that 
 the Age of 30 is the beft Age of Life on which 
 to have an Annuity depending, for that young 
 
 People
 
 [ 142 ] 
 
 icopit under that Age are expofed to tribf^ 
 Hazards and Cafualties than thofe who are ar- 
 rived to fome Maturity of Age and Difcretion; 
 and confequently that the Chance of Vitality in 
 the younger Part of Life is lefs, at leaft more 
 Uncertain) than at the Age of 30, or there- 
 abouts; and that a loofe Calculation of the 
 Chance of the Duration of a Life may fuffice 
 here. But Dr. Halley difcover^d no fuch Thing 
 in the Bills of Mortality for Bnjlaw^ nor can 
 any one difcovcr it in the BreJJaw Tables which 
 he form'd from thence ; for there the Number 
 of Perfons dying in the Period from lo to 20 is 
 63, from 10 to 36 is 67^ and from 30 to 40 is 
 86. If we look into the modern Bills of Mor- 
 tality for London^ and my Extrad from thence^ 
 there we find the Cafe to be the fame in the main^ 
 though not in the fame Proportion, and that the 
 Number of thofe dying in the older Period of 
 Life greatly exceeds the Number of thofe dy- 
 ing in the Younger t and the Account there 
 jftands thus. Out of every thoufand Perfons dy- 
 ing, there die, of the Age between lo and 26^ 
 30 \ of the Age between 20 and 30, 72 ; and of 
 the Age between 30 and 40, 93 : And from 
 hence it appears evidently that this Notion of 
 the beft Age of Life is not a jull one* 
 
 But if there was a better Foundation for the 
 Notion than there appears to be, and if thete 
 was Room for a Latitude in the Calculation' of 
 the Chances of Mortality for the Age between 
 10 and 30, this can avail nothing In the Age of 
 40 or thereabouts ; and yet in the Period of 40 
 
 we
 
 t 143 ] 
 
 we have an improper Proportion in the DifFe* 
 rence of the Term, for that it is too fmail, when 
 compared with the Difference in the Periods on 
 both Sides of it. In this Inftance, indeed, the 
 Deviation is not great ; but if any Difference 
 were made, it Ihould have been greater in this 
 than in the precedent, and lefs than in the fub* 
 fequent Period ; that is, it ihould grow greater 
 and greater gradually, as the Age of Life ad- 
 vances : whereas the Difference is lefs in this Pe* 
 riod than in the Periods on either Side. And 
 fince all that I have urged againfl the Want of 
 Proportion by an over Difference is equally ap- 
 plicable to this Cafe of an under Difference ; I 
 fhall only add, that I believe no Bills of Mor* 
 tality whatever or any Thing in Nature do, 
 and that I am very fure the Bills of Mortality 
 for BreJIaw or for LotKion do not, warrant the 
 one or the other. 
 
 Thefe are the fingle Articles in this Table 
 which are moft exceptionable j but, in Truth, 
 this Dcfed goes through the whole Performance, 
 which I crave Leave to Ihow by an Inftance, 
 which, in my Apprehenfion, demonftrates the 
 Defedl, and that it really is fuch. The Dodof, 
 on the Bills of Mortality for Bre/laWy obfervcs, 
 and I believe all Mankind will agree it to be 
 true in the reft of the World as well as there, 
 that out of a certain Number of Perfons in the 
 Decline of Life, more die in a Year, or any de- 
 terminate Number of Years, than there do out 
 of an equal Number of Perfons in |he Youth 
 and Vigour of Life ; And he has himfelf given 
 
 us
 
 Us the B-re/law Table, as a Scheme of the Id- 
 creafe of Mortality, according to the Advance 
 in Age. Now if the Chance of Mortality in- 
 creafcs, and the Chance of Vitality decreafes, in 
 Proportion to the Advance in Age, as is here 
 alTertcd, and is undoubtedly true ; and if the 
 Value of an Annuity for a Life, or the Number 
 of Years to which a Life is equal, which differ 
 in nothing but the Manner of Exprellion, does 
 depend on the Chance of the Vitality of the 
 Life and on nothing elfe, as moft certainly it 
 does, and fhall be proved beyond Contradi6tion ; 
 i'(- 'evidently and neceflarily follows, that the 
 Value of an Annuity for a Life, or the Number 
 of Years to which a Life is equal, mull decreafc 
 fafter and by greater Degrees in the older than 
 in the younger Stages of Life, and in the fame 
 Proportion as the Chance of Vitality decreafes. 
 
 Let us fee now how this Difference or De- 
 •creafe in the Term will ftand, if we put toge- 
 ther the whole of the Decreafes for the four 
 Periods of younger Life, of middle Life, and of 
 the oldeft Life. The whole then for the firfl Set 
 is a Term of 7-1-46, for the fecond 7-0-15, and 
 for the third 7-1-00 : which, we fee, is diredly 
 contrary to the Rule laid down, and almoU in- 
 ^verts the Proportion. If we would know what 
 is the right and juft Proportion to be obferved, 
 and might depend on the BrefJaw Tables as 
 giving us fuch Proportion, and this Author's 
 Table for the Value of Annuities was framed 
 from thofe Tables; on a Computation from 
 thence, tl|c Cafe, as between the Perfons of the 
 
 youngeft
 
 [ 145 ] 
 
 youngeft and the oldeft Period of Life, flands 
 thus. The Number of Perfons co-exiiling above 
 the Age cf 10 and under 30, which contains 
 the four Periods of younger Life, confifting of 
 twenty Years, is in all 11 875; and the Num- 
 ber of Perfons dying out of them in one Year 
 is 130;" that is, one in 91, or thereabouts. The 
 Number of Perfons exifting above the Age 
 of ^^ and under y^, which contains the four 
 Periods of older Life, conlifting likewife of 
 twenty Years, is in all 3726 ; and the Number 
 of Perfons dying out of them is 204, that is 
 one in 18, or thereabouts. The Proportion 
 here then is plainly five to one ; that is, of an 
 equal Number of Perfons of each Stage of 
 Life, the Number of the older Stage of Life 
 dying in a Year will be five, and of the younger 
 Stage one only : Or if one Perfon only be nomi- 
 nated of each Stage of Life, the Chance of 
 Mortality on the Side of the Perfon of the older 
 Stage of Life againft the Perfon of the younger 
 Stage is five to one. Whether this Proportion 
 of the Increafe of Mortality, as it arifes from a 
 Computation on the Bre/Iaw Table, be a jufl 
 one, I neither affirm nor deny ; but be it right, 
 or be it wrong, this Author's Table for the 
 Value of Annuities for a Life was framed from 
 thence ; and therefore the Difference or Decreafe 
 in the Term, from one Period of Life to ano- 
 ther, and between the Periods of younger and 
 elder Life, (hould bear fuch a Proportion to one 
 another on the Table for the Value of Annui- 
 ties, as the Increafe of the Chance of Vitality in
 
 [ 146] 
 
 one Period of Life bears to that Increafe in ano- 
 ther Period, according to the Bre/law Table : 
 And I may appeal to the Do6tor's own Calcu- 
 lations, whether the Dccreafes given in one 
 Table are conformable to the Decreafes in the 
 other Table, or are agreeable to any Rule which 
 arifes from the Nature of the Life of Man. 
 
 I might reft the Matter here ; but fince it may 
 be urged, although the proportional Decreafc 
 contended for is not preferved, when we com- 
 pare it, as we have hitherto done, with the 
 Scheme of the Difference of the Term ; yet if 
 we compare it with a Scheme of the Deereafe of 
 the Value, and the Table of Annuities is framed 
 by a Computation from the Value and not from 
 the Term, there pofFibly the proper Proportion 
 may be well enough preferved. For this Rea- 
 fon I made the Calculation infcrted in the Fifth 
 Column of the Table N*- III, which contains 
 an Account of the Deereafe of the Value of an 
 Annuity from one Period of Life to another, 
 computed by the Values only ; and we find the 
 very fame Defects here as we had before, and in 
 the very fame Inftances : And indeed, it muft 
 of Neceflity fo come out, fince the Value and 
 the Term do always reciprocally correfpond. 
 
 The fingle Periods of Life which we found 
 Fault with, according to the Scheme in the 
 Terms, were 20 and 40 ; and in thefe two Ar- 
 ticles the fame Error occurs upon this Scheme 
 as we met with on the former. On the Period 
 2.0 we objedled that the Deereafe, when com- 
 pared with the precedent and fubfcquent one, 
 •was too great , and here the fcveral Decreafes of 
 
 thefe
 
 [ 147 ] 
 
 thefe three Periods being, as flated in Decimals, 
 II, 55, 51, it appears plainly on the Face of 
 them that the Middle one, 'viz. that for the Pe- 
 riod 20, is greater than the precedent one by a 
 Difference much too large, and alfo greater than 
 the fubfequent one, when undoubtedly it ought 
 to be fomething lefs. And as for the Period 40^ 
 to avoid Repetitions, I would refer you to the 
 Scheme itfelf ; an Infpedion into which does 
 plainly enough difcover the Defeat ; for the De- 
 creafe in that Period is lefs than in either of thofe 
 on each Side, and therefore cannot poffibly be 
 in a proper Proportion to both of them. 
 
 Theie, indeed, are not very great Miftakes^ 
 or, however, by a fmall Variation might be rec- 
 tified ; but if thefe were fet right, yet the grand 
 Error will Hill remain, 'uiz. Want of Proportion 
 through the whole Scheme. The Decreafe in 
 the four firfl Stages of Life is indeed in Value, 
 computed by Way of Decimals, 1-72 ; and in 
 the latter Stages is 3-pi ; notwithllanding which, 
 in Reality, there is no great Difference in thefe 
 two Decreafes, and what Difference there is lies 
 on the wrong Side, viz. on the Side of I-72, 
 that being the greater Decreafe of the two. To 
 fay that 1-72 Value in Decimals is fomething 
 greater than 3-pi Value in Decimals, feems to 
 be a Paradox, and if faid of them limply and 
 lingly taken, is manifeflly a Contradidion t but 
 then they may be fo placed with Reference to 
 fomething elfe, or may be confider'd as Part of, 
 or taken out of, fomething elfe in fuch a Manner, 
 as that the firft Ihall be a Value greater than the 
 K 2 fecond.
 
 [ 148 ] 
 
 feeond. For Tnftance, i fiinply taken and by 
 itfelf is certainly not {o much as 2 fo taken, but 
 1 conlider'd as Part of 10, or as drawn out of 
 10, is fomething greater than 2 confider*d as Part 
 of 40, or as drawn out of. 40 ; for the one is a 
 tenth and the other a twentieth Part only of the 
 Thing to which it llands related : And that this 
 IS the Cafe here will appear from hence. 
 
 A Term for po Years, Tnterclt computed at 
 6 1, per Cent, is in Value 16-58 ; and the Propo- 
 lition inverted is a true one ; 16-58 in Value is 
 equal to a Term of po Years : Now if 'out of 
 16-58 we draw 41 Decimals, the Remainder 
 will be 16-17, ^"'^ 16-17 ^3 equal to a Term of 
 60 Years, and no more ; fo that a Subllradion 
 of 41 Decimals only leflens the Term here 30 
 Years. A Term for 21 Years, at the fame Rate 
 of Intereft, is in Value 1 1-76 ; and if we draw 
 cut of this 41 Decimals, the Remainder will 
 be li-35, and 11-35 ^^ equal to a Term of 
 lp-2-30: So chat the Subftradion of 41 Deci- 
 mals out of the Ihorter Term leflens that Term 
 only one Year and an half, or little more ; 
 whereas a Subftradiion of 41 Decimals out of the 
 longer Term leflens that Term ^o Years. Frorn 
 hence *tis plainly feen, that one and the fame 
 Value In Decimals, drawn out of a greater given 
 Value in Decimals, and out of the longer Term 
 to which fuch given Value is equal, lefTens the 
 Term out of which 'tis fubllraded in a much 
 greater Degree, than the fame Value in Decimals, 
 drawn out of a lefs given Value in Decimals, 
 and out of the fhorter Term to which fuch given 
 
 Value
 
 [ 149 ] 
 
 Value is equal, lefiens the Term out of which 
 that is fubltra6ted. Therefore it may be true, 
 that 1-72 Value in Decimals, though not half fo 
 great a Value as 3-pi, drawn out of one Value, 
 Ihall leiien the Term out of which it is fubltrac- 
 ted as much, or more, than 3-pi Value in Deci*- 
 mals, though more than double the Value of 
 the other, drawn out of another Value, lliall 
 Icflen the Term out of which that is fubrtradted. 
 If !wc examine out of what thefe two Values 
 1-72, and 3-91, arc refpedtively drawn; it ap* 
 pears that 1-72 is drawn out of the Value 13-44, 
 ■which is equal to a Term of 28-1-00; that is 
 out of the greater given Value, and the longer 
 Term correfponding : and ^-^i is drawn out of 
 the Value p-2l, which is equal to a Term of 
 13-3-3Q, that is, out of the lefs given Value and 
 the fhorter Term correfponding. If we pro- 
 ceed in the Computation, and apply this, it 
 ftands thus. The Value 1-72 fubftraded from 
 13-44, leaves a Value 11-72: the Value 11-72 
 is equal to a Term of 20-3-45 : a Term of 
 20-3-45 fubftradied from a Term of 28-1-00 
 leaves a Term of 7-1-46 ; and this is the T)e^ 
 creafe in the Term for the four Periods of young-? 
 er Life, The Value 3-91 fubftraded from 9-21 
 leaves a Value 5-30 ; the Value S'S'^ '^ equal to 
 a Term of 6-2-30; a Term of 6-2-30, fub« 
 flradted from a Term of 13-3-30 leaves a Term 
 of 7-1-00 ; and this is the Decreafe in the Term 
 for the four Periods of older Life. Therefore 
 qppn the whole, whether the Decreafe here be 
 taken immediiitely from the Ternij or be com^ 
 ■ 1^ 3 pute4
 
 [ 15° 1 
 
 puted from the Value, and fo to the Term, this 
 Decreale in the Term comes out to be one and 
 the fame ; that is, the Dccreafe of the Term in 
 the Stages of younger Life is greater than in the 
 Stages of older Life : and confequently, the 
 gradual Decreafe of the Chance of Vitality, in 
 Proportion to the Increafe of Age, is not pre- 
 ferved : which was the Thing to be dcmonflrated, 
 and I think is fully done. 
 
 This Way of Reafoning may feem, perhaps, 
 to have fome Difficulty in it, but, I believe, will 
 be rendered more intelligible when I have Ihown, 
 as I Ihall have Occafion to do hereafter, the great 
 Difference between adding to, or fubltrading from 
 a Term of Years, and adding to or fubftrading 
 from the Value correfponding to the Term. It 
 will appear then : If a Term and a Value corre- 
 fponding are given, where a Subllradtion is made 
 of a proportionate Part, (fuppofe a Quarter) out 
 of the given Value, and a SubHradtion of the 
 fame proportionate Part out oi^ the given Term, 
 that the Value remaining will not correfpond to 
 the Term remaining on fuch Subflra6tion. If 
 this be the Cafe, which I fhall in a proper Place de- 
 monftrate to be the Cafe ; it may be true, that 
 1-72 being the Value remaining on a Subftra6tion 
 jnade out of one given Value and a Term cor* 
 rcfponding, and 3-5)1 being the Value remaining 
 on a SubHradtion made out of another given 
 Value and a Term correfponding, fhall be equal 
 to one and the fame Term ; or 1-72 the lefs 
 Value fhall be equal or correfpond to a longer 
 T^rm, than 3-91, tho' the greater Value. 
 
 That
 
 [ 151 ] 
 
 That this is the Fad here, is proved beyond 
 Contradidtion by the Way of arguing which 
 follows, which I hope will be clear enough, and 
 I'm fure is right. By the Dodlor's Table for 
 the Value of Annuities for Lives, the Life of a 
 Perfon aged lo is equal to a Term of 28-1-00 ; 
 the Life of the fame Perfon, when aged 30, is 
 equal to a Term of ao-3-45 : the Difference be- 
 tween thefe two Terms is evidently 7-1-46: 
 that is, in 20 Years Time the Life of this Perfon, 
 whim in the younger Periods of Life, is dc- 
 creafed or grown lefs in Computation a Term of 
 7-1-46. The Life of this Perfon, when aged 
 30, is equal, as noted before, to a Term of 
 ao-3-45; the Life of the fame Perfon, when aged 
 50, is equal to a Term of 13-3-30 ; the Diffe- 
 rence between thefe two Terms is 7-0-15 ; that 
 is, in thefe 20 Years Time the Life of this Per- 
 fon, in thefe middle Periods of Life, is impaired 
 or grown lefs upon an Eftimate a Term of 
 7-0-15. The Life of this Perfon, when aged 
 50, is equal, as noted before, to a Term of 
 ^3''3'3^'') the Life of the fame Perfon, when 
 aged 70, is equal to a Term of 6-2-30 ; the 
 Difference between thefe two Terms is 7-1-00; 
 that is, in 20 Years Time in the oldelt Periods 
 of Life, the Lite of this Perfon is grown worfe 
 and diminifh'd, upon the Chance, a Term of 
 7-1-00: So that the Differences or Decreafes in 
 the Term will Hand thus ; for the 20 Years in 
 youngell: Life 7-1-46, for the 20 Years in middle 
 Life 7-0-15, and in the oldeft of all 7-1-00. 
 
 K 4 Now
 
 [ 152 ] 
 
 Now fince this Table for the Value of Annu- 
 ities is conllruded on the Foundation of the 
 Table of Mortality for BreJJaWy it ought, as I 
 obferved before, to agree with and be conform- 
 able to that Table, or there is a Defeat in the 
 Structure ; but on looking into this latter Table, 
 it appears that the Number of Perfons dying in 
 the firft Period, that is between lo and 31, is 
 138 ; in the fecond Period, between 30 and 51, 
 the Number is 196 ; and in the laft Period, be- 
 tween 50 and 71, the Number is 215; and yet 
 the Number of Perfons co-exifting is greater in 
 the younger than in the older Period of Life, 
 and gradually dccreafes as Age increafes : From 
 •whence 'tis plainly feen, in this Calculation as 
 well as in a former, that the Table of Annuities 
 has no fort of Conformity with the Table of 
 Mortality, though the one be built, in the main, 
 on the Foundation of the other. And as this 
 Want of Conformity between thefe two Tables, 
 in a proportionable Decreafe, runs through the 
 Tvhole, To I cannot forbear obferving the foul 
 Work that it makes in one parncular Inflance, fuch^ 
 as moll certainly overthrows the Juftlce of this 
 Table for Annuities, and of the Rule too, by 
 which it was framed. 
 
 The Value of an Annuity for a Life of 10 
 Years Age is by this Table 13-44, which Ts 
 equal to a Term of 28-1-00: and the Value of 
 an Annuity for a Life of 70 is 3-32, which is 
 equal to a Term of lix Years and tvyo Thirds. 
 If weco'nfult the Brejlaiv Table, the even Chance 
 of the Duration of the Life of 10 Years Age is 
 
 41 Years
 
 [^53] 
 
 41 Years and over, and the Chance of the Du- 
 ration of the Life of 70 is fix Years, and aboun 
 two Thirds of a Year. Now that an Annuity 
 for the Life of a Perfon aged 10 Years fliould be 
 in Value 13-445 that is, ihould be equal only to 
 a Term of 2 8-1 -co, when his Life, on the even 
 Chance, is equal to 41 Years and above ; and 
 that an Annuity for the Life of a Perfon aged 
 70 fliould be in Value 5-32, which is equal to a 
 Term of fix Years and two Thirds, when his 
 Life, on the even Chance, is equal only to fix 
 Years and two Thirds : 1 fay, that the Chance 
 of the Duration of the Annuity in one Cafe 
 ihould fall fiiort of the Chance of the Duration 
 of the Life, and fo much a^ thirteen Years ; and 
 in the other Cafe, that the Chance of the Dura- 
 tion of the Annuity and of the Life fhould be 
 fo near an Equality, or the very fame- has 
 fomething in it fo much of the abfurd, that lean- 
 not reconcile it to my Underftanding. And yet 
 if we compute the Value of an Annuity for a 
 Life of 80, and make Ufe of the Rule which 
 the Dodlor prefcribes and made Ufe of in frame- 
 ing his Table for fuch Value, we Ihall difcover 
 fomething ftill more extraordinary: 'viz. that an 
 Annuity for fuch a Life will be in Value 3-84, 
 which is equal to a Term of four Years and an 
 half, when on the BreJIaw Table the even Chance 
 of the Duration of fuch a Life is not full four 
 Years ; which I think is fo palpable a Contra- 
 diction to common Senfe, that nothing can main- 
 tain the Rule by which it was produced. But 
 this Objedion goes to the Rule itfelf by which 
 
 the
 
 [ 154 ] 
 
 the Table was framed, and I meant here to take 
 Notice only of the Defedts which arife on the 
 Table, fo I fliall wave any farther Reflexions on 
 that Point ; intending to refume the Confidera- 
 tion of it in a more proper Place : and conclude 
 that I have made good my firfl; Exception, that 
 a proper and proportional Decreafe in the Term, 
 or in the Value of thefe Annuities is not pre- 
 lerved in thefe Tables. 
 
 A fecond Objedion lies againft Mr. Hayeses 
 and Mr. Richards*s Tables, viz: that, being 
 computed for the feveral Rates of 4, 5, 6, 7^ 
 and 8 per Cent, they give us fuch a Value of an 
 Annuity for a Life, as that one and the fame 
 Life is equal to a different Term for each Rate 
 of Interelt. There is no Room for this Defedt 
 in Dr. Hal/e/s or Mr. Morris's Tables, they be- 
 ing computed for a iingleRate of Intereft only; 
 but in Truth, the Objedion does lie againll the 
 Dodor's Rule for forming thefe Tables ; for 
 that any Table drawn by his Rule, and for more 
 Rates of Intereft than one, will have the fame 
 Dcfctl* This appears plainly enough by Mr. 
 Kichards^s Tables, which were conftruded by 
 this Rule ; and that this of NecelTity muft be 
 the Cafe in all Tables for Variety of Interefts 
 drawn by the Rule, will be demonftrated when 
 I come to fpeak to the Rule itfelf. 
 
 Mr. Morris's Table ftands clear of this Error, 
 and fo indeed would any Table he Ihould frame 
 for Variety of Interefts, if made in the Manner 
 in which I fuppofe his to have been made. He 
 does not diredly tell us by what Method he 
 
 formed
 
 [ 155 ] 
 
 form'd his Table ; but I think I can fee that he 
 made Ufe of Dr. Halley's Table as his Model ; 
 and his Way of Reafoning upon it was thus. 
 An Annuity for the Life of a Perfon, for In- 
 ftance of ten Years Age, by the Do6tor's Tabic, 
 is in Value 13-44; the Value 13-44 at 6 per 
 Cent. Intereft is equal to a Term of 28-1-00, 
 or thereabouts; every Life of ten Years Age is 
 equal to one and the fame Term, at whatever 
 Rate Intereft be computed ; a Term of 28-1-00 
 at 4 /. per Cent, is in Value 16-72, cr thereabouts : 
 Therefore the Value of an Annuity on fuch a 
 Life, where Intereft is computed at 4/. per Cent. 
 muft be 16-72 : and accordingly this is the Sum 
 given in his Table as the Value of an Annuity 
 on fuch a Life at that Rate of Intereft. That 
 this muft have been his Manner of Reafoning 
 and forming his Table, and fure enough 'tis a 
 juft one, is evident from hence ; that the Term 
 to which the Value given in his Tabic corre- 
 fponds at the different Ages of Life varies only 
 a few Days from the Term to which the Value 
 given in the Do6tor*s Table correfponds at the 
 fame Age of Life ; and in two Inftances the 
 Term correfponding to the Value given in one 
 Table is the fame, to a lingle Day, as in the other. 
 As to Mr. Hayes's and Mr. RkbarJs's Tables, 
 that thefe two Tables for every Rate of Intereft 
 give us fuch a Value of an Annuity for a Life as 
 does in Fad make one and the fame Life equal 
 to a different Term of Years, and what that 
 Variance is on each of them, may be beft and 
 moft plainly feen by the fhort Schemes in Tables 
 
 N^'- VIL
 
 [ ^56 ] 
 
 N^- VIL and N^- VIII. The firft of thefe gives 
 us the Value of an Annuity for a Life of 30 
 Years Age, and the Term to which fuch Value 
 is equal at the feveral Rates of Interell, as ftated 
 by Mr. Hayes himfcif ; and *tis marvellous to me 
 how he could make the Computation, and not 
 fee the Abfurdity of the Suppofition on which it 
 mufl be grounded : And the other gives the 
 Value of an Annuity for a Life of 12 Years 
 Age, at the like feveral Rates of Intereft, as 
 ftated by Mr. Richards^ and the Term to which 
 fuch Value is equal, as I compute the fame. 
 
 Now I fay, that the Values ftated in thefe two 
 Tables thus correfponding to and producing a 
 different Term of Years to which one and the 
 fame Life is equal, according to the feveral Rates 
 of Intereft at which the Calculations are made, 
 undoubtedly are, and neceffarily muft be, wrong 
 lor that Reafon becaufe they produce fuch dif- 
 ferent Terms. This being the Cafe on both 
 thefe Tables • and iince Mr. Hayes has not ac- 
 quainted us what Rule he made Ufe of in form- 
 ing his Table, and Mr. Richards informs us that 
 he built on Dr. Hallcfs Hypothefis, and 'tis 
 evident he did fo : And fince one and the fame 
 Defe<fl goes through both Tables, I ftiall apply 
 myfelf chiefly to Mr. Richards's Table, and the 
 Rule by which that was conftrudted : Only I 
 take Notice here, that my Obfervations on one 
 of them, in moft Inftances, and fpecially in that 
 which I have before mentioned, will equally, af-^ 
 fe6t and be applicable to the other, whatever 
 was the Rule by which it was framed, 
 
 I fuppofq
 
 [157] 
 
 I fuppofe no one will controvert thofe Points: 
 That he who has an Annuity for the Life of a 
 Perfon has an Annuity for fuch a Term of Years 
 as fuch Perfon in Fad: fhall live ; and when he 
 buys it, the Term of Years to which any Perfon's 
 Life Ihall be prolonged being uncertain, that he 
 buys it for fuch a Time as there is a Chance or 
 reafonable Probability that the Perfon may live 
 whofe Life is nominated : And I am fure 'tis 
 poffible that five feveral Annuities, at the five 
 feveral Rates of Intereft in thefe Tables may be 
 granted on one and the fame Life, as well as on 
 different ones. To fay then, that in Cafe of one 
 Annuity there is a Chance or Probability that 
 the Perfon on whofe Life the Annuity depends 
 may live for 30 Years, or near it ; in Cafe of a 
 fecond Annuity, that he may live for 29 Years, 
 or thereabouts ; and on a third, on a fourth, and 
 on a fifth Annuity, on each of them, that he 
 can and probably may live for a different Term 
 of Years • is ridiculous and abfurd to the lafl 
 Degree. And if we take a particular Inftance, 
 and fuppofe two Annuities only, at the two dif- 
 ferent Rates of 4 /. and 8 /. per Cent, to be grant- 
 ed to two diftind: Perfons on the Life of one and 
 the fame Nominee j to fay, with Regard to the 
 Annuitant who purchafes at 8 /. per Cent, that the 
 Nominee, on whofe Life the Annuity depends, 
 may or has an even Chance to die in 25 Years \ 
 but with Regard to the Annuitant who purchafes 
 at 4/. per Cent, that the fame individual Perfon 
 may probably live, or has a Chance to live 30 
 Years, is a flat Contradiction j and is, in Effed, 
 
 to
 
 [ 158 ] 
 
 to aflcrt, either that one of the Annuities will 
 continue after the Nominee is dead on whofc 
 Life it depends, or that the other will ceafe 
 whilft the Nominee is living for whofe Life 'tis 
 to continue ; when the Annuides mud both de- 
 termine at one and the fame Time, on the Death 
 of the Nominee. 
 
 Now if Mr. Ricbards*s Tables were the Re- 
 fult of Dr. HaJIeys Obfervations, and were cal- 
 culated by the Method prefcribed by him, or by 
 Mr. Moivre, as I believe they were; yet if the 
 Error I have mention'd proceeds from the Foun- 
 dation or the Method, as moll certainly it does, 
 I apprehend it will prove that the one or the 
 other of them is wrong, much more ftrongly, than 
 the Foundation or the Method will prove that 
 the Calculations are right, notwithftanding the 
 great Authority of the Inventors of them : and 
 therefore, fince the Operations upon them have 
 produced fuch abfurd Effeds, I don't fee they 
 are of any Significance, except to have demon- 
 flrated that the Inftrudtions are wrong. I muft 
 confefs I do not know any Method, and very 
 much quellion whether there be any, to adjuft 
 the Value of an Annuity for a Life, but by com- 
 puting for what Term there is a Chance or Pro- 
 bability fuch Life fhall continue ; and when that 
 is done, the common Tables, which give us the 
 Value of any Term of Years at a flated Inte- 
 reft, give us at the fame Time the Value of an 
 Annuity for the Life fought after at that Rate' 
 of Intereft. But fince Dr. Halky has found out 
 and publifh'd another Rule for this Purpofe, we 
 will confider it. The
 
 [ 159] 
 
 The Rule then which he prefcribes, and which 
 he and Mr. Richards made Ufe of, to form theie 
 their Tables for the Value of Annuities on a 
 iingle Life, is grounded on the Brejlaw Table, 
 and fome Proportions deduced from thence. 
 The Bre/law Table exhibits the whole Number 
 of People fuppofed to be living in that City at 
 a certain Time, and the fuppofed current Age 
 of them all, from the Birth to the Age of 84 : 
 And at prefcnt I will fuppofe, though I do by 
 no Means grant, that this Table gives a true 
 Account of the Perfons living there, and their 
 feveral Ages. To afcertain the different De- 
 grees of Vitality in all Ages of Life, this Au- 
 thor advances this Propofition. If the Number 
 of 'J^erfotis of any Age remaining after one Tear he 
 divided by the Difference between that and the 
 Number of the Jge propo/ed, it Jhows the Odds that 
 there is, that a ^erfon of that Jge does not die in 
 one Tear. For Inftance ; a 'T^erfon of 25 Tears of 
 Jge has the Odds of 560 to 7, or 80 ?o i, that 
 he does not die in a Tear : Becaufe that of ^6y 
 livingy of 25 Tears of Age, there do die no mors 
 than 7 in a Tear, leaving 560 of 26 Tears old. 
 
 Here I have given the Author's Propofitlon 
 and Inftance in his own Words ; but if I may 
 have Leave to exprefs the Meaning of them in 
 my Way, which, perhaps, may tend to explain 
 and illuftrate them, I Ihould fay : That the 
 Number of Perfons living of any Age propofed 
 was the Number of Chances belonging to a Per- 
 fon of that Age ; that the Number of Perfons 
 living of any fubfequent Age, fuppofe at one 
 
 Year's,
 
 [ i6o ] 
 
 Year 3, at ten Years, or at twenty Years ViC" 
 tance, was the Number of Chances with him, 
 or the Number of Chances that he had to live 
 to fuch fubfequent Age; and that the Difference 
 between the Number of Pcrfons living of the 
 Age propofed, and the Number of Perfons living 
 of the fubfequent Age, that is the Number of 
 Pcrfons djing in that Time was the Number of 
 Chances againft him, or the Chances that he 
 ■would die within that Time, or before he ar- 
 rived at fuch fubfequent Age. In the Inftance 
 here given, a Perfon of the Age of 25 has ^6y 
 Chances, fo many Perfons being living of that 
 Age J of the Age of 26, which is the next 
 Year, there are 560 Perfons living, and within 
 the Compafs of that Year there have died, feven 
 Perfjns ; fo that on the Life of a Perfon of 25, 
 there are 560 Chances with him that he lives to 
 the End of one Year, fo many being living at 
 the B'.nd of that Year, and there are 7 Chances 
 againil him that he dies within that Time, fo 
 many being dead in that Time. This Method, 
 if purfued, gives the Chances for him and againft 
 him in like Manner for any other fubfequent 
 Year. For Inftance ; the Number of Perfons 
 living of the faid Age of 25 is ^Sy, the Num- 
 ber of Perfons living at ten Years End, or of 
 the Age of ^^^ is 490; the Difference between 
 thefe two Numbers, or the Number of Perfons 
 dying in that Time, is yy ; therefore the Chances 
 vv'ith him, or the Chances that the Perfon lives to 
 ten Years End, are 490; and the Chances againft 
 him, or the Chances that he dies within thatTime, 
 are yy, I have
 
 [ i6r ] 
 
 I have here avoided the Ufe of the Word 
 Odds, bccaufe 'tis an equivocal Term, and may 
 mean the Odds in the Number of Chances ; or 
 the Odds in the Value of thoie Chances, and in- 
 flead thereof I make Ufe of the ExprefTion, 
 Chances with him or Chances againft him, and 
 ihall do fo hereafter ; for if this Author does 
 never intangle himfelf and his Reafoning with 
 the Ambiguity of that Term, I am fure another 
 does : For I have met with an algebraical Calcu- 
 lation of the Length of two Pages, compofed 
 by the Honourable Francis Roberts a Gentle- 
 man very learned in Figures, to prove a Propo- 
 rtion which he calls an arithmetical Paradox, 
 which, in Truth, is no better than a meer 
 Quibble on the double Meaning of the Word 
 Odds ; for, inftead of the Word Odds in the 
 Gentleman's Propolition, let us infert the Ex- 
 prefTion, Chances for and againft him, and the 
 Paradox is gone, and the Demonftration unne- 
 ceffary. 
 
 The Propofition, as it flands in a Colledlion 
 of Tracts, which are taken out of the Philofo- 
 phlcal Tranfadlions, is this, ^bere are two Lot" 
 teries, in either of whicJo a Gamejier paying a ShiU 
 ling for each ticket \ the firji Lottery^ uponajaji 
 Computation of the Odds, has ^ to i of the Gajne- 
 fler, thefecond has hut ito \\ nevenhelefs^ ths 
 Gamejier has the fame {and no more) Difadvan* 
 tage in playing at the fir fi, as in the fecond Lot" 
 tery. I admit that the Gentleman proves the 
 Truth of his Pofition by a rational Argument, 
 and I doubt not but he proves it by his algebrai- 
 L cal
 
 [l62] 
 
 cal Procefs like wife ; but this, I think, is full as 
 great a Paradox in Arithmetick, wz. that i is as 
 much as 42 ; and yet moll certainly this is a true 
 Propolition, if by a I mean two Pieces of Gold 
 caird Guineas, and by 42 I mean fo many Pieces 
 of SiLver call'd Shillings: But I believe there is 
 never a Cobler in the Kingdom who fings over 
 old Shoes, or Ploughman that whiftles after his 
 Horfes, but could readily clear up this Difficul- 
 ty. And the feeming Paradox here, to be fure, 
 might have been folv'd without fuch a vaft Pro- 
 fufion of Algebra as is expended on this Occa- 
 fion ; fpecially when it had been laid down as a 
 Maxim, that the Value of a Ticket in a Lot- 
 tery is the Sum produced on dividing the total 
 Value of all the Prizes by the Number oi Tickets; 
 for this algebraical Procefs, as long as it is, proves 
 nothing but what was before taken for granted 
 or proved by this Maxim \ and therefore fome 
 will furmife that this is meerly an Impropriety 
 of Language, and that the Procefs in Algebra 
 need not to have been quite fo long to prove 
 that a Man may talk or write improperly, fpe- 
 cially if he fludies to do fo. 
 
 And though this Author has very well proved 
 his main Pofition, that an Adventurer has an 
 equal Difadvantage in either of thefe Lotteries ; 
 yet he has mif-ftated the Cafe in both of them, 
 or has mif-computed the Odds in them, when 
 he tells us, the Odds in one is 3 to i, and in the 
 other a to I ; for the Odds are 2 to I in one 
 Lottery, if 4 Blanks to 2 Prizes makes fuch 
 Odds, and an even Chance in the other Lottery, 
 
 if
 
 [ i63 ] 
 
 if an equal Number of Prizes and Blanks makes 
 an even Chance : For the Number of Tickets in 
 each Lottery is fix, and in the firll there are four 
 Blanks and two Prizes, and in the latter three 
 Blanks and three Prizes. But be that as it will, 
 I do not fee the mighty Difcovery in finding out 
 that he who has only one Chance in three (which 
 I think is a Blunder to call the Odds of 3 to I 
 againll him) to gain a Prize in one Lottery, has 
 or may have the fame Advantage as he that has 
 one in two or an even Chance in another Lottery • 
 for we need only to make the Value of the Prizes 
 in Proportion, and he that has only one Chance 
 in a 1000/. fliall be in as good a Condition as 
 he that has one Chance in two. 
 
 I do not fay Dr. Halley has been guilty of the 
 fame Fault with this Honourable Gentleman, 
 for the long and elaborate Calculations juft men- 
 tioned tend to no Purpofe but to prove fomething 
 which is previoufly fuppofed or proved by a 
 Maxim ; and when a Pofition is fo proved, I 
 don't think a long String of Algebra adds either 
 Weight or Beauty to the Argumentation : But 
 I am afraid, before we come to the End of this 
 Difcourfe, we lli^ll find that the Dodlor, or 
 Mr. Moivre, or both, in the Computation of the 
 Value of Annuities for Lives, have, in fomeln- 
 ftances, brought, tp Account more Chances of 
 ^Vitality on Lives than belong to them, in other 
 ■i^Jnftances, have inferted fuch Chances as do not 
 .and can not belong to them, and in fome Cafes 
 have aflign*d wrong Values to right Chances : 
 And where any of thefe Miftakes are committed, 
 L a all
 
 [ 1^4 ,] 
 
 all the Algebra in the Univerfe will never prove 
 that the Sum total of the Values arifing on fuch 
 Computations will be the true Value of the An- 
 nuities inquired after. 
 
 The Doctor having deduced his Propofitio'n 
 from the BreJJaw Table, and aflerted, that the 
 Valuation of Annuities for Lives depends on it ; 
 and having laid it down for a Maxim, that a 
 Purchafer of an Annuity on a Life ought to pay 
 for fuch Parts only of the Value of the Annuity 
 as he has Chances that his Nominee is living, 
 which is undoubtedly true ; and having told us 
 how the prefent Value of a Sum of Money, pay- 
 able at any future Time, and at any Rate of In- 
 tereft, may be known ; and given us a Scheme 
 for that Purpofe, at the Rate of 6 per Cent. In- 
 terefl: : He alTumes, from the foregoing Propo- 
 fition ; 'That^ as the l^itmBer of 'Ter/om livh/g 
 after a Tea/, or after a7;yNnmher of Tears, are to 
 the Niimherof '■Ter/ons dead within that 'Time^ fo 
 are the Numher of Chaiiccs with him to the Num- 
 her of Chances agaivflhim; that the Perfon nomi' 
 fjated is then 'living : In Confequence of which, 
 the Rule for afcertainitig the Vdlue of an Annu- 
 ity for a given Life will be this. Js the Number 
 of Perfons living of an Jge propofed is to the 'Num- 
 her living after one Tear, or after any Number of 
 Tears", fo the prefent Vdlde -of ihe Sum payaM'e at 
 the End of one Tear, or at the End of the given 
 Numher of Tears, is to the Sum which ought to he 
 paid for the Chance which the Perfon has to enjoy 
 • his Jmmity for that one Tear, or the given Ntm- 
 her of Tears. And in the Conclufion he adds; 
 
 V
 
 t i65 ] 
 
 •If this Operation he repeated for every Tear of the 
 P^rfon's Li fey that the Sum of all theprefent Values 
 of thofe Chances is the true Value of an Annuity 
 for fuch ^Perfon*s Life : And. that by this Rule he 
 framed the Table he gives us, which is for the 
 Value of Annuities on a fingle Life at 6 /. per 
 Cent. Intereft. 
 
 I will not pbfitively fay, that the Author, in 
 forming his Table, did not himfelf obferve his 
 own Rule, becaufe he fays he ufed fome Com- 
 pendia in forming it, and in feeking a fhorter 
 Way he might poffibly mifs the right Way ; but 
 if I underftand his Rule right, as I am fure I do, 
 unlefs the Expreflion, for every Year of the 
 Perfbn's Life, has fome Meaning in it which I 
 cannot find out \ and if my Calculations by thaC 
 Rule are right, as I am confident they are : 
 Thus much I may fay, that the Rule does not 
 in every Inftance produce that Value which is 
 exhibited in the Table as the Value of an An- 
 nuity for a Life. The Ryle, in Ihort, is this : 
 We are, in the firll: Place, to compute what is 
 the Value of an Annuity for one Year of the 
 Life, beginning at the Age of the Perfon nomi- 
 nated, looking upon it as an abfolute Annuity 
 for one Year, allowing and difcounting out of 
 it a Value proportionate to the Chance of Mor- 
 tality ariling in that Year ; and this is done by 
 the firfl Part of the Rule : and then we are to 
 proceed in the fame Manner for every Year of 
 fuch Nominee's Life ; and 'tis aflerted, that all 
 thefe Values put together are the Value of the 
 Annuity on fuch a Life. But then the latter 
 L 3 Part
 
 [ i66] 
 
 Part of the Rule, where it directs that this Ope- 
 ration mull be repeated for every Year of the 
 Nominee's Life, does not dillinftly and exprefs- 
 \y fay for what Number of Years this Compu- 
 tation mull be made : But I cannot find out that 
 'tis capable of any other Meaning than one of 
 thefe two, either that it muft be made for fo 
 many Years as the Nominee has an even Chance 
 to live, or for fo many Years as he has any 
 Chance at all, or a Polfibility to live. How- 
 ever, let the Computation be made in which 
 Way of the two he pleafes, it produces a Value 
 different from what this Table gives ; one of 
 them a Value much under, and the other a Value 
 a little over, the Value exhibited in the Table, 
 and this is the Cafe in more Inftances than one. 
 
 The Value exhibited in the Table for an An- 
 nuity on a Life of ten Years old, Intercft com- 
 puted at 6 /. per Cent, is 1 3-44 ; and if an An- 
 nuity be granted for loooo /. per Jnnmn^ fup- 
 pofe on the Life of jd. of ten Years old, the 
 total Value of it will be 134: 400 /. To Ihow 
 that this is not the true Value of fuch an An- 
 nuity, even on a Computation made according 
 to the Rule here laid down, I have made a Cal- 
 culation purfuant to that Rule, for every Year 
 of fuch a Perfon's Life, from the Age of his 
 Life when nominated, that is, from ten Years 
 old, to the Extremity of old Age, that is to an 
 100 Years old ; and have fet it forth in Table 
 N^' IX. In this Table, and in the fecond Co- 
 lumn, we have the prefent Value of fuch an An- 
 nuity payable at the End of each Year for po 
 Years on an abfolute Term, tranfcribed from 
 
 this
 
 [ i67] 
 
 this Author ; and in the fixth and laft Column, 
 we have the prefent Value of fuch an Annuity- 
 payable at the End of each Year for 90 Years, 
 on the Chance of Mortality on the Life of a 
 Perfon of ten Years Age, form*d exadly, I 
 think, by the Rule given. And becaufe the 
 Chance of Mortality is calculated by the Brejlaw 
 Table, I have inferted that likewife in Table 
 N^' X. ; with an additional Computation for 
 16 Years, from the Age 84 to the Age of an 100, 
 omitted in the former Table. 
 
 From the Table, N^' IX. and the fixth Co- 
 lumn, and by a fliort Calculation added at the 
 End, it appears ; if we compute the Values on 
 the Chance of Mortality for fo many Years only 
 of fuch Perfon's Life as he has an even Chance 
 to live, that is for 41 Years, the Value is no 
 more than 13-02 ; and if we compute fuch 
 Values to the Extremity of Life, that is, for 5)0 
 Years, the Value is 1 3-50. I admit that the Value 
 given in the Doctor's Table, and the Sum ariling 
 here, does not vary greatly ; for 13-44 is not 
 much fliort of 13-50, which in fo large a Total 
 is an inconfiderable Sum : But then I obferve, 
 and it appears from my Table, and the Calcula- 
 tion at the Foot of it, that 13-44 is equal only 
 to a Term of 58 Years, if a Perfon of ten 
 Years old fo long lives, whereas I fuppofe we 
 ought to compute here to the Extremity of Life^ 
 that is for po Years, and the Sum produced 
 ought to be equal to a Term for 90 Years, if 
 the Perfon fo long lives : So that in this Author's 
 Way of computing in this Inilance, which has 
 L 4 produced
 
 [ i68 ] 
 
 produced no more than 1 3-44, there is an Omit* 
 fion of no lefs than 32 Years in the Computa- 
 tion. 
 
 Now I fuppofc it will not be pretended that 
 an Annuity for 58 Years, if a Perfon of ten 
 Years old fo long lives, is of equal Value with 
 an Annuity for 90 Years, if fuch a Perfon fo 
 long lives ; for though the Difference is fmall, 
 fome Difference there is, v and there is no more 
 Reafon for omitting 32 Years in the Computa- 
 tion, than there is for omitting 42 Years, or any 
 other Number of Years whatever ^ therefore 
 13-44, the Value given in the Table, is not the 
 right Value of an Annuity on fuch a Life, even 
 though the Calculation be made according to the 
 Rule here laid down. If it fliould be allcdged^ 
 that this Variation is no other than what might 
 eafily arife from a compendious Way of making 
 the Calculation, which the Author intimates he 
 made Ufc of, and that 'tis not of ConfideratioH 
 enough to overthrow the Rule ; yet I aflert, 
 that neither 13-44, nor 13-50, is the true Value 
 of fuch an Annuity, but that its Value is 15-12 : 
 And if I prove it, this, I fuppofe, will be a fuf^ 
 ficient Demonllration of the Faultinefs of this 
 Table, and of the Rule too by which it was 
 framed. 
 
 To prove this, and from the Doctor's own 
 Pofitions, as I mean to do, and can very readily 
 do, I mull have Recourfe to, and tranfcribe 
 hither, another Propofition advanced by this 
 Author, and which he calls his third Ufe of the 
 Brejlaw Table. If it he inqiiired at what Numher
 
 [ i69] 
 
 of Tears it is an even Lay (or Chance) that 
 a ^erfon of any Age pall die^ this Table readily 
 performs it \ for if the Number of Ferfons living of 
 the Jge prcpo/ed be halfed, it will be found by that 
 Table at what Tear the f aid Number is reduced to 
 Half by Mortality • and that is the Jge to which 
 'tis an even Wager {or Chance) that a ^erfon of 
 the Jge propofed pall arrive before he dies. Js for 
 Inflance ; a Perfon of 30 Tears of Jge is propofed^ , 
 the Number of that Jge is 531, the Half thereof 
 is 26^y which Number is found to be between 57 
 and 38 Tears ; fo that a Man of 30 Tears of 
 j4ge may reafonably espeSi {or has an even 
 Chance) to live between Tj and 28 Tears. This 
 Rule for the Valuation of Annuities upon a Life, 
 founded on the Propofition laft mentioned, is a 
 very obvious and a very natural one, eafiiy to be 
 underflood, and readily to be pradtifed ; and, if 
 applied to the Cafe we put, Hands thus. The 
 Number of Perfons of the Age of 10, of which 
 Age we have fuppofed A. to be, is 661 ; the 
 Half of this is 330 ; this Number we find be- 
 tween 51 and 52, fo that A. has an even Chance 
 to live 41 Years, and fomething over : A Term 
 for 41 Years, Interelt computed at 6 /. per Cent, 
 is in Value 15-12, as may be feen in Table N^* 1.; 
 therefore an Annuity for the Life of A» who has 
 an even Chance to live 41 Years and beyond, 
 muft be worth 1.5- 12, and above. 
 
 If to this Cafe we apply the Do6tor's Maxim, 
 that the Purchafcr of an Annuity on the Life of 
 A. ought to pay only for fuch Parts of the Value 
 of the Annuity as he has Chances that A. is 
 
 living,
 
 [ I70 ] 
 
 living, it will fland thus. If the Annuitant ha§ 
 an even Chance that A. will live above 41 Years; 
 though we admit there are many Chances againft 
 him in that Time, or many Chances that he does 
 not live 41 Years; yet there mull be fo many 
 Chances with him that he docs live to 41, and 
 fo many Chances with him that he lives beyond 
 41, that the Number of all the Chances put to- 
 gether mull be fuch that his Chance upon the 
 whole is an even one that he lives for 41 Years : 
 The Chances with him, fo long as they laft, are 
 Chances to enjoy the whole and every Part of 
 the Annuity ; but thofe Chances with him are 
 fo many, that they mud continue 'till it becomes 
 an even Chance that A. dies, that is^ 'till he has 
 lived 41 Years, and over : And therefore a Pur- 
 chafer having fo many Chances of enjoying the 
 whole and every Part of this Annuity as amount 
 to an even Chance of enjoying it for 41 Years, 
 and upwards, mufl pay for it as for an abfolute 
 Annuity for 41 Years, that is, muft pay for it 
 15-12. Or put it thus : If the Annuitant has 
 an even Lay or Wager, or may reafonably ex- 
 pert, that A. his Nominee will live 41 Years, 
 he may for that Reafon expe6t to receive the 
 whole or all the Parts of the Annuity for that 
 Time ; for he muft receive the whole of it, if his 
 Nominee fo long lives, and that he fliall fo long 
 live is admitted to be a reafonable Expe<Sation ; 
 and then I am fure the Seller of the Annuity on 
 his Part has the fame Reafon to expedt to be paid 
 for the whole Value of the Annuity as an abfo- 
 lute Annuity for 41 Years, as the Purchafer has 
 
 Reafon
 
 [ lyi ] 
 
 Rcafon to expe<5t to receive the whole of the 
 .Annuity for that Term of Years. 
 c^-rl cannot conceive, for my Part, on what Ac- 
 count the Doctor deferts this eafy, plain, and 
 familiar Method of afcertaining the Value of 
 thefe Annuities, fpecially when it has this farther 
 very advantagious Circumflance . attending it, 
 viz, that it determines the Value of fuch Annu- 
 ities for every Rate of Interelt at once, for fo it 
 manifeltly does, by giving the Number of Years 
 to which a Life is equal ; whereas of the Method 
 . he ufes he fays bimfelf, and *tis notorioufly true, 
 that it requires a very long and laborious Calcu- 
 lation ; and 'tis as true, that at beft it gives us 
 the Value of Annuities at one Rate of Intereft 
 only, and for every other Rate of Intereft the 
 fame long and laborious Calculation mull be en- 
 tirely repeated ; and moil other Perfons, all who 
 think as I do, that the Author of it has not hinl- 
 felf purfued his own Method with Exadnefs, 
 will be apt to fay of it, that 'tis an intricate and 
 an abftrufe one. 
 
 I have already oblerved that this Author has 
 advanced two Proportions, from each of which 
 may be deduced a Rule to compute the Value ot 
 this Annuity ; and from one of thofe Rules, 
 which is drawn from the firft Propofition, and 
 which the Author himfelf makes Ufe of, is pro- 
 duced, as we have feen, the Sum of 1 3-44, or 
 rather 13-50, as the Sum to be paid for it, on 
 the Contingency of Mortality ; and from the 
 other Rule, which is drawn from the fecond Pro- 
 pofition, is produced the Sum of 1 3-1 a as the 
 
 Sum
 
 [ 172 ] 
 
 Sum to be paid for it ; in which latter Method 
 nothing is allowed for the Contingency of Mor- 
 tality daring the Term of 41 Years, but the 
 Life of J. being on an even Chance equal to a 
 Term of 41 Years, is ftated as equal to an ab- 
 folute Term for fo many Years. Now if both 
 thefe Proportions are true, one or other of thefe 
 Rules deduced from thence, producing thefe 
 two different Sums, muft be falfe ; for 'tis im- 
 poflible that two Rules, producing two Sums fo 
 widely different, lliould both be right. That 
 both Propolitions are true will ealily be admit- 
 ted, for they are in Effcdt one and the fame, 
 only cloath'd in different Expreffions ; at leaft, 
 the fecond is a Deduction and a neceffary Confe- 
 quence of the firll : And if we can deduce a 
 Rule from a third Proportion which Ihall give 
 DS the fame Value of this Annuity as is produced 
 by the Rule drawn from the fecond Proportion > 
 this will be a llrong Confirmation of the Juftnefs 
 of the fecond Rule, and of the Badnefs of the 
 firfl. 
 
 The third Propofition, which the Doctor calls 
 the fecond Ufe of the Bre/Iaiv Table, and which 
 is framed to find out the Odds (or the Chances) 
 that any Perfon nominated does not die before 
 he attains any propofed Age, is this : That we 
 divide the Number of Perfoiis living of the ■propofed 
 ^ge^ by the Difference between that Ntmiber and 
 the Number of '^erfons living of the Jge of the 
 Perfon nominated^ and the Number produced 
 poivs the Odds [or the Chances) that the Perfon 
 lives to the Jge propofed. J. the Perfon nomi- 
 nated,
 
 [ 173 ] 
 
 nated, is oLthe Age of lo Years, and the 
 "Queftion Is, 'What are the Odds (or Chances) 
 that he lives to the Age of 51, that is, that he 
 lives 41 Years? 'The Number of Perfons living 
 of the Age of 51, the Age propofed, is 335 j 
 the Number of Ps^rfons living of the Age of A. 
 is 661 ; and if out of this Number we fubflra(5t 
 335, the Number left will be "^zd^ which is the 
 Difference between . the Number of the Perfons 
 living of one Age, and of the Perfons living of 
 the other Age 5 or, which is the fame Thing, 
 ~31.6 is the Niimber of Perfons who have died in 
 thefe 41 Years, which comes out to be fomething 
 lefs than a Moiety : The Odds therefore that J, 
 lives 41 Years, or to be 51 Years old, are plainly 
 thefe, as 335 is to 326 \ or, if I may ufe my 
 Manner of Expreffion, there are 335 Chances 
 with him that he lives 41 Years, and 326 Chances 
 ' algainft him, that he does not live fo many Years : 
 "That is, in either Way there is fomething better 
 tha'ri an even Chance that he does live 41 Years. 
 ■■'We have feen already, if we make Ufe of the 
 fecond Propofition, and the Rule deduced from 
 thence, and a very plain and eafy Rule it is; 
 ' &dX. 'tis more than "an even Chance that A, fup- 
 pofed to be ten Years old, will live 41 Years 
 'ahdover, from the Time that the Annuity is 
 granted : TheConfequence of which is evident- 
 ly this, that'"tfhd Vatu^ of the Annuity depend- 
 ing on this LJfe, ' by whichfoever Rule of thefe 
 ■' two we compute the Term of fuch Life, will be 
 "^5-12 :■ An(ff ff-om thefe Premifes, which are col- 
 lected from the Dodtor's own Pofitions, I con- 
 -illcfiJ:,.! .iii.;3:-o , elude
 
 [ 174 ] 
 
 elude thus. Since the two firfl Rules for ad- 
 jufting the Value of this Annuity vary fo much 
 in the Quantum which they refpedlively produce, 
 that one or other of them mufl neceflarily be 
 falfe, for fo one mufl: be, unlefs they concur in 
 producing the fame, or very near the fame Quan- 
 tum ; and lince we fee here a third Rule, drawn 
 from another Proportion of the fame Author's, 
 "which produces the fame Chance of the Durati- 
 on of the Life, as that Rule which I call his 
 true Rule does ; and the Duration of the Life 
 muft be the Meafure of the Duration of the An- 
 nuity; the Prefumption will be a very ftrong 
 one, if it docs not amount to a Demonflration, 
 that the other Rule, which is the Rule the Doc- 
 tor made Ufe of, is not a true one. 
 
 So much for the Proofs of the Falfity of this 
 Rule drawn from the Author's Pofitions ; let us 
 examine now how the Argument Hands on the 
 Effects which the Rule has produced in the fe- 
 veral Tables which have been form'd from thence 
 for the Valuation of thefe Annuities. In my 
 Obfervations on the Dodor's Table for this 
 Purpofe, I took Notice that it does not pre- 
 ferve a Decreafe either in the Value or in the 
 Term to which an Annuity is made equal in a 
 Proportion anfwerable to the Decreafe in the 
 Chance of Vitality from one Period of Life to 
 another. In particular, that on an even Chance 
 a Life of ten Years Age will have a Duration 
 for a Term of 41 Years and above, and the An- 
 ^nuity depending on that Life for a8 Years only, 
 . and about a Quarter over j but that, on the like 
 
 even
 
 [ 175 ] 
 
 even Chance, a Life of 70 Years of Age, and 
 an Annuity depending on that Life, have one 
 and the fame, or very near the fame. Duration. 
 I omitted, but might have taken Notice of the 
 fame Defed in Mr. Richards*s Table for the like 
 Annuities, with this fmall Difference ; that on 
 his Table the Chance of the Duration of the 
 Annuity for the elder Life is made about three 
 Months (hort of the Chance of the Duration of 
 the Life. The even Chance of the Duration of 
 a Life of la Years Age, which is the Age in- 
 ferted in Mr. Kichards's Table, is a Term of 40 
 Years, and the Chance of the Duration of the 
 Annuity for that Life is a little ihort of 28 
 Years ; but on a Life of 72 the even Chance of 
 its Duration is juft 4 Years, and the Annuity 
 depending on it is by his Table made equal to 
 three Years three Quarters. But even this fmall 
 Diiference will be reconciled, and both Tables 
 appear equally and alike defedive in this In- 
 flance, when an Error in the Doctor's Table is 
 redtified, and the Value of an Annuity for a Life 
 of 70 is made only 5-23, as upon a Computati- 
 on made I find it fhould be, inftead of 5-32 : 
 For then the Difference between the Chance of 
 the Duration of a Life of 70 compared with the 
 Duration of the Annuity on it in one Table, 
 and the Chance of the Duration of a Life of 7a 
 compared with the Duration of the Annuity on 
 it in the other Table, will be much one and the 
 fame, mz. about a Quarter of a Year. Here 
 again we have another Error in the Doctor's 
 Table, even on a Computation made according 
 
 to
 
 [ 176 ] 
 
 to his own Rule ; for I have computed the Value 
 of an Annuity for a Life of 70 by that Rule, 
 and 'tis no more than 5-23, when he gives us 
 5-32. Since then this Rule, where it has been 
 made Ufe of by Variety of Hands, concurs in 
 producing the fame or much the fame Effeds, I 
 cannot but think that thofe Effeds are neceflary 
 ones • and if fuch, I leave it to the Reader to 
 make his Conclulions on the Rule which pro- 
 duced them. 
 
 ' Under a former Head I took Notice, that 
 Mr. Richards formed his Table for the Value of 
 thcfe Annuities likewife from this fame Rule, at 
 lead at the Rates of 4 /. 6 /. and 8 /. per Cent, 
 Intereft, and at the Rates of 5 /. and 7 /. by an 
 Equation ; and I obferved there, that for every 
 different Rate of Interelt he has Hated fuch a 
 Value, as the Value of an Annuity on one and 
 the fame Life, that the Term correfponding will 
 be different as the Rate of Interefl: is different • 
 And I affirm'd that fuch Difference gives fuch a 
 Term to which a Life is equal, that 'tis not in 
 jSature poffible that a Life can be equal. I af- 
 fert then, that the Value of an Annuity for a 
 Life depends upon and is governed by the Term 
 to which the given Life is equal, or has an even 
 Chance to continue in Being; as much and as 
 certainly as the Value of an Annuity for a Term 
 of Years does depend upon and is governed by 
 the Number of Years for which that is to con- 
 tinue : And that one and the fame Life cannot 
 poffibly be equal to two or more different Terms, 
 becaufe and when two or more different Annui- 
 ties,
 
 .[ "77 ] 
 
 ties, granted at two or more different Rates of 
 Interell, are depending on fuch Life; for 'tis in 
 Effedl to fay, that a Perfon does exift and does 
 not exift at one and the fame Time ; and that 
 there is one and the fame Chance that he does 
 live, and that he does not live, to a determinate 
 Time. The Confequence of this muft be, if 
 this Difference in the Term to which a Life is 
 equal is a neceffary Refult from this Rule, that 
 it eifedually deftroys the Rule ; I fhall proceed 
 therefore to {how that this Difference, or fome 
 Difference, if not the identical one we find in 
 thefe Tables, does neceffarily refult from the 
 Rule ; and that the Value of an Annuity for a 
 Life at the lower Intereft muft be equal to a 
 longer Term, than the Value of an Annuity for 
 one and the fame Life is equal to at the higher 
 Intereft. 
 
 I affirm then ; if a certain Value and Term 
 correfponding are given, and out of that Value 
 a proportionate Part be deducted ; firft, that 
 the Term correfponding to the Value remaining 
 will not bear the fame Proportion to the Value 
 remaining as the original Term does to the ori- 
 ginal Value : Secondly, that the Term corre- 
 fponding to the Value remaining after a Deduc- 
 tion, at one Rate of Intereft, will greatly differ 
 from the Term correfponding to the Value re- 
 maining, after a Deduction, at another Rate of 
 Intereft ; And Thirdly, that it differs in this 
 Manner ; viz. at the ioweft Intereft, that is at 
 4/. per Cent, that the Difference or Decreafe in 
 the Term is the Icaft, and at the higheft Intereft, 
 
 U that
 
 [ 178 ] 
 
 that is at 8 /. per Cent, the Difference or Dccreafe 
 \h the Term is the greateft. To put this Matter 
 *n the cleareft Light I can ; I will ftate a Cafe, 
 and fuppofe that the Perfon on whofe Life an 
 Annuity is granted is ten Years old, that the 
 Life of fuch a Perfon is equal to a Term of 2.8 
 Years, and at all Rates of Intereft equal to the 
 £ime Term. 
 
 An Annuity,, for 
 the whole Liie, 
 is 
 
 A Moiety of the 
 Annuity^ is 
 
 Value. Term, 
 16-64X28-0-00/ 
 
 0^-321 
 
 10-2-00 
 
 Jntereft at 
 4 per C. 
 
 An Annuity, for 
 the whole Life, 
 
 IS 
 
 A Moiety of the 
 Annuity, is 
 
 11-05 
 
 28-0-00, 
 
 05-52/^07-2-00 
 
 Intereft at 
 8 per C 
 
 On this Cafe, in which the proportionate Part 
 of the Value deducted is a Moiety, the Truth 
 of my firll Pdfition appears ; for that the origi- 
 nal Value and Term were equal to one another, but 
 the new Term is only a third Part, or there- 
 abouts, of the old Term, when the new Value 
 is a Moiety of the old one : And the Second and 
 Third Pofitions are as evident, for that at 4 /. 
 per Cent, which is the lowcft Interefl, the new- 
 Term, or the Term correfponding to the Value 
 remaining after the Dedudion^ is more thaa 
 
 one
 
 [ 179 ] 
 
 one third Part of the old Term, or the Tefttj 
 correfponding to the firft Value ; for the Tcrhi 
 cdrrefponding to the firlt Value is 28-0-00, and 
 ■the Term correfponding to the fecond or re- 
 Tnaining Value is 1 0-2-00 : But at 8 /. per Cef}f, 
 which is the higheft Intereil, the new Term, or 
 the Tefni correfponding to the Value remaining, 
 is not one Third (I fhould rather fay, is little 
 more than a Fourth) Part of the old Term, or 
 the Term correfponding to the firil Value ; foir 
 l:he Term correfponding to the firft Value is 
 ii8-o-06, and the Term correfponding to th6 
 fecond or remaining Value is only 7-2-00. 
 
 From thefe Premifes it neceflarily follows • if 
 the proportionate Part dedudldd out of a give A 
 Value be a twentieth or an hundredth Part only, 
 yet that the Term correfponding to the Vaiufi 
 remaining after the Dedudion muft be different 
 at different Rates of Intereft ; and that at the 
 lowefl Intereft the Difference or Decreafe in th6 
 Term will be the leaft, and at the higheft Inte^ 
 reft that Difference or Decreafe will be the great^ 
 eft : From whence it follows, if we compute 
 the Value of an Annuity for a Life at the feve** 
 ral different Rates of Intereft:,' and compute fuch 
 Values by deducting a proportionate Part of th^ 
 Value of an abfolute Annuity out of each Year 
 of the Life oi^ the Nominee, as the Rule direds 
 us to do : I fay, it neceflarily follows, th^t the 
 Term correfponding to the Value remaining af^ 
 ter fuch Dcduftion made, muft be a fhortcr 
 Term for every Year of the Life, and confe- 
 quently a fhorter Term for the whole Life, in 
 M 2 ■ thofg
 
 [ i8o ] 
 
 thofe Cafes where a proportionate Part of the 
 Value is deduded out of each Year of Life, at the 
 higher Intereft, than in thofe Cafes where a pro- 
 portionate Part of the Value is dedu6led out of 
 each Year of Life, at the lower Intereft : which 
 was the Thing to be proved. 
 
 Since then the Matter of Fa6t, on a Compur 
 tation, is agreeable to the Reafon of the Thing^ 
 as from Mr.Richards's Tables which were formed. 
 by the Rule it appears to be, I might very well 
 reft the Matter here ; for this, as I apprehend, 
 is a Demonftration that the Rule itfelf muft be 
 wrongs fince in Fa£t it does, and necelTarily 
 muH, produce fuch Terms, as the Terrns to 
 which a Life is equal, to which Terms 'tis in 
 Nature impoffible that a- Life can be equal, urv 
 lefs a Life can be in Being and not in Being at 
 one and the fame Time. However, we Ihall go 
 .on to examine the Dirc(5tions of the Rule, and 
 .wc fhall difcover, perhaps, from thence, not 
 only that 'tis erroneous, but wherein the Error 
 of it lies : And this I choofe the rather to do, 
 becaufc the Arguments which tend to difprove 
 Xhe Dodlor's Rule will prove and llablilh mine. 
 r'j To determine the Value of the Annuity for 
 the whole Life, the Rule direds us to adjuli 
 £rll the Value of it for each Year of the given 
 i<ife, and to carry on fuch Computation as long 
 as Life can well be fuppcfed poffible to lall, and 
 then to put together all thefe Values : And 'tis 
 iiffirm'd, that the Aggregate or Sum total is the 
 Value of the Annuity for the given Life. To 
 adjuft the Value of the Annuity for each Year 
 
 of
 
 r 1 8.1 r ^, 
 
 of Life, the Rule diredls us ; of 'the Value of 
 the Annuity, as an abfolute one for a Year, to 
 take fuch Parts only as.correfpond to the Chances 
 that the Nominee lives to the End of the Year;- 
 or, which comes to the fame Thing, out of the 
 Annuity, as an abfolute one for a Year, to de- 
 dud: fuch Parts of it as correfpond to the Chances 
 that the Nominee is dead before the End of the 
 Year ; and *tis affirm'd, that the Parts of the 
 Value which are taken in the firil Cafe, or the 
 Parts of the Value which are remaining in the 
 latter Cafe, which will be one and the fame Sum 
 total, are the Value of the Annuity for each 
 Year of the Life. ■ 
 
 *Tis certain, and will be admitted, that la 
 each Year of the Life of any Perfon there are 
 many Chances arifing that he dies within fuch 
 Year, which we may call Chances of Mortality 
 or lofing Chances, as well as there are many 
 Chances that he lives to the End of thofe Years 
 refpedively, and thefe we may call Chances of 
 Vitality, or gaining Chances : and if the Rule 
 be applied to the Life of J. of ten Years Age, 
 and on an Annuity of loooo/. per jitmam^ the 
 Value correfponding to the gaining Chances 
 arifing in the firft Year of the Life of ^. will be 
 the Sum of 9319/. as the Value of the Annuity 
 for fuch firfl: Year of that Life, as may be feen 
 by my Table N^. IX. \ and the Sum of 1 15 /. 
 •will be the Value corre(ponding to the loiing 
 Chances arifing in the fame Year, and to be de- 
 duced out of the prefent Value of that Annuity 
 taken as an abfolute Annuity for one Year : 
 M 3 Thofe
 
 1 182 ] 
 
 Thofe two Sums 9315)/. and 115/. making to- 
 gether 9434 /. which is the Value of the Annu- 
 ity as "H abfolute one for one Yea?;. Now for 
 thefe loling Chances ariling in each Year I alTert, 
 that no Value at all is to be declud:ed in the Year 
 or Years in which they refpe6tively arife ^ but 
 in another Manner, and in another Place, as I 
 ■will prove prefently. 
 
 Suppoling here, but not admitting, that on 
 
 this Account fome Value were to be dedu6ted, X' 
 
 do deny that 115/. is the right Sum; becaule"'^ 
 
 that Sum is the Value correfponding to the 
 
 Chances that may be loft in each Year, and thole 
 
 Chances are proportionate Parts of the Valqe of 
 
 the Annuity ; whereas the Sum to be dedudcd, 
 
 if any, ought to be the Value correfponding to 
 
 the Term or Number of Days which on the 
 
 Chance of Mortality may be loft in each Year^t 
 
 Novy if thefe two Sums will be. different ; thatf 
 
 is, if the Sum produced as the Value correfJ3ondi- 
 
 ing to the Chances which may be loH will be 
 
 one Sum, and the Sum produced as the Value 
 
 correfponding to the Number of Days in a Ycaif 
 
 which maybe loft will be another Sum, thefe' 
 
 tvyp diiferent Sums will have a different Term 
 
 correfponding : And fince theSum. produced in 
 
 the latter Way la the Value correfponding to 
 
 the Term or Number of Days, fuch Sum muffc 
 
 give us the right Term or Number of Days to 
 
 be deduced, and the former as giving a diffe-' 
 
 rcnjl;,,Term muft give us a wrong one. This 
 
 Way of Reafoning will hold equally good, and 
 
 ii^ t.he ;^m^ Manner, iq the other Cafe j I mean 
 
 if
 
 [ i«3 ] 
 
 if we take the Value correfponding to the gain- 
 ing Chances in each Year as the Value of the 
 Annuity for fuch Year ; when we ought to take 
 the Value correfponding to the Term or Num- 
 ber of Days which may be gainM in each Year ; 
 here likewifc, the Sums produced, and the Terra 
 or Number of Days correfponding will be diffe- 
 rent, and the latter only the right Sum, i ad- 
 mit that the Difference in the Term or Numbec 
 of Days in each Year, arifing from thefe two 
 Ways of Computation, will not be great ; but 
 then it is to be confidered, that this Operation 
 is to be continued through all the poffible Years 
 of Life, that is for po Years in fome Cafes; and 
 if there be fome Difference, be it more, or be it 
 iefs, fuch Difference, though it were only one 
 Day in each Year, and whether it be too much 
 or too little by that one Day, is fatal to the 
 Rule : For if my Pofition is right, that the 
 Value of an Annuity for a Life depends on the 
 Number of Years and Days to which fuch Life 
 is equal, if the Value produced does not corre- 
 fpond to fuch Number of Years and Days, fuch 
 is not the Value of the Annuity, nor the Rule a 
 right one. To prove then, that the Sum corre- 
 fponding to the Value of the Chances of Morta- 
 lity arifing in each Year will be a Sum different 
 from the Sum correfponding to the Value of the 
 Term or Number of Days of Mortality arifing 
 in each Year, I argue thus. 
 
 A Term of Years given correfporids and is 
 
 equal to a certain Value, and a Value given cor- 
 
 r^fponds and is equal to a certain Term of Years. 
 
 M 4 For
 
 [ i84] 
 
 For Inilance ; a Term of 20 Years, at 4 /. per 
 Cent. Intereft, is equal to 13-57 in Value : and 
 on the Reverfe, 13-57 in Value is equal to a 
 Term of lo Years. But, where a Value and 
 Term correfponding are given, if we add to, 
 or if we multiply, or if we fubftra6t from, each 
 of them the fame proportionate Parts, the Value 
 and Term ariiing on the Addition or Multipli- 
 cation, or the Value and Terrp remaining on the 
 Subftradion, will, in none of the Cafes, corre- 
 fpond to one another. Thus ; a Term ot 20 
 Years is in Value 13-57, if we add to the Term 
 a quarter Part, the Term will be 25 Years, and 
 if we add to the Value a quarter Part, that is 
 3-39, the Value will be 16-96 : but a Term of 
 25 Years is in Value no more than 15-60 ; and 
 the Value 16-96 is equal to a Term of 29 Years, 
 or very near. Thus in Subilradion likewife ; 
 if out of that Term and that Value we fubftrad 
 a Moiety of each, the Term remaining will be 
 10 Years, and the Value remaining will be 6-78 : 
 but a Term of 10 Years is in Value 8-09, and 
 the Value 6-78 is equal only to a Term of 8-1 -00, 
 or thereabouts. The Cafe is the fame if we make Ufe 
 of Multiplication, as Mr. Moivre in his Rules fre- 
 quently does, viz. if we multiply a given Term 
 into itfelf, or different Terms into one another, 
 and do the fame by the rcfpedtive Values corre- 
 fponding, the Terms and the Values produced 
 will in no fort correfpond. For Inftance ^ a 
 Terrn of 10 Years multiplied into a Term of 
 10 Years produces an 100 ; and 8-09, the Valqe 
 correfponding to that Term, being multiplied 
 
 into
 
 [ i85 i 
 
 into itfelf produces 65-44: but this Term and 
 this Value produced are out of all Bounds of 
 Proportion to one another. 
 
 I affirm farther ; if we multiply a longer and 
 a ihorter Term, each into itfelf, and do the 
 fame by the Values refpedively correfponding, 
 and out of the Produce of the longer Term and 
 of the greater Value rcfpedively fubftrad the 
 Produce of the Ihorter Term and the fmaller 
 Value, that the Term and Value remaining will 
 not correfpond. A Term of 10 Years multi- 
 plied into itfelf produces an lOO ^ and 8-op, the 
 Value correfponding to that Term, fo multiplied 
 produces 65-44 : A Term of 9 Years multiplied 
 into itfelf produces 81; and 7-42, the Value 
 correfponding to that Term, multiplied in the 
 fame Manner, produces 54-05 : If then we fub- 
 ftrad 81 out of an 100, the Term remaining 
 will be 1 9 Years ; and if we fubllrad: 54-05 out 
 of 65-44, the Value remaining will be 11-39: 
 But the Term remaining and the Value remain- 
 ing do not correfpond, for a Term of 19 Years 
 is in Value 13-12, and the Value 11-39 ^^ equal 
 only to a Term of 1 5-2-00, or thereabouts. 
 From hence is evidently feen, that a Term given 
 correfponds to the Value, and a Value given 
 correfponds to the Term ; but when we add to 
 that Value, or fubftradt from it, or multiply it, 
 that the Value arifing on the Addition or Mul- 
 tiplication, or remaining on the Subftradion, 
 does not give us a Term correfponding which 
 bears a Proportion to the Increafe and Decrcale 
 in the Value rcfpedively : And in like Manner, 
 
 if
 
 [i86] 
 
 if we add to, or fubftrad from, or multiply, a 
 Term given, that the Value correfponding to 
 the new Term will bear no Proportion to the 
 Increafe and Decreafe of the Term refpedtively. 
 
 I cannot forbear obferving here, though it 
 mxy be a little out of Time, that thefe Pofiti- 
 ons, which are demonftrated in Numbers and 
 Figures, do effedlually deftroy the Foundations 
 of Dr. Halle fs and Mr. Moivre*s Rules for cal- 
 culating the Value of Annuities on Lives, not 
 only of thofe for a lingle Life, but alfo thofe for 
 any Number of Lives, in whatever Form and 
 Shape they are combined : For thefe Rules are 
 evidently grounded or depend upon this Suppo- 
 iition, which is manifeftly a falfe one, that to 
 add to, or fubllrad, or multiply, or in any other 
 Form to operate on the Value of an Annuity, 
 is one and the fame Thing as to perform the fame 
 Operation on the Term to which the Annuity is 
 equal : And by this Means they do, in EfFe(5t:, 
 make the Value of the Annuity to be the Mea- 
 furc of the Term to which the Life or Lives are 
 equal ; whereas the Value of the Annuity, as it 
 mull attend and depend upon the Life or Lives, 
 muft be governed by, and follow, the Term, or 
 the Chance of the Term, of the Life or Lives. 
 
 At prefent to apply thefe Politions to the 
 Life oi J. on an Annuity of loooo/. and to the 
 Method of calculating the Value of this Annui- 
 ty on his Life, for the firfl Year of his Life, as 
 prefcribed by this Rule, the matter ftands thus. 
 
 The number of Perfons living of the Age of 
 J. fuppofed to be ten Years old, is 66 1 ;. the 
 
 Number
 
 [ »S7 ] 
 
 Number of Perfons living of the Age fucceeding 
 is 653^ the prefent Value of a loooo/. to be 
 paid at the End of one Year abfolutely is 9434, 
 and the Sum produced by th- Rule, is 9319; 
 that is f-J-^ Parts of 9434 /. is 9319/; and this is 
 the Value, or the Sum to be paid for the firlt 
 Year of this Annuity on the Contingency of 
 Mortality. Now if if Parts, or a Moiety of a 
 Term of 20 Years is one Thing, and produces 
 one Sum for its Value, and f ^ Parts or a Moiety 
 of the Value of fuch a Term is another Thing, 
 and produces a different Sum for its Value: 
 Then ^f f Parts of a Term for one Year mufl 
 be one Thing, and produce one Sum for its 
 Value, and f f f Parts of the Value of fuch a 
 Term mull be another Thing, and produce a 
 different Sum for its Value : But if Parts or a 
 Moiety of a Term of 20 Years does produce a 
 Sum different from the Sum which if Parts or 
 a Moiety of the Value of fuch a Term does pro- 
 duce, as we have already Ihewn : Therefore^ 
 f If Parts of a Term mull produce a Sum dif- 
 ferent from the Sum which f-|f Parts of the 
 Value of fuch a Term produces. The true 
 Value of this Annuity for this one Year, as well 
 as for all the Years of the whole Life, depends 
 upon, and is governed by, the Term or Num* 
 ber of Days of the Year, for which it will con- 
 tinue: But by the foregoing Propofition ■Hi- 
 Parts of 9434/. or 9319 mull be one Thing, 
 and m Parts of a Term of one Year or 365 
 Days mufl be another Thing, and produce a 
 different Sum for its Value : Therefore the 
 
 Value
 
 [ i88 ] 
 
 Value of the Annuity for this one Year, or the 
 Sum to be paid for it for this one Year, muft 
 be another, and a different Sum from 9319/.; 
 the neceffary Confequence of which, is, that 
 the Rule which produces 9319/. as the Value 
 or Sum to be paid for this one Year, is not a 
 right Rule : which was the Thing to be proved. 
 But fince I have allerted that no Sum at all 
 is to be deducted in each Year of Life, in 
 Conlideration of thofe Chances of Mortality, 
 which arife in fuch Year ; 'tis not much material 
 whether the Sum of 115/. or any other Sum, 
 might be the more proper Sum to be deduded y 
 and therefore I fliall proceed to maintain that' 
 AfTertion. I obferve then, that upon the Foun- 
 dation of this Rule, there are two Methods of 
 adjufting the Value of thefe Annuities ^ one by 
 the Value of the gaining or loling Chances arife- 
 ing in each Year of Life, which is the Method 
 which this Author alledges he made Ufe of 
 in forming his Table of fuch Value ; the other 
 is, by the Value of the gainableor lofeablc Days, 
 arifing in each Year of his Life ; fo let us exa- 
 mine how the Computation is made in the one 
 way, and in the other. 
 
 In this Author's Method, we compare the 
 Number of Perfons living ; fuppofe of the Age 
 of 10, with the Number of Perfons living of the 
 Age of 1 1, and one we find to be 65 1, and the 
 other 653 and the Difference to be 8, that is 8 
 have died in that Year ; fo there are 6^^ Chance3 
 with the Annuitant, that his Nominee lives that 
 Year out ; or 653 Chances of Vitality, and there
 
 L 189 ] 
 
 are 8 Chances agalnft him, or 8 Chances of 
 Mortality in the fame Time; in the fecond 
 Year there are 646 Chances with him, and 7 
 Chances againft him ; in the third Year, 640 
 with him, and 6 againft him ; and (6 we proceed 
 for the whole 90 Years, that is to the very Ex- 
 tremity of poffible Life. By a Table given us 
 here, which I have tranfcribed into my Table, 
 N**. IX. we find what is the prefent Value of 
 1 0000/. to be paid at the End of each Year of 
 an 100 Years, where 'tis to be paid as an abfo- 
 lute Annuity, and without any Chance of lofing 
 any Part ; and then we take fuch Parts of the 
 prefent Value of the Annuity abfolute, as cor- 
 relponds to the Value of the Chances of Vitality, 
 or gaining Chances, arifing in each Year, as the 
 Value of the Life Annuity for that Year : In 
 which way of Computation, we leave out or 
 dcdud, in each individual Year, through the 
 whole poflible Life, a Sum correlponding to the 
 Value of the feveral lofing Chances, or Chances 
 of Mortality, in each Year as they refpedtively 
 arife. In the other Method, which differs very 
 little from the former, wc compute how many 
 Days in each Year may be gained by the Chance of 
 Vitality, and how many may be loft in the fame 
 Time, on the Chance of Mortality ; and we take 
 the Number of gainable Days arifing in each Year 
 refpe(SiveIy, as the Term to which the Life is 
 equal in that Year, or the Value correfponding 
 to fuch gainable Days as the Value of the An- 
 nuity for that Year ; and in this Way as well 
 as in the other, in every Year through the whole 
 
 poffible
 
 C 190 ] 
 
 polTible Life, we leave out or dedu6t from tht 
 whole Number of Days in a Year, the lofeable 
 Days arifing in that Year, and from the Value 
 of an abfolute Annuity for one Year, we dedudl 
 the Value corrcfponding to the lofeable Days in 
 the Year : the gaining Chances in one Cafe 
 anfwering to the Days gainable on the Chance 
 of Vitality in the other, and the lofing Chances 
 to the lofeable Days on the Chance of Mortality. 
 
 Now I alfert, that neither the lofing Chances 
 in each Year, if they are taken Simply and with*- 
 out Regard to their Values corrcfponding, nor 
 the Value of thofe lofing Chances when thofe 
 Chances are refolved into the Sums correfpond- 
 ing ; nor the lofeable Days in each Year, nor the 
 Value correfponding to thofe lofeable Days, arc 
 to be dedu<^ed in thofe Years in which they re- 
 fpedively arife ; but that thofe Dedudions arc 
 to be made out of fomething elfe, and in another 
 Manner; and herein lies the Error of the Rule. 
 
 To begin with the lofeable Days in each Year, 
 becaufe the Reafoning on that Head may be the 
 plained and molt clear ; I fay that thofe lofeable 
 Days are not to be deduced out of each Year 
 as they arife, but all the lofeable Days arifing in 
 the whole Life are to be accummulatcd, and the 
 total is to be deduded out of the total Days of 
 po Years, or of fo many Years as the Life cart 
 poffibly laft ; and the Remainder is the Term to 
 which the given Life is equal : Or, which is the 
 fame Thing, only the Method is inverted, all 
 the gainable Days in each Year for 90 Years, 
 "or for all the pofTible Years of the given Life 
 
 arc
 
 L m I 
 
 are to be accumulated, and the Sum total is the 
 Term to which a Life is equal. This Propofi- 
 tion and Method of attaining the Term to 
 which a Life is equal, is fo plain and obvious as 
 to need neither Proof nor Illultration: But to 
 try the Truth of the Method, I have made a 
 Calculation on the Life of a Perfon of 30 Years 
 old, of the Number of gainable and lofeable Days 
 ariling in each Year of his Life for 70 Years ; 
 that is to the Extremity of Life ; and the Sum 
 total of the gainable Days comes out to be 
 10024, that is the Life of a Perfon of 30 Years 
 old is equal to 10024 Days, or 27 Years and 
 i6^ Days; and if we confult this Author's 
 fecond Propofition, and the Rule for computing 
 the Chance of Vitality deduced from thence, he 
 there ufes this Inftance, and makes fuch a Life 
 equal to a Term of 27 Years, and fomething 
 above ; fo that thcfe two Rules may very fairly 
 be faid to produce one and the fame Term, to 
 which a Life is equal. 
 
 This then being the Produce in my Method 
 of calculating the Chance of Duration, or the 
 Value of an Annuity, on the Life of a Perfon of 
 30 Years old, which concurs with that produced 
 by this Author's lecond Proportion and Rule ; 
 now let us fee what is the Produce, if the Value 
 be computed by his firll Rule. If we confult 
 his Table, which was form*d by fuch firfl Rule, 
 the Value of fuch an Annuity there given is 
 11-72, which is equal to a Term for 21 Years, 
 wanting four Decimals, or about half a Quarter : 
 If we inquire how it comes to pafs, that the 
 
 Value
 
 [ 192 ] 
 
 Value or Number of Years in the laft Way of 
 computing, falls fo much fhort of the Value and 
 Number of Years in the former Way of comput- 
 ing, the Reafon is very obvious and plain. In 
 the former Method, all the lofeable Days arifing 
 in the whole Term of 70 Years being accumu- 
 lated, are dedudied out of the whole Term; by 
 which means, all the gainable Days in the Term 
 of 70 Years, are computed as fo many Years 
 and Days in immediate Poflcflion, and all the 
 lofeable Days, as fo many Years in Reverfion, 
 after the Term in PofTelTion, or at the end of 
 Life ; and in the latter Method, the lofeable 
 Days ariling in each Year, are dedu6:ed in each 
 Year ; fo that the gainable Days are not con- 
 nected and all computed, as fo many Years and 
 Days in immediate Poffeffion, there being lofeable 
 Days intervening in each Year, and thofe lofe- 
 able Days are not computed as Days, all of them, 
 in Reverfion at the End of Life, but in Rever- 
 fion at the End of each Year : For Inllance, 
 in the Calculation I made, of the gainable and 
 lofeable Days arifing in each Year, of the Life 
 of aPerfon of the Age of 30 ; (and if the Calcu- 
 lation there, were faulty a Day, more or lefs, 
 'tis not material here,) I found the lofeable Days 
 in the flrfl Year to be 6, in the fecond Year 1 1, 
 and in the third 17, and {o on ; which Days in 
 this Author's Method, are dedu6ted in each 
 Year as they refpcdively arife, or a Value cor- 
 refponding to them ; and in my Method, the 
 total of the lofeable Days are deducted out of 
 the total of 70 Years or the whole Term of 
 
 poffible
 
 [ 193 3 
 
 poffible Life : That is, in one Cafe we take thefc 
 iix lofeable Days out of the firft Year, and in the 
 other Cafe we take them out of 70 Years, and a 
 Value tefpcdtively correlponding : Therefore 
 the Value of the Annuity, and the Number of 
 Years to which a Life is equal, mufl vary greatly 
 according as the Eftimate is made by thefe dif- 
 ferent ways of Computation. From hence we 
 difcover plainly, what it is in thcfe Methods 
 which gives Occafion for the great Variety we 
 find in the Produces of them ; and that very 
 fame Thing, I fay, is the Error in this Author's 
 Method, wz. that it dired:s that thofe lofeable 
 Days arifing in each Year, Ihould be dedu<5led 
 in the Years in which they refpe6lively arife. 
 
 That this is an Error in fuch Method I aflert^ 
 becaufe 'tis grounded on fomething which never 
 does, and never can, in Fad, happen ; for the 
 Life of Man is made up of a continued Series of 
 fo many Years and fo many Days, and not of 365 
 Days wanting 6 Days in the firll Year, and of 
 365 Days wanting eleven Days in the fecond 
 Year, and of 365 wanting 66 Days, fuppofe in the 
 eleventh Year ; and therefore fince a Life cannot 
 be difcontinued at the End of 359 Days, and 
 begin again de iwoo at the End of 365 Days ; 
 and iince the Perfon has a Chance to live many 
 Days in the fubfequent Year, as in the Cafe 
 Hated 'tis fuppos'd he has, he muft of neceffity 
 firft enjoy the fix lafl Days of the precedent 
 Year, before he can enjoy the gainable Days in 
 the fucceeding Year ', and if he does not there 
 jcnjoy them, he never can enjoy them, fo that if 
 N they
 
 [ 194 ] 
 
 ihey are there deduded, we deduct foine Days 
 which he has aRight, as having a Chance, to enjoy. 
 If we make Ufe, of the fame Rule, but^the in- 
 verted Method, of afcertaining the Number of 
 Years. and Days to which the given Life is equal ^ 
 that is, by computing how many gainable Days 
 there are in each Year to the Extremity of Life, 
 and taking thofe gainable Days accumulated as 
 the Years and Days to which fuch Life is equal; 
 this Way of Reafoning will be full as ftrong and 
 perhapsilill more clear. The Number of gain- 
 able Days, or Days which an Annuitant has a 
 Chance to enjoy, on the Chance of the Vitality 
 of his Nominee, in the firll Year of his Life are 
 359, and in the fecond Year, 354 ; and the fuper- 
 numerary Days, that is the 6 Days, which on the 
 Chance of Mortality, are lofeable Days , in the 
 firft Year; I fay, mull be fupplied; from the 
 gainable Days in. the fecoad- Year, and placed. to 
 the Account of the Days in the firft Year, fo as 
 to make up fuch firft Year a compleat Number 
 of 365 Days ; for if they are not brought to the 
 Account as enjoyed in the firft Year, they, can- 
 not be brought to Account, as enjoyed .apy 
 where ; and.- then ry ou .do . not. allow the. Iperfon 
 to have a ^Chance tO;. enjoy jQme,oC. the Days, 
 which the Cafe -ftated has 'adtiiitted him to have^ 
 a Chance ,tp enjo,y. . 
 
 If therefore we , will confider the Life of Man, 
 as fuch a Thing as it, really. is, that is, that it 
 muft continue, fo long as it. has any Being at all, 
 in an interrupted Series of Years, and Days, and 
 of necelfity muft run the Courfe of the precedent 
 
 Year^
 
 [ 195 i 
 
 Year, before it can enter upon a SubfeqUent one ; 
 we mult compute the gainable Days in each 
 Subfequent Year as a Part of the Year precedent, 
 fo as to make it up to its full Complement of art 
 intire Year : or, which comes to the fame Thing, 
 we mull take the gainable Days in each Year for 
 the whole Life, and join them together as in 
 one Link, and the Sum total, which in both Ways 
 mull be exadly the fame, will be the Years and 
 Days to which a Life is equal. The Truth is, 
 that thefe lofeable Days, or the Days which on 
 the Chance of Mortalilty may be loll in each 
 Year, in Fad are not loft, unlels Life itfelf is loft, 
 and Life is fuppos'd not to be loft ; and there- 
 ibre ought not to be computed as loft, as long 
 as the even Chance is that it continues* and 
 lincc Life, fb far as its Tether goes, muft be 
 connected in a continued Chain of Years and 
 Days, we muft compute all the Days of a Man^s 
 Life, as fo many continued Days in immediate 
 PolTeffion. 
 
 Having proved then that the lofeable Days 
 ariling in each Year, are not to be deducted out 
 of thofe Years in which they arife feverally ; and 
 fhown likewife, when, and where they are to 
 be left out, and when and where they are to 
 be brought to Account: I proceed now to 
 make good my other AlTertion, that neither the 
 loling Chances ariling in each Year, nor the 
 Values correfponding to thofe Chances, are to 
 be dedudled in thofe Years in which they rcfpcc- 
 tively arife ^ tho' I think there need not many 
 Words to convince us, that the JLeafoning made 
 '■ ' N 2 Ufe
 
 [ 196 ] 
 
 Ufe of under the former Head, will hold good 
 and be applicable here ; yet, fince this great 
 Author's grand Rule depends on that Point, 
 that thefe Deductions are to be made in this 
 Manner, and at thole Times ; I will take the 
 liberty to repeat the force of my Arguments fo 
 far, as to Ihew that they are applicable here; 
 and fhall make Ufe of the Inftance on the Life 
 of A having before flated the Chances, and the 
 Values correfponding, on his Life. 
 
 The Number of Perfons living of that Age, 
 that is the whole Number of Chances, is 65i \ 
 the Number of Perfons living of the Age fuc- 
 ceeding is 653 ; fo that the gaining Chances in 
 that Year are d^}^^ and the lofing Chances are 8: 
 The Value correfponding to the 66 1 Chances, 
 or, which is the fame Thing, the prefent Value 
 of that Yearns Annuity, taken abfolutcly and as 
 Subjed to no lofing Chance, is 9434/. and on 
 the fame Annuity taken as Subjedt to the Chance 
 of Mortality, the Value correfponding to the 
 gaining Chances is 9319/. and to the lofing 
 Chances 115/. making together 9434^- The 
 Rule then giving for the gaining Chances d^^it 
 and for the Value of them 9319/. and for the 
 lofing Chances 8, and the Value of them iij/. 
 aiTerts that the greater Sum is the Sum gained 
 that Year, and to be accounted as the Value of 
 the Annuity for the firll Year; and that the lefe 
 Sum is the Sum loft, and to be deduced in that 
 Year. My alTertion is, that the Sum of 9434^- 
 .which is the Value of that Year's Annuity, as an 
 .abfolute one, is the Value of the firll Year's Life 
 
 Annuity j
 
 C 197 ] 
 
 Annuity; and that the Sum of 115/. fuppofed 
 to be loft, and which I admitted, on the Chance 
 of Mortality, poffibly may be loft, is not to be 
 deduded in that Year, but at the End of Life. 
 I obferve here, as I did under the former Head, 
 that 'tis in Nature impoffible, that A fliould 
 enter into a fecond Year of his Life, unlefs, and 
 until he has lived all the Days of his firft Year ; 
 and confequcntly that he cannot enjoy the gain- 
 ing Chances of a fubfequcnt Year, until he has 
 enjoyed all the Chances, both gaining and lofing 
 Chances, of the precedent Year : And iince 'tis 
 fuppofed in the Cafe, that he has many gaining 
 Chances in the next and other fubfequent Years, 
 it neceilarily follows that we muft look upon 
 him as enjoying, or having a Right to enjoy, the 
 whole Number of Chances, both gaining and 
 lofing Chances in fuch firft Year ; and then the 
 Sum to be taken as the Value of that Year's 
 Annuity, muft be a Sum equal to the Value of 
 the whole entire Year, that is 9434/. 
 
 It is agreed that an Annuity of looco/. for 
 one Year, and payable at the End of one Year, 
 is worth 9434/. where fuch Annuity is abfolute, 
 -and not fubjed to any Contingency ; and I vvill 
 admit that fuch an Annuity, if for one Year 
 only, on the Chance that A, ftiall fo long live, is 
 •in Value no more than 9319 ; and that the Lofs 
 on the Chance of Mortality in that Year, is in 
 Value 115/.; and therefore that a Purchafor of 
 fuch an Annuity for one Year and no more, 
 muft have an Abatement out of 9434/. in propor- 
 tion to the Lofs polTible on fuch Chance of Mor- 
 N 3 tality.
 
 [ i9« ] 
 
 tality. But for what Reafon and onwhat Foun- 
 dation mufl fuch an Abatement be made ? Moft 
 certainly for this, becaufe on the Chance of 
 Mortality within that Year, there is aPoffibility 
 of a Want of Value, and fuch Want of Value is 
 computed at 1 15/.^ and unlefs this Annuitant has 
 an Equivalent for fuch Want of Value within that 
 Year, he can have it no where clfe, for the 
 whole of his Intcrefl ends with the Year; but it 
 fuch an Annuity be purchafed for- the Life of ^. 
 abfolutely, or for 90 Years, if jJ, Ihall fo long 
 live, which are in Value the fame; in this Cafe 
 the Reafon of the Thing don't hold throughout. 
 It is indeed here, as well as in the former Cafe, 
 fuppofed that there is fome Chance that J. may 
 dye in the firft Year of the po; but in this Cafe 
 there is another Suppolition, which deftroys 
 or fets afidc, all Effe6t of the former ; for 
 'tis fuppofed I ike wife, that there are " many 
 Chances of Life to be enjoyed in the fecond 
 Year, which cannot be had unlefs, and until, he 
 has lived out the whole of the firfl Year. To 
 which I fliall only add, that there is no Neceflity 
 here, though there is in the former Cafe, for 
 the polfiblc Want of Value, on the Chance of 
 Mortality in the firft Year, to make the Abate- 
 ment for it out of fuch firft Year; for that it 
 may be made elfewhere, and I fay ought to be 
 made elfewhere, and the Place where is the very 
 Extremity of Life, and is to be done and readily 
 may be done, by computing the Number of 
 lofing Chances ariifing in 90 Years, and deduit- 
 
 ing
 
 [ 199 ] 
 
 iflg them out of the whole Number' of Chance^ 
 arifing in chat whole Term. - . 
 
 . -The Faulrthe^ of;the Rule,' in .fliort, lies 
 i^ this, m^, in computing the "Value of each 
 Year's Annuity,- it fuppofes that the.loling 
 Chances arifing in the firft Year, and fo fuccef- 
 lively in each Year, are actually loft in fuch Years 
 refpedivcly, and directs a Value, proportionate 
 to fuch Lofs, to be deduded out of the Value of 
 each Year's Annuity, as not an abfolute Annuity; 
 Whereas the Fadt is, that thofc Chances, or any 
 Part of the Value of the Annuity, are not loft, 
 and theref9re ought not to be computed as loft ; 
 for no Part is loft or can be loft, unlefs the 
 whole is loft; and the Cafe itfelf fuppofes that 
 neither the one nor the other is loft'; for it fup- 
 pofes that there are gaining Chances in the fulD- 
 fequent Year, and more than an evert Chance to 
 enjoy the Annuity of the fecond Year; but thofe 
 Chances cannot be had, or the Annuity be enjoyed 
 in fuch fecond Year, without a previous Enjoy- 
 ment of all the Chances, and of the whole An- 
 miity of the firft Year; and this holds good in 
 each Year and on every Year's Annuity fuccef» 
 iively, till we come to that Year in which it is 
 an even Chance, that Life is at an End. The 
 true State of the Cafe then is, that there are no 
 lofing Chances, or any Want of Value in the 
 Annuity, do in Fad'i- or can arife till Life 
 itfelf determines, and the fuppofition is, thaC 
 Bife continues ; and if it continues/ it miift con-? 
 tinue wfthqilt any.Breach or Interruption, 'and fo 
 muft the Annuity which is attendant on it, till 
 N 4 'is
 
 [ 2GO ] 
 
 It ceafes to be an even Chance, that Life has any 
 longer Duration. 
 
 Or take the Matter in this Light ; and pof- 
 fibly in this View we fhall fee the Error of the 
 Rule more plainly, and what led this great Au- 
 thor into it ; and the Error perhaps more pro- 
 perly and ilridtly may "be in this. The Rule 
 then diredls, though not in fo many Words, 
 yet in Meaning and EfFedl, that we compute 
 the Number of gaining Chances arifing feparately 
 in each Year of Life, for fo many Years as there 
 is any poffible Chance that the given Life may 
 be extended to ; then to afcertain the Values 
 correfponding to fuch Chances, and to take all 
 thofe Values put together, as the Value of the 
 Annuity for fuch Life. Now I admit that the 
 Rule gives us all the poffible gaining Chances 
 arifing in Life, and Values correfponding to 
 fuch Chances ; and if we view the Rule in this 
 Light, it may feem to give us the Whole of the 
 Value of an Annuity for a Life; but I fay, that 
 the Values correfponding to thofe Chances, arc 
 not the very Values which the Annuitant has a 
 Chance to enjoy. It mud be admitted, and it is 
 allowed by this Author, that on the Bre/Iaw 
 Table it is an even Chance, an even Wager, or 
 Lay, call it which you will, that J. before 
 named lives 41 Years, and that he lives no 
 longer ; and it being admitted, that this Rule 
 does rightly afcertain the Number of gaining 
 Chances upon this Life, it neceffarily follows, 
 that the Annuitant muft enjoy all thefe gaining 
 Chances within the Compafs of 41 Years, and 
 
 not
 
 [ 201 3 
 
 not afterwards, or we run ourfelves into two 
 contradidory Suppofitions. The one is, that 
 the Annuitant has not a Chance of enjoying this 
 Annuity, which upon the Cafe ftated is to con* 
 tinue for the Life of J.^ at a Time when wc 
 ^ave before fuppofed that he has a Chance that 
 ji. is living, which is fuppofing J.^ to be alive 
 and not alive at the fame Time j and the othe^ 
 IS, that the Annuitant has feveral Chances of 
 enjoying this Annuity, which, upon the Cafe 
 ftated, is to determine on the Death of A at and 
 after a Time, when we have already fuppofed 
 that the even Chance is that J. is dead ; Whicli 
 is fuppofing J. to be living at, and long after, 
 the Time we have fuppofed him to be dead and 
 gone. 
 
 If then all thefe gaining Chances upon the 
 Life of ^. mult be taken in the flrft Part of Life, 
 and in a continued Series, and fo they mull be 
 taken, or they cannot be taken at all ; and if the 
 Values annexed to thefe Chances fo taken, be 
 put together ; then thofe Values put together, 
 the Value of the even Chance on the Life of >^., 
 and the Value of an abfolute Term for 41 Years, 
 will all be the fame, 'viz. 15- 1 2. But if thefe 
 gaining Chances and the Values annexed to them, 
 are to be taken as they are computed fevcrally 
 to arife, in each Year of Life, and thefe Values 
 put together are the Value of the Annuity for 
 this Life, as this Rule alTerts ; Since many of 
 thefe Chances are fuppofed to arife after the Ex- 
 piration of 41 Years, and the Chances which 
 arife in the firft Part of Life or before 41 Years 
 
 expire,
 
 [303 ] 
 
 expire, are of much greater Value, than thofe 
 "which arife in the latter Part of Life, .or sfter 
 ,41 Years ; for Inflancc, if the Value- of all the 
 gaining Chances ariling in the laft ten Years of 
 poflible Life, is not greater than the Value of 
 eight fuch Chances ariling in the firit Year of 
 X.ife; and it we leave out or fubftrad the Value 
 of eight Chances in the firft Year of Life, and 
 in their ftead infert, fuppofe the Value of an 100 
 Chances iq the laft ten Years of I^ife, as this 
 Rule direds, it may very well come out that 
 the Value of the Annuity on fuch a Life, where 
 the Computation is made in this Manner, (hall 
 t)e no more than 13-44, or 13-50, and if made 
 5n the former Maqncr, fhall be 15-12 : But that 
 thofc Chances or the Values correfponding to 
 them are not then and there to.be taken, but 
 when and where they are to be taken, I think has 
 been fufficiently Ihcvvn. 
 
 " To go on therefore to another Pofition, which 
 is a Confcquence of the laft; I affirm that this 
 Sum 13-44 (or' rather 13-50) given in thefe 
 Tables as the Value of an Annuity for the Life 
 of J. fuppofed to be ten Years old, is^the Value 
 of another Annuity, or rather of feveral Annu- 
 ities joined together, different from an Annuity 
 • for the Life of J, abfolutely, or for 90. Years, 
 if J. fo long lives. I fay then, that ijiis. Value 
 Is the Aggregate or Sum total of ihe Value of 90 
 leveral Annuities, granted to^;ppfeyeratPerfons, 
 iiippofe to N^'-"i'..tp,N^''2!^^ andfopnto N^' 90. 
 inade m this Manner,.w^.VtO'lfj'^*-i. for one 
 year, if he Ihaiflive from lO to li Years Age- 
 
 to
 
 [ 203 3 
 
 to N°* 2. for one Year, if he fhall live from ii 
 tq/ii Years -Age ^ ' and fo on fucceffively to* 
 N^Vpo. for a Year, if he fliall live from 99 to 
 ' be an 100 Years old ; each Perfon to be nomi- 
 nated at the Beginning of that Year for which 
 the Annuity . is granted, " determinable by his 
 Death within the Year. ' I affirm then, that this 
 Sum 13-50 is the Value of thofc 90 feveral An- 
 nuities, feparately computed for each Life, and 
 then' put' together"; 'and that the Intereft, fup- 
 pofe of'N^viob. in an AnRuity for the Life of 
 A abibluteiy, orfor^o Ypars^if i^.'fo long lives, 
 is of a different and much greater- Yalue than the 
 Interefl of N'^-i. N''/ 2:' iin'd'fo on to N^* po. 
 all put^ together : The plain and neceflary Con- 
 fequenceof which'is,''that 13-56 is not the Value 
 of- an Annuityfor the Life'of .^»; ' 
 •: My-firi! AlTertibn is,' that the Sum of 13-50 
 is the Value' of thefe fe Veral 96 Annuities put to- 
 gether. This is. a Point that feems to require 
 . little Proof^ fince fiich Sum appears to be no- 
 thing more or other than a Collection of 90 fe- 
 veral Annuities, feparately computed for each 
 Year of a Life for 90 Years, dedudting out of 
 each Year a Value correfponding to the Chance 
 of Mortality' arifing on the Life of a Perfon of 
 the Age of each Year refpedtively. In my 
 Table, ]SI^' IX. we' have a Computation of the 
 Value of an Annuity of looob/. per .Afwum^ 
 calculated on the Chance of 'Mortality for each 
 Year of the Life of a Perfon of ten Years old 
 for 90 Years, drawn accdrciing to the DiredHon 
 of the Doctor's Rule ; the Sum total of all 
 
 which
 
 [ 204 ] 
 
 which Annuities put together, is 1 35000. Now 
 I fay that any one and every one of the Annui- 
 ties for the Year, if N^- 1. N^' a. and fo on to 
 3Sf^* po. Ihall live to the End of that Year for 
 which the Annuity is refpedively granted to 
 them, is of the fame Value, neither more nor 
 Icisy as any one and every one of the annual An- 
 nuities exhibited in that Table ; the Annuity of 
 N^- 1, being the fame as the Annuity for the 
 firil Year of the Life there computed ^ the An- 
 nuity of N''* 2. the fame as the Annuity for the 
 fecond Year ; and fo on fucceflively and corrc- 
 ipondently, the Annuity of N"* 90. the lame as 
 the Annuity for the 90th Year of the Life of 
 the Perfon intended in fuch Calculation. This 
 will appear very evidently, on comparing the 
 Intereft of one with the Intereft of the other 
 Annuity correfponding ; and, for Inflance, we 
 will take the Intcrefl of N^* i. and compare it 
 with the Intereft of the Annuity for the firft 
 Year of the Life on the Table, and the Intereft 
 of N^' 20. and compare it with the Annuity for 
 the correfponding Year, the 20th Year of the 
 Life on the Table. 
 
 The Intereft of the Annuity belonging to 
 K^' I. is the fame as the Intereft of the Annuity 
 for the firft Year of the Life in the Table, and 
 muft, be the fame ; for that both are computed 
 ^d -afccrtain''d by one and the fame Rule, 'viz, 
 by takbg the Value of one Year's Annuity as an 
 abfolute one, after deducing out of fuch Value 
 fo much as is equal to the Contingency of Mor- 
 tality on fuch Life within that Year. The very 
 
 fame-
 
 [ 205 ] 
 
 fame is the Cafe on the Intereft of N^. ao. in 
 his Annuity, and the Intereft on the 20th Year 
 of the Life of the Perfon defcribed in the Table. 
 Both Annuities are in Reverfion, commence at 
 one and the fame Time, in all Events expire at 
 the End of the Year, and both depend on a 
 Contingency of determining within the Year, 
 and upon two Lives of equal Age ; and upon 
 the Foot of fuch Circumftances and fuch Con- 
 tingencies, which are in all Refpeds the fame, 
 the Value of each Annuity is computed. 
 
 My fecond Aflertion was ; that an Intereft of 
 an Annuity for the Life of ^, abfolutely, or for 
 ^O Years, if A, fo long lives, is of better Value 
 than all thefe feveral 90 Annuities put together. 
 This I ihall prove by fhowing, that an Annuity 
 for the firft Year of the Life of A. where 'tis 
 abfolutely for his Life, is more valuable than an 
 Annuity for the Life of N^* i., which is to de- 
 termine with the Year; and in like Manner, 
 that the Annuity for the fecond, the third, and 
 every other Year of the Life of A. fuccellively, 
 'till we come to that Year of the Life of A- in 
 which it ceafes to be an even Chance that his 
 Life has any farther Duration, is of more Value 
 than the Annuity for the Life of N^* 1. N^* 3. 
 and fo on fuccefTively, 'till we come to N^* 42, 
 within which Time it is computed that A, may 
 be dead : Comparing the Annuities belonging 
 to each Proprietor on the feveral Years, as they 
 ' refpe(aively correfpond to each other. 
 
 For Inftance ; the Intereft of N^* I. in h^s 
 Annuity may determine within the Year by his 
 
 Death
 
 [ 206 ] 
 
 Death in that Time, anid miift determine at the 
 End of the Year by the Effluxion of his Term. 
 The Intereft of N^* lOO. in his' Annuity for the 
 Life of -^. may likewife determine within the 
 fame Year' by the Death • of ^. his Nortiinee ; biit 
 docs not determine at; the End of the Year by 
 the Effluxion of the Term, Tor 'tis to continue 
 fot 8p Years "mord,- if '^. fo long lives : That is, 
 to one of the Annuities are annexed two Limi- 
 tations, ' one of which poffibly may,' and the 
 other neceflarily niuft, put an End to it by the 
 Completion of the Time for which it was grant- 
 ed. The other has one Limitation only, viz. 
 the very fame Contingency of Mortality which 
 the other has'; but has no fuch Termination at 
 the End of the Year... Therefore N®* loo. in his 
 Annuity muft have a better Chance to the fame 
 Thing, or a Chance to fomething more or better 
 than N^* i. has in his Annuity. For this Rea- 
 fon, and on this Foundation, the Circumltances 
 of every other of thefe po Annuitants for a Life 
 determining with the Year, at leaft 'till we are 
 come to that Year when the Chance is an even 
 one that J. is dead, will be the very fame ; that 
 is, their Annuities refpedtively will be of lefs 
 Value than the Annuity for the Life of J. in the 
 Year correfponding. ' 
 
 If we enquire what is the Difference, and what 
 makes the Difference, in the Value of thefe An- 
 nuities ; 'tis plainly this, viz, N^- i. N^.' i- and 
 fo on, cannpt, be looked upon to have a Chance, 
 and their Annuities are computed upon fuch 
 Foot as if they had not, and moil certainly they 
 
 have
 
 [ 207 } 
 
 have not a Chanccto enjoy all the Parts of an 
 Annuity for^aiY'eari,for within the Year they.can-. 
 not be.faid^to,|i^ye a Chance to ..enjoy that which'. 
 on thcGontiogency of Mortality they have a 
 Chance to lofe in that Time ; and beyond the Year 
 they have no Chance, or Poflibility of Chance, 
 bccaufe their refpedive Annuities abfblutely, de- 
 termine with the Year. But witbRegard to N^': 
 100, he may very well be look'd upon as having 
 a Chance to enjpy all the Parts of this Year's An- 
 nuity,^ be^caufejiis Annuities no^ determining with 
 the Year, he has many.Chanccs of enjoying, tlae 
 fecond Year's Annuity, and by .that Means hegains 
 in fuch fecond Year what there was a Chance^mights 
 be. loft in, the firft : And this Chance of gain- 
 ing in a.fubfequent Year, what might be loft in 
 the prccedent.Year continuing in the Cafe of N^* 
 lOO. as long as the Chance of the Vitality of Jl» 
 continues to be an even Chance, that is, 'till A. ar- 
 rives to the Age of 51 ; the Annuitant in the whole 
 has fo many and fuch Chances of enjoying the 
 Annuity for 41 .Years as make it equal in .Value to 
 an abfolute: Annuity for that, Term. of Years.-... 
 As'.this,app.ears to .be. the true State. of the; 
 Matter, . w.hen . we confine r. the Cafe of the . An- 
 nuitant-for. the. Life .of .^. abfolutely compared 
 with'the Cafe. of. the feveral Annuitants. for a 
 Year .whpfe Intere.fts, arc determinable .by their 
 X.ives;;the;fame will. come out to be the^State 
 of the Matt,erj,.if.>ye.take a .View of the Cafe of 
 the Grantor .of fugh. Annuities refpediyely," or of 
 .him who is to pay the fame. This is an un- 
 queftionable Truth, that the Chance to pay or 
 
 not
 
 [ 2o8 ] 
 
 not to pay, and the Chance to receive or not tof 
 receive the Annuity, muft be reciprocal and one 
 and the fame ; fo that if the Grantor of the An- 
 nuity has a Chance to pay a greater Sum to the 
 Annuitant for the Life of J. iingly, than he has 
 ^ Chance to pay to the fevcral 90 Annuitants, it 
 plainly follows, that the fingle Annuity is of 
 more Value than all the feveral 90 Annuities put 
 together. With Refped to the Annuitant N^* 
 I. I agree that the Grantor has a Chance not to 
 pay fuch Annuity, and I admit he has jull the 
 like Chance not to pay the Annuity for the firll 
 Year of the Life of A.^ there being within that 
 Year an equal Chance of Mortality both againft 
 the one and the other. With Refpedt to the 
 Annuity to N^* a., I will admit likewifc that the 
 Grantor is on the fame Foot as with Refped; to 
 the Annuity for the fecond Year of the Life of 
 A. \ but then I admit this on a Suppolition that 
 A, is living at the Beginning of fuch fecond 
 Year; for N^* a. is fuppofed to be then living, 
 or rather to be then nominated to take the An- 
 nuity \ for unlefs both are fuppofed to be living, 
 and both Annuities to exilt ftill, there is no Room 
 to make a Comparifon of the Annuities for fuch 
 fecond Year. If then A. is fuppofed to be living 
 in the fecond Year, it neceflarily follows, that 
 the Grantor muft have paid the Annuity for the 
 foil Year of the Life of A. as an Annuity abfo- 
 lute, and without any Defalcation ; and this is 
 the Cafe in every other Year of the Life of A, 
 lb long as it continues an even Chance that A, is 
 living : that is, we can fet ng Value on, or make 
 
 any
 
 [ 209 ] 
 
 any Computation of the Price to he paid for, 
 the Annuity in any fubfequent Year of the Life 
 of A without previoufly fuppoling that A is 
 then living ; the neceflary Confequence of which 
 is, that the Annuity for the precedent Year muft 
 alfo be fuppofed to have been paid, and if paid 
 the Sum paid mufl have been the full Value of 
 fuch Year's Annuity, as an abfolute one without 
 any Deductions. 
 
 The Doctor's Hypothefis, and the Rule form'd 
 upon it, is faulty therefore, in making two Sup- 
 politions which are not confillent one with the 
 other, 'viz. it fuppofes that the Grantor of the 
 Annuity in each Year of the Life of A. has fuch 
 a Chance of the Mortality of ^. as may pofTibly 
 exempt him from paying that Year's Annuity ^ 
 and at the fame Time it fuppofes there is a 
 Chance that A is living in the fubfequent Year 
 without paying the Whole of the Annuity for 
 the precedent Year as an abfolute one. Now 
 'tis poffible that J. may die in the ftrft Year, for 
 without Doubt there is fome Chance he may die 
 in each Year, and in every Day and Hour of 
 fuch Year, and if he does die, the Grantor will 
 fave the Payment of the Annuity for that Year, 
 and all the fubfequent Years ; but fuch Grantor 
 has no Chance of J.'s dying or not dying i,n a 
 fecond or other fubfequent Year, unlefs he has, 
 and until he has, lived out the firlt and precedent 
 Year : Confequently the Grantor can have no 
 Chance of paying or of not paying the Annuity 
 of the fecond or any fubfequent Year, unlefs he 
 has, and until he has, paid the whole of the An- 
 O nuity
 
 C 210 ] 
 
 nuity ibr the firfl and precedent Year : So that 
 if we make the firft Suppofition, there is ncr 
 Room to make the lecond at all, and if we make 
 the fecond the firft is fet alide and made void. 
 
 Where 90 feveral Annuities are granted for 
 90 Years, if fo many feveral Perfons live to the 
 End of thofe Years refpedively for which fuch 
 Grants are made, here the fame Suppolitions are 
 made as before, and here are rightly made. 'Tis 
 evident in this Cafe, that the Grantor of thefe 
 Annuities has a Chance of Mortality of the No- 
 minees in each of thefe Years, fuch a Chance as 
 fhall difcharge him from Payment of the Annu- 
 ity of that Year ; and if the Nominee does die 
 within the Year, the Grantor will fave that Year's 
 Annuity. The fecond Suppofition is likewife 
 true here ; for the Grantor will have a Chance of 
 the Death of the Nominee in the fecond Year, 
 or any other fubfcquent Year, whether the No- 
 minee in the firll, or any other precedent Year, 
 did or did not die within the Year; and he will 
 have the Chance of not paying in the fubfequcnt 
 Years, whether he paid the whole or paid no 
 Part of the Annuity in the precedent Years. 
 Therefore, the Chance of the Grantor of the 
 fingle Annuity and of the feveral Annuities, not 
 being one and the fame, the Value of the An- 
 nuity for the fingle Lil^e mull be a Sum different 
 from the Sum of the Values of the feveral An- 
 nuities collected together, that is, from the Sum 
 of 1350/. 
 
 There is flill another wrong Suppofition made 
 in this Rule, and which, I apprehend, gave Oc- 
 
 caiioa
 
 [ 211 ] 
 
 cafion to the two former, viz. that the Chances 
 of Mortalitj are the fame on the Annuity for a 
 lingle Life as in thefe Annuities for 90 feveral 
 Lives, fuch as I have before defcribed 5 or ra- 
 ther that the Chance of Mortality may as often 
 arife in the one Cafe as in the other. Where 
 there are 90 feveral Annuities depending on fo 
 many feveral Lives, 'tis obvious there is a Chance 
 of Mortality on each Life, and fuch Chance may 
 arife in the Year of the refpedlve Life ; but in 
 the Cafe of an Annuity on a fingle Life there 
 can be but one Chance of Mortality, at leaft, 
 fuch Chance can arife but once, for a Man can 
 die but once only. On this Account it feenis 
 abfurd to compute how many Times in each 
 Year, or in 90 Years, the Chance of Mortality 
 may come up, fince in the whole 90 Years it can 
 come up but once only ; and molt certainly 'tis 
 unneceffary to make fuch a Computation, fines 
 during all the po Years of the pofTiblc Life of 
 ^. the Grantor, in the whole, has but one Chance 
 of being difcharged from paying the Annuity, 
 and in every Year, until that one blank Chance 
 does come up, he muft pay the Whole of the 
 Annuity, as an abfolute one. 
 
 With a View of afcertaining the Time to 
 which the Life of a Perfon may probably be ex- 
 tended, it might not be improper to compute 
 how many Chances of Mortality there are in 
 each Year of fuch Life, and from thence to col- 
 led: in how many Years fuch blank Chance, in 
 all Probability, will come up • and then we may 
 Hate itp that the Life and the Annuity attending 
 O 2 on
 
 [ 212 ] 
 
 on it will continue to that Year in which on fuch 
 Computation the Chances of Vitality and Mor- 
 tality are even, that is, when the Chances of Vi- 
 tality of the given Life are funk to a Moiety ; 
 and from and after that Year, that the Life and 
 Annuity with it will ceafe j but this evidently 
 deftroys the Rule. 
 
 Since then the Chance of the Grantor of an 
 Annuity for the Life of A. is a fingle one, and 
 more than once it cannot come up \ and the only 
 Thing ncccffary to be computed is, within what 
 Time on the probable Chances of Mortality J. 
 may die : And lince the Chance of his living for 
 41 Years and dying before 42 Years is an even 
 Chance, as computed on the Brejlaw Table, and 
 feveral Rules deduced from thence \ and feeing 
 in any one and every one of thofc Years in which 
 the Annuity is paid, the Whole oi the Annuity 
 mult be paid : on this Cafe, and from thefe Pre- 
 mifes, I infer, where an Annuity is granted for 
 the Life of J. abfolutely, or for 90 Years, if J. 
 fo long lives, that the Annuity for any one aiid 
 every one of the Years of the Life of ^. until 
 it becomes an even Chance that J. does no 
 longer cxili, is of more and greater Value than 
 an Annuity for any one and every one of the 
 Years correfponding, where the like Annuity is 
 granted to 90 feveral Perfons for 90 feveral 
 Years, if fuch feveral Nominees Ihall refpec- 
 tivcly live to the End of that Year for which 
 the Annuities are refpedivcly granted. 
 
 The Conclufion, upon the Whole, is this: 
 Since the Sam of 13-44, (or rather 13-50?) 
 
 which
 
 [ 213 ] 
 
 which 18 the Value of an Annuity for the Life 
 of A. when fuch Value as computed by this 
 Rule, is the Aggregate or Sum total of the 
 Value of 90 diflindl and feparate Annuities for 
 a Year for fb many fingle Years as the feveral 
 Nominees in them fhall refpcdtivcly live, and 
 which determine at the End of each Year \ and 
 iince the Value of an Annuity for the Life of ^. 
 abfolutely, or for 90 Years, \i J, fb long live?, 
 is the Aggregate or Sum total of the Value of 
 fo many Years of 90 Years as A. fhall live to 
 the Elnd of, and which do not determine at the 
 End of each Year, but at the End of all the 
 Years, or by the Death of J. and then only : 
 And Iince the Value oi the Annuity in each 
 Year, where it is for a Life abfolutely, amounts 
 to a greater Sum than the Value of the Annuit}', 
 where it determines at the End of each Year, 
 amounts to : The Conclulicn is, that this Sum 
 of 13-44 or 13-50, which is the Sum total of 
 the Value of thofe Annuities which arc lor Lives, 
 and determine at the End of each Year, is not 
 tke Sum total of the Value of the Annuities 
 which are for Life abfolutely, and do not de- 
 termine with the Year, that is, is not the Value 
 of the Annuity for the Life of J. %. E\ D, 
 
 The Doctor, in framing this Rule, and as a 
 Foundation on which he builds it, lays down 
 this Maxim ; that a Purchafer of an Annuity 
 for a Life is to pay for fuch Parts only of the 
 Value of the Annuity as he has Chances that his 
 Nominee is living, which moil cei-tair^ly is true ; 
 but then 'tis as certainly true, that he is to pay 
 6 3 Tor
 
 [ 214 ] 
 
 for all the Parts and the very Parts of the An- 
 nuity which he has Chances to receive on the 
 Life of his Nominee : But whether this Maxim 
 be rightly and thoroughly complied with and 
 preferved in this Rule, is a great QuelHon. In 
 the Cafe before ftated on the Life of ^. 'tis pret- 
 ty plain, that the Rule does not bring to the 
 Annuitant's Account thofe Parts of the Value of 
 the Annuity which he has a Chance to enjoy, 
 fince he has a Chance to enjoy all the Parts of it, 
 as an Annuity abfolute and entire for fo many 
 Years as he lives, whereas the Rule afligns to 
 him fuch Parts only as are anfwerable to the 
 Chances of Vitality arifing in each Year. If we 
 vary this Cafe, and put it, that an Annuity is 
 purchafcd on the Life of B. whom we will fup- 
 pofe to be of fuch an Age, be that what it will, 
 as that the even Chance is that he lives one Year 
 and no rriore ; I apprehend we fliall fee more 
 plainly that the Annuitant, purchafing accord- 
 ing to the Diredion of the Rule, does not pay 
 for all and the very Parts of the Annuity which 
 he has the Chance to enjoy, and muft enjoy, it 
 he enjoys any. 
 
 Wc will take then the Age of So as the Age 
 in which B. has an evea Chance to live one Year, 
 and no more ; and in ftating this to be the 
 Chance of Vitality for fuch Age, we neceflarily 
 fuppofc, and ground our Cafe upon this, that 
 the Chances of Vitality and Mortality ariling in 
 that Year are even. Now if the Purchafer of 
 the Annuity for the Life of B. abfolutely pays 
 po greater Sum for his Interell than a Purchafer 
 
 pays
 
 215 J 
 
 pays for his Tntereft who buys the like Annuity 
 for one Year only, determinable by the Death 
 of 5., 'tis obvious, and I fuppofe it will be ad- 
 mitted, that the Owner of the Life Annuity ab- 
 folute does not pay for all the Parts of bis An- 
 nuity which he has Chances that his Nominee is 
 living. The Proprietor of the Annuity for a 
 Year, determinable by Death within the Time, 
 pays only for fo many Chances of Vitality as 
 arife within the Year, that is half a Year's Value, 
 and in juftice ought to pay for no more, lince 
 he can have no Chances arife beyond the Year; 
 but the Proprietor of the Life- Annuity abfolute, 
 if the Price of it were the fame and no morc^ 
 would have many Chances for which he paj's no- 
 thing. In this Inilance, where there are as many 
 Chances of Vitality to arife beyond the Year as 
 there arc Chances arife within the Year, he 
 would have juft as many Chances of Vitality left 
 at the end of the Year, for which he pays nothing, 
 as he has Chances for which he does pay. 
 
 It being admitted then, that the Purchafer of 
 the Annuity for the Life abfolute, is to pay 
 more than he who purchafcs for one Year only, 
 if B. Co bng lives ; it is faid that thefe Chances 
 of Vitality left at the End of fuch firlt Year, are 
 not totally funk and unpaid for, by the Pur- 
 chafer of the Life- Annuity abfolute; for though 
 they are not brought to Account and paid for as 
 Chances of Vitality arifmg in the firfl Year, yet 
 the Rule direds, that the Chances of Vitality 
 are to be computed like wife for the fubfequent 
 Years, as long as Life can be fuppofcd poiTi- 
 O 4 bly
 
 [ 2l6 3 
 
 bly to laft ; that the Chances arifmg in thofe fubfe- 
 quent Years of poffible Life, are fuppofed in the 
 Cafe to be the fame, in Number, as the Chances 
 left at the End of the firfl Year ; and that fuch a 
 Purchafer muft pay for thofe Chances. I admit, 
 that the Rule does dired that thefe remaining 
 Chances of Vitality are to be paid for, and by 
 that Means, that all the Chances in fome Senfe, 
 ^nd at fome Rate, are paid for ; yet thefe are not 
 to be paid for as Chances ariiing in the firft 
 Year, but as Chances ariiing in the feveral fub- 
 fequcnt Years of polTible Life j and then the 
 Annuitant dpcs not pay for the very Parts of the 
 Value of the Annuity, which he has a Chance to 
 receive on the Life of the Nominee ; for fure 
 enough, the Chances ariliug in the fubfequent 
 Years of pofFible Life, are not thofe very Parts 
 of the Value of the Annuity which he has a 
 Chance to receive on fuch Life. 
 
 If we reckon the whole Number of Chances, 
 in the firll Year of the Life of this Nominee tp 
 be twenty, it being juft an even Chance that he 
 lives to the end of that Year and no longer, ten 
 of thofe Chances in that Year will be Chances 
 of Vitality, and ten of Mortality; fo that at 
 the end of that Year, there will be ten Chances 
 of Vitality remaining. As for the ten Chances 
 pf Vitality arifing in fuch firft Year, there is no 
 Controverfy but thofe are to be paid for as 
 Chances arifing in that Year ; nor can there be 
 any Difpute but that the Sum to be paid for 
 them is one half of the prefent Value of the 
 Annuity, as an abfclutc one. Put as for the ten 
 
 Chances
 
 [ 217 ] 
 
 Chances of Vitality, fuppofed to be remaining 
 after the end of the Year, in my Way of ftate- 
 ing the Value of this Annuity, thefe ten Chances 
 muft be looked upon as all ariling within fuch 
 firft Year ; for none can arifc in any fubfequcnt 
 Year, until and unlefs thefe have firft arifen and 
 that they ought to be paid for, as for Values 
 in prefent PolfelTion ; but in the Way of ilating 
 fuch Value, according to the Rule, thefe ten 
 Chances are to be look'd upon as arilino- all 
 alter the Expiration of the firft Year • foms 
 ariling in the fccond, fbme in the third, and 
 fome in the twentieth Year of poffible Life ♦ 
 and the Sum total of the Values correfponding 
 to thofe Chances fo ariling, is to be taken as the 
 Sum to be paid for thele remaining Chances. 
 Now, lince the Number of Chances of Vitality 
 remaining unpaid for, in cither Way of ftadno- 
 it, is one and the fame, 'viz. ten ; here where 
 thofe ten Chances are all in Reveriion, and 
 fome very diftant, and the Values correfponding 
 reverlionary ones, the Sum total of thofe Values 
 muft be greatly lefs than the Sum total of the 
 Values correfponding to the like Number of 
 Chances, in my Way of Hating it, where the 
 Chances and Values correfponding, are all ia 
 immediate Polfeffion, as fuppofed to arife in the 
 firft Yean 
 
 I am afraid then that a Perfon, who undertakes 
 to pay an Annuity for fuch a Life, will think he 
 has an ill Bargain, if he is to be paid for fuch 
 Undertaking, according to the Rate of fuch re- 
 verlionary and remote Values j and will be apt 
 
 to
 
 [ 2l8 ] 
 
 to furmife, that the Purchafer does not pay 
 for the very Parts of the Value of the Annui- 
 ty, which he will have the Chances to receive. 
 He fays then, and mentions it only, becaufe it 
 has been infilled on already, that on computing 
 the Chances of Life and Death, the even Chance 
 in this Cafe is, and the Nominee on whofs Life 
 the Payments depend, may reafonably expedt, 
 that he Ihall live to the End of one Year, that 
 on the fame Grounds he muft exped: to pay the 
 Annuity for that Year ; that if he does fo, he 
 muft make prefent and immediate Payment : 
 And therefore that it is unreafbnable, that the 
 Annuitant fliould pay him lefs than the prefent 
 Value of the Annuity, payable at the End of 
 the Year, or pay his Purchafe-Money, by a 
 Computation on reverfionary Values, that will 
 fall greatly fhort of the prefent Value of the An- 
 nuity for a Year, and yet fo much the Annui- 
 tant has an even Chance to receive. 
 
 He has this farther to offer under this Head ; 
 that if the Nominee furvives the Year, he £hall 
 be obliged to pay the whole and entire Annuity, 
 for that he knows no Contingency in his Cafe, 
 that will difcharge his Engagements by paying 
 Part only ; for there is but one Chance in the 
 whole, either that the Nominee does, or that he 
 does not furvive the Year ; and here the even 
 Chance is, that he does furvive the Year, in the 
 fecond Year of the Nominee's Life ; the fame 
 Obfervation recurs, that he has no Contingency 
 of performing his Contract by a partial Pay- 
 ment ^ for if the Nominee beyond Expeftation 
 
 Ihould
 
 [ 219 ] 
 
 Ihpuld outlive a fccond Year, he mufl again pay 
 the Whole ; but here the even Chance is, that the 
 Nominee is dead, and that he has nothing to pay. 
 The Gafe will' be the very fame, if the Nomi- 
 nee be young, and in Gourfe of Nature, likely 
 to live many Years, yet there is no Room, .13^0 J 
 there can be no Chance of dividing the Annuity 
 in any one Year, and paying a Part, and faving 
 the refl in that Year. Here indeed, the Chance 
 of Vitality gives room to make a Divifion of 
 the whole Annuity, by the Number of Years 
 of the whole, as of 70, 80, or po Years, to 
 which the given Life may pollibly extend ; and 
 we may, on a Computation, cftimate, that to 
 fo many Years fuch a Life probably will extend, 
 and beyond fo many Years it will not extend ; 
 but of the Annuity in each individual Year, no 
 DiviJion can be made. There being then no 
 Medium, on any one Year, between paying the 
 Whole and paying nothing ; the Right of the 
 Thing feems- to be where the even Chance is, 
 that the Nominee lives to the End of the Year, 
 and fo long as that Chance continues, fo far and 
 for fo many Years, the Purchafer muft pay for 
 the whole and every Part of the Annuity, as 
 an abfolute one; and where the even Chance 
 is, and from the Time when the Chance begins 
 to be fuch, that the Nominee is dead, thatfrom 
 thence the Purchafer is to pay nothing. 
 
 He admits it may be true, that the Annui- 
 tant pays for all the Chances of Vitality on 
 the Life, if by all, is meant all in Number, 
 but he apprehends the Senle of the Maxim laid 
 
 down
 
 [ 220 ] 
 
 down by the Do6tor, to be, that the Chances 
 which the Purchafer is to pay for, are to be 
 equal in Value as well as Number, to the Chances 
 which he has to receive. He has been taught by 
 another Learned A uthor, that a Man who has one 
 Chance in three only, may have as much Advan- 
 tage as he that has one in two ; and I will add, 
 that he that has the fewer Chances may have 
 ten Times, or a thoufand Times a better Chance, 
 than he that has the greater Number of Chances, 
 provided the Sum which he has a Chance to 
 gain, be proportionally greater than the Sum 
 which the other has a Chance to gain. He al- 
 ledges therefore, that it is not enough for him, 
 that for the ten Chances of Vitality, fuppofed to 
 be left at the end of the firll Year, he is to be 
 paid in the Value of fo many Chances, which by 
 a fictitious and groundlcfs Suppofmon, are com- 
 puted to be fo left, and to arife in the fubfe- 
 qucnt Years, Years long fubfcquent to the Time, 
 when in all Probability the Nomineee is dead, 
 and in his Grave ; Since the Value of all thofe 
 Chances fo ariling, at the higheft Calculation 
 that can juflly be made in his Favour, will not 
 exceed a third Part of a Year's Value of the An- 
 nuity, and he on the other lide has an even 
 Chance to pay, nay, is as fure to pay, as he is 
 fure to pay any Thing for thofe ten Chances, as 
 if they were the Value of a Moiety of the Year's 
 Annuity ^ that is, as for fo many Chances of 
 Vitality ariling in the firft Year. In Fad and 
 Reality , thefe fuppofed remaining Chances mull 
 arife in the firll Year, and before any other 
 
 in
 
 [ 221 ] 
 
 in any fubfequent Year do arife, or they can 
 never arife at all, and as fuch they mull: be paid 
 to the Annuitant, or none muft be paid ^ and 
 therefore, as fuch, they ought to be paid for by 
 the Annuitant, or none ought to be paid tor : 
 So that in the Event it comes out, if the Pur- 
 chafe-Money to be paid is calculated by this 
 Rule, the Purchafer will not pay for all, and the 
 very Parts of the Value of the Annuity which 
 in Reality he has the Chances to receive, or has 
 the Chance that his Nominee is living. 
 
 He adds this farther j that his Cafe upon the 
 Life of B. where the even Chance is that he lives 
 to the End of one Year, and no longer, and 
 where the Chances in ail are fuppofed to be 
 twenty, and ten of them are Chances of Vitality, 
 and ten of Mortality, may very well be re- 
 fembied to the Cafe of a Proprietor of a Lottery, 
 and the Life of Man has but too much Refem- 
 blance to a Lottery, in w hich the whole Number 
 of Tickets is twenty, and one Moiety of them 
 is Prizes, and the other Blanks. We will fup- 
 pofe then that the Prizes in this Lottery are each 
 of the Value of 9434/. or the prefent Value of 
 loooo/. to be paid at the End of one Year, 
 and which, according to my Computation, is 
 the Value of the Annuity for the Life of J5. ; 
 and the Qucftion will be only this, How many 
 Tickets an Adventurer in fuch Lottery mult be 
 poflefs'd of, tohave an even Chance to a Benefit- 
 Ticket, and what Price he muft pay for each 
 Ticket ? 
 
 I fay
 
 [ 222 ] 
 
 I fay then ; to have an even Chance for a Be- 
 nefit-Ticker, he muft buy two Tickets, and for 
 each he mull pay 3. Moiety of the Value of a 
 Prize ; that is, for his two Tickets he mull pay 
 the Sum of 9434 /. To prove thcfc AiTertions, 
 it v.'ill hardly be necefTary to refort to a long al- 
 gebraical Calculation, though they may be fo 
 proved, becaufe I believe there is not a Broker 
 in Eschafige^Jlhyy even of the lowelt Form, but 
 knows, if he cannot demonllrate, the Truth of 
 them. The Cafe then of him who purchafesan 
 Annuity on the Life of B.^ and of him who ad- 
 ventures in this Lottery, being in all Refpedts 
 parallel; for the Chance of gaining is equal, and 
 the Sum to be received is one and the fame ; 
 therefore one and the fame Sum is to be given on 
 each Adventure. 
 
 If it be true, as I admit it is, that the Annu- 
 itant may receive no Part of this Annuity, as he 
 will receive none if B, dies within the Year, and 
 this undoubtedly is pofTible ; fo it is true, and 
 mull be admitted, that the Lottery- Adventurer 
 likewife may receive nothing, as will be his Cafe, 
 if both his Tickets come up Blank; and this is 
 poffible too, as well as the other. But as fuch 
 Chance, or PolTibility rather, of both Tickets 
 being Blank, in Reafon, ought not, and to be 
 fure will not prevail on the Proprietor of the 
 Lottery, who is to pay the Prizes, to fell two of 
 his Tickets, which will give the Adventurer an 
 even Chance to a Prize, at a Rate lower than the 
 full Value of a Prize : So the Grantor of an An- 
 nuity on the Life of JB. abfolutely, though there 
 
 is
 
 [ 223 ] 
 
 is a Poflibillty that 6. may die within the Year, 
 and he may have nothing to pay, fince he under- 
 takes to pay the Annuity on the even Chance of 
 one Year's Life of B, cannot, in Juftice to him- 
 felf, accept of a Sum lefs than the prefent Value 
 of one Year's entire Annuity. 
 
 If it be faid on the other Side, that there is a 
 Contingency that both Ticlcets may be Prizes, 
 it may alfo be faid that poffibly B. may live a 
 fecond or a third Year : But fuch Poflibility of 
 having two Prizes, will never be allowed to be a 
 juft Reafon for giving more for two fuch Tickets 
 than the Value of a lingle Prize ; and the Pofli- 
 bility of two or more Years Continuance of the 
 Life of B. will hardly prove that more is 
 to be given for the Annuity, than one Year's 
 prefent Value of it; and to be fure will never 
 prove that lefs is to be given, as the Rule directs, 
 I may therefore fafely leave it to Dr. Halley^ and 
 thofe who follow his Rule, to fhevv in what In- 
 ftance thefe Cafes are not parallel ; or to prdve 
 that the Sum of 9434 L is not the Parts of the 
 Value of the Annuity which the Proprietor hais 
 a Chance to receive, and the Grantor a Chance 
 to pay, on the Life of this Nominee. 
 ' " From hence,. I apprehend, it will follow; 
 cither that the Maxim laid down, that a Pur- 
 chafer is to pay for fuch Parts only of the Value 
 of an Annuity, as he has Chances that his No- 
 minee is living, is not in all Senfes true ; or that 
 in fuch Senfe in which it is true, it is fometimcs 
 wrongly applied by the Rule. In this Senfe of 
 the Maxim, ws, that fuch Parts of the Value of 
 
 the
 
 [ 224 ] 
 
 the Annuity are to be paid, for which the An- 
 nuitant has the Chances to receive, fuppofing the 
 Parts to be paid for, and the Parts to be received 
 are in Number the fame, whether they are in 
 Value the fame or not the fame; in that Senfe 
 the Maxim is not true. In the Senfe in which 
 it is true, viz. that fuch Parts of the Value of 
 the Annuity are to be paid for, as in Value are 
 equal to thofe Parts which the Annuitant has a 
 Chance to receive; in that Senfe the Rule has, in 
 this Inftancc, made a wrong Application of the 
 Maxim ; for thofe Parts of the Annuity which 
 the Annuitant has the Chances to receive, arc of 
 much greater Value, then thofe Parts of it, 
 which by this Rule he is directed to pay for in 
 his Purchafe. 
 
 In the former Inflance, on an Annuity for the 
 Life oi^ Af fuppofcd to be ten Years old, the 
 Maxim is as ill applied as here, though not in 
 the fame Manner. There the Rule applies it to 
 every Year of the Life of Jl., as fcparate ^nd 
 diftind Years, and as feparate and diftindt An- 
 nuities for thofe Years; which is applying it to 
 fomething in which the Annuitant for the Life 
 of X is in no wife concerned ; for his Annuity 
 has no Termination at the End of each Year, 
 but is to continue for 90 Years in an uninterupt- 
 ed Line, unlcfs Death intervenes and breaks it. 
 The Maxim may indeed be applied to this An- 
 nuity, and will be rightly applied thus far, and 
 in this Manner. The Value of an Annuity for 
 a Term of po Years abfblute, is in Value fo 
 much ; of this Term and an Annuity attendant 
 
 on
 
 [ 225 ] 
 
 on ic, the Proprietor of tfie Life Annuity has an 
 even Chance, or the Chances to enjoy fo many 
 Years ; and if we enquire in particular, what 
 !^arts of the Term or the Annuity the Proprietor 
 has the Chances to enjoy, I fay he has fo many 
 Chances that J. his Nominee, will continue in 
 Life, as that the Chances in the whole (hall 
 amount to an even Chance, that he lives for 41 
 Years from the time of the Annuity granted^ 
 and by two Rules of this Authors, which wc 
 have before mentioned, that is computed to be 
 his Chance of Vitality : Confcqucntly that for 
 thefe 41 Years, the Annuitant will receive the 
 whole and all the Parts of the Annuity as an ab- 
 folutc Annuity, and therefore by the Maxim 
 mull pay for it as fuch for that Term ; for it is 
 certain, as he ought not to pay for more than 
 all, fo he ought to pay for all, and fuch as he 
 receives, or has an even Chance to receive. 
 
 I muft obferve here, that this Maxim does not 
 prove, nor has any Tendency to prove, what is 
 the Chance of Vitality on a Life, or what are 
 the Parts of the Value of an Annuity, which an 
 Annuitant has the Chance to receive on the Life 
 of his Nominee; but, it being fuppofcd or 
 proved that this or that is the Chance of Vita- 
 lity, that thcfe or thofe are the Parts of the Va- 
 lue of the Annuity, which the Annuitant has 
 the Chance to receive, it follows by Virtue of 
 the Maxim, and if the Maxim had never been 
 heard of, it would have followed, that the Va- 
 lue of the Annuity would be fo much : But the 
 Chance itfelf, or the Parts g/f the Value to be re- 
 
 P ccivcd
 
 [ 226 ] 
 
 ccived, cannot irom thence be cotkdted and as- 
 certained. In the Inftance upon the Life of ^. 
 it is polTiblc that the Annuitant may receive all 
 the Parts of the Annuity for 90 Years, becaufe 
 it is poffible A. may live for 90 Years, or to be 
 an 100 Years old; it is poffible like wife, that he 
 may enjoy no Parts at all of the Annuity, be- 
 caufe A, may dye within the firil Year ; But the 
 Parts of the Value of the Annuity, which the 
 Annuitant mull receive on the Life of this No- 
 minee, are thofe Parts which arife and grow due,, 
 within the firft Years that Ihall immediately en- 
 fue from the Time of the Grant of the Annuity, 
 be the Number of thofe Years more or lefs. 
 But what that Number of Years will be, is in 
 no fort determined from hence, but mult be 
 colle(fted from the Chance of Vitality on that 
 Life ; and it being calculated by a Rule ot the 
 Dodtor's, which is certainly a right one;, fo far as 
 the BreJJaw Table is a jull one, that the Chance 
 of Vitality on fuch a Life, is a Term of 41 
 Years, it follows, that the Parts of the Value of 
 the Annuity which the Annuitant will, or has a 
 Chance to receive, are thofe Parts which will 
 arife in 41 Years from the Time of the Grant ^ 
 becaufe the even Chance is, that A, lives for 41 
 Years from fuch Time \ and that the Parts of 
 the Value of the Annuity which the Annuitant 
 will not, or has not a Chance ta receive, are 
 thofe Parts which might arife and grow due after 
 the firft 4 1 Years, from the Time of the Grant, be- 
 caufe the Nominee will not, or the even Chance 
 
 is
 
 [ 227 ] 
 
 is, that he does not live beyond thofe 41 Years. 
 I cannot forbear taking Notice here, that the 
 Sum total of the Parts of the Value of an Annu- 
 ity on the Life of J.y which arc computed to 
 arife after the Expiration of the Term of 41 
 Years, and which Dr. Halley fays, are to be 
 taken as Parts of the Value of the Annuity on 
 the Life of ^., and which I fdy, are to be 
 omitted and left out of fuch Value, as arifing 
 after the Expiration of the 41 Years, the Sum 
 total of thefe, I fay, is no more than 4a, or rather 
 45. On the other fide I obferve, that the Sum 
 total of the Parts of the Value of the Annuity 
 on fuch Life, which arife within the faid Term 
 of 41 Years, and which Dr. Halley fays are to be 
 left out and omitted, as being no Part of the 
 Value of fuch Annuity, and which I fay, are to 
 be accounted and infer ted as Parts of fuch Value, 
 becaufe in Fadt, they do arife, and may be re- 
 ceived within fuch 41 Years, the Sum total of 
 thefe, I fay, amounts to 2-09 ; fo that it may 
 very well come out, as in Fadt it does, that the 
 Value of this Annuity in one Way of Computa- 
 tion, fliall be no more than 13-44 (or rather 
 13-50) and in the other Way 13-12. 
 
 That the even Chance of Vitality on fuch a 
 Life as this is a Term of 41 Years, is colleded 
 from the Doctor's Rule on the Bre/Iaw Tabic 5 
 and whether this be the right Chance or no does 
 not affed: the prefent Argument ; however, we 
 have hitherto fuppofed it to be Co : And lince I 
 have fufEciently proved, that the Parts of the 
 Value of the Annuity which the Annuitant has 
 P 2 a
 
 [ 228 ] 
 
 a Chance to receive on this Life, are all thofe Parts 
 of fuch Value which arife fo long as the even 
 Chance of Vitality continues, and no other, or 
 for any longer Time ; all I fliall add farther, 
 and by Way of Conclufion, on this Head, is 
 this. Since it is not only poffiblc but probable 
 and a reafonable Expedation, that the Annui- 
 tant will receive all the Parts of the Value of the 
 Annuity which arife within thefe 41 Years, becaufe 
 the even Chance is, that the Nominee lives thofe 
 41 Years, and in the Nature of the Thing, it is 
 abfolutely nccclfary that he lliould receive 511 
 thefe Parts before he can receive any other Parts : 
 On the other lide, lince it is not probable nor a 
 reafonable Kxpedation, that the Annuitant will 
 receive any Parts of the Value of the Annuity, 
 w^hich may arife after the Expiration of thefe 41 
 Years, becaufe the even Chance is, that the Nomi- 
 nee will be dead before any of them can arife; 
 and from the very Being of the Life of Man, it is 
 impolTible he fhouid receive any fuch Parts which 
 arife after 41 Years, unlefs he does, and before 
 he does recei ve all thofe which arife before 4 1 Years, 
 for the Nominee muft live 41 Years before he can 
 live 42 Years. Upon the whole, I leave it with 
 the Reader to determine, what are the Parts of the 
 Value of the Annuity, which the Annuitant has 
 a Chance to receive on the Life of this Nominee. 
 I have infilled the longer on the Proof of the 
 falfity of this Rule, not only on Account of the 
 Charader of its Author, to whom great Defe- 
 rence is due ; but becaufe the Arguments I have' 
 alledged to difprove his Rule, are of equal Force 
 
 againft
 
 [ 229 ] 
 
 againft all other Rules and Calculations whatever, 
 which have not, and I have met with none that 
 have, a proper Regard to the Chance of the 
 Duration of a Life ; and they tend likewife to 
 prove and llablifh the Rule, which I have of- 
 fered for the Valuation of thefe Annuities ; for 
 the Jufticc of mine depends intirely on that Po- 
 fition, that the Chance of the Duration of a Life 
 is the fole Meafure ol the Duration, and confe- 
 quentiy, of the Value of the Annuity depending 
 on it. The Method then, which I have pro- 
 pofed for adjufling the Value of thefe Annuities 
 is, as appears from my Scheme in Table ISJ^' II. 
 by Eftimating the Duration of every Life, and 
 fuppofing it equal to a determinate Number of 
 Years ; and fuch an Eftimate I have framed for 
 every five Years of Life, from i o to 80 Years of 
 Age. This I lay down as a general or common 
 Rule, but to be fure many Exceptions are to be 
 made out of it, from the greater or Icfs Salubri- 
 ty of the Place of Inhabitation, from the remark- 
 able Longevity in fome Families, and the con- 
 trary in others, and from other particular Cir- 
 eumftances which may attend particular Perfons; 
 and for thefe, Confiderations muft be had, as 
 they refpecftively arife, fince no one Rule can 
 be framed to fuit them ail. 
 
 This Scheme I have drawn, chiefly from a 
 View and Obfcrvations on the Bills of Mortality 
 for London • and to make my Refllcdlions on 
 them more plain and confpicuous, I have made 
 an Abftradt of them for thefe fix Years laft pall; 
 wherein a Diftindion is made of the Ages of 
 P 3 the
 
 [ 230 ] 
 
 the feveral Perfons dying, in the feveral Periods 
 of Life, and have fet it forth in Table N^- XT. 
 In this Table we fee at one View, the Number 
 of Perfons dying in each Period of Life, in each 
 of thefe ilx Years, the Number of Perfons dying 
 in each Period, for all the iix Years put toge- 
 ther, and the Number of them dying in each 
 Period in a Year by a Medium. We have like- 
 wife there, the whole Number of Perfons dying 
 in each of thefe fix Years, the whole Number of 
 Perfons dying in all the Iix Years put together, and 
 the Number dying in each Year by a Medium. In 
 thefe Bills wc have always two Articles ; of Abor- 
 tives and Stillborn,which do not properly come un- 
 4er the prcfent Confederation, and therefore I have 
 put them under a fcparate Head, and have fub- 
 llraded them out of the general Account of 
 Mortality : And in the laft Column, I have in- 
 ferted a Calculation of the Number of Perfons 
 dying in each Period of Life, in every looo dying, 
 I have begun my Computation at a Life of 
 ten Years of Age, and Eilimated it as the bell 
 Life ; though I think one of 7, 8, or 9 Years of 
 Age may be equally good, and indeed, better in 
 thofe Cafes, where the Perfons have got over the 
 Small-pox, Meafles, and fuch like Ailments in- 
 cident to Youth, and have attained fome firmnefs 
 of Body. This I was induced to do from thefe 
 Bills of Mortality, for Lovdof) ; from whence it 
 appears, that the Number of Perfons dying in 
 the Period, between 5 and 10 Years of Age^ 
 exceeds the Number of Perfons dying of the 
 Age between 10 and ao, in the Proportion of 
 38, to 30; and yet the firft Period takes up 
 
 only
 
 [ 231 ] 
 
 only half the Time of the latter. It appears 
 likewife from thence, notwithflanding the Ha- 
 zards and Cafualties which may be attendant Oft 
 Perfbns, between the Age of ten and twenty, 
 and the vulgar Opinion, that the Chances of 
 Mortality is greater in young than in middle 
 Life, yet that fewer Perfons dye in that Period 
 of Life, than in the Period between 30 and 40, 
 though the Number of Perfons coexifting in the 
 younger Period of Life, mull be much greater 
 than the Number of thofe coexiilingin the elder. 
 In the next Period of Life, vi:?i. between 20 and 
 30, though the Mortality is conliderably in- 
 creafed, yet the fame Obfervation holds good in 
 a great Meafure, the Mortality in the two fub- 
 fequent Periods, being Hill higher than in this, 
 and in a great Degree fo, 
 
 Having Ellimated a Life of ten Years of 
 Age as the beft of Lives, I have ftated it as 
 equal to a Term of 28 Years, which will not be 
 thought too high, by thofe who approve the 
 Bre/law Table and Dr. Halkfs Scheme • nor 
 indeed by any one who will form an Eftimate of 
 the Duration of Lives, from the London Bills of 
 Mortality. But in my Table, .1 have made an 
 Allowance at large, for the uncommon Acci- 
 dents of Peftilences, Famines, and civil Wars, 
 to which Regard muft be had, when a Calcula- 
 tion is made for a Number of Years • and being 
 one of thofe who think the City o^ London to be 
 a more healthful Place, than almoft any other in 
 this Kingdom, at leaft more than the Generality 
 of Places are ^ I have fo far complied with that 
 P 4 Notion,
 
 [ 232 ] 
 
 Notion, as to fix the Number of Years lower 
 than thofe Bills feem to require ; for thofe Bills 
 will admit of a Calculation for a Term ot 30 
 Years on fuch a Life, at the lowed Rate of fet- 
 ing it ; and at the higheft Rate, and where you 
 will reckon to the Extremity of the Years, they 
 may reach a Term of ^^ Years. 
 
 On looking into my Abftraft of thefe Bills, 
 V7e find that more than one half of the Perfons 
 dying in Loudon^ are under ten Years of Age ; 
 the Confequence of which muft be, that one 
 half of the Perfons born alive there, dye before 
 they arrive at that Age, and the other half live 
 beyond that Age. If we proceed and take a 
 View of the other Moiety of Perfons dying, it 
 appears that the Number of Perfons dying of 
 the Age of 50 and upwards, exceeds the Number 
 of thofe dying between the Age of 10 and 40 ; 
 and of thofe dying between the Age of 40 and 
 50, if wefuppofe one half to be between 40 and 
 45, and the other between 45 and 50 \ then one 
 half of this fecond Moiety of Perfons dying, is 
 under the Age of 45, and the other half is above 
 45 : And the Confequence in this, will be the 
 fame as in the former Cafe, that of thofe who 
 live to ten Years of Age, one half dies before 
 the Age of 45, and the other lives beyond it. 
 If thefe Conclullons are right, it muft mufl fol- 
 low, if we compute the Chance of Vitality of a 
 Perfon of ten Years old, for the City oi Lovdovy 
 to the utmoft Extent, the Life of fuch a one is 
 equal to a Term of 35 Years, he having an even 
 Chance to live to the Age of 45. When I 
 1 diiw h (iy.
 
 [ 233 ] 
 
 fky^ if we compute to the utmoft Extremity, I 
 mean, if we begin our Computation prccifely at 
 ten Years of. Age, for that we muft not take a 
 Life of 12, 13, or 14, nor a Medium between 
 10 and 15; becaufe on a Life of 15, if we make 
 a Calculation of the Chance of its Duration, in 
 the fame Manner as I have done on a Life of 10, 
 the Chance is very little above 31 Years; and 
 therefore as a Medium between 10 and 15, we 
 cannot take more than 33 Years. 
 
 By the fame Abftradt, and in like Manner, I 
 have I computed the Chance of Vitality for the 
 Period of Life, 30 to ^^ ; and it appears, that the 
 Number of Perfons of 50 and upwards, exceed^ 
 the Number of thofe dying between 30 and 50; 
 from whence I conclude as before, that fuch a 
 Life muft be equal to 20 Years and over ; fincc 
 of thofe who live to the Age of 30, more than 
 one half live to 50 and above, that is above 20 
 Years. I have made a Computation on fuch a 
 Life more particularly, and in another Method, 
 which I fhall mention prefently, becaufe it ftands 
 contradicted by an Eftimate, made by the Au- 
 thor of a Ti'eatije^ called ^ trife EJfimate of the 
 Value of LeafehoJd Eftates^ i3c. who reckons fuch 
 ^ Life as the beft, and Hates it as equal only to 
 14 Years, which is fhort of mine fome Years. 
 This Gentleman has fome very good Thoughts 
 on this Subject, in which I readily concur with 
 him ; but I can by no Means fubfcribe to his 
 Opinion, either that a Life of 30 is the beft Life, 
 or that fuch a Life is equal only to 14 Years. 
 pis Notion of the belt Age of Life, is diredly 
 
 Contradi(Sed
 
 [ 234 ] 
 
 contrad idled and overthrown by what has been 
 already ofTered, and might be farther colledted 
 from the London Bills of Mortality; and his 
 Obfervations and Pofitions do in no fort juftify 
 his Conclufion, that fuch a Life is equal only to 
 a Term of 14 Years ; but rather produce another 
 and a different Conclufion, fuch a one as con- 
 firms my Eilimate ; for which Reafon I Ihall 
 confider them at large. 
 
 The Obfervations he makes, are on the Re- 
 giflers of two Country Parilhcs, which have been 
 regularly and well kept thefe 40 Years laft paft; 
 from which it appears, as he exprefles it, that 
 the whole Numher of Inhabitants above two Tears 
 oldy are buried in 26 Tears or fomethiug above y 
 from whence he concludes, that a Life is equal 
 to little more than 13 Years: And then, making 
 an Allowance of a Year more, for the greater 
 Mortality of Children from 2 till about 10 Years 
 old, than afterwards, he fixes the Term of the 
 bcft Life as equal to 14 Years. 
 
 Admitting that the Author's Obfervations of 
 Facts are true, as there is no fort of Reafon to 
 call in Qucllion his Veracity ; yet his Pofition, 
 that the whole Number of Inhabitants of two 
 Years old in thefc Places, were buried in 25 
 Years, does not prove that the Life of any one, 
 or every fuch Inhabitant of the Age of 30, is 
 .equal only to 13 or 14 Years. 
 
 This Pofition is capable of being underftood 
 |n two different Scnfes \ either that every indi- 
 vidual Inhabitant of two Years old, who were 
 iivlpg fupjjofe in 1700, were all dead and buried
 
 t 235 ] 
 
 by or before the End of the Year 1727; or 
 th^ a Number of Inhabitants of two Years old 
 and upwards, equal to the whole Number o£ 
 thit Age, at any given Time, Hied and were 
 buried in that Compafs of Time. That the 
 firft is the Meaning ot the Polition I can hardly 
 think, becaufe in that Senfe, furcly it could never 
 be true, in thefe or in any other Places ; and in 
 the other Senfe, it may indeed te tiuc ; But in 
 cither Senfe, I conceive that it does not prove 
 his Inference, or rather that it proves fomething 
 which mult have another Conclulion, than what 
 is here made. 
 
 To fliew the Difference of thefe two Senfes^ 
 and that in the firft Senfe, the Pofirion is very 
 unlikely to be true, though poifibly in the fecond 
 it may ; I Ihall give a known and lamiliar In- 
 ftance. The Number of Biihops in England is 
 a6, and the Number of Aldermen for the City 
 o^ London is the fame ; now it may be true, that 
 in 26 Years laft paft, and to be fure in a much 
 lefs Time, a Number of Bifhops, and like wife 
 of Aldermen, equal to the whole Number, that 
 is 16 Bifhops and 16 Aldermen, may have died: 
 But it is not true that the very individual Billiops 
 and Aldermen who were fuch, 26 Years ago, 
 are all dead ; for my Lord oiWorceftcr, has been 
 Bifhop above forty Years, and his Grace of 
 Canterbury about thirty Years ; and I believe two 
 or three at Icaft, of the Senior Aldermen, have 
 been of that Body above 26 Years: And if in 
 fuch Bodies of Men, -where rarely any come to 
 be Members of it, till they arrive at feme Ma^ 
 
 turity
 
 [ 236 ] 
 
 turity of Age, the Aflertion is not true, it is 
 much more likely not to be true, where many of 
 the Perfons are of a very young Age. 
 
 But the Senfe of the Author's Propofition 
 here, may be coUeded with moll Certainty from 
 his own Account and Calculations, and if wc 
 look into his Computations for the lefs Parifh, 
 he Hates the Number of Inhabitants above two 
 Years old, to be i ip, and the Number of Per- 
 fons above two Years old dying annually, to be 
 4 I ; and then having divided the former by the 
 latter Number, and the Produce being 26, he 
 concludes that the whole Number of Inhabitants 
 are buried in 26 Years. Now this is a Demon- 
 Ikation of the Propofition, if underftood in the 
 fccond Senfe ; viz. that in 26 Years Time a 
 Number of Inhabitants, equal to the whole 
 Number of Inhabitants, that is, the Number 
 of 1 1 9 has died ; but is no fort of Proof at all, 
 that the very individual 119 Perfons, taken at 
 any one Time, were dead in 16 Years. And in 
 Truth, the Proportion in the firft Senfe is intirely 
 incredible; for that it muft neceffarily follow, 
 that there was never a Native of either of the 
 Pariihes, living in cither of them in 1728, who 
 was then 30 Years old ; every individual who 
 was living there in the Year 1700, and then two 
 Years old, being dead and buried. 
 
 Our Enquiry in general is. How long it is 
 likely any Perfon of a given Age may live ; and 
 in particular here, to what Time any one of the 
 prefent Inhabitants of thefe Parifhes, between 
 the Age of 30 and 40, may live. For the So^ 
 
 lution
 
 C 237 ] 
 
 lution of this Qucftion wc arc dlreded to make 
 Ufe of this Method • we muft compute in what 
 Time a Moiety of a certain Number of Perfons 
 given of that Age have in Fa6t died in that 
 Place, and take that Time as the Term to which, 
 there is an even Chance that any one Perfon of 
 that Age, and in that Place now living, fhall 
 live. Now this Rule for ellimating the Dura- 
 tion of a Life means, and is applicable to, Indi- 
 viduals only ; and this Author's Aflertion, I 
 fuppofe, is not true, and cannot be underllood 
 to fpeak of Individuals ; for it could not be 
 true that the whole Number of the individual 
 Inhabitants of thefe Parifhes did die in 26 Years, 
 or one Half of them in 13 Years : The Gonfe- 
 quence is, that this AiTertion does not prove that 
 the Life of any one of them of the Age between 
 30 and 40 is equal to 13 Years, and no more, 
 bccaufe the Rule means one Thing, and the Af- 
 fertion another. 
 
 And if the Matter of Fad were fuch, in thefe 
 Parifhes, or in either of them, that it would 
 maintain the Pofition in the firft Senfe, and if 
 the Author could be underllood to fpeak of In- 
 dividuals ; yet I cannot admit that the Pofition, 
 fo underllood, proves that the Life of a Perfon 
 here, of the Age between 30 and 40, is equal 
 only to a Term of 13 or 14 Years, lince it wilj. 
 follow, even from thence, that fuch Life muft 
 be equal to a Term much longer. "-[ 
 
 To make Ufe of the Inftance here produced 
 of the lefs Parifh: The Aflcrtion, if applied 
 here, is this* that of up Perfons fubfiiling in 
 
 this
 
 [ 238 ] 
 
 this Piirilh in the Year 1700, one Moiety, that 
 is 60 of them, died and were buried by the End 
 of che Year 17 13: Now admitting that this 
 Moiety were really a Moiety of the Individuals 
 fubfifting in 1 700, if the Rule for eftimating the 
 Duration of Lives, which we have before men- 
 tioned, is a right Rule, and Dr. Halley makes 
 Ufe of it, this Author makes Ufe of ic, and if 
 this be not, I very much doubt there is no fuch 
 Thing as aright Rule; it neceflarily follows, 
 that the Life of every one of thefe Inhabitants 
 who were in Being in 1700, and then above two 
 Years old, that is of every one of thefe i ipPer- 
 fons, and not of thofe only who were of the 
 Age between 30 and 40, at an Average, was 
 equal to a Term of 13 Years; one Moiety of 
 fuch Perfons living to the End of 1 3 Years. 
 
 If this be the Confcquence, and nothing does 
 fo naturally and fo certainly follow ; then, in 
 Order to make every Life, one with another, to 
 come out to be equal to a Term of 13 Years, it 
 "will be neceffary that we greatly incrcafe the 
 Chance of Vitality in the Lives of that Part of 
 thefe Inhabitants who are in the Period of Lilc 
 between 30 and 40, or it will be impoflible to 
 make up the Deficiency that will arife in the 
 Chance of Vitality for that Part of the Inhabi- 
 tants which is in the bad Periods of Life, that 
 is from 2 to 10 Years old, or towards it, and 
 from 40 upwards, which arc admitted on all 
 Hands to be among the Periods of Life where 
 the Chance of Mortality is greater than in the 
 Period between 30 and 40. 
 
 In
 
 [ 239 ] 
 
 In enumerating the Periods of Life which arc 
 bad, I have omitted the Whole of the Time be- 
 tween lo and 30, becaufe in my Judgment 'tis 
 not fuch, but quite the contrary ^ and yet, ac- 
 cording to this Author's Way of Thinking and 
 Rcafoning, it ought to be inferted, for if the 
 Chance of Mortality in this Period is not greater 
 than in the Period between 30 and 40, a Life in 
 this Period muft necefTarily have the Chance of 
 the longer Duration, or be a better Life, fince 
 it is fo much farther removed from the Decline 
 of Life, than a Life in the other Period is. 
 Leaving therefore this Period out of the Account 
 of bad ones, let us calculate what may be the 
 Increafe of Mortality for the two Periods from 
 2 to 10, and from 40 upwards, which are agreed 
 on all Sides to be bad ones. 
 
 This Author takes Notice of, and makes an 
 Allowance for, the greater Mortality in the Pe- 
 riod between 2 and 10; but then 'ds only by the 
 Subftradiion of one Year from thence, and making 
 an Addition of one Year to the bcft Period of 
 Life, viz. from 30 to 40, and Hating fuch a Life 
 as equal to 14 Years. But this Allowance is 
 furely too fmall, and will certainly appear fo to 
 any one who confults the Bills of Mortality for 
 Lofj^ofj, and finds there, and whoever does con- 
 fult them will find, that the Number of Perfons 
 dymg between 2 and 10 is more than four to one 
 ot thofe dying between 10 and 20, the Propor- 
 tion bemg as 126 to 30, and yet the Period not 
 lo long neither. 
 
 But
 
 [ 240 ] 
 
 But if 'twere admitted that this is a fufficicnt 
 Allowance here ; yet I obferve no Notice at all 
 is taken of the Periods from 40 upwards, nor 
 any Dedu6tion made for the greater Chance of 
 Mortality in thofe Ages of Life ; which yet un- 
 doubtedly ought to have been made, and is very 
 coniiderable. If we would keep up the Ave- 
 rage, and make each of the Lives in thefe two 
 different Periods, that is in the Period between 
 30 and 40, and the Period of 40 and upwards, 
 to be equal to a Term of 1 3 Years one with 
 another, the Increafe of Vitality in one Period 
 mull be juft equal in the Whole to the Decreafc 
 of Vitality in the other : So that the Queftion 
 will be, What may be the Decreafe of Vitality 
 in this older Period of Life. 
 
 I have computed, by Way of Suppofition 
 only, that thefe 119 Inhabitants, in Point of 
 Age, might be thus divided ; that is, between 
 a and ic Years Age the Number might be 30, 
 between 10 and 30 might be 40, between 30 and 
 40 might be 20, and from 40 upwards 30; 
 making in all 120. If this Repartition fhould 
 not be very exad, though I believe *tis not far 
 off the Point, yet a fmall Variation will have no 
 great Influence on my Reafoning, and whatever 
 it be, it will appear on the Computation. From 
 this Author's Scale of the Term of Years to 
 which a Life is equal, I have computed, in fe- 
 veral Ways, to what Number of Years every of 
 the Lives in the oldeft Period, I mean from 40 
 upwards, at an Average, muft be equal ; and 
 making Ufe of that Method of Computation 
 
 which
 
 [ 241 ] 
 
 which produces the longeft Term to which fuch 
 Lives can be equal, it comes out that they can 
 be equal to no more than eight Years at the moft, 
 one with another. If then the Lives of every 
 one of thefe 30 older Perfons are at an Average 
 equal only to eight Years, we mull from every 
 one of fuch Lives take off five Years, eaclx of 
 their Lives wanting fo much of being equal to 
 13 Years ; and if the Number of Perfons of that 
 Period be 30, we mufl: take off in the Whole 5 
 Times 30, that is 150. And fmce this whole 
 Number of 150 Years (^o taken off here muft be 
 added to the Lives of the Perfons in the Period 
 between 30 and 40, and thofe are only 20 Per- 
 fons, 150 divided by 20 producing 7 i, we mull 
 add feven Years and an half to the Life of every 
 Perfon, in this younger Period, one with ano- 
 ther ; that is, we mull add thefe Years to 13, or 
 rather to 14, which we had before; and then 
 their Lives, at an Average, are equal to twenty 
 one Years and an Half. 
 
 If the Number of Perfons of this Parilh in 
 thefe two Periods of Life were equal, that is, if 
 there were as many Perfons living of the Age 
 between 30 and 40, as there were of the Age 
 of 40 and upwards, which is not at all likely to 
 have been the Cafe, either here or any where 
 elfe, then the Number of Years to be added to 
 each of the Lives of the younger Set of Per- 
 fons would be jult the fame as the Number ot 
 Years taken off from each of the Lives of the 
 elder Stty that is an even five Years; but 
 then en that Foot, each of the Lives in the 
 Q^ younger
 
 [ 242 ] 
 
 younger Set would be equal to a Term of 19 
 
 Years. 
 
 From thefe Premifes thefe ConcluHons njay 
 very fairly and very juftly be drawn. If this 
 Author might be underftood to fpeak of Indi- 
 viduals, when he aflercs that the whole Number 
 of Inhabitants died in 26 Years, and a Moiety 
 of them in 13 Years Time, it follows in the firft 
 Place, that each of their Lives, at an Average, 
 was equal to a Term of 13 Years; and from 
 hence it follows, in the fecond Place, and as a 
 neccfTary Confequence of the former Conclulion, 
 that the Lives of thofe Perfons in the bettermoll 
 Period of Life, that is in the Age between 30 
 and 40, mufl each of them be equal to a longer 
 Term than 13 Years, or other wife it will be im- 
 poffible that each Life of the whole Number, 
 which includes thofe between the Age of 2 and 
 10 Years, and thofe of the Age of 40 and up- 
 wards, fliould be equal to a Term of 13 Years : 
 And therefore that the Life of every Perfon ot 
 this Parifh between the Age of 30 and 40 may 
 very well be cftimated equal to a X^^^i of 2o 
 Years and above, even on a Suppofition that the 
 Author fpeaks of Individuals. 
 
 But in Fa<a this is not the Cafe, for this 
 Calculation includes not only all the Inhabitants 
 between 1 and 10, and above 40, who were in Be- 
 ing at the Beginning of the fuppofed 16 Years 
 or 13 Years, but it includes alfo all thofe who 
 from Time to Time during the whole Courfe of 
 thefe 26 Years or 13 Years grew up and fuc- 
 ceeded into the bad Period of Liie between 2 
 
 and
 
 r 243 ] 
 
 and 10 ; which Is a wide Difference, and a Dc- 
 ^c6t of greater Confequence than the former. 
 The Number of Pcrfons above two Years old 
 living in thisParilh, fuppofe in the Year I700> 
 was 119; by the End of the Year 1726 an 
 equal Number of Inhabitants above that Age, 
 that is 119, had died and been buried there; 
 but who were the Perfons dying there in this 
 Compafs of Time? Not the very individual 
 Perfons who were fubfifting in 1700, for that, as 
 we have already feen, is neither credible, nor can 
 be the Meaning of the Expreifion : But the 
 Number is made up partly out of the Individu- 
 als, and the reft out of fuch as did aferwards 
 grow up to the Age of two, of which there 
 muft be annually and conilantly fome through the 
 whole Courfe of the 26 Years. If we would 
 know what the Number of Perfons might be 
 thus growing up and fucceeding into this Lift, 
 the Author's Account here gives us an Oppor- 
 tunity of computing it, and it ftands thus. The 
 Number of Perfons annually born is 6, the Num- 
 ber of Perfons undcjT two Years Age annually 
 dying is i j, therefore the Number of Perfons 
 annually added to the Lift is 4 t? '^nd confe- 
 quently in 26 Years the whole Number added 
 is 1 17 ; fo that in Fad: thefe 1 19 Perfons dy- 
 ing were neither the Individuals, nor out ot the 
 like Number of Individuals, but out of a Num- 
 ber of Individuals almoft double. 
 
 The Rule, and the only unexceptionable Rule, 
 
 to adjuft the Chance of the Duration of a given 
 
 Life, is j to compute in what Time a Moiety of 
 
 0^2 a Numbcs
 
 [ 244 ] 
 
 a Number of Perfons of the fame or the like 
 Age with the given Life, in Fadl has lived or 
 commonly does live, and to take the Time to 
 which fuch Moiety lives as the Term to which 
 'tis probable the given Life may extend. But 
 then we mult make the Cafes in every Inftancc 
 parallel, as we muft do wherever we will argue 
 from one Cafe to another • that is, here we muft 
 take a Set of Perfons of the fame Age or near 
 the fame Age as the given Life, and Ihould re- 
 tain and hold ourfelves clofe to the fame Perfons ; 
 but this Author has not obferved this Method 
 in either Inftance. The Enquiry is, to what 
 Time 'tis probable that Perfons of the Age be- 
 tween 30 and 40 may live ; and to determine 
 this he has taken a Lift of Perfons of all Ages 
 from two upwards, and does not keep to them 
 neither folely, but brings in another Set of Per- 
 fons of an allovv'd bad Age of Life, viz. between 
 2 and 10 Years Age ; and this he does not for 
 once only but in a continued Succcflion for the 
 •whole Time of 26 Years, 
 
 The Truth is, into fuch a Lift not only no 
 new crazy Lives, but even none at all, good, 
 bad, or indifferent, ought to be introduced to 
 fupply the Vacancies that do and muft happen 
 by Mortality ^ for that in Effed brings us back 
 to the former Point, that the Death of a Num- 
 ber of Inhabitants equal to the whole Number of 
 Inhabitants is, in fuch a Computation, the fame 
 Thing as the Death of the Individuals nominated. 
 That this is not right, is plain enough from the 
 Obfervations I have m^dc on this Parifh j but 
 
 will
 
 [ 245 ] 
 
 will perhaps appear more evidently and cer- 
 tainly in the Inftances of the Bifhops of the 
 Kingdom, and of the Aldermen of the City of 
 Lotjdoji ; for my Lord Bilhop of Ji'orcejler has 
 lived to bury his Brethren more than twice over, 
 and the Senior Alderman of the City his Bre- 
 thren twice over or near it ; that is, a Number 
 of Bifhops and Aldermen equal to the whole 
 Number of thofe on each Bench, viz, 26, have 
 died twice over, that is, 52 have died during 
 their Time ; that is, twice the Number of the 
 Individuals, but not the very Individuals them- 
 felves, have died in a determinate Time. 
 
 To apply this to the particular Purpoie which 
 ■we are now confidering ; if an Annuity had been 
 purchafed byonePerfon, fuppofe in 1693, when 
 the now Bifhop of Worcejhr was firft made a 
 Biftiop, to continue for the Lives of the indivi- 
 dual Perfons then on the Bench of Bilhops, and 
 by another an Annuity to continue 'till a Num- 
 ber of Bifhops equal to the whole Number of 
 Bifhops, that is 2(5, fhould have died ; one of 
 the Annuities would be Itill fubfifling, and the 
 other would have determined above twenty Years 
 ago : For that, on a Computation made, I find 
 that in 17 Years Time or thereabouts there 
 are 26 Bifhops die; and the Cafe among the 
 Aldermen of the City of London is much the 
 fame. If we turn our Manner of Reafoning ,- 
 another Way, and fuppofe a Man to buy an • . - 
 Annuity now on the Life of one of the prefent *■ ^" 
 Aldermen, and that the Purchafer was to takc*•**^^ 
 ;h^ Life of one Mean, betwixt the oldcft and the 
 CI 3 youngcfi;
 
 [ 246 1 
 
 youngeft of that Body ; it might be a tolerable 
 good\ule of eftimating the Continuance of fuch 
 Life, to examine the Time in which 13 Alder- 
 men of any determinate Set of Aldermen have 
 died, and to take that Time as the Term to 
 which fuch Alderman now to be nominated 
 might live ; but it would be a wrong Rule to 
 examine in what Time any 13 Aldermen of the 
 Body at large have died, and take that Time for 
 fuch Term; becaufe in one Method 13 is cer- 
 tainly a Moiety of the Lift, and in the other 
 \\s manifeftly not fo, the Lift being enlarged by 
 the continual fupplemental Lives. Or take it 
 thus. In one Way the Chance of Mortality by 
 the Addition of fuch fucceeding Lives, and 
 keeping the Nnmber always full, continues much 
 at a Par for the whole Time ; but in the other 
 Way fuch Chance is gradually decreafing as the 
 Number of Individuals decreafes by Death ; fo 
 that the Death of 13 ftiall happen much iooner 
 in the one Way than in the other ; and this al- 
 ways comes out to be the Fa6t, when Trial is 
 made and applied to particular Cafes. And that 
 this is the F^aa, with Regard both to the Biihops 
 of the Kingdom, and the Aldermen ot Lofidofi, 
 is evident from hence, that from his prefent Ma- 
 iefty's Acceffion to the Crown, and before the 
 ninth Year of his Reign expired, there died 
 ^;- >. ^^ twelve Biihops and thirteen Aldermen, x 
 ^i^U^X^Th\s Author eftimating the Age between 30 
 ^^^'^^^^nd 40 as the beft of Age of Life, and having 
 •/>2C4^«:^^ounded his Calculations on the whole Number 
 If the Inhabitants of the Parifti he mentions, 
 
 ■without
 
 [ 247 ] 
 
 without dlilingulihing their Ages, and not on 
 any certain Set of Individuals of the Pariih, I 
 can make no poiitive and certain Calculation 
 from thence to what Number of Years fuch a 
 J-ife may be equal, and Hiall only fay : Since I 
 have fhown under a former Head, that from his 
 Premifes and in his own Way of computing 
 fuch a Life muft be equal to a Term of 20 Years 
 or above, I may infer, if his Computation had 
 been made from Individuals and not the Inha- 
 bitants at large, the Chance of Vitality on fuch 
 a Life might have come up to the Eilimate which 
 I have put upon it in my Tables, or fomething 
 near it. But this Author has more Obfervations 
 on this Subjed which, I think, tend to confirm 
 my Opinion, that a Life in the beft Stage of it, 
 viz. often Years of Age, may very well be com- 
 puted equal to a Term of 28 Years; and as I 
 took the firft Hint of fuch Eilimation from 
 thence, it is but juilice to acknowledge from 
 whence I received it, and how and by v/hat 
 Means I colled:cd it. 
 
 Ke obferves then, that one Half of the Mais 
 of Mankind dies before ten Years of Age ; 
 which holds true for the City of LoJidon^ and I 
 believe in moft other Places, except in the Tables 
 for the City of Ereflaw : Now if this Gentle- 
 man would give himfelf the Trouble to take 
 another View of the Inhabitants of this Parifh. 
 which he knows fo well, and examine the Re- 
 gifter there once more, which gives an Account 
 of the Ages of the 125 Perfons buried in the 20 
 Years there mentioned, he might readily find the 
 0^4 Tern?
 
 [ 248 ] 
 
 Term of Years to which any Perfon, of that 
 Parifli at leaft, has a Chance to live ; for the 
 Age of the Perfon dying, whofe Age was the 
 Middle between the oldell and youngeft Perfon 
 dying, always excluding out of the Account 
 Children under ten Years old, the Age of fucli 
 Perfon, when we have dedudted out of it fo 
 many Years as are equal to the Age of the Per- 
 fon whofe Life we enquire after, gives us the 
 Term of Years to which the Life enquired after 
 is equal. 
 
 The Way of Reafoning is this. One Moiety 
 of the Mafs of Mankind dies before or about 
 the Time of ten Years of Age ; or, which is 
 the fame Thing, a Moiety of the Mafs of Man- 
 kind lives to the Age of ten Years, or there- 
 abouts; therefore every individual Perfon born, 
 and when born, has a Chance to live ten Years* 
 Of the Perfons who live to ten Years or up- 
 wards, one Moiety dies before fuch a Time, or 
 one Moiety lives to fuch a Time ; therefore 
 every Individual, who arrives to ten Years of 
 Age, has a Chance to live to fuch Time. To 
 determine what this particular Time is, we muft 
 confult Regillers and Bills of Mortality which 
 diftinguilli the Ages of Perfons dying ; and 
 from thence we mull colled, as near as we can, 
 the Age of the Perfon dying, whofe Age was' 
 the Middld or Mean between the oldcfl and 
 youngeft dying; and in doing this, the larger 
 the Number of Perfons is, and the longer the 
 Period, the more exa6t the Medium riiuft Come 
 out j becadfe fome Years and fome Seafons pro- 
 duce
 
 [ 249 ] 
 
 duce DIftempers which carry off more Perfbns irx 
 Proportion of the younger or middle Age of 
 Life, and other Years and different Seafons fhall 
 ]Dear hardeft on thofe in the Decline of Life : 
 and the Age at which the Perfon of the middle 
 or mean Age died is the Age to which the Per- 
 fon enquired after has a Chance to live. 
 
 To fhow better the Meaning and Ufe of this 
 Rule, we will apply it to a Perfon of the Parilh 
 fo often mention'd, and to one of ten Years of 
 Age ; from whence it will appear, that the Life 
 of fuch an one may very well be eftimated as 
 equal to a Term of a 8 Years. The whole Num- 
 ber of Perfons, all Children included, dying here 
 in 20 Years Time, was 125, and a Moiety of 
 them, that is 62, died before ten Years of Age, 
 fo that the only Enquiry neceffary is of the other 
 Moiety. If we begin then at the youngell of 
 the Perfons above 10 Years old who was buried 
 during this Period, and count forwards, or be- 
 gin with the oldefl: who was buried, and count 
 t)ackwards, the Age of the Perfon dying when 
 we come to a Moiety of 62, that is to 31 ; this 
 is the Age we enquire after, or the Term we 
 want, deducing ten out of it. I appeal then to 
 this Gentleman's farther Review of this Parilh, 
 and for the Period he mentions or any other, 
 whether there were not more Perfons buried of 
 the Age of 38 and upwards, than there were of 
 the Age between 10 and 38. If that comes out 
 to be the Cafe, as I verily believe it will, it fol- 
 lows, that even in this Spot of Ground, which 
 does not feem to be amongft the moft healthful, 
 
 a Perfon
 
 [ 250 ] 
 
 a Perfbn of ten Years Age has an even Chance to 
 live to be 38 Years old, and his Life confequent- 
 \y is equal to a Term of 28 Years ; that being 
 evidently the Term to which fuch a Perfon has 
 a Chance to live. 
 
 That this, which I have furmifed, would come 
 out to be the Cafe in this Country Parifh, is only 
 a Conje(5ture, though not an ill grounded one j 
 but that this is the Cafe in the City of Lojjdon is 
 more than Conjedure. On the Bills of Morta- 
 lity for that City for thefe hift iix Years, I ob- 
 ferve, that the Number of Perfons d) ing of the 
 j\ge of 40 and upwards exceeds the Number 
 of thofe dying of the Age between 10 and 40 ; 
 and the Proportion of one to the other is as 202 
 to ip5, as may be collected from my Abitrad of 
 thofe Bills in the Table N^- XI. : And. if we 
 take the Number of thofe only dying between 
 10 and 38 on one Side, and on the other Side, 
 thofe dying from the Age of 38 upwards, the 
 Exccfs is much greater, and the Proportion is 
 220 to 177, as near as I can ftate it. The Truth 
 is, as I have obferved before on thcfe Bills, that 
 the Number of Perfons dying in that City of the 
 Age of 45 and upwards is full as great as the 
 Number of thofe dying of the Age between 10 
 and 45 ; and confequently, in LovdotJ^ a Life of 
 the Age of ten muft be^ellimated as equal to a 
 Term of 30 Years, or abo^e, and in other Places 
 equal to a Term of 28 Years. 
 
 On this Abftrad of the Bills of Mortality for 
 London I obferve farther, that the Number of 
 Perfons dying there of the Age between 10 and 
 
 20,
 
 [ 251 ] 
 
 20, to the Number dying there between 30 and 
 40, is in the Proportion of 30 to ^^ : So that 
 whatever might be the State of the Country Pa- 
 rifh here mentioned, or whatever Reafons this 
 Author, or any other, might have for flating a 
 Life of the Age between 30 and 40 as the beft 
 Life, either in this or any other Place, fuch an 
 Eilimate will by no Means fuic with the Chances 
 of Vitality for London. 
 
 From thefe Premifes, and the Calculations 
 even on the State of this Village, as far as the 
 Knowledge of it is come to our Hands, as well 
 as on the Bills of Mortality for London^ I think 
 •we might fafely conclude, that the Period of 
 Life between 30 and 40 is not the bell Stage of 
 Life ; but whether it be the bell or be not the 
 bell, yet a Life in that Stage may be equal to a 
 Term of 20 Years, and above ; and on the other 
 Side, that a Life of the Age of 10, or there- 
 abouts, is the belt Life, and in a general Way 
 may be Itated as equal to a Term of 2.8 Years : 
 But I have fomething to add farther under both 
 thcfe Heads. 
 
 I have look'd into the Lift of a certain Soci- 
 ety which conlillcd of 70 Perfons, and was in 
 Being in 1700, the Perfons and Ages of whom 
 I knew, and know who arc now living and who 
 are dead, and the Times of their Death ; all of 
 them pretty exactly. If the whole Society were 
 divided into two Parts, one Moiety was of the 
 Age from 18 to 30, or thereabouts, and the other 
 Moiety of the Age from 30 to 45, or there- 
 abouts ; and upon Examination I find, that one 
 
 Half
 
 [ 252 ] 
 
 Jialf at leail of the elder Moiety were living at 
 the End of ao Years, and one Half of the 
 younger Moiety at the End of 26 Years ; and of 
 the elder Moiety there are now, in 1 736, only fe- 
 ven or eight, at moll, living ; and of the younger 
 Moiety cither 15 or 16. The Confcqucnce is, 
 that a Life of any Perfon of the elder Moiety 
 Ihould be equal to a Terrn of 20 Years, and 
 one of the younger Moiety to a Term of 26 
 Years ; thofe being the Terms to which a Moi- 
 ety at leaft of each Set refpedively lived ; and 
 that a Life of the Age between 30 and 40 is noc 
 fo valuable as one of a younger Age. 
 
 To fhow what is the Chance of Vitality on 
 Lives, where the Parties interefled in them are 
 at Liberty to pick and choole their Lives, and 
 there we are to fuppofe that they will take the 
 befl:, and fuch at leafl: as, in their Judgment, and 
 to common Appearance, are the bell ; I Ihall 
 add one Thing more, which has been commu- 
 nicated to me by a Friend fmce writing this, viz. 
 an Account of the Time to which a Moiety of 
 thofe Perfons lived, whofe Names were put into 
 the Annuities on Survivorfhip which were grant- 
 ed by an A61 of Parliament of the late King 
 William and Queen Mary. The Perfons for whofe 
 Lives thofe Annuities were to be granted, were 
 nominated at or before Midfummer 1693 ; at 
 Mtdfummer 1728, that is at the End of 35 Years, 
 one Moiety were living, but the Number ex- 
 ceeding a Moiety was then fo fmall, that at 
 Chriiimafs following it was funk under a Moiety ; 
 from whence^ upon the Rule formerly laid.dawa, 
 
 it
 
 C 253 1 
 
 it will follow, that the Lives of any dne, and 
 every one, of the Pcrfons nominated in the Or- 
 ders of thofe Annuities in Purfuartce of this 
 Ad of Parliament, were equal to a Term of '^^ 
 Years. 
 
 On this I remark, that the Term of ^§ Years 
 is the Term to which I have before afTerted, 
 that the Lives of Perfons living in London of th6 
 Age of 10, might be computed to be equal, if 
 we would carry the Computation to the utmoft 
 Extent. But then on the City of London y I 
 have taken notice that it is more healthful thatx 
 moft other Places, in my Opinion at leaft: Thus 
 much however is certain, that People in London 
 generally live to a greater Age, or more Perfons in 
 Proportion die of a great Age there, than al- 
 molt any where elfe in this Kingdom : And upon 
 the length of the Term, to which a Moiety of 
 the Perfons lived, who where named in the An- 
 nuities on Survivorfhip, every one will readily 
 obferve, that thofe Lives were picked and 
 chofen Lives- for which Reafons, I think that 
 the Duration of the Lives of the Inhabitants of 
 the City of London^ or of the Nominees in thefe 
 Annuities, will not be a good Rule in all Cafes, 
 to meafure the Duration of Lives for that Part 
 of the Mafs of Mankind, which is in the belt 
 Stage of Life; I mean, of the Age between 10 
 and 15. Thus much however I may conclude, 
 that thefe Accounts will fufficiently juftily any 
 Eftimate, which ftatcs fuch Life as equal to a 
 -Term of 28 Years and no more; and all the 
 
 Doubt
 
 [ 254 ] 
 
 Doubt remaining with me is, whether thev 
 don't require that it fhould be fet higher. 
 
 As for the other feveral Periods of Life, I 
 have made the Eftimates by a gradual Decreafe, 
 in a Way and Proportion, which I believe will 
 appear to be a realbnable and a juil one ; that is, 
 I have made the Decreafe in each fubfequent 
 Period, greater than the Decreafe in the prece- 
 dent Period, by the Addition of one Quarter of 
 a Year to each ; except only, that in the Periods 
 from 60 to 65, 6^ to 70, and 70 to y^ ^ to each 
 I have made an Addition of two Quarters inftead 
 of one. I fhould take Nodce too, that in the 
 two 1 alt Periods of Life, I have not made the De- 
 creafe in any Proportion to the reli ; partly becaufc 
 I found it not ealily pradicable, for that Perfons 
 who are arrived to that Age, generally fpeaking, 
 are paft mofl of the Diftempers of Life, except 
 that of old Age, and a decay of Nature, which 
 ihows itfelf and can be judged of, only on Indi- 
 viduals ; and partly, becaufe I faw no Neceffity 
 of being very exad: in the Eftimates of thofe 
 Lives, they being of fmall Confidcration, and 
 inferted only for a particular Purpofe. 
 
 To the Remarks already made on the Bills 
 of Mortality for Loudon, I crave leave to add 
 one or two more 5 the firft, to remove an Objec- 
 tion which is made againft them, that they are 
 an improper Rule to determine the Chance of 
 the Duration of a Lite; and a fecond, which 
 tends to Ihevv the Healthfulnefs of that City, fo 
 far as the Longevity of Perfons living there, 
 flaows it; and with thefe I Ihall conclude this 
 
 Head.
 
 [ 255 ] 
 
 Head. It is alledgcd then, tliat the whole 
 Number of Perfons dying there, are not included 
 in thefe Bills, for that none are inferted but thofe 
 who die in the Communion, or are buried by 
 the Service, of the Church of £/;^/<^;/^; and that 
 the Number of Perfons who are only cafual or 
 temporary Inhabitants, is very great and incer- 
 tain, and yet, if they dye there, and there arc 
 buried, that they are reckoned in thefe Bills: 
 Therefore fuch Bills, not being colledted for all 
 the Inhabitants, or for the conltant Inhabitants, 
 cannot be a good Meafure to compute the Chance 
 of Mortality for that City. 
 
 Admitting thefe Allegations to be true, as 
 moft certainly they are, I agree the Rule is noc 
 quite fo certain and exa6t as if the Numbers 
 were intire, and there were no Acceffion of 
 Foreigners; but then I fay, that Dillenters from 
 the Church of Englamd, of whatever Sed: or 
 Religion they may be, and whatever Exemp- 
 tions they may claim from the Law of the 
 Land, they can have none from the common 
 Courfe of Nature; but are fubjed: to the fame 
 Laws of Mortality with other People, and in the 
 fame Place, or in the fame Country and Climate, 
 dye much at a much, in the fame Proportion 
 as other Mortals of the like Age, and of a dif- 
 ferent Religion ; therefore, be the Number of 
 thofe who on this Account are omitted out of 
 thefe Bills, greater or lefs^ yet I doubt not but 
 in the like Periods of Life, the Number of thofe 
 who die and arc buried out of the Pale of this 
 Church, and are left out of the Bills, is in the
 
 fame, 01* neai* the fame Proportion, as the Num- 
 ber of thofe who live and die within fuch Pale, 
 and are included in the Bills. The fame Way of 
 Reafoning, in great Meafure at leaft, will hold 
 good, with Regard to the great and continual 
 Refort of Strangers to that City, and their 
 Death there; for even thefe, taking them all to- 
 gether, young and old, fane and infanc, are lia- 
 ble to much the fame Chance of Mortality, with 
 the natural Inhabitants. 
 
 Thefe Allegations will indeed have great 
 Weight, where thefe Bills of Mortality are made 
 Ufe of, to compute what is the whole Number 
 of Perfons living in that City, or to ftate what 
 are the feveral Ages of the Perfons living there ; 
 and even, if thefe Bills were in thefe Retpedts 
 intire and compleat, I cannot difcover that fuch 
 Bills would be a fure Rule for making fuch 
 Calculations; though, believe me, they may give 
 us a far better Account than the wild Guelfes 
 and vague Computations, which fome People 
 give us. To fix the Number of Souls in that 
 City, or in any other Place, otherwife than by 
 a diltind Numeration of the Individuals, is a 
 Matter of much Difficulty, and, notwithltanding 
 the Attempts of fome great Perfons, remains yec 
 under great Incertainties. To do this ; it is 
 not fufficient to know folely the Number of Per- 
 fons dying in each Period of Life, but we muft 
 know, or be able to llate, out of what Number 
 of Perfons living in each Period, they did fo die; 
 but no Bills of Mortality, that I have fecn or 
 heard of, have ever yet carried their Computa- 
 tions
 
 [ 257 ] 
 
 tions fo far ; or, we mufl fix the Proportion 
 which the whole Number of Perfons dying 
 bears to the whole Number living: but the Com- 
 putures of this Kind, which have been made by 
 different Hands, to compute the Number of 
 Inhabitants of a given Place or County, have 
 been fomc of them fo wide and fo diftant from 
 one another, that the one Computure fhall give 
 you near double the Number, which the other 
 does ; for I have feen one that computes one to 
 die annually, out of 21 living; and another, 
 one out of 40 living. 
 
 Now, on the other fide, to fix the Chances of 
 Mortality on any Perfon named, or between Per- 
 fons of one Age and another, I fay it isfufficient 
 to know the Proportion, which the Number of 
 Perfons dying of one Age bears to the Number 
 dying of another Age: And thus much thefe 
 Bills for Loudon J or the Improvements I have 
 made upon them, have done. For Inftance : Of 
 the Perfons dying there in a Year, or in feveral 
 Years collected together, we fee that one Moiety 
 or above, is under ten Years old \ therefore on 
 any Perfon or every Perfon born alive, the even 
 Chance is, that he lives to be ten Years old, or 
 thereabouts : Of the Perfons of ten Years old 
 and above, dying there in one Year, or in a 
 Number of Years by a Medium, we may difco- 
 ver from thence, that one Moiety or thereabouts 
 is above 45, and the other between 10 and 45; 
 therefore on a Perfon of 10 Years old, the even 
 Chance is, that he lives to be 45 Years old ; con- 
 fequently his Life is equal to a Term of ^^^ Years ; 
 
 K. Again >
 
 [ 258 ] 
 
 Again ; of a looo Perfons dying, ^ arc of the 
 Age between 90 and 100, or, which is the fame 
 Thing, one in 200 dying, is of the Age between 
 90 and lOO, therefore one in 200, muft firft 
 live to the Age of 90, or one in 200 could not 
 poffibly, die, of, or above that Age : Therefore, 
 iince theie Bills do give us, or are a fufficient 
 Guide to us, to difcover thefe Proportions, I 
 conclude that they are a good Rule, to deter- 
 mine the Chances pf Vitality and Mortality, for 
 the Place where. 
 
 The other Obfervation which I make on thefe 
 Bills, is, that upon a Computation from thence, 
 it appears, that in London y one in 27 S6 lives to 
 the Age of an loO, one in 200 to 90, one in 33 
 to So, one in 14^ to 70, one in 8 to 60, one in 
 5 to 50, and one in 4 to 45 : And as a Moiety 
 or more of the Perfons dying, is under io Years 
 of Age, and a Moiety or lefs is above 10 Years 
 Age ; then out of the Perfons of ten Years Age 
 and above who die, one in a Moiety, of the 
 former Numbers refpedively, lives to the feveral 
 and refpedtive Ages before-mentioned. And if 
 ■we take another Age, fuppofe 50, and enquire 
 what is the Chance that fuch an one lives to be 
 100 Years old : The whole Number of Perfons 
 dying in fix Years Time, of the Age of 50 and 
 above, being 30242, and the Number of thofe 
 dying of the Age of an 100 and above, being 
 54, the former Number being divided by the 
 latter, leaves 560; that is, of the Perfons of 50 
 Years Age and above, dying one in 560, is of 
 the Age of 100 j or one in 560 of jo Years 
 
 ■Age,
 
 [ 259 ] 
 
 Age, lives to the Age of lOO. Now, I believe, 
 if we were to fearch all the Regifters in the 
 Kingdom, we ihall hardly find one where thcfe 
 Propofitions are exceeded, or even attained. 
 In the little Country Parifh, fo often before 
 fpoke of, the Gentleman, who has known the 
 Inhabitants for many Years, and has examined 
 the Regifter there ; tells us, that of 1 25 buried 
 in 20 Years laft pad. Children included, but 3 
 have arrived at 70 Years, and 2 at 80 ; from 
 whence, if there be no Miftake in the Numbers, 
 he concludes very righdy, that in that Place and 
 in that Period of 20 Years, one only in 41, ar- 
 rived to 70 and upwards ; and one only in 62, 
 to 80 and upwards ; which very great Difference 
 from the Chance of Vitality for London^ evidently 
 fhows the Longevity of People in that City, at 
 leaft beyond the Inhabitants of this Gentleman's 
 PaHfti; and I am very inclinable to think, 
 beyond thofe of moft other Places of this 
 Kingdom. 
 
 E?2d of the Second B oo;c. 
 
 Ri jBOOIC
 
 [ 26o ] 
 
 BOOK III. 
 
 ^Jn ESSAY to Eftimate the Chance of 
 the Duration oft-wo or more gwen Lrjes^ 
 and to afcertain the Value of Jiimdties 
 
 and Leafes for fuch Lwes^ &c. 
 hail -z-j'.-M. 
 
 ^E are come now to the lafl: and moll 
 difficult Task, to afcertain the Va- 
 lue of Leafes and Eftates for two or 
 more Lives; and as this is the moft 
 difficult Part of the Work, fo it has 
 |)een the moft weakly performed by the feveral 
 Authors, who have treated this Subjed ; and 
 Mr. Richards's Performance is no better than 
 any of the reft, notwithftanding he ufhers it into 
 the World with great Pomp and Solemnity ; and 
 tells us in his Preface, that his Tables, fuch of 
 them as are new, are built on the Foundation 
 laid by Dr. Halley^ and formed by the Method 
 prefcribed by Mr. Moivre\ and in his Book 
 gives us to underftand, that his Tables, in parti- 
 cular for the Value of Annuities, for two and for 
 three Lives, meaning furviving Lives, are framed 
 by a Rule itt out by Mr. Moivre, and by him 
 proved Mathematically, to be a true one for 
 that Purpofe« 
 
 Not
 
 [ 26i ] 
 
 Not to enter into that Point at prefent, whe- 
 ther the Foundation be wrong, or the Rule be 
 wrong ; or whether the Author of the Tables 
 did not underftand, or did not purfue the Rule ^ 
 this I am fure of, and doubt not but I Ihall be 
 able to make it evidently appear, that his Tables 
 for the Value of Annuities for two Lives and for 
 three Lives, are wrong: And the Errors in 
 them are, I think, fo grofs and fo very extrava- 
 gant, that they will bear hard upon the Method 
 which produced them, let who will be the Father 
 or Grandfather of it, and whatever Support it 
 may have from pretended Demonftrations. 
 
 Thefc Tables being fo Solemnly introduced, 
 and alledged to be framed by a Mathematical 
 Rule, gave me great Expectations of finding a 
 right Eilimate of the Value of Annuities for 
 thefe Lives j but upon firfl looking into them I 
 faw, and I wonder Mr. Richards^ who feems to 
 underftand and to be very well verfed in Figures, 
 did not fee, that his Value of Annuities for three 
 Lives, of II Years Age, and at 5/. -per Cent* 
 Intereft, muft certainly be falfe ; fince it appears 
 at firft View to be very near equal to the Value 
 of the Inheritance. It was this particular Efti- 
 mate, and its being the Value of Annuities for 
 three Lives, at the bcil Age of Life, and made 
 at the legal and moft common Rate of Intereft, 
 which gave Occaiion and firft induced me to 
 enter into this Matter; and after much Thought 
 and many Calculations on this Subject, made at 
 different and diftant Times, as I had leifure, a. 
 Method occurred to me of making thefe Efti- 
 B- 5. m^tes,^
 
 [ 26? ] 
 
 mates, which I think will anfwer the Purpofe 
 more efFedually, and more nearly than any 
 other yet made public, or that I have feen : But 
 this I mufl fubmit to the Judgment of the 
 Reader. 
 
 Before I give you the Refult of my Thoughts 
 and my Pains, it may not be amifs to fhew the 
 Defedts of former CalcuUtions; for if any of 
 them are right, mine is certainly unnecefTary, 
 and poffibly wrong. The Calculations of that 
 Author, who has Sir Ifaac Newton's Name and 
 Approbation prefixed to his Book, fo far as they 
 relate to thefe Eftates, are built on fuch wild 
 and groundlefs Suppofitions, that they do not 
 deferve, and in Truth, they have not the Sancr 
 tion of, that great Man's Authority, whatever 
 Appearance it might be intended they fhould 
 have of it. He does indeed approve the Method 
 of making the Calculations, and the Calculations 
 there exhibited- but he fays not a word of the 
 Application of them to the Purpofe, and in the 
 Manner in which this Author there applies them? 
 And as for the Method of computing the Value 
 of Annuities for two, or for three Lives, by 
 Hating the Value of an Annuity for the firft Life, 
 as equal to a certain Number of Years, and an 
 Annuity for the Subfequcnt Lives by decreafing 
 fuch Term by one Year only, for every additir 
 onal Life, it is fo far from the Truth, whoever 
 was the Maker or the Approver, that it feems to 
 me not to be even guelTing what might be the 
 xeal Value of an Annuity for fuch additional 
 Life or Lives, but merely ftudying a Mcafure 
 
 that
 
 [ 263 ] 
 
 that might depreciate the Value of the Annuity 
 for the firft Life, or the Life in PoiTeflion, and 
 to inhance the Value of the Annuity for the Life 
 or Lives in Reverlion, or to be added, purely 
 for the fake of advancing the Fines, on adding 
 or renewing a Life or Lives, 
 
 With this View, he Hates the firft Life or 
 Life in Pofleffion, as equal to a Term of IQ 
 Years only ; becaufe he fuppofes two Lives, or 
 three Lives, or any other Number of Lives, to 
 be equal to a Term certain ; and if we allot a 
 ihort Term only for the firft Life or Life in 
 Poflelfion, there is better Room to make an Ap-<- 
 portionment of a larger Term for the Life or 
 Liyes in Reverfion, or to be added ; and confc- 
 quently an Appointment of a larger Sum for the 
 Finej and unlefs he had this Reafon, he had 
 none, to make a Life in PofTefFion, equal only to 
 a Term of 10 Years. If a Life were equal to np 
 longer Term than this, an Annuity for a Life, 
 where Intereft were computed at %l,per Cent, or 
 higher, would not have been worth feven Years 
 Value; and they muft have been moft abfurd 
 Computers, who could compute a third Life, or 
 a Life to be added to two in Being, worth 
 feven Years Value in Reverfion, and at the fame 
 Time compute the firft Life or Life in Poiieflion, 
 not worth feven Years Value in Pofl'effion : And 
 yet they muft have computed the Value of a 
 third Life at this Rate, who ftated the Fine for 
 renewing or adding a third Life to two in Being", 
 at one Year's Value. Having fixed the Term 
 for the firft Life at ten Years, be computes a 
 R 4 fecond
 
 [ 264 ] 
 
 fecond or additional Life at nine Years, a third 
 Life at eight Years ; and fo he proceeds to de- 
 creafe each additional Life by one Year. 
 
 Suppoling now the firft Life to be equal to a 
 Term of 10 Years only ; a fecond and a third 
 Life, whether they be added to the former, by 
 way of original Purchafe, or by Renewal, can 
 be eflimated equal to no longer a Term than lo 
 Years, if we conlider them lingly and feparatcly 
 by themfelves, and as Lives in immediate Poflef- 
 fion : But, as the firft additional Life is a Life 
 in Reverfion, to commence after ten Years, it is 
 pretty extraordinary to compute fuch Life equal 
 to nine Years, but it is more fo to compute the 
 fecond additional Life, which is in Reverfion 
 after 19 Years ; as equal to eight Years. But the 
 greateli Defed in this V/ay of computing, lies 
 in this : A Perfon who purchafes an Annuity 
 or Leafe by renewing, or adding a fecond or 
 third Life to another in Being, is to pay no 
 more for fuch fecond or third additional and 
 concurrent Life or Lives, than a third Perfon 
 would do for the fame ; and they are really of 
 no more Value to the one than to the other : But 
 fuch third Perfon ought to buy, them and pay 
 for them as Annuities or Leafes on a Contingen- 
 cy only, as well as Annuities or Leafes in Re- 
 verfion; fince with Regard to him it may fo hap- 
 pen, that one of them, or neither of them may 
 ever take Place. Now the Chance between the 
 firft and fecond Life, being one in two, or an 
 even Chance, that the fecond does pot furvivc 
 the firft Life, and between the firft and fecond 
 
 Life
 
 [ 265 ] 
 
 Life on one fide, and the third Life on the other- 
 fide, the Chances that the third Life docs not 
 furviv'e both the other Lives, are two in three ^ 
 it becomes an even Chance that the firlt additi- 
 onal Annuity never takes Place, and two Chances 
 in three, that the fccond never does: And there- 
 fore, the firll additional Life is equal only to a 
 Moiety of nine Years, and the fecond only to a 
 third Part of eight Years ; the two Terms of 
 nine Years and eight Years, being the Terms to 
 which thefe two Lives were computed to be re- 
 ipedivcly equal, confidered merely and folely 
 as Lives in Reverfion, fo that the Term in all 
 t'he three Lives put together, would be little 
 more than 17 Years. But to this Point of the 
 Chances of Survivorihip, between two or more 
 Lives, and how they are to be ellimatcd, I 
 mean to fpeak at large, under a more proper 
 Head of Difcourfe, fo fliall prefs it no farther 
 here. 
 
 Mr. Haje*s Rule for valuing thefe Annuities 
 13 ft ill more extravagant than the laft Author's; 
 for he allows no Decreafe at all, in the Term for 
 the fecond or the third Life; his Diredtion being, 
 to put together the Term to which each Life, 
 fingly taken, is equal, whether two, three, or, I 
 fuppofe, more Lives, and to take the refpedtive 
 aggregate Terms, as the Terms to which the 
 two Lives, the three Lives, or all the Lives, 
 whatever the Number may be, are equal. Very 
 few Words will fiiow the great Abfurdity of this 
 jRule. Where Intereft is computed at 4/. per 
 Cent, a Life of 30 is equal to a Term of 25 
 
 Years,
 
 [ 266 ] 
 
 Years, according to this Author's Computet-" 
 tion; a fecond and a third Life of the fame Age, 
 muft therefore each of them be equal to a Term 
 of 25 Years ; the Sum total of the Term on all 
 the three Lives, will be y^ Years ; each of the 
 three Perfons are fuppofed to be 30 Years old, 
 when the Annuity is granted ; therefore one of 
 the three, mull be fuppofed to live to be an 105 
 Years old. 
 
 Mr. Morris's Rule for this Purpofe, is merely 
 ^n Extradt from Mr. Moivrc, and fince Mr, 
 Richards*s Tables for the Value of thcfe Annui- 
 ties, were conftru6ted by the fame Rule, this 
 will be confidered under the following Head ; for 
 I mean to examine thofe Tables more at large; 
 partly becaufe they have the Appearance of 
 Regularity, and of being fupported by Ma- 
 thematical Demonftratiops; and pardy becaufe 
 fuch Examination will, in fome Meafure, let 
 us into the Underftanding and the Reafon of 
 the Rule, which I mean to offer for the Valua- 
 tions of thqfeAnnuities. The Method which Mr. 
 Richards made Ufe of^ for Computing and 
 Forming thefe Tables, under Mr. Moivres Di- 
 rediions, is this : He ftates firfl the Value of 
 Annuities for a fingle Life, for two Joint-Lives, 
 and for three Joint-Lives; and then adding fome 
 farther Calculations, grounded on thefe feveral 
 Values, he colledts the Value of thefe Annuities 
 for furviving Lives. Thefe Tables thus framed, 
 I aitirm, do not give us the true Value, or any 
 Thins; that is near the true Value, of Annuities 
 for two or for three Lives^ and the Survivor;
 
 [ 267 ] 
 
 aad therefore, either Mr. Richards does not 
 fairly reprefent Mr. Moivre, which I fee no 
 Reafon at all to fuppofe, or he does not purfue 
 his Method, which upon repeated Experiments, 
 I am fatisfied, he docs purfue ; or the Method 
 prefcribed, or the Hypothefis on which the 
 Method is founded, or both, are wrong : And 
 this I principally aim to ihew, and I think is 
 very fairly deducible, from the Errors in thefc 
 Tables, if thofe Errors are the genuine and 
 neceflary EfFeds of the Method. 
 
 I have obferved under a former Head, that 
 this Author's Table for the Value of Annuities 
 on a fingle Life are faulty, on Account that they 
 make one and the fame Life equal to a" different 
 Term of Years, according to the various Rates 
 of Intereft, at which the Calculations are made; 
 and this, for the Reafons there given, I have 
 aflerted, cannot poffibly be the right Value, and 
 yet that fuch Value is the necellary Produce of 
 the Hypothefis. And lince this Table, for the 
 Value of an Annuity for a fingle Life, is the Foun- 
 dation of the Table, for the Value of an An- 
 nuity for two Joint-Lives, and for three Joint- 
 Lives, and thofe Tables for the Value of Annui- 
 ties, for two and for three Joint-Lives, with 
 feme farther Calculations and Dedu6tions from 
 them, are the Principles, from which the Tables 
 for the Value of Annuities, for two and for 
 three Lives and the Survivor, are framed ; the 
 Confequence is, that thefe lafl Tables necellarily 
 muft; and in Fad they do, partake of the fame 
 
 Imperfedions,
 
 [ 268 ] 
 
 Im perfections, , which the precedent Tables, 
 from whence they are derived, labour under. 
 
 However, that is not the Error which I 
 chiefly infill on here, though an Error undoubt- 
 edly it is, and an indcfcnftble one ; but if I 
 fhould admit, that this Table for the Value of 
 Annuities for a fingle Life was right, at any one 
 Rate of Intereft, and fince every Man takes 
 himfelf to be at Liberty to ellimate a Life, to 
 be equal to fuch a Number of Years as he likes 
 belt, the Table may be right at fome one Rate 
 of Intercft, or at every Rate of Intereft, con- 
 iidered iingly and uncompared with the Table 
 at any other Rate of Intereft; and admitting, 
 that the Tables for the Value of Annuities for 
 two Joint-Lives, and for three Joint-Lives are 
 right, and at prelent, I do neither affirm or' 
 deny them to be fo ; yet I fay, that thefe Tables 
 for the Value of Annuities, for two and for 
 three Lives, and the Survivor, do not give us 
 the true Value ; and from thence it will follow, 
 if the Method for conftruding thofe Tabl-es 
 has been rightly purfued, that fuch Method is 
 ii wrong one. 
 
 To examine flrll the Table for the Value of 
 Annuities for two Lives, let us fee whether it 
 gives us fuch Values as in Fadl doe, or pofTibly 
 can belong to fuch Lives, If we look into this 
 Table for the Value of an Annuity for two 
 Lives, each of 22, and compare it with the 
 Tables for the Value of an Annuity for th^ee 
 Lives, two of 22 each, and a third Life of 82 ; 
 they ftand thus^ on this Table. 
 
 Intereft
 
 [ 269 ] 
 
 /. /. /. /. f. 
 
 Intereft at 4 s_ ^_ " 2 ^ 
 
 The two Lives 20-46 1| 17-48I! 15-08II 13— 04II ii~8t 
 The 3 Lives 20 - 46 jj 1 7—46 |i 1 5 - 07 |1 1 3—04 i! 1 1 — 79 
 
 It appears here evidently, in two of the In- 
 ftances, that the Value of an Annuity for two 
 Lives and the Survivor, each of 22, is equal to 
 an Annuity for three Lives and the Survivor, two 
 of 22, and one of 82 ; and in the other three 
 Inftances it appears, that an Annuity for two 
 Lives and the Survivor, each of 22, is of greater 
 Value than an Annuity for three Lives and the 
 Survivor, two of 22, and one of 82 ; v^^hich is 
 impoffible to be right, unlefs it be right to fay, 
 that two and two make four, and that two and 
 two, after an Addition of fomething more, make 
 Hill but four, and fometimes lefs than four;* 
 which I apprehend may be contrary to a received ^ 
 Maxim in the Mathematicks, and I am fure is 
 contrary to common Senfe. It will not, in this 
 Cafe, be to any Purpofe to alledge, that a Life 
 of 82 Years of Age is of final I Value; for if *tis 
 not plus nothing, I am fure 'tis not minus no- 
 thing, as this Calculation makes it ; or, in other 
 Words, if an Addition of a thi.d Lite of 82 to 
 two Lives of 22, makes no Addition to the 
 Value of an Eftate held by thofe Lives, yet 'tis 
 impoffible it can make a Subflradion from fuch 
 Value. 
 
 This Error is fo apparent that it cannot, I 
 think, be denied ; and fo grofs as not to be de- 
 fended ; and yet it does not arife from any De- 
 icdi in Mr. Richards' s Table for the Value of 
 an Annuity on a fingle Life ; for if fuch Table 
 
 at
 
 [ 270 ] 
 
 at any one Rate of Interefl is right, and if hia 
 Tabic for the Value of Annuities for two joint 
 Lives, at a correfponding Intereft, is likewife 
 right ; and if the Values in the Tables, which I 
 have here tranfcribed, both for the Annuities on 
 the two Lives and three Lives, and the Survivor, 
 are form'd according to Mr. A/biwe's Diredbions, 
 as I am fatisfied they are, having made the Ex- 
 periment in all the Inftances^ the Conclufion 
 muil be, that this Error is to be imputed to feme 
 Defeat in the fubfequent Calculations, grounded 
 on thofe previous Tables, and in the Method 
 prefcribed. 
 
 Another Error which occurs in this Table for 
 Annuities on two Lives and the Survivor, is, 
 that two fuch Lives, at 5 /. per Cent, Intereft, arc 
 made to be equal to a longer Term than at any 
 other Rate of Intereft, even longer than when 
 Intereft is computed at 4 /. per Cent. This ap- 
 pears plainly in the Inftance of two Lives, each 
 of 12 Years Age; where at 4/. pit Cent, the 
 Value given in the Tables is 21-3^, which is 
 equal to a Term of 49 Years only, or a little 
 above, and at 5/. per Cent, the Value given is 
 18-41, which is equal to a Term of 51 Years, 
 or near it. 
 
 This Error, again, could not arife from Mr, 
 "Richards* s Tables ; either that for Annuities on 
 a fingle Life, or that for Annuities on two joint 
 Lives, and thofe only are made Ufe of here to 
 compofc this third Table. It could not arifc 
 from the Table for the Value of Annuities on a 
 lingle Life, becaufe the Value of an Annuity for 
 
 a finglc
 
 I 271 ] 
 
 a fingic Life is fet highet in fuch Table, and the 
 Term to which Value correfponds is longer^ if 
 Intereft be computed at the Rate of ^l. per Cent, 
 than if computed at the Rate of 5 /. ; and con- 
 fequently, the Term to which two fuch Lives 
 and the Survivor are equal, Ihould have been 
 longer, where the Eflimate is made at the Rate 
 of 4/. per Cent, than where 'tis made at the Rate 
 of 5/. per CefJt.\ and I lay fuch Term would 
 have bpen longer, if feme fubfequent wrong Cal- 
 culations had not intervened and prevented it. 
 
 Neither could this Error arife from the Tabic 
 for the Value of Annuities on two joint Lives \ 
 becaufe the Calculation which Mr. Moivre di- 
 rects to be made here, with Regard to the Value 
 of Annuities for two fuch Lives, muft necellarily 
 have produced quite a different EfFedt. By his 
 Diredions, whether Intereft be computed at 4 /. 
 per Cent, or at 5 /. this fame Value of Annuities 
 for two joint Lives is to be deduded out of the 
 Value of the two Lives fingly taken and put to- 
 gether ; and the Value of each of thefe Lives 
 ilngly taken is greater, and the Term to which 
 fuch Value correfponds is longer, when Intereft 
 is computed at 4 /. per Cent, than when 'tis com- 
 puted at 5/. per Cent.y and confequently the 
 Value of thefe two Lives, when put together, 
 muft be greater, and the Term to which fuch 
 Value correfponds muft be longer, according to 
 the different Rate of Intereft. Now if two 
 Sums total are given, the one a greater and the 
 other a lefs Sum, out of each of which a Sub- 
 ftradion is to be made ; if we fubftrad a greater 
 
 Sum
 
 [ 272 i 
 
 Sum out of the lefs Sum total, and a lefs Sunt 
 cut of the greater Sum total, the Sum given 
 which was the lefs Sum total at firft, muft ftill 
 continue the lefs Sum total after fuch Subllrac- 
 tion. 
 
 To apply this to the prefent Cafe, The Term 
 ■which correfponds to the Value of thefe two 
 Lives, lingly taken and put together, is ^g-l-c^ 
 if Intereft is computed at 4/. per Cent. '^ if com- 
 puted at 5/. the Term is 58-2-88; as may be 
 feen by my Table ISl^* VII. : And thefe are the 
 Terms out of which the Subllradlion is refpec- 
 tively to be made. The Term which corre- 
 fponds to the Value of thofe two joint Lives, 
 calculated by Mr. R/V^<^r(:/j's Tables, is 18-3-7, 
 if Intereft be computed at 4 /. per Cent. ; and if 
 computed at 5/. is 18-5-54 : And thefe are the 
 Terms which are refpedtively to be fubftradted. 
 But 18-3-7, which is the lefs Term to be fub- 
 ilradted, being fubftraCled out of 59-2-0, which 
 is the greater Term total, leaves a Remainder, 
 40-2-84; and 18-3-54, which is the greater 
 Term to be fubftraded, being fubftraded out of 
 58-2-88, which is the lefs Term total, leaves a 
 Remainder 39-3-34 ; confequently, 40-2-84 
 fhould be the Term to which the two Lives and 
 the Survivor are equal, where Intereft is com- 
 puted at 4 /. per Cent. ; and 39-3-34 fhould be the 
 Term to which the fame two Lives are equal, if 
 Intereft were computed at 5 /. per Cent. 
 
 From thefe Tables for the Value of Annuities 
 on two joint Lives it appears ; if an Annuity be 
 granted for two joint Lives, and Intereft is com- 
 puted
 
 [ 273 ] 
 
 puted at 4 /. per Cent, the Chance is that one of 
 the Nominees will die in 18-3-7 \ ^^^ ^^ granted 
 for the fame Lives, and Intercft be computed at 
 $L. per Cent, then the Chance is that one of the 
 Nominees will not die 'till 1 8-3-54. Upon which 
 I cannot forbear remarking the great Abfurdity 
 of this Calculation ; and yet this is ilill aggra- 
 vated by fomething which follows, 'viz. that if 
 an Annuity be granted on any of thefe Lives 
 lingly taken, and Intereft were computed at 4/* 
 per Cent, then the Term to which fuch fingle Life 
 is equal, comes out to be longer than if Intereft 
 ■were computed at 5/. J)^?* Cent.: And from thence 
 it neceffarily follows, that thefe Tables for the 
 Value of Annuities for joint Lives are. erroneous 
 as well as the reft; for it can never be true, that 
 a fingle Life of a given Age, Interefl computed 
 at 4/. per Cent,^ Ihould be equal to a Term longer 
 than a fmgle Life of the fame Age, Intereft com- 
 puted at 5 /. per Cent, ; and at the fame Time be 
 true that two joint Lives of the given Age, In- 
 tereft computed at 5/. per Cent.^ are equal to a 
 longer Term, than the very fame Lives would 
 be equal to if Interefl were computed at 4 /• 
 per Cent, 
 
 I will now flate a Cafe, which I think will per- 
 fedly clear up this Matter, and fhow plainly, 
 that the Table, which we are now confidering, 
 is falfe throughout, and in every Inftance, unlefs 
 by Accident ; and if this be the Cale, as indeed 
 it mult neceffirily be the Cafe, if the Method of 
 framing it is wrong, I think this will be a very 
 Itrong Proof of my Polition, that the Method is 
 S wrong.
 
 [ 27+ ] 
 
 wrong. We will fuppofe then an Annuity of 
 loo/, ftv Annum ^ and at 4/. per CefJt.^ to b(i 
 granted to N^. i. for the joint Lives of J, and 
 5. who are both of the Age of 12 Years; a 
 fecond to be granted to N^. 2., to commence 
 from the Death of the firll of thofe two who 
 dies, and to continue for the Life of the Survi- 
 vor of them ; and a third to N^. 3. for the Lives 
 of A and B, and the Survivor. 'Tis evident 
 here, that the Intereft of N^. 3. is juft equal to 
 the Interefts of N^. i. and N®. 1. put together; 
 for their feveral Intcrefts here commence and de- 
 termine at one and the fame Time. By this 
 Table, the Value of an Annuity for two fuch 
 Lives, and the Survivor, at 4/. per CetJt. is 21-39 ; 
 if the Annuity be 100/. per Annum ^ the Worth 
 in Money wall be 2139/. and is equal to a Term 
 for 49 Years, or thereabouts ; and this is the In- 
 tereft of N^. 3. The Intereft of N^. i., which 
 is in the joint Lives of A. and B. by the Tables 
 for thofe Eftates, is in Value 13-ci ; in Money 
 is 1301 /., and is equal to a Term of 19 Years, 
 wanting fome few Decimals, or near a Quarter of 
 a Year. The Intereft of N^. 2., which is to 
 commence from the Death of any one of the two 
 joint Nominees, and to continue for the Life of 
 the Survivor of them, is a Reverfion only, and 
 will come into Poffeffion on the Death of the 
 Nominee who Ihall die firft. And the Chance 
 being, as computed by thefe Tables, that one or 
 other of the Nominees will die in 19 Years, or 
 fomething lefs Time, the Intereft of N'='. 1, is a 
 Reverfion to take Place at the End of 19 Years ;
 
 [ 275 ] 
 
 at the End of 19 Years, A. or B. or whoever 
 is the Survivor of them, will then be 31 Years 
 old \ fo that the Intereft, and the whole of the 
 Intcrefl of N*'. a. is in a Life of 31 to commence 
 after a Term of 1 9 Years. The Life of this 
 fingle furviving Perfon, being of the Age of 31, 
 by this Author's Table for the Value of Annui- 
 ties for one Life, is in Value 15-18, and is equal 
 to a Term of 24 Years wanting feme few De- 
 cimals, or near a Quarter of a Year ; and a 
 Term for 24 Years in Reverlion after a Term 
 "for 19 Years, is in Value 07-40, and in Money 
 is worth 740/. / So that the Intereft of N''. i. is 
 1301 /. and the Intereft of N^. 2. is 740 /. ; both 
 put together is 2041. Or take it thus ; the In- 
 tereft of N^. I. is near equal to a Term of ip 
 Years, the Intereft of N^. 2. is near equal to a 
 Term of 24 Years, and both put together are 
 near equal to a Term of 43 Years ; and a Term 
 for 43 Years in an Annuity of 100 /. pr Jnnum^ 
 at 4/. pT Cent. Intereft, is worth in Money 2041 /. 
 All thefe Computations are made from Mr. 
 Richards*s Tables ; and fince his Tables for two 
 Lives and the Survivor give us a Term of 49 
 Years as the Term to which the Intereft of 
 N^. 3. is equal, and the Sum of 2139/. ^^ ^^^ 
 Value of this Annuity in Money ; and fince his 
 Tables for two joint Lives and a fingle Life give 
 us only a Term of 43 Years, as the Term to 
 which the Interefts of N^. i. and N^.2. put to- 
 gether are equal, and the Sum cf 2041 /. only 
 as the Value in Money of both theip Annuities 
 put together; if the Intereft of N''. i. andN<^.2. 
 S 2 put
 
 C 276 ] 
 
 put together is jull equal to thelnterefl of N^. 3. 
 by itfclf, as moil: certainly it is ; if the Tables 
 for a Tingle Life and two joint Lives were right, 
 it mull unavoidably follow that his Tables for 
 t^o Lives and the Survivpv arp falfe,: and if 
 Mr. Richards undcrftood Viv. Moivres Method 
 rightly, and there is no Room to make aDoutajt 
 of it, andpurfued fucli Method in framing thele 
 Tables, as by repeated Experiments I am fa- 
 tisfied he didv <^he Gonclufion muft be, that the 
 Method itfelf is wrong. < 
 
 In ftating this Cafe^ and making the Compu- 
 tations upon it, one plain pefe6t has occurred to 
 me; and it lies in- this. In my Ci^Iculations of 
 the Intereft of N". 1., which we will fuppof? to 
 be in the Life of B. as the Survivor, and where 
 the Lives arc equal 'tis no Matter which is fup- 
 pofed to be the Survivor, I have computed fuch 
 Interell as a Reverlionary one, and- to commence 
 from the Death of J. ; and fo fir I fuppofe this 
 j\Tethod of computing may fall in with mine : 
 But then I go farther, and compute the Life of 
 3» to be equal to fuch a Term only as the Life 
 of B. is equal to at the Time when the Intereft 
 of N^. 2. in fuch Life commences ; and undoubt- 
 edly the Life of 5., with Regard to N°. 2. is 
 not equal to a longer Term than this, and B. at 
 the Commencement of the Intereft of N*^. 1. in 
 his Life is really 31 Years old, ip Years of his 
 Life having elapfed during the Life of J., and 
 a Life of 3 1 is equal only to 24 Years. But 
 now, in the Computations made here, according 
 to Mr. Mohre's Diredions, the Intereft of N'^- a. 
 
 in
 
 [ 277 ] ■ ' 
 
 in the Life of JB., the fuppofed Survivor, is 
 made equal to fuch a Term as the Intercft would 
 be equal to, in Cafe no Part of the Life of B* 
 were elapfed, arid as if i?. were no more than 12 
 Years old at the Commencement of the Intereft 
 in his Life, and his Life were then equal to a 
 Term of 30 Years. This evidently appears to 
 be the Cafe in the Inftance juft now mentioned ; 
 for the Intereft of N^. i. in the joint Lives of 
 ^. and B. is equal only to 19 Years, and fmce 
 thefe Computations muft make the Interefts of 
 N^. I. and N''. 2: put together to be equal to a 
 Term of 49 Years, or otherwife they will not 
 be equal to the fame Term to which the two 
 Lives and the Survivor are equal ; it neceffarily 
 follows, that the Intereft of N^. 2. in the Life of 
 B. the Survivor muft be here computed as equal 
 to 30 Years: And in other Ages of Life I have 
 made the Experiment, and they all appear to 
 have been computed much on the fame Foot. 
 From whence one may very well conclude, that 
 Mr. A^oi-vre*s Mathematical Demonftrations, how- 
 ever elaborate and accurate they may be, and 
 whatever it is they do prove, do not, and cer-p 
 tainly cannot, prove and ftabliih a Method of 
 Calculation which produces fuch evident and 
 grofs Errors. But whether Mr. Moivre miftook 
 the Point to be proved, or Mr. Richards has 
 miftaken his Meaning, that I Ihall leave at pre-? 
 fent to them to accommodate between them- 
 felves, as they think fit ; but as I Ihall have an 
 Opportunity of refuming the Confideration of 
 S 3 this
 
 [ 278 ] 
 
 ,this Matter, I will then attempt to fliow on 
 which Part the Millake lies. 
 
 From this Cafe, fo far as we have already 
 gone in it, I think it appears pretty plain that 
 this Table does not give us the true Value of 
 Annuities for two Lives and the Survivor; and 
 if we carry it a little farther, I believe we fliall 
 fee to what Sort of Annuities on Lives, or to 
 what Sort of Lives thefe Calculations may more 
 juilly be applied : I mull add in this particular 
 Inftance; becaufe I apprehend they cannot right- 
 ly be applied to any Lives as an univerfal Rule ; 
 and when they are applied in this Inftance, we 
 Diall find them to be Lives much different from 
 two concurrent and furviving Lives. 
 
 To put the Cafe then, as before, I will fup- 
 pofe that N^. I. N^. 2, and N^. 3. have Grants 
 for the Lives of A. and 5., and on the fame 
 Terms as in the former Cafe ; and farther, that 
 !N^. 4. has a Grant for the fame Lives and the 
 Survivor of them, with this Addition to his 
 Grant, that on the Death of A. whom we will 
 fuppofe to die firil, he has a Liberty to renew 
 his Grant, and to infert another Life inllead of 
 JB. the Survivor. Let us examine now what is 
 the Value of thefe two Literefls belonging to 
 !N*^. 4., t^i^- his original Grant, and his Right to 
 renew and exchange a Life. His original Grant 
 is, in Effed, the fame Thing as a Grant for the 
 joint Lives of A, and B., and is therefpre equal 
 to a Term of 19 Years, or thereabouts ; but the 
 Value of his Right of renewing and exchanging 
 a Life depends upon the Terms which are or 
 
 may
 
 [ 279 ] 
 
 may be put upon him, as to the Age of the Life 
 whom he has Power to nominate on fuch Ex- 
 change. If he has Liberty to nominate at large, 
 and does nominate a Perfon of the beft Age of 
 Life, that is about lo or 12, then the Life of 
 fuch Nominee will be equal to a Term of 30 
 Years in Reverfion after ip ; and then the Inte- 
 refls of N^. 4. taken together, will be equal to 
 a Term of 49 Years. 
 
 But as thefe Cafes are feverally dated, can any 
 Pretence be made, that the Intereft of N^. 3. is 
 equal to the Intereft of N0.4.? It is impoffible ; 
 for at the End of 19 Years, when A. is fuppofed 
 to die, the Intereft remaining to N^ 4. is equal 
 to a Term of 30 Years, that being the Term to 
 which the Life of the Perfon is equal whom he 
 nominates to fucceed to A. \ whereas in the Cafe 
 of N^ 3., at the End of ip Years, when A is 
 fuppofed to die, the Intereft remaining to N°. 3^ 
 is only in the Life of 5. ; at the Death of A, 
 the Age of B. will be 31, a Life of the Age of 
 3 1 is equal only to 24 Years ; therefore the In- 
 tereft remaining to N°. 3. is equal only to 24 
 Years, at the fame Time as the Intereft remain- 
 ing to N". 4. is equal to 30 Years. 
 
 Since then thefe Tables give us a Term, of 49 
 Years, as the Term to which two Lives and the 
 Survivor are equal ; but two fuch Lives and con- 
 current Lives as are comprized in the Grant to 
 N^. 3. are not equal to fuch a Terrri, or any 
 Thing near it \ and two fuch Lives and a fucce- 
 ^aneous Life as are mention'd in the Grant to 
 J^°. 4. ai:e equal to fuch a Term, or thereabouts • 
 S, 4 we
 
 [ 28g] 
 
 we fee to what two Lives thefe Tables happen 
 to be applicable in this particular Inftance. The 
 Relult of the whole is, that Mr. Richards has 
 taken a great deal of Pams, as in Truth he has, 
 to calculate the Value of Annuities for two Lives, 
 intending thereby Annuities for two Lives and 
 the Survivor ; but his Tables, if they give any, 
 give us the Value of Annuities for two joint 
 Lives and a fingle fucceeding Life : So that, if 
 Mr. Moivn\ Dcmonftrations prove the latter, 
 which I very much queftion, yet even then they 
 prove fomething that is not at all to the prefent 
 Purpofe. 
 
 Let us fee now what the Rule is which Mr. 
 Moi'vre has prelcribed for adjufting the Value of 
 Annuities for thefe Lives, and examine it parti- 
 cularly ; for perhaps we may difcover fomething 
 in it belides the enormous Errors which it has 
 produced, which will convince us, that the Rule 
 itfelf muft neceffarily be a falfe one, notwith- 
 Handing the Allegation that the Truth of it has 
 been dcmonftrated : For the Demonftrations, 
 whatever they are, may have proceeded on Prin- 
 ciples or Suppolitions, for which there are no 
 juil Foundations in the Reafon or Nature of the 
 Thing. The Rule then is this ; to put together 
 the Value of the Annuity for the two given Lives 
 iingly taken, and from that Value to fubftrad: 
 the Value of thofe Lives jointly taken, and the 
 Sum remaining, he fays, is the Value of the An- 
 nuity for the two Lives and the Survivor. I fay, 
 on the contrary, that fuch Remainder is not, and 
 cannot be the Value ; and that the Cafes which I 
 
 have
 
 [ 28l ] 
 
 have already produced, and anotlicr or two which^" 
 I. crave Leave to itate, do put the Matter be- 
 yond all Gontroverfy. 
 
 To apply this in an Inftance or two; we will 
 firft make Ufe of the Lives of J. and 8. ; fuppofc ' 
 them to be each 1 2 Years old, that B. becomes 
 the Survivor, and that the Intereft is 4 A per Cent, 
 as before. 
 
 An Annuity for the Life of /. fingly'^^ 
 taken, is in Value j 
 
 One for the Life of B, fingly taken, is\ 
 
 in value j ' 
 
 Thefe Values put together make 34-4^ 
 
 An Annuity for their joint Lives is in Value i ^-oi 
 This laft Value, fubilradted from the ") 
 precedent Value, leaves a Remain- (^ 
 der, as the Value of the Annuity forf" ^^'39 
 the two Lives and the Survivor, j 
 
 This Sum 21-39, which comes out to be the 
 .Remainder, and to be the Value of an Annuity 
 for thefe two Lives and the Survivor, is equal to 
 a Term of 49 Years, and above ; but that this, 
 "which is the Term produced by Mr. Mohrcs 
 mathematical Operation, is not the true one, I 
 affirm ; and I think, what I have already offer'd, 
 and what I fhall farther add, will put it paft Con- 
 tradidlion. 
 
 On the Cafe thus flated, we will fuppofe Nc. i. 
 has the Intereft in the Life of A fingly taken ; 
 N°. 2. in the Life of B. fingly taken ; N°. 3. in 
 jthe Joint-Lives of J. and J5. \ and N°. 4. in the 
 
 Lives
 
 [ 282 ] 
 
 liives of J* and B, and the Scrvivor. The 
 Value of the feveral Intcrefts of N°. i. N''. i. 
 and N'^. 3. are before ftated and agreed upon * 
 and we want to know what is the Value of the 
 Intereft of N°. 4. This Author's Rule aiferts, 
 that this Value is adjufled barely by deduding 
 the Intereft of N°. 3. out of the Intereft of W. i. 
 and N°. 2. fingly taken and put together : and 
 I sfTert the contrary. Th^t the Intefeft of N^ 
 4. is not fo valuable as the Interefls of N> i. and 
 N''. a. put together is admitted on both Sides; 
 and it muft be admitted likewife, that fuch Want 
 of Value arifes from hence, that the Intereft of 
 2>J"°. 4. in the two Lives, fo long ^s both of them 
 laft, is combined, and with Regard to him and 
 his Annuity both the Lives are as one only; fo 
 that the Enquiry is, what Prejudice or Lofs ac- 
 crues X.0 him from fuch Combination ; and whe- 
 ther any and what after fuch Combination ended. 
 
 If we admit that we may adjuft the Value of 
 an Annuity for thefe tvyo Lives and the Survivor, 
 by deduding fomething out of the Value of the 
 two Lives fingly taken and put together ; fuch 
 Deduction mull be of fomething and every 
 Thing, whatever that be, which does come to,^ 
 or may be enjoy 'd, cither by N'^. i. or by N°. 1. ; 
 but does not come to, or can in any Sort, or by 
 any Chance, be enjoy 'd by N°.4. in the Whole 
 or in Part. 
 
 I agree then in this Cafe, that N^. i. and N^. 2. 
 have each of them a Chance to enjoy their An- 
 nuities for the Joint-Lives of A and i^., that is, 
 fox a Term of 18-3-07, or thereabouts; an4 
 
 tha^
 
 ^ [ 283 ] 
 
 that each of them, during that Time, will be 
 entitled to 1 3- 1, as the Value correfponding to 
 fuch Term ; and that N°. 4. has a Chance to 
 enjoy that Term and that Value no more than 
 once only during thofe Joint-Lives : And 'tis 
 from thence urged, that one of thofe Terms and 
 one of thofe Values is loft to N°. 4., fince he has 
 not a Chance to enjoy them both, and confe- 
 quently one of them is to be deduced out of 
 the Intereft on the two Lives lingly taken and 
 put together. If we grant this to be true, a9 
 moft certainly it is, we muft then compute all 
 jthe Lofles accruing to No. 4. as well as this, all 
 I mean that Hand on the fame Foot with this, 
 whether arifing during the Combination of the 
 two Lives, or after it is ended : And if we do 
 fo, believe me, before we come to an End of 
 our Computation, we Ihall meet with fo many 
 Lofles and Dedudions, that we fliall leave an 
 Annuity for two concurrent Lives and the Sur- 
 vivor, of no greater Value than an Annuity for 
 any one of the Lives fingly taken. 
 
 In the Cafe ftated 'tis fuppofed, that the Inte- 
 reft of No. I. in the Life of J^. is in Value 17-20, 
 and equal to a Term of 29-3-00 ; and that the 
 Intereft of N°. 2. in the Life of B, is the fame ; 
 and 'tis fuppofed likewife, that at the End of 
 18-3-00 one of the two dies, and that the In- 
 tereft of No. 3, being an Intereft for thofe Joint- 
 Lives, then determines ; and it not being mate- 
 rial which dies firft, we have fuppofed B, the 
 Survivor. From hence it follows, that N^. 4. 
 has not a Chance to enjoy fo much on the Life 
 
 of
 
 [ 284 ] 
 
 oi A. asN''. I. has ; for the Chance of N". 4. on 
 that Life is only a Term of 18-3-00, in Value 
 13-01 \ whereas the Chance of N^. I. on that 
 Life is a Term of 29-3-00, in Value 17-20: So 
 that on this Life there is a Lofs of a Term of 1 1 
 Years, and 4-19 in Value. The Dcdudions 
 then out of the Value and the Term of the two 
 Lives, iingly taken and put together, will Hand 
 thus. For the Lofs during the Joint- Lives in 
 the Term 1.8-3-00, for the Lois on the Life of 
 ^., he being dead, 1 i-o-oo in the Term; in all 
 29-3-00 : For the Lofs during the Joint-Lives 
 in the Value 13-01, for the Lofs on the Life of 
 ^., he being dead, in the Value 4-19; in all 
 17-20: That is, jull the Term and the Value 
 of one of the Lives fingly taken is loft to N^. 4, 
 by this Way of computing. 
 
 That this Term of 11 Years, and this Value 
 4-19, muft be looked upon as loft to N'^. 4., and 
 be dedua:ed out of the Term and the Value of 
 the two Lives fingly taken and put together, 
 might be thus proved. If this be good Rea^ 
 foning, 'oiz. N^. i. and N^ 2. have each of them 
 a Chance to enjoy a Term of 18-3-00, in Value 
 13-01, during the Joint-Lives of ^. and jB.; 
 but N'^. 4. has a Chance to enjoy one of the 
 Terms and one of the Values only, therefore one 
 of them is to be deducted out of the Term, and 
 the Value of the two Lives fingly taken and put 
 together, as fomething loft to N^ 4 ; which 
 niuft be Mr. Moi^Ye\ W^ay of Reafoning : Then 
 this alfo muft be good Reafoning, 'dIz, N°. i. 
 tand N°. 2., befides the Term 18-3-00, and the 
 
 Value
 
 [ 28s ] 
 
 Value 13-01, Incurred and grown due during 
 the Joint-Lives, have each of them a Chance 
 likewife to a farther reverfionary Term of 1 1 
 Years, in Value 4-19 ; for the Chance of each of 
 them is in the Whole a Term of 29-3-00, in 
 Value 17-20 ; but N°. 4. has a Chance to enjoy 
 this reverfionary Term of 1 1 Years, in Value 
 4-19, once only, for the very fame Reafon that 
 he cannot, during the Joint-Lives, enjoy more 
 than one Term and one Value, for that his An- 
 nuity is a fingle one only, and only one Nomi-* 
 nee living : one of thefe Terms therefore, and 
 the Value correfponding, Ihould be dcduded 
 out of the Value and the Term of the two Lives 
 iingly taken and put together, as fomething loft 
 to N°. 4. 
 
 Now it is obvious, that both thefe Lofles 
 muft not be computed, and deduced as fuch 5 
 for that this Method makes the Value of an 
 Annuity, for two Lives and the Survivor, to be 
 equal only to an Annuity for one of the two> 
 Lives, fingly taken : But then I affirm, that 2 
 bare Subftra6tion of the Value of the Annuity for 
 the Joint-Lives, that is the Subftradionof 13-or, 
 which is the Value of the Annuity for thefe 
 Joint-Lives, out of 34-40, which is the Value 
 of the Annuity ; for thefe two Lives fingly taken, 
 and put together, does not and cannot leave us 
 the Value of the Annuity for the two Lives and 
 the Survivor. If out of this total Value 34-40, 
 we make no other Subftnidion than the Value 
 13-01, we manifeflly leave in fuch Sum total 
 4-19; the reverlionary Value on the Life of A'y 
 
 and
 
 i 286 ] 
 
 and another 4-ip, the reverlionary Value on the 
 Life of B. and once 13-01, fuppofed to be due 
 and received during the Joint-Lives : For 4-19, 
 4-ip, and 13-01, put together, make 2.1-39; 
 and the Rule does plainly enough fo dired. 
 Now N^. 4 has no Right to both thefc rever- 
 fionary Values, but upon a Suppoiition that A 
 lives to the End of 29-3-00, or has a Chance to 
 live fb long ; and that B. alfo lives, or has a 
 Chance to live, to the fame Time ; and that there 
 is the fame Chance that both of them, when 
 jointly confidered, may live to the End of that 
 Term, as there is on each of them, when fingly 
 confidered ; which is not only not true, but is 
 alio contrary to the former Suppofition, that 
 one or other of them dies at the End of 18-3-00 : 
 And we mull fuppofe like wife, thatN°. 4. has two 
 -Annuities, when the Cafe Itates it as a (ingle one 
 only. 
 
 To avoid thefe Inconfillencies, I mean thcle 
 on one fide, if at the End of 18-3-00, we de- 
 duct the Whole of one of thefe reverfionary Va- 
 lues, then we make an Annuity for two Lives 
 equal only to an Annuity for a fingle Life ; on 
 the other fide, if we dedu6t neither of thefc re- 
 verfionary Values, we fhall make the Chance of 
 the Duration of both Lives, when Joint, to be the 
 lame, as when each of them is Single, and equal 
 to 29 Years ; and befidcs out of one Annuity, 
 we Ihall carve two : To avoid this Difficulty, we 
 need only to take the Fad, and the Chance, to 
 be fuch as it is ; viz. that the Annuity is a fingle 
 one, and the Chance to be, that one or other 
 
 of
 
 [ 287 ] 
 
 of the Nominees will be dead at the End of 
 iS-3-00; and if we compute the Value of the 
 Annuity for the Life of the Survivor, on this " 
 Foot, it will come out to be more than that 
 Value 4-ip once taken, and fomething lefs than 
 that Value twice taken : But of this Point, and 
 of the Manner in which the Life of the furviving 
 Nominee, and the Annuity depending on it, is 
 to be eftimated, I (hall have Opportunity to 
 conlider under another Head. 
 
 As this Way of Reafoning, may fcem per- 
 haps a little abftrufe, and not to (hew clearly 
 and diHindly enough, that the Value of the An- 
 nuity for the Joint-Lives is not the proper or 
 the fole Dedudion to be made; and fince this 
 Rule is fupported by a Claim to a Mathematical 
 Demonftration, and a Mathematical Demon(lra- 
 tion if rightly applied, is an unconteftable Evi- 
 dence; the Reader will excufe me, if I produce 
 another Inftance, in which this Error is (till 
 more grofs and more palpable; and indeed lies 
 fo open, that one would wonder how the Con- 
 flrudtor of the Tables could mi(s difcovering it, 
 when he made his Calculations. From hence it 
 will be fcen, and when pointed out, it will be 
 acknowleged by every Man, who has common 
 Underftanding in Numbers and Figures, evi- 
 dently to appear; that, in Order to find out the 
 Value of the Annuity for the two Lives, and the 
 Survivor, the Value of the Annuity for the two 
 Joint- Lives, is not the fole Value, or the proper 
 Value, to be dcdudted out of the Value of the 
 
 Annuity
 
 [ 288 ] 
 
 ■Annuity for the two Lives, fingly taken and 
 put together. 
 
 The Cafe is upon two Lives, each of the Age 
 of 8a, and Hands thus. 
 An Annuity for the Life of ^. aged 82,^ 
 
 is in Value j ^ 
 
 One for the Life of 5. aged 8a, is in Va-^ _ ._ 
 
 lue J 
 
 Both Values put together, are a-82 
 
 An Annuity for their Joint- Lives, is in\^ 
 . Value /_lf 
 
 Tl?elall Value deduded out of the prece-"^ ^__^ 
 
 dent one, leaves j 
 
 This remainder then, is the Value of an Annuity 
 for thefe two Lives, and the Survivor of them, 
 ias.it pomes out upon the Operation, made by 
 Mr. Moivris Rule, and is the fame as is given 
 in Mr. KicPjardis Tables. The Lives of ^. and 
 B. being equal, and the Chance of dying, con- 
 fequently equal, we will fuppofe that J. dies 
 firft, and the Intereft of an Annuitant in thefe 
 two Lives and the Survivor, is plainly this, and 
 no more than this, 'viz. in their Joint-Lives, 
 during the Life of A^ and after his Death in the 
 Life of B. furviving. An Annuity for their 
 Joint-Lives, is in Value 0-55, an Annuity for 
 the Life of B. the Survivor, when the Intereft 
 on both Lives commences, is only 1-4 1 ; there- 
 fore, the whole of this Annuitant's Intereft, can- 
 not poffibly be more than 55 and i-44> both put 
 together i-p6. 
 
 That the Intereft in the Joint- Lives of ^. and 
 By and the Intereft in the lingle Life of B. 
 
 the
 
 [ 289 ] 
 
 the Survivor, are all the Interefls which this 
 Annuitant has, in the Lives of A> and B. and 
 the Survivor, is paft all Controvcrfy : Thefe two 
 Interefts put together are, as before coIle6ted> 
 no more than 1-5)6 in Value; and it is impofTible 
 they (hould be more, unlefs we fuppofe the Life 
 of JB. the Survivor, at the Time he becomes the 
 Survivor; that is, at a Time when he is (ix or 
 feven Months older, than when the Intercft firll 
 commenced; is become a better Liie now, when 
 it is imglc, than it was at firft, when it was 
 lingle. But this is a eontradidion to common 
 Senfe and Reafon, and to fpeak a little in the 
 Mathematical Language, is contrary to the Pof- 
 tulatum at firft laid down, that fuch a Life is in 
 Value 1-41, and no more; therefore the Annui- 
 ty for thefe two Lives and the Survivor, cannot 
 be more than 1-96 in Value. 
 
 If in this Cafe, as well as in the former, wc 
 make only a fingle Subftradion, and take a-27 
 as the Value of the Annuity on thefe two Lives 
 and the Survivor, here occurs the fame Excep- 
 tion that we took before, and appears in a 
 ftronger light ; that the Term in the Life oiB. 
 fuppofed to be the Survivor, will be much too 
 long, and the Value of the Annuity much too 
 great. There, if we made only a finglc Subllrac- 
 tion, we left the Term in the Life of the Survi- 
 vor, the fame at the Time when he became the 
 Survivor; that is, when he was 36 or 31 Years 
 old, as the Term of his Life was, when the In- 
 terell firft commenced, that is, when he was 1 2 
 Years old only. Kere, if we make only a lingle 
 T Subftradtion,
 
 [ 290 ] 
 
 Subftradlion, the Value of the Annuity for the 
 Life of the Survivor will be greater, and the 
 Term correfponding to fuch Value, will be 
 longer, at the Time he becomes the Survivor, 
 than the Value and Term of the fame Life was, 
 even at the Commencement of the Intereft. 
 At the Commencement of the Intereft, the 
 Value of the Annuity for the Life of B. was 
 1-41, and no more; and at the Time when B. 
 becomes the Survivor, the Value will be 1-72; 
 for if out of 2-27, which is the Value given of 
 the Annuity for the two Lives and the Survivor, 
 ■we Subftrad 00-55, which is the Value of the 
 Annuity for the tw^o Joint-Lives ; the Value re- 
 maining, will be 1-72, as the Value of the An- 
 nuity for the Life of B. when he becomes the 
 Survivor, and the Term in the Life, will corre- 
 fpond to fuch Value. Here the Defed is more 
 vifible than before, and from hence it neceffarily 
 follows, where we are to adjult the Value ot the 
 Annuity for thefe two Lives and the Survivor, 
 that the Value of the two Joint-Lives, viz. 0-55, 
 is not the fole Value to be dcdndied out of the 
 Sum total, of the Value of the two Lives lingly 
 taken and put together, 'viz, out of 2-82 ; for 
 fuch a Value deduced, if nothing more is de- 
 du6ted, leaves 2-27, as the Value of fuch an 
 Annuity ; which is fo great a one, as we 
 fee cannot polTiblj be the true Value of this 
 Annuity. 
 
 Nay we may go farther, and aflert that it fol- 
 lows from hence, not only that 2-27 is not the 
 Value of the Annuity for tbcfe two Lives and 
 
 the
 
 [ 291 ] 
 
 the Survivor, but alfo that it is not the Value of 
 two fuch Lives of 8a, and a fuccedaneous Life ; 
 I mean, where the Annuitant upon the Death ot 
 hin) who dies firlt, has the Privilege in the room 
 of the Survivor to nominate a SucceiTor of the 
 Age of which the Survivor was, at the Gom- 
 «ienccment of the Intereft, that is, of the Age of 
 82 : For it appears from hence, that an Annuity 
 for two fuch Lives, is in Value 1-96 and no more, 
 ilnce the Intereft on the Joint-Lives, is c-55, and 
 the Intereft in the fingle fucceeding Life, fup- 
 pofed to be of the Age of 82, is 1-41, and thofe 
 make together i-p6 and no more. 
 
 And lince it is plain, that an Intereft for thefe 
 two concurrent Lives, and the Survivor of them, 
 is not fo valuable as an Intereft in thefe two Lives 
 taken as Joint-Lives, and an Intereft in a fingle 
 Life to fucceed and be nominated in the Man- 
 ner, and at the Time I have mentioned ; it fol- 
 lows farther, that we muft make a Dedudion 
 even out of this l-p6, before we can come at the 
 Value of the Annuity, for thefe two Lives and 
 the Survivor. Upon the Whole, I very much 
 queftion, whether Mr. Moivre or Mr. Richards 
 can affign any certain, or determinate two Lives 
 whatever, of 82, to which an Annuity of the Value 
 of 2-27 will correfJDond : But whether they 
 can or they cannot, I think this Cafe gives us a 
 convincing Proof, that a Subftra6tion folcly, of 
 the Value of the Annuity for the two Joint- 
 Lives, out of the Value of the Annuity for the 
 two Lives fingly taken and put together, will 
 not leave us the Value of the Annuity for thole 
 T 2 two
 
 [ 292 1 
 
 two Lives and the Survivor; for that the 
 Value 2.-27, which is the Produce of fuch 
 Operation, is not, nor can poffibly be, the 
 Value of an Annuity, for the two Lives and 
 the Survivor. 
 
 To make this Matter flill more clear, I crave 
 leave to produce one Inftance more, where the 
 two Lives are of different Ages ; and I mean 
 here, toconiider the Rule in another Light, and 
 to examine what are the Parts of the Value of 
 fuch an Annuity, which by this Rule are directed 
 to be brought to the Annuitant's x^ccount, as 
 belonging to him. And here I affert, that more 
 and other Parts of the Value of fuch an Annuity, 
 are by the Rule brought to his Account, as be- 
 longing to him, or more Parts are left in the 
 Acco&nt of the Value of the Annuity, for the 
 two Lives lingly taken and put together, than 
 the Annuitant has a Chance to enjoy : And 1 
 will attempt to affign, the particular Values 
 or Parts of Values, fo wrongly brought to 
 Account. 
 
 In an Inftance on the Life of ^. aged ll, and 
 B. aged 42 ; Intereft computed at 6/. per CeiJt,, 
 the Account liands thus. 
 
 An Annuity for the Life of A fingly"^^ 
 taken, is in Value j -^ -^ 
 
 One for the Life of B. fingly taken, is^^ 
 in Value / 
 
 Both Values put together, are 113-60 
 
 A«i
 
 [ 293 ] 
 
 An Annnuity for their Joint-Lives, is^ 
 
 inValue / ^^ 
 
 This laft Value dcdudled out of the^ 
 
 precedent Value, leaves j ^' 
 
 V/hich Sum of I4-7<S, is the Sum produced 
 by an Operation made, according to Mr. A'foi'vres 
 Method, and is the Value given in Mr. Richards's 
 Tables, as the Value of an Annuity for thefe 
 two Lives and the Survivor. We have before 
 obferved, that the Reafon for deducing the 
 Value of the Joint-Lives out of 23-60, the total 
 Value of both Lives put together, is this; be- 
 caufe in that Sum total, the Value of the Annui- 
 ty on the Life, both of J. and of B., is included, 
 as Parts of the Annuity to be received by the 
 Annuitant, whereas during their Joint-Lives, 
 he receives the Value of one only ; fo that dur- 
 ing that Time, one or other of the two Lives, 
 is no better than a dormant and barren Interell, 
 producing no other Value to the Annuitant, 
 than the Life of his Companion produces : And 
 therefore unlefs 8-84, which is the Value of the 
 Joint- Lives, and is twice brought to Account in 
 the total Value, is once fubltradted out of it, 
 the Annuitant will be computed to receive 
 that Value twice, when he can receive it once 
 only. 
 
 If this be a good Reafon for fubftrading the 
 Value of the Joint- Lives once, and not eftimat- 
 ing fuch Value as Part of the Annuity belonging ' 
 to the Annuitant, as undoubtedly it is; then 
 the fame Reafon will hold good, to dedud or 
 T 3 leavQ
 
 [ 294 ] 
 
 leave out of this fame total Value, any and every 
 other Value which ftands in this Account, as a 
 Sum to be received, if any there be, which in 
 Fadt cannot be received. And moft certainly in 
 this total, there Hill refts a Value, or a Part of a 
 Value, which can never be received ; for at the 
 Death of the Nominee who dies firll, be he 
 which we will, the Deduction of 8-84 already 
 made, does not make a Dedudion of the intirc 
 Value of the Annuity, either on the Life of ^. 
 or of -B. ; for the one is 13-36, and the other is 
 10-24. But after the Death of one of the No- 
 minees, fuch Part of the Value of the Annuity 
 on the Lite of the Perfon dying, as remains un- 
 received at his Death, never can be received ; 
 iince the Annuitant has neither Title nor Polli- 
 bilty to receive it in the Life-time of the Survi- 
 vor. Title to receive he has none ; for the In-^ 
 tereft in that Life is now more than dormant, 
 for it is abfolutely cxtind: and gone j and to re- 
 ceive the Remanetof the Annuity on the Life of 
 the Nominee dead, and at the fame Time to re- 
 ceive the Remanet of the Annuity on the Life of 
 the Nominee living, is not poffible ; for here oc- 
 curs an infuperable Difficulty, that we make 
 two Annuities going on together, and a double 
 Receipt, where we fuppofe but one, and there 
 can be no more than one. 
 
 I have before admitted, that this my Way of 
 computing and deduding, will make an Annui- 
 ty for thefe or any other two Lives, to be of no 
 greater Value than an Annuity for one of the 
 fame two Lives; and yet the Reafon is full as 
 
 flrong
 
 [ 295 ] 
 
 llrong for" making the one Dedudlion, as for 
 making the other ; it being equally true, that the 
 Annuitant docs not and cannot receive his An- 
 nuity double, during the Life of the furviving 
 Nominee, as that he cannot receive it double 
 during the Joint-Lives of both the Nominees: 
 And I do not urge this as an Argument to prove 
 that my Way of computing is right, for I 
 know it to be falfe, and from thence argue, that 
 both of them are falfe. 
 
 To make an Annuity on thefe two Lives, to 
 be of greater Value than on any one of them 
 fmgly taken, as moft certainly it is; we mull 
 reckon the Intereft of the Annuitant in the Life 
 of the Survivor, as of more Value than the Re- 
 manet on his particular Life, at the Time he be- 
 comes the Survivor, though not equal to the Re- 
 manet on both Lives. I mean thus : The whole 
 Value of the Annuity for the Life of A is 13-36; 
 the Value on their Joint-Lives 8-84, which be* 
 ing deducted, theRemanet on the Life of ^. will 
 be 4-52 : The whole Value on the Life of -B. is 
 10-24; and 8-84, the Value on the Joint- Lives, 
 being deduded, the Remanet on the Life of B, 
 will be 1-40. We muil admit then, and the 
 Fad is fo, that the Intereft of the Annuitant in 
 the Life of J.^ if he becom.es, and when he bc^ 
 comes the Survivor, is of greater Value than 
 4-59, the Value then remanent on his Life; and 
 in like Manner, if 5. becomes the Survivor, that 
 the Intereft in his Life will then be greater than 
 jr40, the Value then remanent on his Lite : But 
 T d neither
 
 [ 296 ] 
 
 neither of them will be equal to the Values re- 
 manent on both their Lives. 
 
 If the Queftion be asked, in what Manner 
 the Quantity of this greater Value of the Intcreft 
 in the Life of the Survivor is to be adjufted, 
 which I promifcd before to confider; my Anfwer 
 is, that it mult be done by looking upon the 
 Life of the Survivor as equal to fuch a Term, as 
 his Life will be equal to, at the Time he be- 
 comes the Survivor, or has the Chance to be<p 
 come fuch; and as fuch it may very well be 
 looked upon by him, and fo it would be looked 
 upon by any third Perfon, who fhould then pur* 
 chafe an Annuity on his Life, fmce then and 
 from thence it is a Jingle Life ; and the Interelt 
 of the Annuitant, either the old or the new one, 
 depends now on the Contingency of Mortaiity 
 on that furviv'ino; Life onlv : And the Term on 
 that Life fo computed, mull be taken with regard 
 to either of the Annuitants, as a Term in Re-^ 
 verfion, to commence from the Determination 
 of the Term, in the two Joint-Lives. 
 
 To this Quefiion, the Anfwer of Mr. Moivre 
 by his Rule is, that the remanent Value of the 
 Annuity on the Life of the Nominee dead, mull 
 be added to the remanent Value on the Life of 
 the Nominee furviving ; and that thefe two 
 Values put together, give us the Intcrell oi^ the 
 Annuitant in the Life of the Survivor : To ap- 
 ply ihis to the Lives of ^. and B : If ^. be the 
 Survivor, the remanent Value on his Life is 
 4-52 ; to this we muft add 1-40, the remanent 
 Value on the Life of ^j thefe two put together 
 
 will
 
 [ 297 ] 
 
 will be 5-91 ; and this will be the Value of the 
 Intereft of the Annuitant, in the Life of J. the 
 Survivor : To this if we add 8-84, the Value 
 received during the Joint-Lives, the whole will 
 be 14-76 ; the intire Value of the Intereft of the 
 Annuitant in thcfe two Lives and the Survivor. 
 In the other pollible Cafe, that B, becomes the 
 Survivor ; to the remanent Value on his Life, 
 which is 1-40, we mull add the remanent Value 
 on the Life of J.y which is 4-52 ; thefe put to- 
 gether will be S'9'^'> ^"'^ ^^ '^^^ Intereft of the 
 Annuitant in the Life of B. the Survivor : And 
 to this, when we have added 8-84, the Value 
 received during the Joint-Lives, the Whole of 
 the Intereft in the two Lives, and the Survivor, 
 comes out to be 14-76 ; the fame as was before 
 computed on the Life of Ay if he were the Sur- 
 vivor. 
 
 This, indeed, is not the Diredion of the Rule 
 in fo many Words, but 'tis the Scnfe of it, and 
 in the Event it comes to this : for if we put to- 
 gether the Value of an Annuity for two Lives 
 iingly taken, and out of this total Value wc de- 
 duct the Value of an Annuity for the Joint-Lives 
 once, fuppofed to be loft to the Annuitant du- 
 ring fuch Joint- Lives, what are the Values then 
 left in the Account ? Moft plainly the Values 
 left in the Account are the Value of the Annu- 
 ity for the Joint-Lives fuppofed to be taken du- 
 ring the Joint- Lives, for in the Sum total this 
 Value is twice included ; and the Surplufage or 
 Remanet on each of the Lives Iingly taken, not 
 yet fuppofed either to be taken or loft : For here 
 
 the
 
 [ 298 ] 
 
 the Value of the Annuity for the Joint-Lives 
 fuppofed to be taken is 8-84 ; the Remanct on 
 the Life of J. is 4-52, and on the Life of iJ., 
 1-40; all which put together make 14-76, the 
 total Value of the Annuity for the two Lives. 
 If this then be the Rule cr the Efted: of ir, that 
 the Remanet on both Lives put together is to 
 be accounted as the Value of the Annuity for 
 the Life of the furviving Nominee, let us take , 
 a View of it, as fuch Value has Corrcfpondencc 
 to the Term to which fuch furviving Life is 
 equal, and we fhall fee the abfurd Confequences. 
 I fay then, that the Age of the Survivor, at the 
 Time he becomes the Survivor, is the Meafure 
 by which we mult judge of the Duration of fuch 
 a Life, and the Meafure of his Life is certainly 
 the Meafure of the Annuity, or of the Parts of 
 the Annuity, which will be received during fuch 
 Life. Mr. Moivre {ays othervvife, and that we 
 mull take the Value of the Annuity on the Life 
 of J. and the V^alue of the Annuity on the Life 
 of B.y not received during their Joint-Lives, 
 and thefe two Values put together are the Value 
 of the Annuity during the furviving Life. Now 
 in this Way of computing fuch Value, we 
 plainly take a Part of the Value of an Annuity 
 on two Lives put together, as a Meafure of the 
 Duration of one cf the two Lives; when a Part, 
 or fcveral combined Parts, of the Value of an 
 Annuity are not fo much as a Meafure of the 
 Duration, even of that very Life on which the 
 Annuity depends ; on the contrary, the Durati- 
 on of the Life is the Meafure of the Value of 
 
 the
 
 [ 299 ] 
 
 the Annuity ; much lefs can a Part of the Value 
 of an Annuity on one Life, and a Part of fuch 
 Value on a fecond Life, combined together, give 
 us the right Term to which fuch fecond Life is 
 equal, or the proper Parts of the Annuity re- 
 maining and to be received during fuch. Life. 
 Nay in this Cafe the Life of the Survivor is 
 in Effed a third Life : For though the Perfbn . 
 furviving mult be one and the fame with one of 
 the Nominees, yet his Life fhould be and muft 
 be conlidered as a different Life now from what 
 it was before, lince he is fuppofed to be advanced 
 in Age about thirteen Years. To which may 
 be added, as fomething more extraordinary, that 
 thefe remanent Parts of the Value of the Annu- 
 ity on thefe two Lives put together are made the 
 Meafure of the furviving Life, or, as I may more 
 properly call it, a third Life, let which will of 
 the two be the Survivor, and let the Ages of the 
 two Nominees differ never fo widely. 
 
 And farther; if wc could fuppofe that the 
 fevcral Parts of an Annuity on a linglc Life, 
 where fuch feveral Parts have been calculated 
 annually, on the Chance of Mortality, thro' the 
 whole poffible Term of a given Life, and when 
 put together, do give us the true Value of the 
 Annuity for fuch Life ; and if we could fuppofe, 
 that a Part of the Value of an Annuity on one 
 Life added to a Part of fuch Value on a fecond 
 Life would give us the true Value of the Annu- 
 ity for the furviving Life of thefe two, which I 
 call a third Life; and that this will hold good, 
 let the Age of the furviving Life be what it will : 
 
 Which
 
 [ 3^o ] 
 
 Which Suppofitlons we mufl: make where wc 
 follow the Diredions of this P.ule ; though t 
 contefs I can .reconcile neither of them to my 
 Underllanding: Yet, even upon thefe Suppofi- 
 tions, as ill grounded as they are, the Values 
 here fpecified, that is, the Valuer not received 
 during the Joint-Lives, cannot be the Values 
 refting to be received, it being impofTible for the 
 J^nnuitant to receive thofe individual Values, as 
 will be obvious when we confider what are thofe 
 Values unrcceived. The Value of the Annuity 
 during the Joint-Lives, ^nd which is received, 
 138-84, and is equal to a Term of 13 Years, 
 \^ery little over ; fo that at the End of 13 Years 
 there remains on the Life of J, 4-52, as the' 
 Value to be received on his Lif?, which is equal 
 to a Term of 14-3-00, or thereabouts; and on 
 the Life of B. there remains 1-40 as the Value 
 to be received on his Life, which is equal to a 
 Term of 3-1-00, or very near it ; both which ' 
 Terms are Terms in Revcrfion after the Expi- 
 ration of the 13 Years incurred during the Joint- 
 Lives, and the Values are correfpondent. 
 
 Now the Annuitant can't poffibly receive, or 
 have a Chance to receive, both thefe Sums of 
 4-52 and 1-40, becaufe in the Cafe ftated we 
 have fuppofed that the Annuity is only a finglc 
 one, and that neither of thefe two Values do 
 arife and grow due 'till after the Expiration of 
 13 Years, and the Determination of the Intereft 
 for the Joint-Lives, that is, after the Death of 
 .one of the Nominees : So that to put the Annu- 
 itant in a Capacity of receiving both thefe Sums, 
 
 we
 
 [ 301 ] 
 
 we muft ftippofe there arc two Annuities, which 
 is contrary to the Cafe ftated ; and we muit fup- 
 pofe that A and B. do both furvive the Term 
 of 13 Years, and each of them one another, 
 which not only contradids the former Suppofi- 
 tion that one of them is dead, but common Scnfe 
 too. 'Tis certain there is a Chance that J. may 
 furvive 5., and a Chance hkewife, that B. may 
 furvive ^., but the Annuitant can't have both 
 thcfe Chances come up Benefit-Tickets ^ for 'tis 
 in Nature impoflible, and a Contradiction in 
 Terms, that A. does in Fad furvive £., and B, 
 at the fame Time furvive A: So that there is a 
 Certainty that the Annuitant will enjoy the Sum 
 attending one of thefc Chances, but an Impolfi- 
 bility that he Ihould enjoy both: And yet this 
 Method of Mr. Moivre's does necelfarily fuppole 
 that he is entitled to, and will receive, both 
 Annuities ; and I doubt Dr. HaHefs Hypothefis 
 fuppofes that both Nominees will become the 
 Survivor. 
 
 This Rule may be confider'd in another Light, 
 and exprelVd in different Terms, and yet, in 
 Effedt, remain the fame ; a^id in this Light, per- 
 haps, we lliall fee better what are the Parts of 
 , the Value of this Annuity which do not belong 
 •tofuch an Annuitant, and yet are here brought 
 to his Account. The Rule will then ftand thus. 
 To take the Value of the Annuity for any one 
 
 • of the Lives, intire as it is ; and out of the 
 
 • Value of the Annuity for the other Life to fub- 
 ilrad the Value of the Annuity for thofe two 
 
 ^ Lives as Jaint^LivcS; and to add the Remainder 
 
 to
 
 [ 302 ] 
 
 to the Value of the firfl Life : And to take thofe 
 two Values put together as the Value of the 
 Annuity for thofe two Lives and the Survivor. 
 If we ftate the Rule in this Manner, it is in Sub- 
 Itance the fame as in the former. There we de- 
 duct the Value of the Annuity for the Joint- 
 Lives out of the Value of the Annuity for both 
 Lives fingly taken and put together j here we 
 keep the Value of the Annuity for the two Lives 
 feparate, and out of the Annuity for one of the 
 Lives feparately and fingly taken we deduct 
 the Value of the Annuity for the Joint-Lives : 
 And then we put together the Value of one of 
 the Annuities intire, and fo much of the Value 
 of the other as remains after the Value of the 
 Annuity for the Joint-Lives is deducted : And 
 therefore we deduct the fame Value out of the 
 fame Value in both Cafes, and the Sum total re- 
 maining will be the fame. 
 
 If the Rule be look'd upon in this View, we 
 fee more clearly what Values we have ; viz. the 
 Value of the firft Life intire, and fo much of 
 the fecond Life as remains after deducting the 
 Value of the Annuity for the Joint- Lives, Now 
 here I admit that 'tis right to take the Value of 
 the Annuity for the firft Life intire as it is; 
 and the Quellion is folely, Whether the Annui- 
 tant for the two Lives and the Survivor has a 
 Right or a Chance to thofe Parts of the Value of 
 the Annuity for the fecond Life as are here af- 
 fign*d to him, and I conceive not • and for this 
 Reafon. The Parts of the V^alue of the Annui* 
 ty which the Annuitant has a Right to on chc 
 
 fecond
 
 [ 3^3 ] 
 
 fecond or additional Life, are the Parts of the 
 Value of an Annuity in Reverlion after the De- 
 termination of the Annuity on the firft Life ; 
 but the Parts of the Value of the Annuity here 
 affign'd to him, are the Parts of the Value of an 
 Annuity in PofTellion ; for the Parts of the Value 
 deduded here are deduded out of a Value in 
 Polfeffion, and confcquently the Parts of the 
 Value remaining, which are the Parts allotted 
 to the Annuitant, are Parts of a Value in Pof- 
 fellion. 
 
 An Inllance will explain and illuftrate this 
 Poficion. To afcertain the Value of an Annui- 
 ty for the Lives of J, and B. both of 12 Years 
 Age and the Survivor, we are to take the Whole 
 Value of the Annuity for the Life of u4. which 
 is 17-20, and of the Value of the x^nnuity for 
 the Life of B., which is likewife 17-20, we are 
 to take fo much as remains when the Value of 
 the Annuity for their Joint-Lives is deduced. 
 The Value of the Annuity for thefe Joint-Livea 
 is 13-01, and if we deduct 13-01 out of 17-20, 
 the Remainder is 4-19, which 4-19 being added 
 to 17-20, the Value of the Annuity for the firll 
 Life, gives us 21-39;. which is the Value of the 
 Annuity for thcfe two Lives and the Survivor, 
 produced by MT.Moivre's Method ^ and given in 
 the Tables. It appears from hence very plainly, 
 that this Value 4-19 are Parts of a Value of an 
 Annuity in PofTeflion, for they are Parts of the 
 Value 17-20, which is the Value of an Annuity 
 for 29-3-co in immediate Poffeflion : And *tis as 
 plain, that the Annuitant haa a Right only to- 
 
 th-
 
 [ 3^4 ] 
 
 the Parts of the Value of an Annuity in Rever^ 
 lion, and that on fuch fecond Life they are Values 
 in Reverfion after the End of 29-3-00. 
 
 And if we look upon this Value 4-19 as Parts 
 of a Value in Reverlion, as in fome Refpcds it 
 may be look'd upon, and examine what thcfe 
 Parts are ; we fhall difcover that ftill they are 
 fuch Parts as the Annuitant has no Right or 
 PofTibility to receive. The Rule, in the Manned 
 I have now flated it, dircds us to take the whole 
 Value 17-20, which is equal to a Term of 
 29-3-00 on the Life of /^., as the Value of the 
 Annuity for that Life ; and for the Annuity tor 
 the Life of B. the Survivor to take fo much of 
 the Value of the Annuity for his Life as remains 
 when the Value of the Annuity for the Joint- 
 Lives is dedudted thereout. I admit that the 
 Annuitant has a Right to fomething over and 
 befides the Value and the Term on the Life ot 
 ^. ; but 'tis plainly a Value in Reverlion after 
 17-20, and after a Term of 29-3-00 : But the 
 Value taken by Virtue of this Rule is a Value in 
 Reverfion after 13-01, and after a Term of 
 18-3-00; and this evidently makes the Reverfi- 
 on to fall into PofTelfion before in Fad it docs, 
 and makes this particular Value to be twice 
 brought to Account; for 'tis already once brought 
 to Account in the Life of Ay {o that we have 
 two Annuities fubfilling for this Space of Time, 
 inftead of one : Whereas the farther or additi- 
 onal Value which the Annuitant has a Right to, 
 over and above the Value of the Annuity for the 
 firft Lifcj is a Value and Term in Reverfion after 
 
 tlie
 
 [ 305 ] 
 
 the Determination of the firft Value and Term, 
 that is after 17-20 and the Term of 29-3-005, 
 /tii^ the Rule itfelf, as ftated by me, does fuppofe 
 fuch remaining Intereft on the fecond Life to be 
 no other or greater. From whence *tis appa- 
 rent that the Value 4-19, which is here afTign'd 
 to the Annuitant on the Life of B. the Survivor, 
 is not the right one ; for the Value which he is 
 intided to is the Value of a Term in Reverfion 
 after the End of a Term of 29-3-00, whereas 
 the Value here afiign'd is the Value of a Term 
 in Reverfion after the Determination of the In- 
 tereft for the two Joint- Lives, that is, after a 
 Term for 18-3-00 : And confcquently, the Parts 
 of the Value of the Annuity not belonging to 
 this Annuitant, which are here brought to his 
 Aceount, are the Difference between the Value 
 of an Annuity for li Years in Reverfion after a 
 Term of 18-3-00, and the Value of an Annu- 
 ity for 1 1 Years in Reverfion after a Term of 
 29-3-00; or the Sum by which the former Value 
 exceeds the latter. 
 
 By Way of Digreffion a litde; let us try what 
 will be the Effe6t of this Rule, which was framed 
 for adjufting the Value of this Annuity for the 
 Life of A. and B. and the Survivor, if we apply 
 it to adjuft the Term to which fuch an Annuity 
 is equal : And if the Rule be a right one to af- 
 certain the Value of this Annuity, I fee no Rea- 
 fon why it fhould not be a right one toafcertain 
 the Term. I Ihould rather fay, that I fee great 
 Reafon why it Ihould be a good Rule to deter- 
 mine the Term in the Annuity, though a very 
 U bad
 
 [ 3o6 ] 
 
 bad one to determine the Value. The Value of 
 the Annuity for the Life of A. being 17-ao is 
 equal to a Term of ap-3-00, and the Annuity 
 for the Life of B. is the fame ; theVahie of the 
 Annuity for their Joint-Lives is 13-01, and is 
 equal to a Term of 18-03-00, or thereabouts; 
 and if we dedudt this Term out of 2^-3-00, the 
 Term of the fecond Life, the Term left will be 
 1 1 -00-00 ; which being added to 29-3-co, the 
 Term on the firfl Life makes 40-3-00, as the 
 Term to which the Annuity for both the Lives 
 and the Survivor is equal. 
 
 Now this, for any Thing I fee to the contrary, 
 may be the true Term to which this Annuity is 
 equal, I am fure at leaft that *tis very near the 
 true one \ for in my Way of computing fuch 
 Term, if each Life were fuppoied to be equal 
 to a Term of 29-3-00, as here they are, the 
 Term of fuch an Annuity would be 40 Years and 
 above \ and on a Computation by Mr. Richards's 
 Tables, it will come out to be very little more. 
 If we take the Term of this Annuity from the 
 Value of it given in his Tables for thofe Annui- 
 ties, there indeed the Term comes out much 
 longer ; for the Value in thofc Tables is 21-39, 
 which is equal to a Term of 49-0-00, or above : 
 But this Value and Term is plainly equal to the 
 Value and Term of an Annuity for two Joint- 
 Lives and a fucceedaneous Life, to be nominated 
 when the Annuity for the Joint-Lives is deter- 
 mined and one of the Nominees is dead, and 
 not of an Annuity for two Lives and the Sur- 
 vivor of them ; as I have elfewhere Ihown. But 
 
 if
 
 r 307 ] 
 
 if we fuppofe the Survivor, at the Time when 
 his Companion dies, that is at the End of 
 18-3-00, to bej as really he is, of the Age of 30 
 and above, and if we take the Value and the Term 
 of a Liie of 30 Years old, and the Annuitant 
 moll certainly has not a Right in a Life of any 
 better Value and Duration ; the Value of fuch a 
 Life is about 14-66 in Value, which is equal to 
 a Term of 22-2-00; which being added to 
 18-3-00 makes 41-1-00, as the Term to which 
 thefe two Lives are equal. 
 
 This Inftance Ihews the Truth and Juflice of 
 my Rcafoning and Obfervations in another Place, 
 where I allcdged, that an Addition to, or a Sub- 
 Itradion from, a Value given was not the fame 
 Thing as an Addition to, or a Subftradtion from, 
 a Term given : For that on a Value and Term 
 correfpon^ent given if an Addition were made 
 to fuch Value of a Quarter, or any other pro- 
 portionable Part, and an Addition of the like 
 proportionable Part were made to fuch Term, 
 the new Value and Term arifing on fuch Addi- 
 tion would not correfpond ; and the Cafe is the 
 fame, and of NeceiTity mult be the fame, on a 
 Subftra6tion. And the Reafon of the Thing is 
 plainly this ; that in the Cafe of the Term, what- 
 ever is added or fubftradted is a Term wholly in 
 Reverlion, but in the Cafe of a Value, what is 
 added is a V^alue in Polfelfion, or Part of a Value 
 in Pofleifion. From hence I infer; iince the 
 Value of Annuities on Lives depends on the 
 Time or Term for which any given Lives en- 
 dure, or on an even Chance may be computed to 
 U 2 endure ;
 
 [ 3o8 ] 
 
 endure ; and fince a Part of a Term, with an 
 Annuity attending it, will not be the fame Thing 
 as a like proportional Part of the Value of the 
 Annuity ; that this Rule, alligning on fuch fe- 
 cond Life a proportionable Part of the Term of 
 the whole Life as the Term to which a Life 
 which depends on the Chance of Survivorlhip is 
 equal, may very well give us the true Term to 
 which fuch Life fo depending on the Chance of 
 Survivorlhip is equal ; but where it affigns a 
 proportionable Part of the Value of the Annui- 
 ty for the whole Life as the Value of an Annu- 
 ity for a Life which depends on the Chance of 
 Survivorfhip, there the Value fo alFign'd Ihall 
 not be the true Value of the Annuity for the 
 Life fo depending on the Chance of Survivor- 
 fhip. 
 
 To return to the Point. As Dr. HaJkys Rule 
 for the Valuation of Annuities on a fingle Life is 
 faulty, in alligning Values in Reveriion to the 
 Annuitant on a Life inllead of, and where? 
 Values in Poffeflion, if any Values at all, do be- 
 long to fuch Annuitant ; and by that Means, 
 the Term to which an x\nnuity for a Life is 
 equal is funk and deprefs'd to a fhorter Term 
 than the Life nominated has an even Chance to 
 endure : So the Method here is faulty on the 
 other Side, in affigning Values in PoffelTion, or 
 Values in Reverfion, as taking Place in Poffefli- 
 on earlier than in Fa(5i: they do take Place, when 
 Values in Reverfion only, or fuch as are at a 
 more remote Diftance of Time than is named, 
 do belong to the Annuitant j and by that Means 
 
 the
 
 [ 309 ] 
 
 the Term to which an Annuity for two Lives is 
 equal, is extended beyond the Term to which it 
 can in any Reafon be fuppofed to continue. . 
 That this muft be the Cafe will appear by 
 comparing the Chance of the Duration of a fingle 
 Life and of two Lives and the Survivor, with 
 the Value and Duration of an Annuity for fuch 
 lingle Life, and for fuch two Lives : F'or the 
 Do6tor, as I have before obferved, and thofe 
 who follow his Rule, make a Difference between 
 the Chance of the Duration of a fingle Life and 
 the Chance of the Duration of the Annuity de^- 
 pending on the Life • and fo they do likewife in 
 the Cafe of two Lives and the Survivor. If we 
 compute the Chance of Vitality by the Breflaw 
 Table on a Life of ten Years of Age, fuch a 
 Life is equal to a Term of 41 Years, but by the 
 Doctor's Tables for the Value of Annuities on 
 Lives, an Annuity for fuch a Life, at 6 /. per 
 Cent. Tnterefl, is equal only to a Term of 28-i-co, 
 and on Mr. Richards's Table much the fame. 
 By the fame Bre/law Table I have made a Cal- 
 culation of the Chance of Vitality on two Lives 
 of ten Years Age, and of the Survivor ; and 
 the Chance of Vitality of one or other of the 
 Lives reaches to 52 Years, or thereabouts ; and 
 the Value of the Annuity for the two Lives and 
 the Survivor, at the like Interell on Mr. Rich" 
 ^r^j*s Table, form'd by this Rule of Mr. Morjre'sy 
 is 15-59 7 anfwering to a Term of 48 Years, or 
 near it. The Remark which I make upon this 
 is, that on a fingle Life the Chance of the Du- 
 ration of the Annuity falls fhort of the Chance 
 U3 of
 
 [ 3IO ]. 
 
 of the Duration of the Life about 13 Year, but 
 on two Lives fuch firil Chance falls ihort of the 
 fecond about 4 Years only, when undoubtedly 
 it ought to exceed the former : And this Difpro- 
 portion, I fay, can no Ways be accounted for, 
 but by fuppofing that in computing the Value 
 of the Annuity for the two Lives, by Mr. Moivres 
 l^ule, more Values or greater Values are infert- 
 ed and brought to the Annuitant's Account than 
 do belong to him. 
 
 But this Matter may be carried farther ; for 
 I am of Opinion, not only that Mr. Moivre*s 
 Rule for calculating the Value of an Annuity for 
 two Lives and the Survivor, but even that the 
 Foundation of his Rule, n.Hz, Dr. Halleys Hy- 
 pothecs, for afccrtaining the Chance of the Du- 
 ration of two Lives and the Survivor, is not a 
 right one : And if Air. Mohre^ Method for cal- 
 culating the Value of Annuities for two Lives 
 and the Survivor depends on the Juftice of the 
 Hypothefis, as I take it for granted it does, and 
 I fuppofc is the Thing which he has undertaken 
 to demonflrate ; then, if the Foundation is 
 wrong, the Method and all the Tables conflruc- 
 ted thereby, will be wrong alfo. It may be true, 
 for any Thing I fna^l at prefent fay to the con- 
 trary, that Mr. Moivre has demonflrated that his 
 Method is right, on aSuppofition that Dr. iJ^/- 
 ley's Hypothcfis, upon which it is grounded, is 
 right ; but thefe Tables of Mr. Richards fhow 
 cither that the Hypothefis is wrong, or that the 
 Method is wrong ; for that I may be confident 
 that thefe Tables conftruded by the Method are 
 
 not
 
 not right ; and therefore that Mr. Moivre*s Dc- 
 monftratioris, if they prove that the Method is 
 agreeable to the Hypothefis, prove that the 
 Hypothefis on which the Method is founded is 
 wrong. My Way of Reafoning hitherto has 
 been principally thus. Thefe Tables are wrong ; 
 therefore, either the Method by which they were 
 conftructed, or the Hypothefis on which the Me- 
 thod is founded, or both, are wrong. Here I 
 fhall invert the Argument, and fay ; the Hypo- 
 thefis and the firft Foundation is wrong, there- 
 fore the Method is wrong, and confequently the 
 Tables which were conftrudted by it are wrong 
 alfo. 
 
 Before I enter into the Difcuffion of that Point, 
 I muft premife, that the Method which Mr. 
 Richards made Ufe of to form his Tables for 
 the Value of Annuities for two and fo for three 
 Lives and the Survivor, is that which Mr. Moivre 
 prefcribes, and not that which Dr. Halley pre- 
 scribes, for that Purpofe ; which may perhaps 
 be thought the fame, but are really very diffe- 
 rent. The Doctor, in calculating the Value of 
 Annuities for two Lives, and fo for three Lives, 
 direds the fame Method which he dire6ted and 
 made Ufe of for calculating the Value of Annu- 
 ities for a fingle Life. His Way is this ; he cal- 
 culates the Value of the Annuity as an abfolute 
 one for each Year of the Life of the Nominee, 
 deducing out of each Year's Value a Sum an- 
 fwering to the Chance of Mortality arifing in 
 fuch Year, and this he repeats for every Year of 
 the poffible Life of fuch Nominee ; and all 
 U 4 thofe
 
 [ 312 ] 
 
 thofe Values, fo colleded and put together, he 
 llates as the Value of the Annuity for that Life. 
 The very fame Method he prefcribes to calculate 
 the Value of Annuities for two or more Lives, 
 by taking the Values arifing in each Year of the 
 Life of the Nominees, or of any one of them, 
 through all the pollible Years of the Life of all 
 or any one of them ; and then putting all thefe 
 Values together, he flates them as the Value of 
 an Annuity for fuch Lives. In this Way of 
 computing the Value of Annuities for one or 
 more Lives, no Regard is had, or intended to 
 be had, of the even Chance of Vitality on the 
 fingle Lite, or on any of the Lives if more than 
 one ; but the Computation is to be made folely 
 on this Foot, viz. fo long 'tis polBble that the 
 fingle Perfon, and if more than one, fo long *tis 
 poffible both or one of them may live ; there- 
 fore Co long the Computation muft be continued* 
 Let us fee now how Mr. Richards framed his 
 Tables for the Values of thefe Annuities. In 
 forming thofe for the Value of Annuities for a 
 fingle Life, he made Ufe of the fame Method 
 which Dr. HalJey prefcribed and made Ufe of to 
 form his Tables for the Life Annuities : But in 
 dating the Value of Annuities for two or more 
 Lives and the Survivor, there Mr. Richards de- 
 ferts the Doctor's Method ; and no Wonder, 
 iince the Operation would prove fo excefiively 
 tedious and long, and reforts to Mr. Moivre*s, 
 This, as we have before obferved, is to put to- 
 gether the Value of the Annuity on each of two 
 Lives fmgly taken, and out of that Produce to 
 
 deduct:
 
 [ 313 3 
 
 dedud the Value of the Annuity for thofe two Lives 
 as Joint-Lives, and to take the Sum remaining as 
 the Value of the Annuity for the two Lives and 
 the Survivor. Now this Method is manifeflly 
 founded on Dr. Halleys Hypothefis, by which 
 he calculates the Chance of the Duration of two 
 or more Lives and the Survivor, and not that by 
 which he calculates the Value of Annuities for 
 two or more fuch Lives : But even this Hypo- 
 thefis for calculating the Chances of the Durati- 
 on of two or more Lives is, as I faid before, in 
 my Apprehenficn and Judgment, a wrong one. 
 
 In a Matter of fo much Intricacy and Per- 
 plexity as this is, whether I fhall be able to exprefs 
 my Sentiments and Reafons in fuch a Manner as 
 fatisfy the Underftanding of others, and to prove 
 my Pofition, I muft leave to the Reader's Judge- 
 ment; however, I Ihall attempt it. The Doctor's 
 Rule for afcertaing the Chance of the Duration 
 of a fingle Life, is this ; to take the Number of 
 Perfons living of the Age named, which Num- 
 ber is given in the BreJJaw Table, as the Chances 
 of that Perfon'sLife; to compute in what Num- 
 ber of Years thofe Perfons are reduced to an half, 
 and the Year wherein thofe Perfons are fo reduced, 
 is the Time to which fuch Perfon has an even 
 Chance to live, and beyond that Year he has not an 
 even Chance to live : And the Number of Perfons 
 living in the firft or any fubfequent Year, I would 
 call the Chances of Vitality on fuch Perfons Life, 
 and the Number of Perfons dead within fuch 
 Year or Years, the Chances of Mortality. The 
 Number of Perfons living, that is the Chances 
 
 of
 
 [ 314 ] 
 
 of Vitality on the Life of a Perfon of ten Years 
 of Age is 66 1 ; tHeilalf of thir is 330; in 41 
 Years, or thereabouts, the Number of Perfons 
 living is reduced to 330; that is, the Chances of 
 Vitality and Mortality are even : Therefore the 
 even Chance is, that a Perfon of fuch an Age 
 lives for 41 Years, and not beyond. 
 
 The Rule for afcertaining the Chance of Du- 
 ration of two Lives and the Survivor, is built 
 upon the fame Foundation ; but with this Differ 
 rence. On a fingle Life, the Number of Perfons 
 living of the Age given, is the Chances of Vi-i- 
 tality of that Perfon's Life ; here on two Lives, 
 we are to multiply the Number of Perfons living 
 of the Age given by the fame Number, if both 
 Perfons named are of the fame Age ; and if they 
 are of a different Age, we mull multiply the 
 Number of Perfons living of one Age, by the 
 Number of Perfons living of the other Age; 
 and the Produces are the Chances of Vitality on 
 the Lives of thofe two Perfons. In like Manner 
 for the Chances of Mortality ; we are to multi- 
 ply the Number of Perfons of the Age given, 
 who have died within any ftated Time, by the 
 fame Number, if both Perfons named are of the 
 fame Age ; and if they are of a different Age, 
 we mult multiply the Number of Perfons of the 
 one Age given, who have died within the ftated 
 Time, by the Number of Perfons of the other 
 Age given, who have died within the fame 
 Time ; and the Produces are the Chances of 
 Mortality on thofe two Perfons for the Time fo 
 Itated. And here, if we enquire to what Time 
 
 'tia
 
 ' [ 315 ] 
 
 *tis an even Chance that two Perfons named, or 
 one of them, fhall live, and beyond which they, 
 or either of them, will not live ; we ftate firil 
 the Chances of Vitality, on the two Lives as be- 
 fore dire^ed, and then we compute the Chances 
 of Mortality on the fame Liveij to a ftated Time*, 
 'till we come to the Time or Year wherein the 
 Chances of Mortality are a Moiety of the Chances 
 of Vitality as before fix*d, at which Time both 
 Chances, in EfFed:, are even., and this is the 
 Time, or Year, to which 'tis an even Chance 
 that one or other of the Nominees will live, and 
 beyond which they, or either of them, will not 
 live. 
 
 To make the Rule and my Objedions more 
 intelligible, I will apply it in an Inftance. If 
 we take the Life of ^. and B., both often Years 
 Age, the Number of Perfons living of the Age 
 of 10 is 65i ; 66i multiplied by 66 1 produces 
 436921, and that Produce is the Chances of Vi- 
 tality on thofe two Lives. The Number of Per- 
 fons of that Age who have died, fuppofe in <c 
 Years, is 469 ; 469 multiplied by 469 produces 
 218761, and this Produce is a Moiety of the 
 former Produce, and a little over, that is in 55 
 Years, the Chances of Vitality and Mortality 
 on thefe two Lives, are come near to a Par ; and 
 therefore the even Chance is, that A and A, or 
 one of them, will live for SS Years, or near it, 
 and neither of them beyond that Time. I deny 
 then that this Calculation gives us the true Chance 
 of the Duration of thefe two Lives and the Sur- 
 
 vivorj
 
 [ 3i6 ] 
 
 vivor, or that A, and B., or either of them, 
 have a Chance to live for ^^ Years. 
 
 My Objedtion to the Rule is this. The Num- 
 ber 66 1 is the Number of all the Chances of 
 Vitality through the whole poflible Life of A.y 
 and of all the Chances of Vitality through the 
 whole polTible Life of £., that is for 90 Years 
 on each of them ; and confequently 661 +661, 
 or 436921, is the Number of all the Chances of 
 Vitality on the whole poflible Lives both of A, 
 and of B., that is for 90 Years both on the one 
 Life and the other. But I fay, an Annuitant, 
 or other Perfon, having an Intereft in the Dura- 
 tion of the Lives of A. and B. and the Survivor 
 has not a Right to the Chances of the Duration 
 of the Lives both of A. and of B. through the 
 whole poffible Lives of each of them, that is 
 for 90 Years on each of them. Therefore this 
 Calculation affigning all the Chances of Vitality 
 on both Lives, and through their whole poflible 
 Lives, allots to fuch an Annuitant having Inte- 
 reft in their Lives more Chances on fuch Lives 
 than belong to them. 
 
 The firft and third Propofitions are fo obvious 
 and clear as to need no Proof; and the fecond, 
 ^Iz. that an Annuitant, having an Intereft in thefe 
 Lives, has not a Right to the Chances of the 
 Duration of the Lives both of A. and of B., 
 through the whole poflible Lives of each of them, 
 that is for 90 Years on each of them, which 
 only has any Difficulty in it, may thus be proved. 
 If we compute the Chance of Vitality on the 
 Lives of A. and h, fingly and feparately taken, 
 
 and
 
 [ 317 ] 
 
 and make Ufe of the Brejlaw Table for that Pur- 
 pofe, the even Chance on each of their Lives is 
 equal to a Term of 41 Years, or thereabouts; 
 and if we compute fuch Chance on thofe Lives 
 as Joint- Lives, and by the fame Tables, in my 
 Way of making fuch Computation, the even 
 Chance on their Lives as Joint-Lives is equal 
 to a Term of 30 Years, or thereabouts. Now 
 at the End of 30 Years, or whenever, in any 
 Way of Computation, the even Chance is that 
 cither J. or B. are dead, from thence the Annu- 
 itant has the Chance of the Duration of one of 
 the Lives only, the other being from that Time 
 to be confider'd as extindt and gone ; and con- 
 fequently, fo many of the Chances of Vitality as 
 are computed to arife on the Life of the Nomi- 
 nee fo dying, after the Expiration of 30 Years, 
 and when one or other of the Perfons is on an 
 even Chance computed to be dead, do, in Fa6t, 
 not belong to fuch Nominee, and therefore ought 
 not to be inferted in fuch firll general Produce 
 or Account. I admit, that in calculating the 
 Chances of Mortality on thefe two Lives, we 
 compute the Chances on both their Lives, and 
 through the whole polTible Lives of each, and 
 dedudt them out of the Chances of Vitality ; 
 but it will not from thence follow, as perhaps 
 may be urged, that the Calculation of the one 
 as well as of the other Chances ought to be 
 made on the fame Foot : For the Cafes I fay 
 differ ; and when we obferve how they differ, 
 that Difference will Ihow us that the one Calcu- 
 lation
 
 [ 3^8 ] 
 
 ktion ought to be made and is juftly made on 
 that Foot, and that the other ought not. 
 
 We will fuppofe then, at prefent, that 'tis an 
 even Chance that at the End of 30 Years either 
 ^. or B. is dead, yet I fay we muft continue our 
 Computation of the Chances of Mortality from 
 thence in the fame Manner as before, bccaufe the 
 Chance of both dying remains the fame at leaft 
 after the Death of one, as whilll both are living. 
 On the other Side, after the End of thefe 30 
 Years, and the fuppofed Death of one, the 
 Chances of Vitality do not remain the fame as 
 before ; and if we continue our Computation of 
 thofe Chances through the whole poUible Lives 
 of both, that is for 90 Years, as the Rule mani- 
 feflly does, we compute as if both were poffibly 
 living for a Term of po Years, when for Part of 
 that Time we have fuppofed one to be dead^ 
 which is, in EfFed, to fuppofe it poffible that 
 both are living when one is dead. The Rule 
 here does plainly take it for granted that the 
 Chance of both being living, and the Chance of 
 both being dead, go equally through the whole 
 Courfe of the Lives of both, or for 90 Years, 
 which is certainly a wrong Prefumption ; for m 
 any given Number of Years there is always a 
 Chance that both are dead, but not always of 
 for the whole Time a Chance that both are 
 living, becaufe one may be dead, and for great 
 Part of the Time the even Chance is that one is 
 dead, living the other. 
 
 For Inftancc, at the End of 50 Years there is 
 a Chance that A. and £. are both dead, and the 
 
 Chance
 
 [ 319 ] 
 
 Chance that both are dead is hot the lefs, be- 
 caufe one of them is computed to have been dead 
 fome Time ; but at the End of 50 Years there 
 is no Chance that A. and JB. are both living, for 
 the even Chance is that one of them has beea 
 dead thefe twenty Years ; and therefore the 
 Chance on the Lives of A. and B, at the End of 
 50 Years mull be a Chance different from and 
 lefs than a Chance that both are living, for the 
 Chance is that they are not both living : And 
 yet the Rule does plainly fuppofe that both arc 
 living, or poiTibly may be living \ for the Sum 
 436921 contains all the Chances through the 
 whole poflible Lives of both, and for that Year 
 gives us the Chances of Vitality anfwerable. And 
 if we carry this Matter the fartheft it can be 
 carried, and admit that there is a poffible Chance 
 of both being living at the End of thefe 50 
 Years; yet the even or probable Chance is that 
 one of them is dead, and we cannot put the 
 polfible Chance of both being living on the fame 
 Foot with the even or probable Chance that one 
 is dead, without fuppofing that a poflible Chance 
 is as ftrong and as good as an even or a probable 
 one. 
 
 The true State of the Chances here is this ; 
 the Chance of Mortality both of A and i?., 
 through all the Years of the poflible Life of A, 
 and B. on the one Side, and on the other Side, 
 the Chance of Vitality of A. and B, for fo many 
 Years as A* and JS. have a Chance both of them 
 to live, and after that, the Chance of Vitality 
 through all the remaining poflfible Years Qf the 
 
 Life
 
 [ 320 ] 
 
 Life either of A. or B, whichfoever fhall become 
 the Survivor. That this is a true Reprefenta- 
 tion of the Chances of Mortality, I fuppofe 
 will be admitted, becaufe 'tis agreeable to the 
 Diredion of the Rule ; and that thefe, and no 
 other or more than thefe, ought to be taken as 
 the Chances of Vitality, is evident from this Ar- 
 gument. No other or more Chances of Vitality 
 do or can belong to thofe or any other Lives, 
 than fuch as arife whilft thofe Lives are in Beings 
 or poflibly in Being, but if we take fuch and io 
 many as the Rule directs, or any more than I 
 have aflign'd, we neceffarily fuppofe that A, and 
 JB. are both in Being or poiTibly in Being, after 
 the Time when we have fuppofed one or other 
 of them to be dead. 'Tis indeed pollible that 
 ^. may extend his Life to 50 or to 90 Years, 
 and poflible that B* may extend his to the fame 
 Time, and therefore that we may fuppofe it 
 poflible that both may be in Being at the End of 
 50 Years : But this Suppofition, that 'tis pof- 
 iible that A, and B, are both living and in Being 
 at the End of po or of 50 Years, is at an End 
 and muft ceafe when the other Suppofition takes 
 Place, that one of the two is dead : Or elfe wc 
 Ihall make contradictory Suppolitions, that J, 
 furvives B. and B. likewife furvives A.^ and that 
 both are living when one is dead. 
 
 When we compute the Chancesof Vitality on 
 the Angle Life of A.^ we fuppofe it poflible that 
 he may live for po Years, or to be an 100 Years 
 old ; but when on a Computation made we have 
 ilated that the even Chance of the Duration of 
 
 his
 
 [ 321 ] 
 his Life is equal to 41 Years, wc have no farther 
 Regard to the firft poflible Chances on his Life : 
 So if we compute the Chances of Vitality on the 
 Lives of A' and B, and the Survivor, from thef 
 End of 30 Years, when J. is fuppofed to be; 
 dead, fuppoilng he dies firil, from thence we arc 
 to have no Consideration of the original polTible 
 Chances on his Life, for they are at an End • 
 and after they are at an End, I fee no Sort ci 
 Reafon why we fliould make any Account of 
 them in this Cafe, any more than wc do of the 
 pofhble Chances of Vitality on his Life remain- 
 ing after the End of 41 Years, where we calcu^ 
 lated the Chance of the Duration of his Life on 
 the Foot of a iingle Life, All that refts to be 
 done after this, is to compute the Chances of 
 Vitality on the Life of B. the Survivor as a jingle 
 Life, and this Computation muft be taken from' 
 the Time of his becoming the Survivor, and be 
 carried on through all the Years of his poffible 
 remaining Life. And that the Dodor's Hypo- 
 thefis and Rule for afcertaining the Chance of 
 the Duration of thefe two Lives and the Survi- 
 vor brings to the Account of the Chances of Vi- 
 tality on their Lives as well all the Chances pof- 
 fibly remaining to arife on the Life of J. after 
 the End of 30 Years, that is afrcr the fuppofed 
 Death of ^., as the Chances polTibly remaining 
 on the Life of B. fuppofed to furvivd J. and the 
 Term of 30 Years, is evident from hence; for 
 that in the firft general Account of the Chances 
 on both Lives are inferted all the original poiTible 
 Chances on both Lives fmgly taken. From 
 X whence
 
 [ 322 ] 
 
 whence I colledt and conclude ; fince In the 
 Number 436921 are included ail the Chances 
 which arife on the whole Life of J, and the 
 whole Life of 5., as well thofe which arife whilit 
 both are living, as thofe which may arife on the 
 Life of B. the Survivor, and thofe which are 
 computed to arife on the Life of J. after 30 
 Years, that is after he is dead, or fuppofed to be 
 dead • from hence I infer, that fuch and fo many 
 Chances are inferted in that Sum total more than 
 ought to be, as are computed to arife on the Life 
 of J.^ after he is fuppofed to be extind and 
 gone. 
 
 On this Conclufion naturally arifes this Obfer- 
 vation, that the Dod:or*s Rule for afcertaining 
 the Chance of the Duration of two Lives and 
 the Survivor, and Mr. Moivre's Method for cal- 
 culating the Value of an Annuity for two fuch 
 Lives, have one and the fame Defed:, grounded 
 on one and the fame wrong Suppolition, that 
 both the Nominees are living when one is dead, 
 or computed to be dead. The Rule for afcer- 
 taining the Chance of Vitality fuppofes, after 
 the End of 30 Years, and the Expiration of one 
 of the Lives, that the Chance of Vitality on 
 both Lives Hill remains ; for the original Com- 
 putation of thofe Chances is made on the Foot 
 that both of them have a Poifibility of Exiitence 
 for 90 Years ; and the Method of valuing an 
 Annuity for thofe Lives does the very fame 
 Thing. Agreeably to this, when we compute 
 the Value of an Annuity for thefe two Lives and 
 the Survivor, the Method is this. We firft take 
 
 the
 
 [ 323 ] 
 
 the Value of the Annuity for the Joint- Lives of 
 J. and 5., and afterwards fo much of the Value 
 of the Annuity on each of the Lives fingly com- 
 puted as remains after we have dedudled out of 
 each of them the Value of the Annuity for the 
 Joint-Lives : We put together thefe three Values, 
 the Value of the Joint-Lives once taken, and 
 fuch Part of the Value on each of the Lives fingly 
 computed as remains after fuch Deduction of 
 the Value o^ the Annuity for the Joint-Lives, 
 and the Sum total is the Value of this Annuity 
 for the two Lives and the Survivor. Juft in the 
 fame Manner, where we compute the Chances 
 of Vitality on thefe two Lives and the Survivor, 
 we carry to Account firft the Chances of Vita- 
 lity which have arifen during the Lives of both 
 of them, and afterwards when one is fuppofcd 
 to be dead, or ought to be fuppofed dead, wc 
 add to fuch Account fuch and fo many of the 
 original poflible Chances on both thefe Lives as 
 remain after deducing the Chances arifen during 
 the Joint-Lives ; or, which is the fame Thing, 
 fo many of thofe firll Chances as have not arifen 
 during the Joint-Lives, and are not already 
 brought to Account. 
 
 Since I have not feen Mr. Moivre*s Treatife oti 
 this Subjed, it may feem a Prcfumption in me to 
 furmife what it is he there demonftrates, or un- 
 dertakes to demonftrate ; but I apprehend he 
 only attempts to prove, and it fliould feem fuffi- 
 cient for his Purpofe to prove, that his Method 
 for calculating the Value of Annuities for two or 
 more Lives was confonant to Dr. Halley's Hy- 
 X 2 potheiis
 
 [ 324 ] 
 
 pothelis for afcertaining the Chances of the Du- 
 ration of fuch Lives ; and I very much queflion, 
 the Reafons of which we Ihall fee hereafter, 
 whether he has proved even fo much as this. 
 However, whether he has or has not demon- 
 flrated it, I think, from what I have ofFer'd, it 
 is evident and certain, that the Method of one 
 is, in fome Inftances at leaft, confonant to the 
 Kypothefjs of the other : The neccffary and un- 
 avoidable Confequence of which is; fince the 
 Hypothecs itfelf is fa!fe, that thofe very De- 
 monftrations which were brought to ftablifh the 
 Method do themfelves deflroy it. 
 
 I am very fenfible that Dr. Halky endeavours 
 to explain and fupport this Hypothefis by a 
 mathematical Scheme, as well as by Numbers 
 and Figures \ but his Attempt this Way is as 
 fruidefs and ineftedual as in the other, fince both 
 are grounded on wrong Suppolitions, and on 
 fomcthing which in Fad: is not true, nor can 
 poflibly ever happen ; which, perhaps, will be 
 fcen more plainly on a Scheme than in Numbers* 
 
 N^.I. 
 
 D. 
 
 B. 
 
 N^. II. 
 
 A. 
 
 S' 
 
 d. 
 
 h. e. h. 
 
 In the Scheme N^. I. we will fuppofe the Line 
 A. B. to reprefent the Chances of Vitality on 
 the Life of A. aged ten Years, and the Line 
 A, C. the Chances of Vitality on the Life of B» 
 
 of
 
 [ 325 ] 
 
 of the fame Age ; and the Line a. h, in the 
 Scheme N^. II. to reprefent the Chances of Mor- 
 tality on the Life of J.^ and the Line a. i. the 
 Chances of Mortality on the Life of B. If En- 
 quiry is made, what is the Chance of Vitality 
 for the firft Year of the Life of J. and of £., 
 each iingly and feparately taken ; if we fuppofe 
 the Lines a. b. and a. c. to reprefent the Chances 
 of Mortality for the fir ft Year of thefe two 
 Lives refpedtively \ the Anfwer will be, A. B. 
 minus a. h. is the Chance on the Life of A, and 
 A. C. minwi a. c. the Chance on the Life of B, 
 If we would be informed in what Time the even 
 Chance is that A. may die, we compute what are 
 the Number of Perfons of his Age dying, or 
 the Chances of Mortality ariling, in each Year 
 of his Life, beginning at the Age ten, and of 
 the Line a. h. we take in each Year a- Part in 
 Proportion to fuch Chances, 'till we come to 
 that Year in which the Line by fuch Takings 
 becomes (^i half the Length of the Line A. B. ; 
 that is when the Number of Perfons of his Age 
 living and the Number of thofe dead, that is 
 the Chances of Vitality and Mortality, are even ; 
 and that is the Year to which 'tis an even Chance 
 that A* will live, and not beyond it. 
 
 If we carry our Enquiries to the Lives of A. 
 and JB., and the Survivor of them, and would 
 know what are the Chances that both are, or 
 one of them is, living at the End of a given 
 Number of Years ; and what is the Year in 
 which the even Chance is that both are dead ; 
 we proceed thus, according to the Doctor's Rule. 
 X 3 The
 
 [ 326] 
 
 The Line A. B. reprefents the Chances of the 
 Life of J.y and A. C. reprefents the Chances of 
 the Life of B. : A. B. +- B. C. produees the 
 Square A. B. C. D. , and this Square reprefents 
 the Chances of Vitality on both their Lives. The 
 Lines a. h. and a. i. reprefent the Chances of 
 JVlortality on the Lives of A. and B. refpedtive- 
 ly ; and if we fuppofe, as we did before, that 
 the Lines a. h. and a, c. reprefent the Chances of 
 Mortality for the firfl Year of each of their Lives, 
 then a. h. -|- a. c. will give us the Square a. h c. d,\ 
 and that Square reprefents the Chances of Mor- 
 tality on both their Lives for fuch £ril Year, I 
 mean the Chances that both are dead : And conr 
 fequently A. B. C. D. — a. h, c. d. is the Chance 
 that both are, or one of them is, living at the 
 End of one Year. To fix the Year in which 
 ^tis an even Chance that both are dead, we com-r 
 pute what are the Chances of Mortality ariling 
 in each Year of the Life of J. beginning at the 
 Age of ten, and of the Line a. h.^ we take fucb 
 a Part as bears Proportion to the Chances of 
 Mortality arifing in each Year refpedively, and 
 the fime we do on the Life of B.^ and take a 
 like proportionable Part of the Line a. i. ; and 
 this we repeat 'till we come to the Year in which 
 the Parts of the two Lines fo taken, being mul- 
 tiplied one into the other, produce a Square 
 which is half the Magnitude of the Square A. B. 
 C. D., and that is the Year in which "tis an even 
 Chance that J. and B. are both dead. Now if 
 we fuppofe the Lines A. B. and A. C to be in 
 Length each of them 66 1 Inches, which 66 1 is 
 
 the
 
 [ 327 ] 
 
 the Number of Perfons living of the Age of A* 
 and 5., and of the Chances of Vitality on each 
 of their Lives ; 661+661 produces 436921, 
 and the Square of the two Lines will be A. B. 
 C. D. If we fuppofs the Line a. e. to be the 
 Parts of the Line a. h. which are taken (fuppofe 
 in ^^ Years) as the Line of Mortality on the 
 Life of J.^ and the Line a. f. to be the Parts of 
 the Line a. /'., which are taken as the Line of 
 Mortality on the Life of B. (fuppofe for the 
 fame Time,) and that thofe Lines are in Length 
 each 469 Inches ; 469 -I- 469 produces 21 9961, 
 and the Square of the two Lines will be a. e.f.g. 
 The Sum 219961 is a Moiety of 436921, and 
 fomething over, and the Square a. e. f, g. is half 
 the Magnitude of the Square A. B. C. D. and 
 fomething more : But 55 Years, or near it, mud 
 pa(s before we have fo many Chances of Morta- 
 lity arife on the Life of J.^ and fo many on the 
 Life of B.^ as (hall amount to 469 Chances on 
 each, and before we take fo many Part3 of the 
 Line a. h. and fo many Parts of the Line^. /. as 
 •Ihall be equal to 469 Inches on each : And there- 
 fore 55 Years muft pafs before we Ihall have the 
 Number 2 1996 1, and the Square a. e. f. ^; 
 therefore 55 Years mufl pafs before 'tis an even 
 Chance that J. and B. are both dead. 
 
 This is the Doctor's Reprefentation of this 
 Matter, as to the Senfe and Subliance oi it, tho' 
 pofTibly not in his Expreifions, his Treatife not 
 being now before me ^ but is framed, I fay, on 
 a Suppoiition which ought not, and cannot, in 
 this Cafe, be made. If we examine and Hate 
 X 4 what
 
 [ 328 ] 
 
 what thefe three Squares A. B. C. D., a, h. c. d.^ 
 and a. e. f. g,^ are compounded of, we Ihall fee 
 with Certainty what each of them contains. The 
 Square A. B. C. D. is made by multiplying the 
 Line A. B. by the Line A. C. ; the Line A. B. 
 contains all the poflible Chances of Vitality on 
 the whole Life of J.^ that is for 90 Years ^ and 
 the Line A. C. all the like Chances on the Life 
 of 5., and for the fame Term : Therefore the 
 Square A. B. G. D. contains all the poffible 
 Chances of Vitality on both their Lives for a 
 Term of po Years. The Square a. h».c, d. is 
 made by multiplying the Line a. b. by the Line 
 a. c.\ the Line a.b, contains the Chances of 
 Mortality arihng on the Life of" J. in the firfl: 
 Year, and the Line a. c. the like Chances on the 
 Life of B. and for the fame Time ; therefore 
 the Square a. h. c. d, contains the Chances of 
 Mortality on both their Lives for fueh firft Year. 
 The Square a, e, f. g. is made by multiplying 
 the Line a. e, by the Line a.f. ; the Line a. e, 
 contains the Chances of Mortality ariling on the 
 Life of A. in ^^ Years Time, and the Line a,f, 
 the like Chances on the Lite of B, for the fame 
 Time: Therclbre the Square a.e.f.g. contains 
 all the Chances of Mortality on both thefe Lives 
 for a Term of 55 Years. The Hypothelis then 
 aflerts, that as the Square A. B. G. D. is to a 
 irnaller Square, made by multiplying fuch Part 
 of the Line a. b. by fuch Part of the Line a. /. 
 as anfvvers to the Chances of Mortality arifing 
 in any given Year, fo are the Chances of Vitality 
 to the Chances of Mortality on thefe Lives for 
 
 that
 
 [ 329 ] 
 
 that Year ; or, which is the fame Thing, if wc 
 dedud in a-ny Year the fmaller Square belonging 
 to that Year out of the greater Square, that the 
 Square or Area fo left rcprefents the Chances of 
 Vitality on thefe Lives for that Year : For Tn- 
 ftance ; as A. B. C. D. is to a. e.f. ^•., the Square 
 anfwering to the 55th Year on thefc two Lives, 
 fo are the Chances of Vitality to the Chances of 
 Mortality on thofe Lives for that Year; or 
 A. B. C. D. — a. e. f. g* is the Chances of Vi- 
 tality for fuch 55th Year on thole two Lives • 
 
 And therefore A. B. C. D. a. e. f. g., or 
 
 the Area left, being in Magnitude rather lefs 
 than the Square a. e,f.g., 'tis lefs than an even 
 Chance that cither of them are living at the End 
 of ^^ Yearfi. 
 
 I affirm then, firll, that an Annuitant who has 
 an Intereft in the Lives of ^. B. and the Survivor, 
 has not a Chance to gain the Whole of the greater 
 Square A. B. C. D. in each Year for a Term of 
 S^ Years ; much lefs for a Term of 90 Years : 
 And, fccondly, that on the other Side he has a 
 Chance to lofe the fmalleft Square, t;/^. a. k c. d,y 
 in the firft Year of the Lives of A. and B.^ and 
 in every fubfequent Year a larger Square in Pro- 
 portion to the Chances of Mortality arifing in 
 each Year 'till the 55th Year ; and in that Year 
 he will have a Chance to lofe the Square a. e,f. g. : 
 Therefore a Deduction of the fmaller Square of 
 each Year out of the greater Square for the 
 Term of ^!y Years, will not leave to the Annui- 
 tant for each Year for ^^ Years fuch a Square as 
 hs has a Chance to j coafequently, a Dedudion 
 
 of
 
 [ 33^ ] 
 
 of the fmallcr Square a. e. f. g.y which is the 
 Square of Mortality for the 55th Year, out of 
 the greater Square A. B. C. D., will not leave to 
 the Annuitant fuch a Square or Area as he has a 
 Chance to for fuch 55th Year. 
 
 The Conclulion here is a neceflary Confequence 
 of the Prcmifcs ; the fecond Polition isfuppofed 
 in the Hypothefis, and if not fo, is apparent 
 from hence ; for that the Squares of Mortality 
 to be dedud:ed in each Year contain the Chances 
 of Mortality arifing in each Year on the two 
 Lives, neither more nor lefs ; and the firlt Poli- 
 tion I have already proved under the former 
 Head, and here fhall add only thus much far- 
 ther to confirm it. 
 
 If we admit, that for the firfl: and every other 
 Year of the Life of A. and B. for 30 Years, or 
 during the Joint- Lives of ^. and B., a Deduction 
 of the fmaller Square, the Square of Mortality, 
 out of the greater Square, may leave to the An- 
 nuitant a proper Square for fuch Year ; the Rea- 
 fon of that is, becaufe during thofe Years the 
 Annuitant may have a Chance to the greater 
 Square ; and this Chance he may have, becaufe 
 j4. and B. are fuppofed on an even Chance both 
 to live fo long ; but when that Reafon ceafes, 
 and the Annuitant has not a Chance that both 
 are living, which after the End of 30 Years he 
 has not, from that Time he has not a Chance to 
 the greater Square A. B. C. D. From that 
 Time, I fay, fuch Annuitant has not a Chance 
 that both are living, becaufe it contradidts the 
 former Suppolitionj that on the even and pro- 
 bable
 
 [ 331 ] 
 
 bable Chance one or other of them is dead : 
 And if wc fliould admit that there is a poffible 
 Chance, tho' not a probable one, that both may 
 live to be an lOO Years old, fuch AdmilTion muft 
 be grounded on this ; that becaufe when we con- 
 iidcr each Life linglv and fcparately wc fuppofe 
 it poffible that A. may live fo long, and poffible 
 that B. may live To long ; therelbre, when we 
 take the Lives of A. and 5. jointly we may fup- 
 pofe that both of them may poffibly live to that 
 Age. Now I will not fiy that *ris in Nature 
 ,abfolutely impoffible that both fhould live to fuch 
 an Age ; but if thia Way of Reafoning is right 
 in the Cafe of two Lives, by the fame Rule we 
 may proceed to three or any other Number of 
 Lives ifi hifuitiitn'^ for where the Poffibility ends, 
 or the Impoffibility is to begin, cannot be fix'd : 
 And if any one can make fuch extravagant Sup- 
 politions, and will make Account of Chances 
 arifing on fuch remote and imaginary Poflibilities, 
 he muft go on as far as he pleafes, but I delire 
 to leave him. In Truth, this Hypothefis for 
 calculating the Chance of Survivance on two or 
 more given Lives does really fuppofe that all 
 the Lives conjun(2:ly taken, whatever the Num- 
 ber of them may be, may poffibly live to the 
 Extremity of Age, that is to an lOO Years old; 
 and in the Cafe of three Lives does in Fadt take 
 it for granted that all of them poffibly may live 
 to that Age, and on that Foot makes a Compu- 
 tation of the Chance of Survivance of one ot 
 them: But there I can precifely and determi- 
 jiately fix the very^ Chances of Vitality which 
 
 on
 
 [ 332 ] 
 
 on this groundlefs Suppofition arc allotted, but 
 do not belong to thofe Lives, or any of them y 
 as ihall be made appear in its proper Place. 
 
 But I confefs I don't fee why any Regard 
 Ihould be had to the Squares of the one Sort or 
 the other, after the End of the 30 Years, when 
 one of the Lives is extindt, or fuppofed to be ex- 
 tind: ; on the contrary, I fee very good Reafon 
 why no Regard Ihould be had to either of them, 
 and fo will every Man who fees the abfurd Gon- 
 fequences of continuing them to any longer 
 Period. From that Time, one of the Lives, 
 fuppofe jl.y and one of the Lines, vfz, A. B. 
 is cut ihort; fo that there is now no Chance 
 left on the Life of A^ nor any Line A B, now 
 in Being, whereby to frame a Square ; and it is 
 therefore very abfurd to carry on the Calculations 
 toa Time, beyond which J is not living, and 
 the Line not exifting. On the other fide, when 
 A* is dead and gone, no Chances of Mortality 
 can in llri6tnefs arife on the Life of A, for none 
 can happen de futuro'^ and with Regard to the 
 Line a. h.^ the Line of Mortality on his Life, if 
 we take any Parts of it for the Chances of Mor- 
 tality, we mufl now make a Taking once for all 
 and take all, for A. by his Death has filled up 
 the Meafure of his Life and his Line. And if 
 after the fuppofed Death of ^,, we might con- 
 tinue our Computations on the fame Foot as it 
 A. were living as well as B., and with Regard to 
 the Line A. B. and A. C. might take the Square 
 of them as the Chances of Vitality on them both, 
 as ftill fubfifting, which is greatly abfurd j what 
 
 mull
 
 [ 333 ] 
 
 muft we doc with Regard to the Lines a. h. and 
 a. i. the Lines of Mortality ; muft we go on to 
 ' take a Part of each of them, and put fuch Part 
 on the fame Foot likewife, and continue the 
 Squares of them too ? mofl certainly we muft ; 
 for if we fuppofe J. and B. both to be living, 
 we do eo ipfo fuppofe neither of them to be dead; 
 and ytii though we do not fuppofe J. to be al- 
 ready dead, fure enough we may fuppofe he will 
 die, and the Chance of his dying may proceed 
 pari ^qfiiy with the Chance of B.'s dying; and 
 then, if we go on in every Year fubfequent to 
 the 30th, to take a Part of the Line a. h. in the 
 fame Proportion, as we do of the Line a. /, and 
 from thefe Parts fo taken, we form a Square 
 for the Chance of Mortality for that Year, and 
 if we carry on this Procefs till the Square fo to 
 be formed, is of half the Magnitude of the 
 greater Square, which will not be the Cafe till 
 1 ^ Years after the End of 30 Years, or till -^ 
 Years from the Beginning of our Computation: 
 If we proceed thus, and the Hypothecs does 
 manifeftly fo intend and fo pradlfe, we certainly 
 and evidently fuppofe fomething, which in Fa6t, 
 is not true, or can ever be fuppofed ; for then 
 we fuppofe that neither J. nor B. is dead till the 
 End of 55 Years, and eo ipfo we fuppofe that 
 both are living at and to the Time, to which we 
 have computed that one of them only is living. 
 And fince we have fuppofed J, and B. to be of 
 the fame Age, and polTibly born of a Day, let 
 us even fuppofe too, as they fay is fometimcs 
 
 the
 
 [ 334 ] 
 
 the Cafe of Twins, that they both die of a 
 Day. 
 
 It Confideration be had of this Square of 
 Vitality, that is the Square A. B. C. D, after 
 the Expiration of the 30 Years and the Death 
 of ^., fuppoling him to die firll, another abfurd 
 Confequcnce following from thence, and which 
 makes the llrongeft againfl it, is this; that the 
 Original poflible Chances on the Life of ^., will 
 be brought to Account twice, once during the 
 continuance of the faid Term, and again after it 
 is ended. That they are brought to Account 
 after the Term of 30 Years is ended, is obvious 
 from hence; for that in the Square A. B. C. D. 
 are contained, all the poffible Chances arifing on 
 the Life of ^. as well as of B.y not only for the 
 Term of 30, but of po Years. And that thefe 
 fame Original Chances which polTibly may arife 
 on the Life of J. after the End of 30 Years, 
 are all brought to Account during the Term of 
 the ^o Years, is proved from hence ; for that 
 in computing what is the probable Chance of the 
 Continuance of both Lives as Joint-Lives, we 
 have taken into Confideration all the pollible 
 Chances of both Lives through their whole 
 Lives, and for a Term of 90 Years, and confe- 
 quently all the pofTible Chances arifing after, as 
 well as thofe ariiing in the Term of 30 Years ; 
 and on comparing the Number of them with the 
 Number of Chances of jMortality, arifing in the 
 fame Time we have collected, that in 30 Years 
 Time the Number will be equal, and confe- 
 quently the even Chance is, that both of them 
 
 may
 
 [ 335 ] 
 
 may live for 30 Years, and one only beyond that 
 Time. As in afcertaining the Chance of the 
 Duration of the fmgle Life of J.^ we take all 
 the Chances through his whole Life, to the Ex- 
 tremity of old Age, in which are included all 
 the poflible Chances arifing after the End ((uppole 
 of 41 Years) as well as thofe which arifc during 
 that Time ; in the fame manner on the Joint- 
 Lives of J, and B.y and on comparing the 
 Chance when one or other will die, we take into 
 our xA.ccount all the Chances through their whole 
 polTible Lives, and to the Extremity of the Age 
 of both, and therein are included all the pot- 
 lible Chances ariling after the End (fuppofe of 
 30 Years) as well as thofe which have arifen 
 during that Time : And from thence, we coIle6t 
 and ftate the even or probable Chance on the 
 Life of ^., to be a Term of 41 Years, and on 
 j4. and B. as Joint-Lives, to be a Term of 30 
 Years : Therefore the Original poifible Chances 
 on the Life of J.y are here twice brought ta 
 Account. 
 
 From hence and upon the Whole : Since the 
 Square of A. B. C. D, contains all the poffible 
 Chances of Vitality through the whole Lives 
 both of J. and B. that is, for 90 Years ; and the 
 Hypothefis diredls us to take this Square, de- 
 ducing thereout the Square of the Chances of 
 Mortality, for a Term beyond "^o Years, to a 
 Term of ^^ Years, and in fo doing, fuppofes 
 this Square and confequently the poffible Chances 
 of both Lives, which . compofe fuch Square to 
 have an Exigence after the End of 30 Years 
 
 and
 
 [ 336] 
 
 and to §S Years, and affirms that Square, de- 
 ducting as aforcfaid, taken for a Term of 55 
 Years, is the Chance of Vitality for the Lives of 
 A' and B. and the Survivor ; And fmce, after 
 the End of 30 Years, when J. Is fuppofed to be 
 dead, this Square is determined or can be no 
 longer continued ; and fince all the poffible 
 Chances on the Life of J, through his whole 
 Life, are brought to Account during the Term 
 of 30 Years: From hence I infer that the Rule, 
 in computing the Chances of Vitality for thefe 
 two Lives and the Survivor, brings to the Ac- 
 count fo many more Chances than belong to thofe 
 Lives, as are fuppofed, after the Expiration of 
 30 Years, and one of the Lives, to arife on that 
 cne Life fo determined. 
 
 As this is the Cafe on Dr. Halkys Hypothefis, 
 for afcertaining the Chances of the Duration of 
 two Lives and the Survivor, and on Mr. Moivre*s 
 Method for adjufting the Value of an Annuity 
 for two fuch Lives, which is founded upon, 
 and is purfuant to the Hypothefis ; the fame we 
 Ihall find it to be, where the Rule is applied to 
 Particulars, as in Mr. Kichards's Tables, and 
 that they all agree in one and the fame Defe(5t. 
 
 That Defed which I mean here, and which I 
 fay is agreeable to, and proceeds from, the Rule,, 
 is this ; that thefe Tables make the Life of the 
 Survivor of two Lives, after he becomes the 
 Survivor, to be equal to a longer Term than 
 they make the fame Life to be equal to, when 
 the Intereft on the two Lives firft commenced, 
 and even after fuch Survivor is advanced in Age^ 
 
 fome
 
 [ 337 ] 
 
 fome feventeen or eishceen Years. We will here 
 make Ufe of the fame Lives o{ J. and B. as 
 before, and inltead of 4/. per Cem. which we 
 ufed before, here we will compute after the 
 Rate of 8 /. per Cent. ; becaufe the Dcfed: is mofl 
 vifible and apparent at that Rate of Intereft, 
 though at all Rates it continues in fome Meafure. 
 The Value of an Annuity for thofc two Liv^cs and 
 the Survivor, is 12-02, which is equal to a Term 
 of 43-3-00 ; the Value of an Annuity for thofc 
 two Lives as Joint-Lives, is 9-32, which is 
 equal to a Term of 17-3-00; therefore to make 
 the Term in the two Lives and the Survivor to 
 be equal to 43-3-00 in the whole, we muft com- 
 pute the Life of the Survivor to be equal to a 
 Term of 26-0-00 ; for no lefs Term will make 
 17-3-00, the Term for the Joint-Lives, to be 
 in the whole 43-3-00. But the Value of an An" 
 nuity for the Life either of ^. or 8. fingly taken, 
 and taken when they are 11 Years old, is no 
 more than 10-67, ^vhich is equal only to a Term 
 of 25 Years ; and the Survivor, at the Time he 
 becomes fuch, is advanced in Age near 18 Years, 
 and is near 30 Years old ; and therefore his Life 
 fhould then be equal to a Term Ihorter, and it 
 is impolfible it fliould be equal to a Term longer, 
 than it was at firft. This Error is fo apparent 
 and fo grofs, thut nothing to be fare can defend ; 
 and iftheTable in this Inltance is rightly conftrufl- 
 ed according to the Rule, as upon Experiment I 
 find i t is, and if thisValue is the necefFary Produce of 
 theRulcasmoil:ccrtainlyitis,theRu1e,hasthef^me 
 Fault, and is as indcfenlible as the Produce of it- 
 
 Y If
 
 C 338 ] 
 
 If vvc carry our Inquiries a little farther, and 
 examine how it comes to pafs, that the Term in 
 the Annuity for the Joint-Lives is fo Ihort, and 
 the Term in the Annuity for two Lives and 
 the Survivor is fo very long, we fhall difcover 
 what makes this Variance, and that the Defedt 
 in the Rule is the fame, and no other than that 
 which I have afiigned. I have before taken 
 Notice, that if in our Computation we make 
 Ufe of 4/. per Cent, as our Intereft, then the 
 Term in the Annuity for the two Joint-Lives is 
 equal to J 8-3-00, and if 8/. per Cent., then the 
 Term will be equal tc 17-3-00, according to 
 Mr. Richards^s Way of Hating the Value of fuch 
 Annuities. And I have aflerted, and it will be 
 •admitted, that neither 17-3-00, nor 18-3-00, 
 iire the Terms of the Duration of thefe two 
 Joint-Lives; but that the even Chance is, where 
 we compute by the BreJIaw Table, which makes 
 each of the Lives ilngly taken equal to a Term 
 of 41 Years, that then both Lives will continue 
 in Being for a Term of 30 Years or thereabouts: 
 The Chance then of the Duration of both thefe 
 Lives, being equal to a Term of 30 Years ; if 
 we inquire why the Chance of the Duration of 
 the Annuity for thofe Joint-Lives, is a Term 
 fo much Ihorter as to be equal only to 1 8-3-00 
 at one Rate of Litereft, and at another Rate, 
 no more than 17-3-00; the Reafon is obvious. 
 To compute the Value o^ Annuities either on a 
 single 'or on tvvo or more Joint-Lives, the Rule 
 direds that in each Year of Life, the Value cor- 
 rcfponding to the Chances of Mortality arifmg 
 
 in
 
 [ 339 ] 
 
 in fuch Year, be deduced out of the Value of 
 the Chances of Vitality, and fo for every Year 
 of poffible Life ; and by that Mean the Term to 
 which the Duration of the Life is equal, and the 
 Term to which the Value of the Annuity is 
 equal, cannot co-in«dc ; but the Term in the An- 
 nuity is Ihorter than the Term in the Lite. I 
 ihall not here difpute the Juftice of the Rule, 
 which makes the Term to which a given Life or 
 Lives is equal to be one, and the Term to which 
 the Value of an Annuity which depends on thofe 
 Lives is equal to be another, but Juft or not Juft, 
 fo the Fad is. Accordingly, in Dr. Halle/s 
 -Table for the Value of Annuities on a finglc 
 Life, we fee the Value of an Annuity on the 
 Life of a Perfon o£ ten Years of Age, is equal 
 to a Term only of 28 Years and a little over; 
 and yet in his Way of computing the Chance ot 
 Vitality of fuch a Perfon, by the Bre/IazvTdbky 
 and the Way is undoubtedly a right one, the 
 even Chance is, that he lives for 4X Years and 
 above. The fame, as we have feen, is the 
 Cafe on Mr. Richards's Table for the Value of 
 Annuities on a fingle Life, and at all Rates of 
 Intercil ; and fo it is on his Tables for Annui- 
 ties on Joint-Lives. 
 
 The Difficulty then which arifcs here, ani 
 wants to be accounted for, is this ; fince in An- 
 nuities for a fingle Life and two Joint- Lives, 
 the Term in the Annuitiy is much fhorter than 
 the Term in the Life or Lives, what it is that 
 gives occalion that in Annuities for two Lives 
 and the Survivor, the Term in the Annuities i?? 
 Y 1 in
 
 [ 3+0 ] 
 
 in fome Tnftanccs longer than the Term in the 
 Lives ; at leaft in the Cafe of the Life furviving, 
 the Annuity; for that Life is computed to laft 
 longer than the Life on which it depends. The 
 Account that mult be given of this Matter is, 
 lor fmce the Operator has performed his Part in 
 Mode and Figure, no other Account can be 
 given but this ; either that the Hypothefis in 
 Calcuhiting the Chances of Vitality on tv^o Lives 
 and the Survivor, or the Method in fettling the 
 Value of Annuities for thofe two Lives, have 
 Jnferted more Chances of Vitality, or more or 
 greater Values than do belong to fuch Lives; or 
 rather, that both of them are refpedively guilty 
 of the Error. 
 
 To fettle the Value of an Annuity for the 
 Life of X and B. and the Survivor, Mr. Moivre 
 direds us to take the Value of the Annuity for 
 their Joint-Lives once, and afterwards and bc- 
 iides, that fo much of the Value of the Annuity 
 for the Life of ^. fmgly taken, and wholly taken, 
 as remains after dedudin^ thereout the Value of 
 the Joint-Lives, and to doe the fame on the 
 Life of B., and to Hate all thofe three Values 
 put together, as the Value of the Annuity for , 
 thofe two Lives and the Survivor: And do not 
 Nve, in this, fuppofe that the Annuity continues 
 on the Liie of J.^ and continues likewife on the 
 Life of i?., after the Time when the Interell in 
 the Joint-Lives is determined, and when one of 
 the Lives on which the continuing Annuity de- 
 pends is extinft ? To compute the Chance of 
 the Duration of thcfc two Lives and the Survi- 
 vor,
 
 [ 341 ] 
 
 vor, Dr. Halley directs us to take all the Chances 
 of Vitality on the Life of X for his whole poifi- 
 blc Life, that is for 90 Years, and the fame on 
 the Life of B., which are found by Multiplying 
 the one by the other, to dedu6l thereout the 
 Chances of iMortality, which arife on the Life 
 both of ^L and of E., for a certain limited 
 Time ; and to look upon the Chances remaining, 
 as the Chances of the Duration of thofe Lives, 
 or the Survivor of them, for fuch limited Time: 
 And do not we here, in the firft Inftant, and in 
 every Year, through which we carry our Com- 
 putation, neceffarily fuppofe that A. and B., are 
 both then living or poffibly living; for the 
 original Sum out of which we make our Deduc- 
 tion, is the fame through all the Years of the 
 Computation; and when we compute the Chance 
 of the Duration of thofe two Lives and the Sur- 
 vivor, do not we continue this Computation for 
 ^^ Years, when the even Chance is, that one or 
 other of them is dead in 30 Years, and his 
 Chances all extindt ? mod certainly, and mod 
 evidently this is the Cafe, both on the Hypothe- 
 cs and on the Method, and there I reil the 
 Matter, leaving it to the Authors to make them 
 good. 
 
 Upon looking into Dr. Hallsys Scheme for 
 the Chance of Survivance of two or more Lives, 
 I find I have not reprefented it in the fime 
 Manner and Expreffions, which he has done; 
 but I fee no Reafon to retradl what I have ad- 
 vanced, being almoll: confident that the DoBoYy 
 on a- Revifal of his own Reprefentation, will 
 
 Y 3 admit
 
 [ 342 ] 
 
 admit that I have rcprefented the Matter, in the 
 Manner in which he Ihould have done, though 
 DOt in the Manner in which he has done. In the 
 Cafe ftated by me, which is on two Lives of 
 equal Age, the Number of Chances of one Life, 
 multiplied by the Number of Chances on the 
 other Life, produces a Square Number; and 
 a Line reprcfcnting the Number of Chances 
 on one Life, miiltiplied by a Line reprefenting 
 the Number of Chances on the other Life, pror 
 duces a Square: And the Number produced by 
 fuch Multiplication of the Figures, is the Num-r 
 her of Chances on both Lives, and the Square 
 produced by the Multiplication of the two 
 Lines, is a Square anfwering to the Chances on 
 both Lives. Upon the Cafe ftated by the 
 Doclor, which is on two Lives of unequal Age, 
 the Number of Chances on one Life, multiplied 
 by the Number of Chances on the other Life, 
 produces the Number 298900, which may be 
 called a Parallelogram Number ; and a Line re- 
 prefenting the Number of Chances on one Life, 
 multiplied by a Line reprefenting the Number 
 of Chances on the other Life, produces an Area, 
 which is a Parallellogram. 
 
 To make my Reafoning on the Dolors 
 Scheme more intelligible, I have here inferred it. 
 P. E. C. 
 
 I 
 
 a 
 
 F. 
 
 B. 
 
 H. A. 
 
 The
 
 [ 343 ] 
 
 The Do^or in the fubfequent Part of his Re- 
 prefentacion, fubllitutes Nn in the Room of, and 
 as equal to the Number 298900; and fuppofes 
 the Line A B to reprefcnt the Number of 
 Chances on one Life, and the Line A. C. to re- 
 prefcnt the Number of Chances on the other 
 Life, and that thofe two Lines multiplied one 
 into the other, will produce the Parallelogram 
 A. B. C. D, and that A. B. C. D. will be N n, 
 or the Product of the two Numbers reprefenting 
 the Number of Perfons of the two Ages given. 
 Agreably to this ; the Inquiry he makes being, 
 what are the Chances that both are dead in 8 
 Years, he multiplies the Number oi Perfons of 
 one Age, dead in eight Years Time, by the 
 Number of Perfons of the other Age, dead in 
 the fame Time, and the Produce is 3650; and 
 here, as before, in the Room of the Number 
 ^6^0^ he fubftitutes Y y as equal to that Num- 
 ber. He goes en and fuppofes the Line C. E. 
 to reprefent the Number of Perfons of one Age, 
 dead in eight Years, and the Line C. F. to re- 
 prefent the Number of Perfons of the other Age, 
 dead in the fame Time : And I add, though he 
 has omitted it, that thefe two Lines multiplied 
 one into the other, produce the Area C. E, F.G, 
 Thus far the DocJor's Reprefentation is undoubt- 
 edly right; but the Part which follows, is un- 
 doubtedly wrong, and wrong for that very Rea- 
 fon, becaufe the firft Part is right, and it is im* 
 poffible both fhould be right. Where he fays, 
 that the Redangle F. E. is the Produdl of the 
 deceafedj or Yy an equal Number of both dead; 
 Y 4 hQ
 
 [ 344 ] 
 
 he is miftakcn : For Y y is not the fame or equal 
 to the Redangle F. E„ but to the whole fmaller 
 Area C. E. F. G.; for Yy reprefents the Num- 
 ber 3650, and that Number is made up by 
 multiplying the Number of Perfons of one Age 
 dead, by the Number of Perfons dead of the 
 other Age, in eight Years ; whereas F. E. is no 
 more than a meer lingle Line pur into fuch a 
 Poiition, as to form a Red:anglc, and can repre- 
 fent only the Number of Perfons of one Age 
 dead in eight Years, added to the Number of 
 Perfons of the other Age, dead in the fame 
 Time. And whereas he fays, that as the whole 
 Redangle A U or N n is to the Gnomon F A 
 B D E G or N n — Y y? fo is the whole Num- 
 ber of Perfons or Chances, to the Number of 
 Chances, that one of the two Perfons is living ; 
 this again is certainly wrong : For N n is not 
 the fame, or equal to the Redangle A D, but 
 to the whole Parallelogram A B C D, and he 
 himfelf fo ftates it ; and, as before obferved, 
 Y y is not the fame or equal to the Re6tangle 
 E F, but to the whole Area C E F G. But the 
 worit Part of the Reprcfcntation is in this Part, 
 viz. that the Gnomon F A B D E G is intend- 
 ed to be, and manifeftly is an Area, and the 
 Redangle A D is no other than a meer Line, or 
 two Lines joined, formed in fuch a Manner as 
 to create a right Angle; and we can make no 
 Comparifon, or frame an Idea of any Proportion 
 between a meer Line and an Area, whether a 
 Square or a Parallelogram. In the next Sentence 
 he repeats the fame Millake, when he fays j as 
 
 the
 
 [ 345 ] 
 
 the Redangle A D or N n is to (the Reftangle) 
 F E or Y y, Co arc all the Chances, to the 
 Chances that both are dead. 
 
 Mow, I apprehend, that in all thefe Places^ ,^ 
 and in fome others in the fame Paragraph, wc 
 fhould read recSangular Parallelograms, inftead .^ 
 of Redtanglcs ; and if we read fo, then our Re- . 
 prefentations of this Matter will agree, faving ^ 
 only, that his Inftance is of Lives of unequal 
 Age, and the Lines multiplied muft produce Pa- 
 rallelograms, and mine is of Lives of equal Age, and 
 the Lines mull produce Squares. And that we ' 
 ought fo to read, is paft all doubt, by his Reprefen- 
 tation ot the Cafe on three Lives. There he makes 
 all his Products Parallelopipedes, as well the feveral 
 Chances of Survivorfhip, as the Chances offome be- 
 ing living, and all being dead : Now we cannot by a 
 lingle Multiplication, of the Lines of a Re(5tangle, 
 immediately form any Solid, but of necellity we 
 muft make an intermediate Operation, and firft by 
 a Multiplication form an Area, and a Multiplica-* 
 tion afterwards, will in this Cafe produce a Paral- 
 lelopipede ^ but here, unlcfs the intermediate Chan- 
 ces on two Lives are fuppofed to be Area's, we 
 form Solids from meer redangular Lines. For 
 this Reafon on the Cafe of two Lives, and be- 
 caufe the whole Number of Chances is in the 
 firft Inftant call'd, and rightly call'd, a Paralle- 
 logram, I conclude that the feveral Chances on 
 thofe Lives, ihould all have been called rectangu- 
 lar Parallelograms, and not merely Rectangles. 
 
 However, be the Reprefentation right or 
 wrong, I have aflerted, and do infill on it, that 
 this Rule, whether it be confider'd in the Shape 
 
 of
 
 [ 346] 
 
 of Numbers or of Lines, does not give us the 
 true Chances of the Duration of two Lives and 
 the Survivor, for that the Proportion is not as 
 the whole Number of Chances is to the Chances 
 of both being dead, or as the Area of the whole 
 Square or of the whole Parallelogram is to the 
 Gnomon of the Square or of the Parallelogram* 
 or as the greater Square or as the greater Pa^ 
 rallclogram is in Magnitude when the lefs Square 
 or lefs Parallelogram is dedudted thereout. I 
 have already given my Reafons for this Allerti- 
 on, and fl^all not repeat them, but crave Leave 
 to add here a Proof fomething of a different 
 Kind, to confirm it. 
 
 I obferve, that the Method here prefcribed by 
 the Do(5tor for computing the Chance of the 
 Duration of two or more Lives differs from that 
 made Ufe of on the Bre/Iaw Table ; for I have 
 made a Calculation of the Chance of the Dura- 
 tion of the Lives of two, of three, and of four 
 Perfons, all fuppofed to be of the Age of ten 
 Years, to difcover to what Age the even Chance 
 is that one of the two, one of the three, and 
 one of the four, refpedlively may live, or with- 
 in what Time the even Chance is that all will be 
 dead. On fuch Computation it appears that of 
 the two Perfons one or other of them has a 
 Chance to live for ^^ Years or near it ; one or 
 other of the three for 6l Years or near it ; and 
 one or other of the four for 6^ Years ; that is, 
 one or other of them may live to be of the re-? 
 fpeClive Age of 65, 71, and y^. But if we make 
 Ufe of the Bre/Iaw Table as a Rule to adjult 
 
 thefe
 
 t 347 ] 
 
 thcfe Chances on thcfe feveral Lives, the even 
 Chance will be that one or ocher of the two may 
 live for 52 Years, one or other of the three for 
 58 Years, and one or other of the four for 61 
 Years; that is, one or other of them may live 
 to the refpe6tive Ages of 62, 68, and 71. The 
 necefTary and obvious Inference from hence is, 
 that the one or the other of thefe Methods foT 
 adjufting fuch Chances cannot be right. 
 
 If we compute, by the fame Bre/law Table, 
 how many Perfons of the A ge of ten Years live 
 to 6^^ yi^ and J^^ which arc the feveral Ages 
 to which the Survivor of thefe Perfons refpec- 
 tively may live according to the Doctor's Me- 
 thod of calculating the Chances of Survivance, 
 which we are now confidering, it appears there 
 are ip2 live to 6^^ 132 live to 71, and 88 to 
 y^. The Number of Perfons living of the Age 
 of ten being 661 ; if we divide that Number 
 feverally by 192, by 132, and by 88, the feve- 
 ral Quotients will be 3 i, 5, and 7 7; that is, 
 of 661 Perfons of the Age of ten Years one 
 only in 3 i lives i^ 6^^ one in 5 lives to 71, and 
 one in 7 i to 75 : Therefore 2 ^ in 3 7 die in 
 ^^ Years, 4 in 5 in 61 Years, and 6 i in 7 3 ia 
 6^ Years. 
 
 My Way of reafoning and computing on the 
 Bi-eJIaw Table is this. Two Perfons in three of 
 a given Age die within fuch a Time, three Per- 
 fons in four within fuch a Time, ^() in an 100 
 within fuch a Time ; or, on the Reverfe, one 
 Perfon only in three of a given Age lives to fuch 
 a Time, one only in four to fuch a Time, and 
 
 one
 
 [ 348 ] 
 
 one only in an ico to fuch a Time ; thercfare 
 if a Perfon would have an Annuity on Lives 
 adequate to fuch Terms refpedively, he muft 
 have fuch and fuch and fo many Lives. And 
 this is the very Method which Dr. Halley pre- 
 fcribes, for adjuiling the Chance of the Duration 
 of a lingle Life, when the Bre/Iaw Table is 
 made Ufe of; ^viz. one Perlon in two of a given 
 Age, dies within fuch a Time, or, on the Re- 
 verfe, one only in two lives to fuch a Time ; or 
 of a greater Number of Perfons, one Moiety 
 lives to fuch a Time, and the other Moiety dies 
 within the Time ; therefore the even Chance is, 
 that a lingle Perfon of the given Age, may live 
 to fuch Time ; and not beyond that Time. 
 
 If we apply this to the prefent fuppofed Per- 
 fons of ten Years Age ; two in three die in J2 
 Years, three in four die in 58 Years, ami four 
 in five in 61 Years; and therefore, if a Perfon 
 would have an Annuity on Lives adequate to 
 fuch Terms ; for the one he muil have two Lives, 
 for the other three Lives, and for the third four 
 Lives, all of ten Years Age : But if he expeds 
 an Annuity for the feveral Terms of 55, 61, and 
 6$ Years, which are the Terms refpedively, to 
 which two Lives, three Lives, and four Lives, 
 are equal , in the Dodtor's Method of 
 calculating the Chance of the Duration of 
 two or more Lives, now under Confideration ; 
 in the one Cafe, he muft have two Lives and an 
 half, in the fecond four Lives, and in the laft lix 
 Lives and a half, otherwife he will not have an 
 even Chance, that his Annuities will continue 
 
 for
 
 [ 349 ] 
 
 for the fcveral Terms intended ; for that any 
 Number of Lives lefs than thefe, will not leave 
 the Annuitant an even Chance, that any one of 
 his Nominees Ihall furvive, to the End of the re- 
 fpe6tive Terms. 
 
 That this Way of computing the Chances of 
 Survivance on the BrefJaiv Table, is a right one, 
 fo far as the Table itfelf is a juft one, I make no 
 doubt: But I think the Computation here made, 
 is a ftrong Indication that this Table is not a 
 juft one, or at leaft that it will not fuit any 
 other Place, if polTibly it may that City. It is 
 very improbable, and I believe contrary to all 
 Experience elfewhere, that of the Perfons who 
 live to ten Years of Age, one in three lives to 
 the Age of 62, one in four to the Age of 68, 
 and one in five to the Age of 71 ; or, which is 
 pretty much the fame Thing, that of the Perfons 
 born aliv^e, one in lix arrives to the Age of 62, 
 one in eight to 68, and one in ten to 71 : And 
 more nearly, onein 5 7, one in 7 i, and one in y 
 ~ to thofe Ages refpedtively. I cannot to be furc 
 affirm, that this is never the Cafe in any indivi- 
 dual Place, in this or any other Kingdom ; but 
 whether it be fo or no, I appeal and muft leave it 
 to every Man's own Obfervation, and any Man 
 may make the Obfervation, and with great eafe. 
 If it be true that one Perfon (fuppofe in 10) in 
 any given Place, lives to be of the Age ot 71, 
 it is grounded upon this, or this muft be the 
 Confequence, that one Perfon in 10 ot that 
 Place, at any given Time in general, is of the 
 Age of 71 or upwards; and this, is eafily known 
 
 and
 
 C 350 ] 
 
 and computed in Places not over great. If In- 
 quiry were made for this Purpofe, 1 believe we 
 Ihould more readily find one or more Places 
 where there are not one in 50 living of that Age, 
 than we ihould find one Place where there was 
 one in 10 living. And yet, as if this Compu- 
 tation were not fufficicntly extravagant, the 
 Doctor's Method for calculating thole Chances, 
 gives us the Ages of 6^, 71, and 75, as the 
 Ages to which one in three, one in four, and 
 one in five, does refpedively live. And fince I 
 cannot, in my Undcrftanding, reconcile and 
 make confiftenc, thefe different Produces of the 
 Chances of the Duration, of two or more Lives, 
 which arife on thefe two Methods of computing; 
 otherwife than by fuppofing, as I before con- 
 tended, that in the Dodor's Hypothefis, more 
 Chances of Vitality, or fewer Chances of Mor- 
 tality than belong to thofe Lives, are fomewherc 
 infertcd ; I think the great excefs in the Com- 
 putation in the Hypothefis, muft be accounted 
 for in that Way, and can be accounted for no 
 other Way. 
 
 After all, it may very well be made a Queftion, 
 whether the original Pofitions about the Chances 
 of Vitality and Mortality, and which are the 
 Grounds 01 the fubfequcnt ones, are themfelves 
 right or no : I mean the Pofitions, that the 
 Number of Perfons living of one Age, multi- 
 plied by the Number of Perfons living of the 
 fame, or of another Age, docs give us the 
 Chances of the Lives of two fiich Perfons; and 
 thac the Line A B reprefenting the Numbrr of 
 
 Chances
 
 [ 351 ] 
 
 Chances on one Life, and the Line A C repro- 
 fenting the Number of Chances on a fccond Life, 
 that if thefe Lines arc multiplied one into the 
 other, which will produce the Parallelogram 
 A B C D, that fuch Parallelogram will reprefent 
 the whole Number of Chances on both thofe 
 Lives. The Cafe is the very fame, with Re- 
 gard to the Chances of Mortality, fo I fhall not 
 repeat them. Thefe Portions which in Effe6t 
 and in Senfe, arc one and the fame, only one is 
 exprefi'd in Numbers, and the other by Lines, 
 the Do6tor lays down without Proof, at leall I 
 fee none that he offers to that Purpofe ; and an 
 Affertion in the Negative, is jull as good as one 
 in the Affirmative, unlefs it be fupported by 
 Argument ; which in my humble Opinion, is 
 wanting in this Cafe. Sure I am, if the Doctor's 
 Reprefentation of this Matter, in the latter 
 Part of it, in that Part I mean, with which I 
 before found Fault, be true and a right one ; it 
 can be fuch on that Foot folely, and upon a Sup- 
 pofition, that the Redlangle A D, which is a 
 meer Line made by adding two Lines together, 
 and not the Parallelogram, which is formed by a 
 Multiplication of the two Lines, does truly 
 reprefent all the Chances on both thofe Lives. 
 
 However, as I have fuppos'd that this Part 
 of the Reprefentation is erroneous and ought 
 to be re6tified, I do not infift on that : But fay, 
 that there is nothing in the Life of ^., or of jB., 
 or in any other Number of Lives whatever, or 
 In the Line A B, as rcprefenting the Life of J.y 
 or in the Line A C, as rep rcfenting the Life 
 
 of
 
 [ 352 ] 
 
 of 5., which can be a fufficient Ground for fup- 
 poling thofe Lines to be, or forming them into, 
 a Plane; whether Square, Parallelogram, or any 
 Area whatever. The Line A B is fuppofed to 
 be, and is a meer ttraight Line, and the Line A G 
 h fuppofed to be fo likewife ; and it being afcer- 
 tained, as far as the even Chance goes, that the 
 Line AB, when conlidcred by itfclf, will be ex- 
 tended to fuch a Length, and the Line A C to 
 luch a Length, and there being a Chance that 
 one or other of the two Lines, when conlidered 
 as combincd,"will be extended to a greater Length 
 than any one of the Lines fingly taken j the In- 
 quiry is folely what fuch greater Length is. To 
 difcover this, we are direded to form thefe two 
 Lines into Squares, or Parallelograms, as the 
 Cafe happens, to be of two equal or unequal 
 Lives and Lines; and^from thence 'tis fuggefted, 
 that we may collect, what will be the Length of 
 the Life and the Line of the longer liver. Now 
 here we form theie Lines into Squares and Pa- 
 rallelograms, which yet we cannot do, without 
 fuppofing them to be fuch Things as they are 
 not, nay fuch Things as we are not concerned 
 in, for our Inquiry is upon meer Lines ; and to 
 what Purpofe do we this ? When we have in- 
 troduced thefe Squares and Parallelograms, we 
 can frame no Proportion between the Length ot 
 a Line, and the Magnitude of a Plane of any 
 Sort or Kind, and if we could conceive any fuch, 
 thefe Squares and Parallelograms are made to re- 
 prefent fomething, to which they have no Sort ot 
 Refemblance, for the Lives of two Perfons can 
 
 have
 
 [ 353 } 
 
 have no Rcfemblance to an Area of any Denc= 
 mination whatever. The Length of the Lrfe 
 of J. and of 5., are meerly the Space of Time^ 
 through which the one or the other may live • 
 and if fuch fpace of Time may be reprefented 
 by two Lines, as well enough it may, yet we 
 cannot form thefe Lines into Areas, without 
 fuppofing the Lines and the Lives to be fome- 
 thing which they are nor, or ever can bcj and 
 that they are to be Meafurcd in Breadth as well 
 as Length. And by the Way, I wiih that this 
 were the only Inftance, in which Mathematicians 
 form Lines into Areas, without fufficient Reafon, 
 or any Foundation in the Nature of the Thing,- 
 for fo doing. 
 
 If thefe fame Lines do not reprefcnt the Lives 
 of thefe Perfons, but the Number of Chances 
 belonging to their refpeclive Lives, yet the 
 Difficulty is not removed, only fliifted off from 
 one Pofition to the other; for it wants to be 
 proved and ftill remains to be proved, that the 
 Chances of one Life, multiplied by the Chances 
 of the other Life, give us the Chances on both 
 Lives. This I fay ought to be proved, for it 
 is more natural, and in my Judgment more ra-* 
 tional to fay; if the Number of Perfons living 
 of one Age, are added to the Number of Per- 
 fons living of the other Age, that the Sum pro- 
 duced, is the Number af Chances on both 
 Lives ; than it is to fxy, if the Number of Per- 
 fons living of one Age, are multiplied by the 
 Number of Perfons livins; of the other A^e 
 that the Sum produced, is the Number of 
 
 7u Chances
 
 , [ 354 ] 
 
 Chances on both Lives. The Chances on the 
 Life of one Perfon named, are the Perfons 
 living of the fame Age as the Perfon named 5 
 and if the Perfon named, fuppofe A.^ as in the 
 Do6tor*s Inftance, be of the Age of 18, the 
 Number of Perfons living, and confequently the 
 Chances on his Life will be 6jo; to this if we 
 add the Life of a fecond Perfon, fuppofe J3., of 
 the Age of 3 ^, the Number of Perfons living of 
 that Age are 490, and thofe are the Chances on 
 the fecond Life. Admitting then, as we do 
 admit, that 610 are the Chances on the Life of 
 ^., and 490 the Chances on the Life of J5. ; if 
 thefe two Sums are added together, the Produce 
 will be 1 100, and if they are multiplied one in- 
 to another, 610 + 49° is 298900 ; and the 
 Confequence is as eafy and as juft, therefore 
 1 1 GO are the Chances on both Lives, as is the 
 other Confequence; therefore 298900 are the 
 Chances on both Lives : Nay the Advantage is 
 manifeflly on the fide of the former, becaufe 
 1100 is the real Number of the Perfons living 
 of both Ages, and 298900 is not fo, and the 
 Number of Perfons living, is admitted to be the 
 Meafure of the Chances of Vitality on each of 
 their Lives, when fingly taken. 
 
 And as I fee no juft grounds in the Nature of 
 the Life of Man, to multiply thefe Numbers or 
 to form thefe Lines into Areas, fo neither do I 
 fee any occafion for making fuch Multiplications; 
 for if thefe Numbers and thefe Lines added to- 
 gether only, will afcertain what is intended to 
 be done by the Numbers multiplied, or by the 
 
 fuppofed
 
 [ 355 ] 
 
 fuppofed Areas ; that is, will difcover what arc 
 the Chances of fiirvivorfliip on thele two Lives ; 
 thefe Transformations are unneceflary. But if 
 ihefc Numbers, which are the redl Numbers of 
 the Chances on thefe Lives, and thefe Lines which 
 truly reprcfent thefe Lives or their Chances, will 
 not make fuch difcovery ; it is very unlikely that 
 thefe multiplied Numbers, which are not the 
 real Number of Chahces, or that thefe Areas,^ 
 which do in no fort refemble thofc Lives, fhould 
 give us any Affiltance towards fuch difcovery : 
 And the Thing perhaps, is not difcoverable by 
 either of tfie Ways. 
 
 ,_ To purfue this Matter sl little farther in the 
 Dod:or's Inftance and in his Manner ; the Lives 
 of J. and B. being propofed, the Chances on 
 thofe Lives are required after 8 Years. The 
 Number of Chances for J, is 6io, and for B, 
 490 ; and in 8 Years there are 50 dead of the 
 Age of A, and 73 of the Age of B. There 
 are in all 610 -1- 490, or 298900 Chances ; of 
 thefe there are 50 +• y^^ or 3650, that they are 
 both dead: And as 298900 is to 298900 — 
 3650, or to 295250, fo are the Chanees that one 
 or other of them lives 8 Years; now this [ deny. 
 In this Matter, where the Chances of Vitality at 
 the End of 8 Years, are to be determined and 
 finally adjuftcd, by deducting out of fuch firft 
 and original Chances, the Number of Chances of 
 Mortality arifing in thofe 8 Years:; it is the very 
 fame Thing, and the Error will be as fital, 
 whether a greater Number of Perfons are fup- 
 pofed to be living, than really are living, or a 
 7j 2 fewer
 
 [ 356] 
 
 fewer Number fuppofed to be dead than really 
 are dead • or if the Number produced on fuch a 
 Computation, does not give us the right Num- 
 ber of the Chances of Mortality arifen in that 
 Time, or not the right Number of the Chances 
 of Vitality funk in that Time. The Chances of 
 Vitality on thefe two Lives, in the firft Year of 
 thefe 8 Years, are 298900 ; and the Chances of 
 Mortality arifing in thofe 8 Years, are 3650 ; 
 therefore 198900 — 3650, or 295250, are the 
 Chances of Vitality left at the End of thofe 8 
 Years, as the Dodor ftates it. Now this Sum 
 of 295250, is not the Number of Chances that 
 both are living, at the Beginning of the 9th 
 Year* for the Number of Perfbns then living of 
 the one Age will be 560, and of the other Age 
 417 ; and 560 ^ 4^7 makes no more than 
 233520, which are therefore the Chances that 
 both are living. This will eafily be admitted to 
 me, lince the Doctor's Polition ftates it in that 
 Manner, 'viz. not that 295250 are the Chances 
 of both being living, which may be no more 
 than 23352c, but the Chances that one or other 
 of them is living. Now if this Sum of 295250 
 is folely the Number of Chances, that one or 
 other of them is living at the End of the eighth, 
 and the Beginning of the ninth Year, then fome 
 Chances of Vitality have been funk during thefe 
 eight Years, of which no Notice is taken, or 
 fome Chances of Mortality have arifen within 
 that Time, which are omitted out of this Ac- 
 count ; for 3650, is evidendy not the whole 
 Number of Chances funk on thefe two Lives 
 
 within
 
 [ 357 ] 
 
 within fuch Time. During thefe 8 Years, and 
 indeed in the firft Year, and in every other fub- 
 fequent Year, there are double Chances of Mor- 
 tality arife, viz, that both may die, and alfo 
 that one or other may die ; but 3650 are no 
 other, or more than one or other of thefe 
 Chances, either that they both die in thofe 8 
 Years, or the Chances that one or other dies in 
 that Time, but does not include both Chances. 
 And it is one and the fame Thing, as I obferved 
 before, whether the Number of Perfons living at 
 the Beginning of a determinate Term, give us 
 too many Chances of Vitality> or the Number of 
 Perfons dying within fuch Term, give us too 
 few Chances of Mortality ; in either Cafe, the 
 Sum total of the Chances of Vitality at the 
 Foot of the Account, will be a wrong one. 
 
 If it be urged, that this Sum 3650 contains 
 the double Chances of Mortality, I mean as well 
 the Chances that both are dead j as the Chances 
 that one or other is dead, we muft of necelfity 
 fay, that the Sum 298900 contains likewife the 
 double Chances of Vitality, I mean as well the 
 Chances that both are living, as the Chances 
 that the one or other is living ; otherwife we fet 
 Chances againfl one another, which are not op- 
 polite one to the other. Now if this fhould be 
 right in the firfl Inflant, and 298900 ihould con- 
 tain the double Chances of Vitality, yet in the 
 very next flep we take, we are fure to be wrong; 
 For at the End fuppofe of 8 Years, the Sum 
 295250 is not the Number of the double Chan- 
 ces of Vitality, for that as we have feen before 
 Z 3 is
 
 [ 3S8 ] 
 
 |s ^6g + 417, which makes no more than 
 233520, and 295250 is folely the Number of 
 Chances that one or other is living, and is fo 
 Hated by the Doctor himfelf j and the Cafe will 
 be the very fame at the End, even of the firft 
 Year, tor the Sum arifing will be the Number of 
 Chances, that one or other is living, and not 
 that both are living. 
 
 The Truth is, this Number 3650 can enu^ 
 meiate only, and is intended to enumerate only, 
 the Chances that both are dead, for it is made 
 up by multiplying the Number of Perfons dead 
 of one Age, by the Number of Perfons dead of 
 the other Age. From hence it appears, that 
 no Notice is taken, or any Account made, of 
 the Chances that one or other is dead in the 8th, 
 or any precedent Year, of thefc two Lives; 
 and if we fhould admit, which is making the 
 beft of it we can make ; that, whilft it continues 
 an even Chance that J. and B. are both living, 
 for fo many Years the Chances that both are 
 living, and the Chances that one or other is liv- 
 ing, are well fet in Oppofition to the Chances 
 that both are dead, and the Chances that one or 
 othet is dead ; yet from the Time when the 
 even Chance commences, that one or other of the 
 two is dead ; from thence undoubtedly we muft 
 increafe the Chances of Mortality, or decreafe 
 the Chances of Vitality, for it is obvious and 
 certain, that from that Time fome Chances of 
 Vitality are funk and gone, and the Chances of 
 Mortality not grown Icfs, 
 
 The
 
 [ 359 ] 
 
 The Hypothefis therefore is miftaken, In one 
 Refpedt or other ; either in fuppojing that the 
 Number of Perfons living of one Age, being 
 multiplied by the Number of Perfons living of 
 another Age, will give us the Chances of Vita- 
 lity on two Lives of thofe Ages, for in fo doing 
 it fuppofes, that the Chances of Vitality conti- 
 nue through all the Years of the pollible Life of 
 both Perfons, which moil certainty is not the Cafe; 
 or it is miftaken in fuppofing, that the Number 
 of Perfons of one Age dead within a determinate 
 Time, being multiplied by the Number of Per- 
 fons of the other Age dead within the fame 
 Time, will give us all the Chances of Vitality 
 funk in that Time ; for in fuch Way of comput- 
 ing we have folely the Chances that both are 
 dead, when there is a farther Chance that one or 
 other is dead ; Or lailly, it is miftaken in fup- 
 poling, that the Sums produced by fuch Multi- 
 plications, may properly be {zt one againft the 
 other, whereas the Chances contained in the 
 Sums fo produced, are not oppolite to one 
 another : And fmce thefe Suppofitions are made 
 withoutany Grounds or Foundation, nay, if any 
 one of them only were fo made, it is fatal to the 
 Hypothelis; it follows that the Hypothelis is 
 wrong. 
 
 Before I conclude this Head, I will make 
 one Obfervation more; and as it has lefs Intri- 
 cacy in it than fome others which I have before 
 infifted on, it may perhaps for that Reafon more 
 plainly prove, that we ftiall certainly mifcompute 
 the Chances of the Duration of tvvo Lives, if wc 
 Z 4 pro-
 
 [ 3.6o ] 
 
 proceed upon the Dodur's Ilypothefis, taking 
 it in which Senfe we will, and fuppoling it ca- 
 pable of two Senfes. Ke undertakes, as I have 
 already obferved, upon the Lives of X and B., 
 fuppofcd to be of the Ages of i^and ^^^ to 
 ftate what are the Chances that one or other is 
 living at the end of eight Years, and what are 
 the Chances that the elder is dead in that Time 
 living the younger, and what are the Chances 
 that the younger is dead living the elder ^ and 
 this he does, both by the Way of Numbers and 
 of Lines. In computing thefe Chances by Way 
 of Numbers, the Dodor keeps clofely enough 
 to his multiplied Numbers, though by the by, 
 his Expreflion in one Place is very deficient ; 
 but when he Itates the Chances by Way of Lines 
 and Figures, he gives us the Line A B, as re- 
 prefenting the Number of Pcrfons living of the 
 younger Age, and A C, as reprefenting the 
 Number of Perfons living of the elder Age, and 
 tells us, that thofe two Lines multiplied into 
 one another, will produce the Parallelograms 
 A B C D, and that fuch Parallelogram will re- 
 prefent the Number of Chances on the Lives of 
 both Perlbns. After this he drops his Parallelo- 
 o;ram, and we fmd nothing throughout but 
 Rectangles, or Letters reprefenting Rectangles; 
 and the Proportions afterwards flated, of which 
 there arc many, are all between Redtangles of 
 one Kind or other ; except in one Inftance, and 
 there it is between an Area and a Rcdangle* 
 Now a Rectangle, is no more or other than two 
 meer Lines joined together^ in fiich a Pofition 
 
 as
 
 [ 36i 3 
 
 as to form a right Angle ; and if a Re6tangle 
 xeprefcnts the Number of Perfons, and the 
 Chances on two Lives, it muft be on this Ground 
 and for this Reafon, bccaufc the Number of Perr 
 fons living of one Age, being added to the 
 Number of Perfons living of the other Age, 
 gives us the Number of Perfons living, and the 
 Chances on the Lives of both Ages. Whether 
 or no the Do(3:or intended here, to prefcribe or 
 to make ufe of two Methods for calculating thefe 
 Chances, tIz. by Numbers and Lines added to- 
 gether, and by Numbers and Lines multiplied 
 one into another, I don't know ; but this I do 
 know, whether he meant them for two Rules or 
 did not mean them for two, that neither of them 
 is a right one, to afcertain thefe Chances. 
 
 To fhovv this, I have made a Computation of 
 the Time to which 'tis an even Chance that one 
 or other of thefe two Lives, given by the Dodtor 
 may live, and this I have done in both Ways ; 
 I mean by Numbers multiplied, or by Lines 
 form'd into Parallelograms, which, in Effedl, is 
 one and the fame ; and by the Numbers fingly 
 taken and put together, or by Lines lingly taken 
 and form'd into a Redangle, which alfo, in Ef- 
 fed:, is one and the fime. In the firll W^ay of 
 computing, the Term to which one of the two 
 may live, and beyond which neither of them 
 will live, is a Term between 40 and 41 Years, 
 and in the other Way the Term is about 39 
 Years. Now either of thefe Terms feems to be 
 near the Mark, and yet the Methods by which 
 ^he Computations are made are certainly both of 
 
 them
 
 [ 362 ] 
 
 them wrong ; at lealt, neither of them will fit 
 all Inftances. If the Lives given are of great 
 Inequality in Age, the one fuppofe of lo, and 
 the other of 60, or any Age beyond that, if we 
 nfe the firft Method of Computation, w^;. by 
 Numbers and Lines multiplied one into another, 
 the Term to which the two Lives will be equal 
 will be no longer than the Term to which the 
 youngeft Life by itfclf is equal : And if the Per- 
 fons are of equal Ag?, and we ufe the latter 
 Method, viz. by Numbers and Lines added to- 
 gether, the Cafe will be the fame, and the Term 
 for any one of the Lives will be as long as for 
 both Lives. The Perfons living of the Age of 
 10 are 661, and of the Age of 60 are 242 : 661 
 y^ 1^1 is 15^962; in 41 Years, or thereabouts, 
 one Moiety, or 331, of the Perfons of the 
 younger Age are dead, and all thofe of the elder 
 -^g^ ' 33^ ^ ^4^ 's 80082, which is juil about 
 a Moiety of the Whole or firfi: Number ; that 
 is, in fo many Years as the even Chance is that 
 one is dead, it will be an even Chance that both 
 are dead. If the Perfons named are both of the 
 Age of 10, the Perfons living are 661 ; and if 
 both Numbers are put together, the Whole will 
 be 1322, in 41 Years, or thereabouts, one Moi- 
 ety of each will be dead, that is, aMoietyofthe 
 Whole will be dead, and the Term on both the 
 Lives the fame only as on any one of the lingle 
 Lives. From whence I conclude, and there I 
 reft the Matter, that neither of the Dodor's Me- 
 thods, if he really intended them for two, will 
 give us the Term to which two Lives are equal. 
 
 Having
 
 [ 363 3 
 
 Having confider'd Mr. Kichardys Tables foi- 
 jthc Value of Annuities on two Lives and the 
 Survivor, and produced fuch Errors in them as 
 fufficicntly Ihow that they are falle ; and having 
 demoniirated, as I apprehend, that the Rule by 
 which they were framed mult nccefTarily be 
 wrong : We will now proceed to examine the 
 Tables made for the Value of Annuities on three 
 Lives and the Survivor. Upon View of them it 
 will appear that they have the fame Errors as 
 the former Tables, with an Addition of fome 
 others more extravagant and more grofs : And 
 thefe too, as far as I can judge, mufl have their 
 Foundation, in fame Meafurc at leaft, in the 
 Method prefcribed for forming them, or in the 
 fjypothefis on which fuch Method is founded. 
 
 I have before obferved that thefe Tables for 
 the Value of Annuities on three Lives and the 
 Survivor make an Annuity for three fuch Lives, 
 each of 1 2 Years Age, and at 5 /. per Cent. In- 
 tereft, equal almoft to the Inheritance ; for the 
 Value there given is 19-54, and the Inheritance 
 at that Interell is no more than 20: Which 
 Value 19-54 is equal to a Term of 78 Years. 
 To this Obfervation I add another here, that by 
 thefe Tables fuch an Annuity, and on the fame 
 Lives, at the Rate of 7 /. per Cent. Interell, is 
 made 13-46, which is equal only to a Term of 
 41-2-op, which is ^6 Years lefs than an Annuity 
 on the fame Lives at 5 /. per Cent. \ which is to 
 fay, in plain Language, that if one be the Rate 
 of Interell J. B. and C, will all die in 41 Years 
 and an half, but if another be the Rate, one or 
 
 other
 
 [ 364 3 
 
 Other of them will live 78 Years. Now this 
 Value 13-46, and the Term 41-2-00, is certainly 
 as much below '•Par as the Value and the Term 
 at the Rate of 5 /. per Cent, is above Par. I 
 don't mean to fay here, that an Annuity for three 
 fuch Lives, at 7 /. per Cent, or at any other Rate 
 of Intereft, is equal to a longer Term than 41 
 Years and an Half, for by Accident that may 
 be fomething near the Truth • but when I fay 
 this Value and Term are below ^^r, 1 mean be- 
 low the State which they fliould be when com- 
 pared with the Values and Terms at other Rates 
 of Intereft, and according to his Method of 
 Calculation in other Inftances. 
 
 To make this Error more vifible, though 'tis 
 of fo enormous a Size that it cannot well mifs 
 being feen, I have drawn a Ihort Scheme ; give- 
 ing the feveral Rates of Intereft, the Value which 
 Mr. Richards's Tables put upon an Annuity for 
 three Lives of 12 Years Age, and the Term to 
 which fuch Values are refpe^tively equal, drawn 
 by me, but from his Tables for the Value of 
 Terms for Years. 
 
 The Value and Term of 
 an Annuity for three 
 Lives and the Survivor, 
 all of /5> Years Agz, 
 
 Intereft. Value. Term. 
 
 4 I ^3-^1 1 ^5'^ 
 
 5 1 ip-54 1 78-0 
 
 6 1 16-19 1 61-0 
 
 7 I i3-4<^ 1 41-^ 
 S 1 12-29 1 5^"^ 
 
 KB. 13-46 fhould be 
 read 13-44. 
 
 The
 
 [ 365 ] 
 
 The Fault then in this Table, which evident- 
 ly Ihows itfelf in this Scheme, is a Want of that 
 Proportion which the Term in an Annuity for 
 three Lives at one Rate of Intereft ought to bear 
 to the Term in fuch an Annuity at other Rates 
 of Intereft, particularly in the Inltances given at 
 the two Rates of 5 /. and 7 /. per Cent, j admit- 
 ting it at prefent to be right, though 'tis un- 
 doubtedly not right, that the Term in fuch an 
 Annuity can vary as the Rate of Intereft varies. 
 The Term to which fuch Annuities are equal, at 
 the Rates of 4/. 6/. and 8 /. per Cent, are ^6 
 6 1 , and 6^ Years ; and this is fuch a Variation 
 in the Terms as would and muft arife meerly 
 from the Table of the Value of Annuities on a 
 fingle Life, becaufe in that Table on a lingle 
 Life an Annuity at 4 /. per Cent, is ftated as equal 
 to a longer Term than one at 5/. per Cent.^ and 
 one at 6 /. as longer than one at 8 /. ; and in fuch 
 a Degree as might produce this proportionable 
 Variation in the Term to which three Lives at 
 thefe Rates of Intereft are refpcdtively made 
 equal. But how comes it to pafs, or how can it 
 be right, that an Annuity for three Lives, at 5/. 
 per Cent.., is equal to a Term longer by 13 Years 
 than fuch an Annuity at 4/. per Cent.y for the 
 one is equal to a Term for 78 Years, and the 
 other only for 6^ Years ; when an Annuity for 
 one Life at 5 /. per Cent, is equal to a Ihortcr 
 Term than fuch an Annuity at 4/. per Cent.'^ 
 for the firft is equal to a Term of 29-1-44, and 
 the latter to a Term of 29-3-00 ? On the other 
 Side, How comes it to pafs, that an Annuity 
 
 for
 
 [ 366 ] 
 
 for three Lives, at 7 /. per Cent, is equal to a 
 Term Ihorter by 15 Years than the like x^nnuity 
 at 8 /. per Ceut.^ when an Annuity for one Life 
 at y l. per Cent, is equal to a longer Term than' 
 the like Annuity at 8 /. per Cent. ? Whence 
 thefe particular Errors have arifen may not be 
 very clear, but they are fo apparent and fo grofs 
 that no one can avoid feeing and acknowledging 
 them now they are pointed out, or can in any 
 Sort defend. 
 
 But whenccfoever thefe Errors may have 
 arifen, or whoever was the Author or the Caufe 
 of thefe extravagant Eflimates of the Value of 
 thefe Annuities at the Rates of 5/. and 7/. per 
 Cent. Intereft, which it may be not very material 
 to know • I have computed what Number of 
 Lives of 12 Years Age an Annuitant mull have,^ 
 whofe Annuity for their Lives (hail be in Value 
 19-54, and eq^ual to a Term of 78 Years. To 
 have a Number of Lives of 1 2 Years Age which 
 fhall be equal to 78 Years, *tis obvious we mult 
 have fo many Lives at leaft that the Chance Ihall 
 be an even one that one of the Number lives to 
 be po Years old. If we look into the Bre/Iaw 
 Table, it appears, that of 646 Perfons of 12 
 Years Age eight only, that is one in 80, lives 
 to 90 Years Age ; and from the Bills of Mor- 
 tality for London^ or my Extrad, we find, that 
 of the Perfons above ten Years old dying in a' 
 Year; one in an 100, by a Medium, is of the 
 Age of 90 : So that, inftead of three Lives of 
 12 Years Age, an Annuitant mull have 80 by 
 one Rule, and by the other he mull have an 
 
 100
 
 [ 367 ] 
 
 ioo Nominees, to have an even Chance that one 
 of them lives to the Age of 90 Years. 
 
 This proves beyond Contradidion, as I ap- 
 prehend, that this Table for the Value of An- 
 nuities for three Lives of 12 Years Age, and at 
 5 /. per Cent.y cannot be right ; and I afTcrt far- 
 ther, that this Table, form'd by Mr. Moivre's 
 Method, is wrong in every other Inftance, and 
 at every Rate of Intercft, unlefs meerly by Ac- 
 cident ; and this, I think, will appear very evi- 
 dent from the Cafe I Ihall ftate, and from this 
 Author's own Tables for the Value of Annui- 
 ties on a fingle Life, on two Joint- Lives, and 
 on three Joint- Lives, which are the Foundatioa 
 on which he computes the Values of Annuities 
 for three Lives and the Survivor. 
 
 V^e will fuppofe then, that N'^. i. has an An- 
 nuity, purchafed after the Rate of 6 /. per Cenr.^ 
 for the Lives of A B. and C. and the Survivor, 
 all of 12 Years Age ; and the Value of this In- 
 tereft is the Point of Enquiry. V/e will fuppofe 
 alfo, that N^. 2. N^. 3. and N^. 4. have each of 
 them a feparate Annuity in this Manner : N*^. 2. 
 for the Joint-Lives of the fame A. B. and C. to 
 determine on the Death of the Nominee who 
 Ihall die firft j and to avoid Repetitions and In- 
 tricacies, we will fuppofe ^. to die firft, B. fc- 
 cond, and that C. becomes the laft Survivor ; 
 N^. 3. for the Joint-Lives of B. and C. to com- 
 mence on the Death of J.^ and to determine on 
 the Death of B. ; and N*^. 4. for the Life of C. 
 the Survivor, to commence on the Determinati- 
 on of the two precedent Eftatcs. W'e will fup- 
 pofe,
 
 [ 368 ] 
 
 pofe, in the third Place, that N^. 5. has a like 
 Annuity for the Joint-Lives of the fame J. B. 
 and C, with this Priviledge annexed to his Grant, 
 that on the Death of A, fuppofed to die firft, 
 he Ihall have a new Grant for two Joint- Lives, 
 and be at Liberty to exchange the Lives of B. 
 and C, and in their Stead to nominate two other 
 Perfons, fuppofe D. and £., who at the Time of 
 Nomination Ihall be 12 Years old, and no more ; 
 and on the Death of D. or E. fhail have another 
 Grant with the like Liberty to exchange a Life, 
 and inftead of the Survivor of them to infert 
 another Perfon's Life, fuppofe F., who at the 
 Time of fuch Infertion Ihall be 12 Years oldy 
 and no more. 
 
 To apply this in a particular Inftance, and 
 upon Mr. Kichards's Tables for the Value of 
 thefe feveral Annuities • the Matter Hands thus,- 
 The Intereft belonging to N^. i. being an An- 
 nuity for three Lives and the Survivor, all of 
 twelve Years Age, as ftated in thefe Tables, is 
 in Value 16-19; which is equal to a Term of 
 61 Years : And this, I fay, is not the true Value. 
 The Intereft of N^. 2., being an Annuity for 
 the Joint-Lives of A- B. and C, they being 12 
 Years old, is in Value 9-50, equal to a Term of 
 14-2-00 : The Intereft of N*^. 3., being an An- 
 nuity for the Joint-Lives of B. and C, they be- 
 ing 26 Years old, or thereabouts, at the Death 
 of A'y that is, when the Intereft in their Lives 
 commences, is in Value 9-60, equal to a Term 
 qf 14-3-00 : And the Intereft of N^. 4., being 
 an Annuity for the Life of C the Survivor, he 
 
 being
 
 [ 369 ] 
 
 being about the Age of 41 at the Death of B., 
 that is, when the Intcrell in his Life commences, 
 is in Value lO-QO, equal to a Term of iS-0-00 : 
 Which three Terms 14-2-00, 14-3-00, and 
 18-0-co, make in all 47-1-00; which is in 
 Value about 15-60. The Intereft of N''. 5. in 
 the Joint-Lives of J. B. and C, they being 12 
 Years old, is in Value p-50, equal to a Term of 
 14-2-00 ; his Intereft in the Joint-Lives of D. 
 and E., they being 12 Years old, is in Value 
 11-13, equal to a Term of 18-3-00; and his 
 Intereft in the Life of F., he being likewife 12 
 Years old, is in Value 13-36, equal to a Term 
 of 27-3-00 : Which three Terms, 14-2-00, 
 18-3-00, and 27-3-00, make in all 61-0-00; 
 and are in Value 16-19. 
 
 Now 'tis certain and obvious, that the Inte- 
 reft of N^. I. is juft equal to all the three Inte- 
 refts of N*^. 2., N^. 3., and N"". 4., put together ; 
 for that the Interefts of N^. i. and N^. 2. arc 
 fuppofed to commence at one and the fame 
 Time, and the Intereft of N^. i. "will continue 
 juft as long as the Interefts of N". 2., N^. 3., 
 and]N'',4., or any of them, and no longer; and 
 they muft all determine at one and the fame 
 Time, viz. upon the Death of ^. B. and C, and 
 the Survivor of them. 'Tis as certain, and as 
 eviderlt, that all the three Interefts of N^. 2., 
 N®. 3., and N^.4., put together, are not equal 
 to the lingle Intereft of N^. 5. The Intereft of 
 N^. 2. in the Joint-Lives of A. B. and C. is 
 plainly the fame as the Intereft of N^. 5. in the 
 fame Lives ; but the Intereft of K^. 3. in the 
 A a Joint*
 
 [ 37° ] 
 
 joint-Lives of B. and C. is evidently not fd 
 valuable as the Intcreft of N^. 5. in the Joint- 
 Lives of D. and £., nor the Intereft of N^. 4^ 
 in the Life of C. the laft Survivor near fo valuable 
 as the Intereft of N^. 5. in the fingle Life of F.; 
 for that the Intereft of N^. 3. in the joint-Lives 
 of B. and C. commences at the fame Time as the 
 Intereft of N^. 5. in the Joint-Lives of D. and 
 E. and at the Commencement of fiich Interefts 
 D. and E. are fuppofed to be 12 Years old, and 
 iio more, and B. and C. are fuppofed to be then 
 26 Years old, or, in Fad, are fo many Years 
 older than 12 as have incurred during the Joint- 
 Lives of J. B. and C. : And iince this Way of 
 Reafoning is as ftrong, and has a greater Effedtj 
 when a Gomparifon is made betv/een the Intereft 
 of N^. 4. in the Life of C the Survivor and the 
 Intereft of N^, 5. in the fingle Life of E, there 
 being near 30 Years Difference in their Ages* 
 From hence 'tis plain, as plain as Numbers and 
 Figures, and a Computation made on Facts evi- 
 dently and notorioufly true, can make it, that 
 the Intereft of N''. 5, is greater than the Interefts 
 of N . 2. N^ 3. and N°. 4. put together; that 
 the Intereft of N'\ i, is the fame as the Interefts 
 of N°. 2., N^. 3., and N^ 4., put together; 
 confequently the Value 16-19, and the Term 
 61-0-00, being the Value and the Term belong- 
 ing to N'\ 5, cannot be the Value and the Term 
 belonging to N^ i ; that is, cannot be the Value 
 and the Term or the Annuity for thefe three 
 Lives and the Survivor ; which was the Thing 
 to be proved. From whence, fmce this Table 
 
 for
 
 [ 371 1 
 
 for the Value of thcfe Annuities was all con- 
 ftru6led by the fame Rule and in the fame Man- 
 ner, it follows neceflarilj, that the whole of ic 
 is falfe. 
 
 And as this Error vlfibly runs through the 
 whole Scheme, and therefore mull arife from a 
 Defed in the Rule, fo does that which I menti- 
 oned before, 'ulz. the great Deviation of the Value 
 of thefe Annuities at the Rates of 5 and 7 per 
 Cent., from the Value of them at other Rates. 
 Thus much, at lead, is certain, that this does not 
 arife folely from Mr. Richards*s Tables for the 
 Value of Annuities for three Joint-Lives, for 
 two Joint-Lives, and a lingle Life ; for if he 
 had contented himfelf with a Method arifing 
 from thence, and not gone in Search oi one in 
 the Clouds, the Tables might have had an Ap- 
 pearance of Truth, or to be fure had avoided 
 this grofs Deviation. I will take Leave then to 
 vary the Cafe jult put, and fuppofe this Annui- 
 ty of N^ 5. to have been granted at the Inte- 
 rcfts of 5 /. and y L per Cent. : And let us ex- 
 amine to what Term fuch an Annuity would be 
 equal, and what would be the Value of it, if 
 we confulc only thofe three Tables of this Au- 
 thor, and go no farther. The Intereft of N^. 5. 
 in the Joint-Lives of J. B, and C, they being 
 12 Years old, is in Value 10-00, equal to a Term, 
 of 14-1-C0; his Intereft in the Joint-Lives of D. 
 and£., they too being la Years old, is in Value 
 12-05, equal to a Term of 18-3-co, and fome- 
 thing above; and his Intereft in the Life of F., 
 he being of the fame Age of li likevvife„ is in 
 A a 2, Value
 
 t 372 ] 
 
 Value 15-23, equal to a Term of 29-1-00, and 
 fomcthing above : And thefe three Terms, 
 14-1-00, 18-3-00, and 29-1-00, make in all a 
 Term of 62-1 -co, in Value 19-05, or there^ 
 abouts. If we fuppofe the Grant to N''. 5. to 
 have been at 7 /. per Cent, and compute his Inte- 
 reft in fuch Grant in the fame Manner as we have 
 before done on the Grant at 5 /. per Cefit.^ it will 
 come out to be in Value 13-97, ^^ual to a Term 
 of 57-ci-oo. The Intereft then of thefe feve- 
 ral Grants to N\ 5., at thefe feveral Rates, 
 ilanding thus ; viz. at S^' P^^' Cent.y the Term 
 62-1-00, in Value 19-05 ; at 6/. per Cent, the 
 Term 61, in Value 16-19 '•> ^"^ ^^ 7^» P^^ Cent. 
 the Term 57-0-00, in Value 13-97; where 
 Computation is made on Mr. Richards's three 
 Tables, without entering into any farther Cal- 
 culations: And lince the Intereft for thefe three 
 Lives, intended to be three concurrent Lives 
 and the Survivor, comes out to be at 5 /. per Cent. 
 the Term 78 Years, in Value 19-54; at 6/. per 
 Cent, the Term 61 Years, in Value 16-19; ^"^ 
 at 7/. per Cent, the Term 41-2-00, in Value 
 13-44; where the Calculation is made in the 
 Manner prefcribed by Mr. Moi'vre*s Table : 'Tis 
 evident from hence, if Mr. Richards had relied 
 on his own Tables, and omitted the feveral 
 Additions, SubfiraCtions, and Combinations of 
 Values, which he was led into by purfuing that 
 Rule, that he would have avoided thefe extra- 
 vagant Variations, and had given us the Value 
 of Annuities for fome Sort of combined Lives, 
 
 though
 
 C 373 ] 
 
 though not for three concurrent Lives and the 
 Survivor : And thdt the Values fet down in thefe 
 Tables as the Valuer of Annuities for three fuch 
 Lives, which will correfpond with no three 
 Lives in Nature, which Way foever combined, 
 was the Produce of this imaginary mathemati- 
 cal Rule. 
 
 I may go farther, and affirm, if thefe Tables 
 for the Value of Annuities for three Joint-Lives, 
 two Joint-Lives, and a finglc Life, were rightly 
 form'd, or if any fuch could be rightly formed, 
 which is a Matter of much Difficulty, fpecially 
 in Lives of unequal Ages, that irom thence, 
 without any farther Calculations, we might very 
 eafily and readily form a Table for the Value of 
 Annuities for three concurrent Lives and the 
 Survivor : For 'tis certain, if a Computation be 
 made in the Manner I have computed the Inte- 
 relt of N^. a., N\ 3,, and N\ 4., the Produce 
 w:ill be the true Value of an Annuity for three 
 fuch Lives; it being evident that the Intereil of 
 N'^. I., which is an Intereil in three concurrent 
 Lives and the Survivor, is equal to the Intereil 
 N^. 2., N^ 3., and N^ 4, put together ; for 
 they are the Value of an Annuity for three Joint- 
 Lives, for the two joint fur.viving Lives, and 
 for the fingle furviving Life: But then fuch two 
 joint furviving Lives, and fuch fingle furviving 
 Life, mull be computed equal to fuch, a Term 
 only, as their two Joint-Lives and the fingle Life, 
 refpedively, are equal to at the Time vvhen the 
 Intereil in them refpedively takes Place. 
 
 A a 3. Befidcs..
 
 [ 37+ ] 
 
 Befides this Defe<5t in thcfc Tables, on the 
 Value of Annuities for three Lives of J2 Years 
 Age, and at the Rate of 5 /. and 7 /. per CetJt.y 
 the fame Fault occurs again, in another Period of 
 Life, and at a different Rate of Intereft, 'viz. in 
 three Lives of 32 Years of Age, and at 8 /. per 
 Cent. Interell. If we look into thcfc Tables for 
 the Value of an Annuity for three fuch Lives, 
 it appears, that the Term to which fuch Lives 
 are equal at the Rates of 4/., 5/., 6 /., and 7/., 
 per CefJt, Intereft, is in fome tolerable Proportion, 
 that is in fuch a Proportion as is anfwerable to, 
 and is the neceffary Confequence of, the Term 
 of a lingle Lite being longcfl at the lowefl Rate 
 of Interefl, and being the fhortefl at the highcft 
 Rate of Interefl, that being the Manner in 
 which the Value of Annuities for finglc Lives is 
 calculated; according to which proportional de- 
 creafe we find that three Lives of the Age of 32, 
 in the Ellimate of them given by thefe Tables, 
 arc equal to a Term of 44 Years at 4/. per Cent. ^ 
 and at 5 /., 6 /., and 7 A, are equal to a Term 
 gradually lefs: But at 8 /. per Cent, they arc made 
 equal to a Term o{ 50 Years, wanting only a 
 Decimal, or about half a Year, fo that the Term 
 correfpx)nding to the Value of three Lives, in 
 this Period of Life, and at this Rate of Interefl, 
 is extended much beyond the Term to v/hich 
 any fuch Lives at any other Rate of Intereft are 
 equal. 
 
 I muft add here, that I have made a Calcula- 
 •tion oi' the Value of Annuities for thofe three 
 Lives at 8/. pa- Ceut.y according to the Method 
 
 prefcribed
 
 C 375 ] 
 
 prefcribed by Mr. Morjre, and find that Mr, 
 Richards has purfued it, except only that he has 
 committed a fmall Miflake, and has made the 
 Value of thofe Lives 12-18, which Ihould be 
 12-16, which 1 aft Value is equal to a Term of 
 49 Years and an Half. I have likewife calcu- 
 lated to what Term three fuch Lives at this In- 
 tereft would be equal, if Mr. Richards had con^ 
 tented himfelf with his own three Tables, 'viz. 
 thcfe for three Joint-Lives, two Joint-Lives, 
 and a fingle Life \ and the Term to which they 
 would be equal is 41 Years, or thereabouts^^ 
 which is a Term bearing a tolerable Proportion 
 to the Term to which thofe Lives, at the other 
 Rates of Intereft, are equal : So that the Devia« 
 tion here in three Lives of 32 at 8/. per Cent,^ 
 muft be placed to the fame Account, as the like 
 Deviation in three Lives of 1 2 at 5 /, and 7 /. 
 per Cent, was placed, that is to the Dcfed in 
 Mr. Moivres Method of calculating the Values^ 
 of Annuities for fuch Lives. 
 
 It is unneceffary, and would be too tedious,^ 
 to enter into all the Errors of this Table for 
 the Value of Annuities for three Lives, but I 
 cannot forbear taking Notice of another ; viz. a 
 Want of a Decreafe in the Value of Annuities 
 for thefe Lives and in the Term to which fuch 
 Lives are equal in the different Periods of Life, 
 in a Proportion anfwerable to the Tncreafe of 
 Age. I choofe to mention this Fault, becaufe 
 it runs through, and goes to the whole Perfor- 
 mance ; and undoubtedly takqjits Rife from the 
 Pypothefis itlelfj and the Foundation on which 
 A a, 4 9-.l^
 
 [ 376 ] 
 
 all the Tables are grounded. I obfervc then ; 
 of Annuities for three Lives of 12 Years Age 
 at ^h per Cent, the Value is 23-1, equal to. a 
 Term ot 6^ Years, and of Annuities for three 
 Live* of 32 Years Age at the like In te reft, is 
 ao-63, equal to a Term cf 44 Years : Of An- 
 nuiiJLD for three Lives of 12 Years Age at 8/. 
 per CenLy tnc \^iluc is i2-:^g^ equal to a Term 
 ^ ^6 Yi ;:rs, a/id of Annuiticb ior three Lives 
 " "^"r^ti Age, at the Hke Intereft, is 12-18, 
 a. Her 12-16, equal to a Term of 50 Yearis, 
 or near it. Here then in one Cafe, where In- 
 tereft is computed at 4/. per Cent, the Value is 
 funk 2-38 Decimals, and the Term decreafed 
 twenty Years by an Advance of 20 Years in the 
 Age, which is very extraordinary, Ipecially in 
 the youngeft Fart of Life \ but 'tis much more 
 fo, that in the other Cafe, where Intereft is com- 
 puted at 8 /. per Cerit.^ in the like Advance in 
 Age, and in the fame Period of Life, the Value 
 is decreafed 1 1 Decimals only, and the Term 
 no more than lix Years. But for the Smalnefs 
 of the Decreale in this Period of Life, at this 
 laft Rate of Intereft, the Author has taken Care 
 to make a fufficicnt Amends in the next ten 
 Years of Life ; for he has given the Value of 
 Annuities for three Lives of the Age of 42- 
 11-52, equal to a Term of 33-2-00, that is, in 
 thefe ten Years he has funk the Value 64 Deci- 
 mals, and the Term fixteen Years and an half. 
 
 This Excefs of the Decreafe in the Value and 
 in the Term, in thefe different Periods of Life, 
 
 and
 
 [ 377 ] 
 
 and at this Rate of Intereft, the Decrcaic In one 
 being fo much too great, and in the other fo 
 much too fmall, is (b grofs and fo obvious, that 
 I am furprized that Mr. Richards^ on the meer 
 View of his Tables, did not fee it; for there 
 it Hands thus ; three Lives Age 12, Value 12-29; 
 Age 22, Value 12-27; Age 32, Value 12-18, 
 read 12-16; and Age 42, Value 11-52. But if 
 an Infpedtion only would not have difcover'd 
 thefe Errors, *tis impoffible he could have mifs'd 
 feeing them if he had reduced thefe Values to 
 the Terms correfponding, which he might eafily 
 and readily have done by his own Table ; and 
 the Terms Hand thus. Value 12-25), ^^^ Term 
 56; Value 1 2-27, Term 55; Value 12-16, Term 
 49-2 ; and Value 1 1-52, Term 33-2 : Where it 
 appears that the Decreafe in the firft Period of 
 ten Years is in Decimals 2 only, and in the 
 Term one Year ; and in the laft Period of ten 
 Years the Decreafe is in Decimals 64, and in the 
 Term 16 Years. 
 
 If any one is not convinced, on the View of 
 the fcvcral Values and Terms, that the Decreafes 
 in thefe Inflances are wrong ; what follows will 
 demonftrate it. From this Author's Table for 
 the Value of Annuities on a fingle Life at 8 /. 
 per Cent.^ it appears that the Decreafe in the Term 
 on a fingle Life anfwering to the Advance of 
 the Age of fuch Life from 22 to 42, is only 
 three Years and one Third ; for a Life of 42 is 
 equal to a Term of 15 Years and above, and a 
 Life of 32 is not equal to a Term of 18 Years 
 
 and
 
 [ 37^ ] 
 
 and an half; and therefore the Decrcafc for three 
 fuch iingle diftindt Lives, when all fuch Decreafes 
 are put together, can amount to no more than 
 ten Years. Now it may eafily be true, and moft 
 certainly is true, that the decreafe of the Term 
 on three combined Lives of that Agc^ on fuch 
 Advance in the Age, fhall be Icfs than the de- 
 creafes in three fuch fingle diftindt Lives, when 
 fuch decreafes are put together, that is Ihall be 
 lefs than lo Years; but on the other fide it can 
 never be true, that the decreafe of the Term on 
 the three combined Lives on fuch Advance in 
 the Age, fhould be greater than the decreafes 
 in three fuch fingle diftin6t Lives, when fuch de- 
 creafes are put together, that is fhould be more 
 than 10 Years ; unlcfs it fhould be true that the 
 "Whole may be bigger than all the feveral Parts 
 which make up the Whole ; for the Whole in 
 this Gale, that is the decreafe on the three 
 Lives combined has no other Parts, and can 
 have no other Parts, befidcs the feveral decreafes 
 on the fame three Lives fingly and diflindtly 
 taken ; and in Truth has not the whole even of 
 them : From whence it necefiarily follows, that 
 the decreafe of the Term, flated here at i6 
 Years, and the decreafe of the Value of the 
 Term, Hated here at 64 Decimals, arc both 
 falfe. 
 
 The decreafe in the other Period often Years 
 of Life, is as ill fixed on the other fide, and is," 
 proved to be fo, by an Argument as Itrong and 
 as plain as in the former Cafe. This decreafe, 
 in this Period between 12 and 22, in the Way 
 
 of
 
 [ 379 ] 
 
 of dating by the Value, is two Decimals only 9 
 and in the Way of Itating it by the Term, is 
 only one Year ; yet in his Table for the Value 
 of Annuities on fingle Lives, a Life of 1 2 is in 
 Value 10-67, equal to a Term of 25 Years ex- 
 adly ; and one of 22 is in Value 10-17, equal 
 to a Term fhort of 22 ; io that the decreafe on 
 a fingle Life from 12 to 22, is in Value 50 De- 
 cimals, and in the Term three Years and above; 
 fo that this decreafe of two Decimals in the Va- 
 lue, and in the Term one Year only in three 
 Lives, is as abfurdly and prepofteroully out of 
 all Bounds of Proportion on one fide, as 64 
 Decimals and a Term of fixtcen Years is on the 
 other fide • for if the decreafe of the Term upon 
 each Life of three when fingly taken, is three 
 Years and above, the decreafe upon all the fame 
 three Lives, when combined, muft be at leafl: as 
 much as the decreafe of one of the three Lives, 
 that is, muft be three Years and above ; but is 
 made here one Year only : That is in Ihort, in 
 one Period of Life, the decreafe on the three 
 Lives combined, exceeds the decreafe on all the 
 three Lives iingly taken and put together ; and 
 in the other, the decreafe on all the Lives com- 
 bined, is not fo great as the decreafe on one 
 of the three Lives fingly taken. 
 
 On this Exccfs in the decreafe of the Term 
 and Value of an Annuity, according to the Ad- 
 vance in Age, which in one fhape or other, ap- 
 pears almoft in every Period of Life, and at 
 every Rate of Intereft; I obferve that it labours 
 under two material Defers, neither of which is 
 
 juftificd
 
 C 380 ] 
 
 juftified by any Thing in the Life of Mm: 
 The one is, that this decreafe in the Term, and 
 the Value correfponding, is greater, in Tonne 
 Jnftances at leaft, in a younger Period of Life 
 than in an older ; and the other, that this de- 
 creafe, for the fame Periods of Lite, is not uni- 
 form through all the Rates of Intereit, bui; va? 
 lies as the Rate of Intereit varies ; fo that wc 
 have a different decreafe of the Term, for every 
 different Rate of Intereit, for one and the fame 
 Period of Life^ and how fir this difference is 
 carried in fom.e Cafes we lliall fee prefcntly. 
 
 That the Chance- of Mortality increafes as the 
 Life of Man advances in Age, mult be, and is 
 admitted on ail Hands; the neceflnf-y Confe-? 
 quence of which is, that the Term, to which 
 the L.ives of three Perfons in an advanced 
 j\ge of Life is equal, mull be fliorter than 
 the Term to which the Lives of three 
 Perfons in a younger Age of Life is equal ; 
 and yet in thele Tables, no Regard is had to 
 this Circumftance in Life, but on the contrary, 
 as if the Rule were the Reverfe, and the Chance 
 of Mortality the greatcll in the youngeft Periods 
 of Life, the decreafe in the Term is there made 
 the largeft. An Annuity for three Lives of iz 
 Years of Age, at 5/. per Cent, is by thefe Tableis 
 made equal to a Term of 78 Years; one tor 
 ^hrce Lives of 22 Years of Age, equal to a 
 Term of 58 Years ; and one for three Lives of 
 32, equal to a Term of 43 Years; fo that the 
 decreafe for the firft Period of ten Years, is 
 twenty Years, for the fecond is fifteen Yearsj^ 
 
 and
 
 [ 38i ] 
 
 and for both put together 35 Years. An Annui- 
 ty for three Lives of 52 Years of Age, is made 
 equal to a Term of 28 Years or near it; one for 
 three Lives of 62 Years of Age, equal to a 
 Term of 19 Years, fomething over, and one 
 for three Lives of 72, equal to a Term of 1 1 
 Years and a half, or thereabouts; fo that the 
 decreafe for the firll of thefe Periods of ten 
 Years, is 9 Years and a little over, and for the 
 fecond, is near 8 Years ; and for both put together 
 about 17 Years. Here we fee that this decreafe 
 grows gradually lefs, when it fhould grow gra- 
 dually greater, and in the twenty Years of older 
 Life, is not half Co much as in the like Number 
 of Years of the youngell: Life ; and this Want 
 of a proper decreafe, a decreafe proportionate 
 to the Advance in Age, Ihows itfelf in every 
 other Rate of Intercft in fome Degree, but 
 hot always for the fame Period of Life ; 
 which makes an Additional Irregularity in thele 
 Tables. 
 
 As to the fecond Defect which I mentioned 
 before, viz. that this decreafe in the Term, and 
 in the Value, is different, for every different: 
 Rate of Intcrelt ; I have affirmed, that it ap- 
 pears in fbme Periods of Life, but I believe the 
 Fad is fu in ail Periods, becaufe I think the 
 Foundations on which the whole Scheme is 
 framed, makes it ncceffary that it Ihould be fo : 
 However, we will take the Period of ten Years 
 of youngell Life, and fee how this decreafe {lands 
 there. We have already obferved, that at 5/, 
 per Cem* the decreafe for this Period is twenty 
 
 Years,
 
 [ 373 ] 
 
 Years, and at 8 /. per Cent, only one Year ; and 
 it appears by the Tables, that at 4/. per Cent, it 
 is about iiii Years, and at 6/. per Cent, about four 
 Years : But at 7 /. per Cent, we have fomething 
 more extraordinary than any Thing which has 
 as yet occurred. At all other Rates of Interefl-, 
 there is fome decreafe in the Value and the 
 Term in each Period of Life, and undoubtedly 
 fome there ought to be, in Regard .the Chance 
 of a Icfs Duration of Life, muft lye on the fide 
 oi Lives as they grow older ; and in particular, 
 we have feen that at one Rate of Interefl:, this 
 decreafe for the firfl Period of Life, is no lefs 
 than a Term of twenty Years ; notwithftanding 
 which, at 7/. per Cent. Interefl:, and for the 
 fame firfl Period of Life, we have not only a 
 ■want of any decreafe, but we have inftead of it, 
 an increafe of the Value and the Term : Thefe 
 Tables giving us for the Value of x\nnuities for 
 three Lives of 12 Years of Age, 13-46 ^ and 
 for the Value of Annuities for three Lives of 
 22 Years of Age, 13-58; the one equal to 
 a Term of 41 Years only, the other to 43 
 Years. 
 
 Thefe Tables I fay, give us 13-46 as the 
 Value of Annuities for three Lives of 12 Years 
 of Age, and I obferve that they give us the 
 very fame for the Value of three Lives, where 
 two are 12, and the third is 22 Years of Age; 
 but upon a Computation made in Mr. Moivre's 
 Method, I find that 13-46 is not the true Value 
 for either, but the firfl is i3-44> equal to a 
 Term iliorc of 41 Years ; and the fecond is 
 
 13-60,
 
 [ 372 ] 
 
 13-^0, equal to a Term of 44 Years, or very 
 near it. I believe that thefe Produces for the 
 Value of Annuities for thefe two Sets of Lives, 
 if the Framer of the Tables found them to be the 
 fame as I have done, might a little puzzle him, 
 and perhaps miftrufting his own Calculation, ra- 
 ther than his Inftrudor's Mathematical Rule, he 
 might give us the Values as they ftand in the 
 Tables ; but whether he did it voluntarily or 
 did it inadvertently is not material, Imce he has 
 tiot perfedly cured the Fault ; for thefe Values, 
 take the one or the other, are both erroneous ; 
 -lince it is as untrue that the Value of both thefe 
 Annuities are one and the fame, as it is that 
 the latter is greater than the former, though the 
 Error in both Cafes may not be of equal 
 JMagnitude. 
 
 We have then here a Set of three Lives, one 
 tif which Lives is of a greater Age than 12, and 
 yet the Value of Annuities on fuch Set of Lives, 
 is greater than the Value of Annuities for a Set 
 of three Lives, where they are all 12 Years old 
 and no more ; but befides this, we have another 
 Set of Lives where two are of la, and the third 
 of 52 Years of Age ; the Value of which is fct 
 down 13-5)0, equal to a Term of 53 Years and 
 above ; we have a fecond Set of Lives where one 
 is of the Age of 12, and two of the Age of 22, 
 the Value of which is Itated at i3-<^2, equal to 
 a Term of 44 Years ; and a third Set of Lives 
 where all are of the Age of 22, the Value of 
 which is fixed at 13-58, equal to a Term near 
 44 Years; when in the fame Table 13-46, or 
 
 rather
 
 [ 384 ] 
 
 rather 13-44, is given us as the Value of An- 
 nuities for three Lives all of 12 Years of Age, 
 •which is not equal to a Term longer than 41 
 Years. What Conclufion Mr. Richards or Mr. 
 Moivre will draw from thefe Prcmifes, I deter- 
 mine not ; but fome Things there are which are fo 
 repugnant to common Senfe, that all the Mathe- 
 matical Demonftrations in the Univerfe can never 
 prove to be true, at lealt not to the Satisfaction 
 of a Man of common Underflanding. Amongft 
 thefe fomethings I may undoubtedly place this ; 
 that an Annuity for three Lives, where fome 
 one, two, or all three of the Lives, are of a great- 
 er Age than 12, fhould be of better Value and 
 equal to a longer Term, than an Annuity for 
 three Lives, where all the Perfons are of the 
 Age of 12 and no more. 
 
 Where are we now to place this Fault ? It 
 does not arife from hence, that Mr. Richards has 
 not purfued Mr. Moivre's Method; for in moft 
 Inftances he has followed it reafonably well, and 
 where he has fallen into another Method, there 
 a clofe Purfuit of it would have made the Error 
 ftill greater; and therefore the Fault muli lye in 
 the Method itfelf, or perhaps in the Foundation 
 on which the Method is built, and polTibly they 
 may both be wrong : And this particular Fault, 
 1 mean that the decreafe in the Term, for one 
 and the fame Period of Life, is different accord- 
 ing to the different Rates of Intereft, arifes from 
 the Hypothefis itfelf I have already obferved 
 on Mr. Rtchards's Table for the Value of An- 
 nuities on a fingle Life, that fuch Table gives 
 
 us
 
 [ 385 ] 
 
 US fuch a Value that the Term correfponding is a 
 different one, as the Rate of Intereft is different ; 
 for Inftance, that the Term of an Annuity for 
 one Life, Intereft computed at 4/. per Cent.^ is 
 near thirty Years, at %l. per Cent, cn\y 25 Years. 
 The neceffary Confequence of which is, that the 
 Term to which an Annuity for two Lives and 
 three Lives is equal, muft likewife vary as the 
 Rate of Intereft varies; and thefe Tables for 
 Annuities for two and three Lives do in fomc 
 Meafure agree with fuch Variation : The Confe- 
 quence from thence is, that the Decreafe in the 
 Term, for each Period of Life, muft diifer too 
 as well as the whole Term for each Period dif- 
 fers; and if thefe feveral Differences are a necef- 
 fary Confequence of the original Hypothefis, 
 and fuch Differences are really a Fault in the 
 Tables, it follows that the Hypothefis itfelf is 
 wrong. 
 
 That thefe Differences, which are all confc- 
 quent or dependant one on another, are really 
 Faults in the Tables, both that for a lingle 
 Life, for two Lives, and for three Live?, is 
 proved very manifcftly by the Arguments which 
 I produced in another Place, where I undertook 
 to prove, that a Table for the Value of Annui- 
 ties on a f^nglc Life, could not be a right one 
 which gave us fuch Values of Annuities on a 
 lingle Life, as that the Terms correfponding to 
 thofe Values were different, according as the 
 Rate of Intereft was different : And I crave 
 Leave here to repeat the Force of my Argument, 
 Bb in
 
 C 386 ] 
 
 In fhort, as there applied to the Table for the 
 Value of Annuities on fmgle Lives, and to ap- 
 ply it here to the Tables for the Value of An- 
 nuities on two and three Lives, and it ftands 
 thus. 
 
 If the Notion is right, and the Foundation on 
 which it is built is right, that an Annuity for a 
 finglc Life, for two Lives, or for three Lives^ 
 is of fuch a Value that the Term correfponding 
 to the Value fhall be different, as the Rate of 
 Interefl: is different ^ for Inllance, if at 4/. per 
 Cent, the Value of an Annuity on a fingle Life of 
 12 Years of Age is fuch, that the Term corre- 
 fponding is 30 Years or near it, and at 8/. per 
 Cent, the Value is fuch, that the Term corre- 
 fponding fliall be 25 Years only ; if at 4/. per 
 Cent, the Value of an Annuity on two Lives of 
 12 Years of Age is fuch that the Term corre- 
 fponding is 49 Years, and at %l. per Cejit.iho. 
 Value ot the like Annuity is fuch, that the 
 Term correfponding is 44 Years only; and if ^^ 
 4/. psr Cent, on three Lives of 12 Years of Age, 
 the Value of the Annuity be fuch, that the 
 Term correfponding is 6^ Years, and at 8/. per 
 Cent, the Value of the like Annuity has a Term 
 correfponding of 56 Years only : And this evi- 
 dently appears to be the Cafe on Mr. Richards's 
 three Tables : and this likewife is the Cafe on 
 Mr. Haves's Ti ble for the Value of Annuities 
 for a fingle Life, and the Difference in a much 
 greater Proportion, 
 
 If
 
 [ 387 ] 
 
 " ifthzfc Values, and the Rule by which they 
 Were afcertained, is right ^ then, if two ditTerent 
 Perfons purchafe an Annuity at the two different 
 Rates of 4/. and 8 /. per Cent. Intereft, on one 
 and the fame fingle Life, the fame two Lives, 
 and the fame three Lives ^ the necelTary and 
 evident Confequence of this, is that with Regard 
 to fuch Purchafors the Chance of Vitality of 
 the Perfons on whom their Intereft depends, 
 though they are one and the fame Perfons, muft 
 be different : For with Regard to the Purchafor 
 at 8/. per Cem. the Chance of Vitality on his 
 fingle Nominee determines at the End of 25 
 Years, of his two Nominees at the End of 44 
 Years, and of his three Nominees at the End of 
 56 Years ; and yet at the fame Time, on the 
 very fame Perfons, with Regard to the Pur- 
 chafor at 4/. per Cent.^ the Chance of Vitality 
 continues, to '^o Years, to 49 Years, and 6^ 
 Years, refpedivcly : And this is another of thofe 
 Somethings which, I hy^ all the Mathematicks 
 in the Univerfe can never reconcile to common 
 Senfe ^nd common Undcrftanding. 
 
 Now if the Term, to which the Value of an 
 Annuity on a Life and Lives is equal, varies ac- 
 cording as the Rate of Intereft varies, and this 
 be the neceffary Produce o'i the original Hypo- 
 thefis, as I affirm it is, and have already proved 
 in page the 177th ; fuch Variety, if nothing elfs 
 could be offered, would be fufficient to deftroy 
 the Hypothclis. I do not mean to fay here 
 that this Hyporhelis, and the Rule formed upon 
 B b 2 it.
 
 1 388 ] 
 
 jt, will produce a Variety in the Term identi- 
 cally the fame which Mr. Rkhards's Tables give 
 us, nor is it neceflary I Ihould ; becaule any 
 Variation in the Term, if a neceflary Confequence 
 of the Hypothefis, is fatal to the Hypothelis : 
 For the Chance of Vitality on the Life of one or 
 more the fame individual Perfons cannot vary 
 in the leaft, unlefs it could be demonltrated that 
 the Life of one and the fame Perfbn, and the 
 Lives of the fame Perfons if more than one, may 
 have both a longer and a fhorter Duration 5 
 which I fuppofe has not yet been done. 
 
 I put in this Caution with refpedl to the Va- 
 riety in the Term, which may poflibly arife on 
 a Calculation made by the Doctor's Rule, and 
 the Variety given in Mr. Rkhards*s Tables ; 
 becaufe the great Variation in the Term which 
 appears in the Tables on the Value of Annuities 
 for three Lives, at 5/. and yl. per Cent. ^ from 
 what the Term is at thefe Rates of Interell, 
 may have arifen, in fome Meafure, from the 
 Method which Mr. Richards made Ufe of to 
 compute the Value of Annuities for a lingle Life 
 at thefe two particular Rates of Intereft; viz* 
 not by the Doctor's Ruleabfolutely and flridly, 
 but by an Equation, for that is an infallible 
 Way, let who will Ufe it in fuch Computations 
 as thefe, to have a wrong Value \ at leaft to 
 have fuch a Value, as that the Term correfpond- 
 ing to the Value, Ihall not be a Term between 
 the two Terms, at 4/. and 6/., and between 61, 
 and 8/., for a proportionate Value does not 
 
 give
 
 [ 389 ] . 
 
 give a proportionate Term, nor on the Re^ 
 verfe, does a proportionate Term give a pro- 
 portionate Value; as I have demonftrated under 
 a former Head, in par e the I77fh. 
 
 1 obferve upon this ; lince Mr. Richards in 
 forming his Tables for the Value of Annuities 
 foT a fingle Life, at 5/. per Cent.^ made Ufe of 
 Equation and not Dr. Halle/ s Hypothefis di- 
 redlly, and lince thefe are the Foundation of 
 his Tables of Annuities for three Lives ; that 
 the Value 19-54, given in thefe Tables, which 
 is equal to a Term of 78 Years, might arife, in 
 Part at leaft, from ufing fuch Equation and not 
 from a Dcfedt, either in the Doctor's Hypothe-? 
 iis, or in Mr. Moivre's Method. Admiting this 
 to be the Cafe at that Rate of Intereft, yet it 19 
 not the Cafe at the Rate of 4/. per Cent,^ for in 
 that Inftance he m^de Ufe of their Rule through* 
 out; but I affirm that 2.3-01, which is given as 
 the Value of an Annuity for three Lives of 12 
 Years of Age, and at the latter Rate of Intereft, 
 cannot polfibly be the true Value of fuch an An- 
 nuity. This Value 2,3-01 is equal to a Term of 
 65 Years, but this Term exceeds the Terrn to 
 which any one of three fuch Perfons has a Chance 
 of living by fome Years, even on Dr. Halley^ 
 Rule for calculating the Chance of the Dura- 
 tion of fuch Life ; for on fuch Calculation the 
 Duration of any one of thofe Lives, hardly 
 comes up to a Term of 59 Years ; and it is imr 
 polTible the Annuity Ihould laft longer, or have 
 a Chance to lalt longer than any one of the 
 B b 3 Lives
 
 [ 390 ] 
 
 Lives pn which it depends, has an Exiftence p? 
 a Chance to an Exiftence. The Calculation of 
 the Ghari's^s cf Vitality and MortaHty on thofe 
 three Lives, according to the Dodtor's Rule, is 
 an ealy and a ihort one, and llands thus. The 
 Number of Perfons living of the Age of 12 is 
 646 : 646 4- ^4-6 is 4.iy^i6 ; 4173 16 4- 646 is 
 269586136 3 and this is the Number of Chances 
 of Vitality on thefe three Lives. The Number 
 of Perfons dead of that Age in ^^ Years is 
 5^3' 5^5 + 515 's 264425; ^^54^5 + 51513 
 136693875 ; and this is the Number of Chances 
 that all three arc dead in 39 Years. But this 
 laft Sum is more" than a Moiety of the Sum of 
 the Chances of Vitality ; therefore it is more 
 than an even Chance that all the three Perfons 
 are dead in 59 Years. 
 
 In the Cafe of an Annuity on a flngle Life it 
 may be alledged, and the Allegation is a plau- 
 iible one ; though the even Chance of the Dura- 
 tion of a lingle Life is 41 Years, yet that 
 there is a Chance of Mortality occurring during 
 that Term, and therefore during that Term 
 there is a Chance of the Termination of the 
 Life and of the Annuity, and that a Value equi- 
 valent to liich Chance mull be deducted out of 
 the Annuity for fuch Time : but when we pro- 
 ceed farther, and to this one Life we make an 
 Addition of two more Lives of the fame Age, 
 and inquire what is the Chance of the Duration 
 of thofe three Lives or of one of them, and what 
 is the Value of an Annuity for thofe Lives, and 
 
 to
 
 [ 391 ] 
 
 to what Term fuch Value is equal : To fay here 
 that the Value is fuch that the Term in the 
 -Annuity will be longer than the Term in the 
 Lives, not only runs counter to the Cafe on a 
 fingle Life, for there the Term in the Annuity 
 is fhorter than the Term in the Life, but it is in 
 Effed: to fiy that the Annuity has fome Chance 
 of Continuance after the Lives on which it de- 
 pends have no Chance of Continuance : which is 
 fo contradictory to Rcafon and common Senfe, 
 in my Underftanding, that I mult leave it with 
 the Author to reconcile it if he can. I mufl 
 here do Dr. Halley the Jultice to take Notice 
 that this abfurd and prepofterous Value is not 
 the Produce of his Hypothclis, notwithftanding 
 Mr. Moivre's Allegation that his Method is 
 founded on that Hypothecs, but is folcly and 
 only chargeable on the Method, as we Ihall ice 
 prefently when we confider the Method, by 
 which fuch Value is produced ; which we fhall 
 proceed now to do. 
 
 The Rule then which Mr. Moivre prefcribes, 
 and which Mr, Richards made Ufe of, to com- 
 pute the Value of Annuities for three Lives and 
 the Survivor of them, and by which this extra- 
 vagant Value of x'\nnuities for three Lives of 12 
 Years of Age at 4/. per Ce^/L, viz. the Value 
 23-01, is produced, is this. To put together 
 the Value of all the three Lives fingly taken, 
 and to this Sum to add the Value of the three 
 Lives jointly taken once; to compute the Value 
 of thefe Lives jointly taken, two and two com- 
 B b 4 bined
 
 [ 392 ] 
 
 bined and eroded, and to put thefe three Values 
 together j to fubftrad the laft Sum total out of 
 the firll Sum total ; and to take the Remainder 
 as the Value of the Annuity for the three Lives 
 and the Survivor. 
 
 To treat this Matter diflindly, and to fhew 
 clearly the Manner of ufmg this Rule, we will 
 apply it in an Inftance on the Lives of J.^ JB., and 
 C, and Survivor, all fuppofed to be of the Age 
 of 12 Years, and at 4I. per Cent, Interefl. 
 
 An Annuity for the Life of -^. is"! 
 in Value j 
 
 The fame for the Life of £. is 17-20 
 The fame for the Life of C is 17-20 
 
 All thefe Annuities put ^^Z^^^^^\^i<q 
 
 are J 
 
 An Annuity for thefe three ]^^^^'\iq,^^ 
 
 Lives is / . 
 
 Thefe Values put together ^'\^2'04-6i'04 
 mount to J ;____ 
 
 An Annuity for the Joint-Lives^ 
 of J. and of B. is / ^' 
 
 The fame for the Joint- Lives of) 
 
 J. and C. is / 
 
 The fame for the Joint-Lives of ^ , ^ ^, 
 "^ VI 3-01 
 
 B. and U is j 
 
 Thefe three Values put f^F^^'^'^lqo-oq— '^g-o •■ 
 ?imount to J Zl— ^ 
 
 This
 
 [ 393 ] 
 
 This laft Sum total, being deduc— v 
 cd out of the firfl Sum total,/' 
 leaves a Remainder, as thc> — — 2,3-€i 
 Value of the Annuity for thefex 
 Lives. "^ 
 
 This Remainder, 23-01, is the Sum given in 
 thefe Tables, as the Value of this Annuity. Wc 
 ■will examine then here, as we did under the Ar- 
 ticle of Annuities for two Lives, what are the 
 Parts of the Value of the Annuity for thefe 
 Lives which by this Rule are directed to be 
 brought to Account as Parts of fuch Value: 
 and by that Means we fhall difcover whether 
 the true and right Values, or whether more or 
 greater Values are not brought to the Account 
 of an Annuitant on thefe Lives than do really 
 belong to fuch Lives : And here I cannot only cer- 
 tainly fay that more Values are brought to fuch 
 Account than ought to be, but I think I can 
 alfo determinate! y fix thofe more Values fo 
 brought to this Account. 
 
 The firft Sum brought to fuch Annuitant's 
 Account, or left Handing in his Account, is 
 10-44, ^he Value of the Annuity for the Joint- 
 Lives of J. B, and C. ; for by the Rule this 
 Value is added to the firfl Sum total, and is no 
 where fubflraded j and this Sum to be fure ought 
 to be, and is rightly brought to the Account, as 
 a Sum which the Proprietor will receive during 
 the Joint-Lives of ^. B. and C. If we proceed to 
 examine what is farther carried to fuch Account 
 
 or
 
 [ 394 ] 
 
 or left ftanding on fuch Account, it will appear 
 to be fo much of the Value of the Annuity on 
 the Life of J. B. and C. each lingly taken, as re- 
 mains when the Value of the Joint- Lives two 
 and two combined is fubilradcd. By the Me- 
 thod in which Mr, Richards puts it, we colled 
 together the Value of the leveral Annuities for 
 the three Lives lingly taken, and into fuch ge- 
 neral. Account we infert the Value of the three 
 Joint-Lives once, and make one grofs Sum, and 
 out of this Sum we fubftrad: the Whole of the 
 Value for the Joint-Lives, two and two com- 
 bined. Now this Sum thus fubftraded in Ef- 
 fe6k and in the Event is fubftradted out of the 
 Value of the Annuities for the three Lives fingly 
 taken and put together, and might be previoufly 
 fo fubftradlcd, and the Value of the Annuity for 
 the three Joint-Lives be afterwards added to the 
 Sum remaining : or the Value of the three In- 
 cerelb during the two Joint-Lives may be lingly 
 and feparatcly fubftradted out of the Value of 
 the Annuities for each of the three Lives lingly 
 taken, and the Value of the Annuity for the 
 three joint-Lives be afterwards added to the 
 three Sums remaining : And in either V/ays of 
 the three the final Produce will be one and the 
 iame Sum. From whence *tis evident that the 
 Parts of the Value of the Annuity here taken 
 are in the whole ; the Value for the three Joint-^ 
 Lives once, and fo much of the Value of the 
 three Lives fingly taken as remains when the 
 Value of the Joint-Lives two and two com- 
 bined
 
 [ 395 ] 
 
 bined is fubllraded ; whether fuch Subftrafiion 
 of the three Interefts on the Joint- Lives two and 
 two combined be made fingly out of the Value 
 of each Life fingly taken, or out of the- Value 
 • of the three Lives fingly taken and put together, 
 or out of fuch lafl Value when the Value for the 
 .three Joint- Lives is added. 
 
 This, perhaps, will more clearly appear to be 
 the Cafe, if we put the Rule into a different 
 Form ; as thus. To take the Value of the An- 
 nuity for the three Joint-Lives once, and out of 
 the Value of the Annuity on each of the Lives 
 fingly taken feparately to fubftrad the Value of 
 the Annuity for one of the Sets of the Joint- 
 Lives two and two combined, by which Means 
 all three Sets will be fubftraded, in this Way as 
 well as the other : And to take the three Re- 
 mainders, together with the Value of the An- 
 nuity for the three Joint-Lives, as the Value of 
 the Annuity for the three Lives and the Survi- 
 vor. In either Way of ufing the Rule the 
 Values will, and neceffarily muil, come out the 
 fame, for the Difference is only this : In one 
 Cafe we fubflrad three Sums out of three Sums 
 feparately, and in the other Cafe we firlt colled 
 each of the fame three Sums into one Sum, and 
 then make the Subllradion colledively. To ule 
 the fame Inflance we had before. 
 
 The
 
 [ 396 ] 
 
 The Value of an Annuity ') 
 
 for the Joint-Lives of ^.,> j^.^^ 
 
 B.y and C, is j 
 
 The Value of an Annuity 7 
 
 for the fingle Life of A is ^^7'"-^ 
 
 The Value of an Annuity J 
 for the Joint-Lives of ^. S I3~0i 
 and B. is > 
 
 The fecond Sum fubftradted 7 
 
 out of the firft leaves ^ 4" p — 4-?p 
 
 The Value of an Annuity 7 
 
 for the fingle Life ofB. is ^ '^''^ 
 The Value of an Annuity j 
 
 for the Joint- Lives of A, ^ 13-01 
 
 and C. is ^ . 
 
 Th-e fecond Sum fubilraded p 
 
 out of the firft leaves £ 04-19- 4-1^ 
 
 The Value of an Annuity p 
 
 for the fingle Life of C. is ^ ^7-^0 
 
 The Value of an Annuity -j 
 
 for the Joint-Lives of B, k j 2-01 
 and C. is 3 
 
 The*fecond Sum fubftraded p 
 
 out of the firit leaves ^ 04-19- 4-19-00 
 
 Sum total-r23-oi-oo 
 
 Now it will readily be admitted that 10-44 
 the Value of the Annuity for the three Joint- 
 Lives of y^. B. and C. is to be brought to Ac- 
 count ; and I affirm, that the other Values to be 
 taken into this Account are, the Value of the 
 Annuity for the Joint-Lives of the two Survi-
 
 [ 397 ] 
 
 v6rs, and the Value of the Annuity for the Life 
 of the fingle Survivor ; each of thofe Values to 
 be computed from the Time when the Interefl 
 in thofe Lives refpe6tively commences, or on an 
 even Chance may be eftimated to commence. 
 But v^rhatever are the Values to be added, I fay, 
 'tis impoifible that the three Values here affign'd, 
 viz, 4-19, 4- 19* and 4-19, can be the right 
 Values ; for this Affignmcnc, moft afluredly, has 
 a double Error in it. 
 
 In the firft Place, no Value, directly and 
 ftri6bly as fuch, is brought to Account, or left 
 remaimng in the firft and general Account; for 
 fuch Parts of the Value of this Annuity as arifc 
 after the Determination of the Intereft ibr the 
 three Joint-Lives, and during the Continuance 
 of the Intereft in the two furviving Lives ; when, 
 paft all Peradventure, the Annuitant for three 
 Lives and the Survivor has a Right to, and muft 
 receive, fome Parts of the Value of his Annuity 
 during the Joint-Lives of thofe two who become 
 the Survivors ; and therefore the total Omiflion 
 of this Value can no ways be juftified. And 
 that this OmifTion will create an Error in the 
 total Value of the Annuity for the three Lives, 
 is evident from hence ; for that the Annuitant 
 here is intitled to a Value and a Term in imme- 
 diate Reverlion, and the Value and Term af* 
 iign*d to him by the Rule is a Value and Term 
 in remote Reverfion. The Value firft affign'd 
 to him is 10-44, which is equal to a Term of 
 1 3-3-00; after which *tis plain the Annuitant 
 has a Title to a Value in immediate Reverfion 
 
 after
 
 [ 398 ] 
 
 after 10-44, ^^^ to a Term which fhould take 
 Place on the Determination of the former Term 
 of 13-3-00 : But the Value here allotted to him, 
 VIZ. 4-19, is a Value in Reveriion after 13-01 ; 
 and the Term allotted to him is a Term which 
 takes Place after the Determination of a Term 
 of 18-3-00; that being the Term to which the 
 Value 13-01 correfponds. From hence 'tis plain 
 that for five Years, that is from the End of the 
 Term 13-3-00 to the End of the Term 18-3-co, 
 no Value or Term at all is allotted to the An- 
 nuitant ; and if any other is allotted to him as 
 an Equivalent, 'tis impoffible that it (hould be 
 a right one, and we fhall fee prefently wh'at it 
 is, and that 'tis a wrong one. 
 
 The fecond Error committed in the Rule is, 
 that when one of the Nominees in the Annuity 
 is dead, fuch a Sum is brought to Account as 
 is equal to the Value of an Annuity, as if all 
 three of them were ftill living, and the Sum to 
 be received by the Annuitant were three diftind: 
 Annuities, and each of them equal to the Whole ; 
 when 'tis fuppoied in the Cafe, that one of the 
 Nominees is dead, and that the Annuity is no 
 more than a lingle one. 'Tis indeed polTible that 
 j^. may be the Survivor of the three, and poflible 
 that B. may, and polfible that C. may ; but that 
 every one of them fliould in Fadl become the Sur- 
 vivor, is Nonfcnfc in Terms, and in Nature In:i- 
 poffible. And as to the Parts of the Value of the 
 Annuity to be received in the Life of the two 
 Survivors, that is when the Interell in the Ellate 
 for the three Joint-Lives is determined, there can 
 
 be
 
 [ 399 ] 
 
 be no more Reafon to bring the Value of the 
 Annuity to Account oftner than once during 
 fuch furviving Lives, than there is for bringing 
 it to Account oftner than once during the three 
 Joint- Lives; and yet this Rule brings it to Ac- 
 count but once only during the Term of the 
 three Joint- Lives, and for the Reft of the Time 
 brings it to Account three Times. 
 
 If we look back and obferve what are the 
 Parts of the Value of an Annuity, which by 
 this Rule are brought to Account in the Valua- 
 tion of an Annuity for two Lives and the Sur- 
 vivor, we fhall find, that there the Value of the 
 Annuity for the two Joint-Lives once taken, 
 and fo much of the Value of the Annuity on 
 both the Lives fingly taken as remains when one 
 only is furviving; that thefe Values put toge- 
 ther are ftated as the Value of fuch an Annuity. 
 Here it appears, that the Value of the Annuity 
 for the three Joint- Lives once taken, and fo 
 much of the Value of the Annuity on the three 
 Lives fingly taken, as remains when two are fur- 
 viving ; that thcfe Values put together are ftated 
 as the Value of the Annuity for three Lives. 
 This being the Cafe, and the Rule proceediflg 
 on the fame Foundations here as before, and 
 having fhown, I think demonftrated, that other 
 and more Parrs of the Value of an Annuity for 
 two Lives are brought to Account there, than 
 do or can polTibly belong to fuch an Annuitant, 
 I might very well refer myfelf to my Reafoning 
 under the former Head ; but fince I can deter- 
 minately fay what are the particular Parrs of the 
 
 Value
 
 [ 400 ] 
 
 Value of the Annuity here which by the Rule 
 arc brought to the Annuitant's Account, but da 
 not belong to him, I will go on, and try to fix 
 thofe Parts by the Inftance following ; on the 
 Lives of A. B. and C, each of 12 Years Age. 
 We will fuppofe then, that N^. 20 grants an 
 Annuity to N^. I. for thofe three Lives and the 
 Survivor of them : and that N^. 40. grants a 
 like Annuity to N^. 2. for thofe three Joint- 
 Lives ; to N^. 3. the like Annuity for the Life 
 of A* if A' furvives either B. or C, and to 
 commence on the Death of B. or C, whichfo- 
 ever of thofe two dies firft ; to N^. 4. the like 
 Annuity for the Life of B. if B. furvives either 
 ^. or C, and to commence on the Death of A. 
 or C, whichfoever of thofe two dies firft ; and 
 to N*^. 5. the like Annuity for the Life of C. if 
 C. furvives either A' or B., and to commence on 
 the Death of A' or jB., whichfoever of thofe 
 two dies firft. The Value of the Annuity 
 granted by N''. 20. to N*^. i., as given in the 
 Tables and computed before, is 23-01. Let 
 us compute, in the next Place, what is the Value 
 of the four Annuities granted by N''. 40. to 
 N^. 2., 3., 4., and 5. The Value of the An- 
 nuity granted to N^. 2., being an Annuity for 
 the Joint-Lives of A. B. and C, is in Value, as 
 by the Tables, 10-44. ^^^e Value of the three 
 Annuities granted to N^. 3., 4., and 5., are 
 evidently the fame ; becaufe they are all to com- 
 mence en an equal Contingency, and on an even 
 Chance are to commence and determine at one 
 and the fame Time : So that a Computation of 
 
 the
 
 [ 401 ] 
 
 the Value of one gives the Value of each. The 
 Annuity granted to N^. 3. is for the Life of J, 
 in Reverfion after the Death of E. or C, and in 
 Cafe he furvivcs either of them, that is, after 
 the Determination of the Joint-Lives of B. and 
 C An Annuity for the Life of J.^ abfolutely 
 and in the Whole, is 17-20, and anx\nnuity for 
 the Joint-Lives of B. and C. is 13-01, as given 
 in the Tables and the Computation above • and 
 if 13-01 be deducted out oi 17-20, the Sum re- 
 maining will be 4-19, which is the Value of the 
 Annuity granted to N^. 3. The Annuity be- 
 longing to N^. 4. and N'^. 5. being each of the 
 fame Value as that belonging to N^. 3. mull 
 each be 4-1 p. If we put together all thefe four 
 Annuities granted by N*^. 40., they ftand thus. 
 
 The Annuity to N^. 2. 10-44 
 
 That to N^. 3. 4-19 
 
 That to N^. 4. 4-19 
 
 That to N*^. 5. 4-iy 
 
 Total Value 23-01 
 
 This being the total Value of all the four 
 Annuities put together, and the Value of the 
 fingle Annuity belonging to N^. i,, as flated in 
 the Tables, being the fame, "uiz. 23-01 ; let us 
 fee how the Cafe ilands with Regard to the re- 
 fpcdlive Grantors, N^. 20. and N^. 40. : And^ 
 to avoid Confuiion, we will fuppofe A. to die 
 firfl, B.fecond, and C. to be the Survivor. Now 
 during the Joint-Lives of A B. and C,y N*^. 20, 
 G c and
 
 [ 402 ] 
 
 and N". 40. are on the fame Foot, that is, each 
 muft pay one Annuity ; on the Death of J. the 
 Annuities to IS ^.4. and N'^. 5. are both to com-^ 
 mence, and N^. 40. will have two Annuities to 
 pay ; but N^. 20. will have one only to pay; 
 and on the Death of B. each of the Grantors 
 will be again on the fame Foot, and each of 
 them have one Annuity only to pay. This In- 
 ftance demonftrates, that the Value of the four 
 fcparate Annuities is no more than 23-01 \ that 
 the Value of the fingle Annuity for the three 
 Lives and the Survivor cannot be fo much ; that 
 during the Joint-Lives of B. and C, the two fur- 
 viving Lives, two Annuities are in the Compu* 
 tation calculated to be coming to that Annui* 
 tant, when in Fad: only one belongs to him 1 
 Confcquently, that one of thefe two Annuities 
 is brought to his Account more than belongs to 
 him. 
 
 To confirm the Juftice of this Calculation, if 
 it needs any Confirmation ; or rather to (how, 
 if Mr. Moivre's Method of computing the Value 
 of Annuities for three Lives and the Survivor is 
 purfucd, that the Cafe is and neceflarily mull be 
 fuch that this particular Chance of a double An- 
 nuity during the Continuance of the two furvi- 
 ving Lives is always inferted and brought to the 
 Annuitant's Account, I crave Leave to make Ufe 
 of the Inftance given by Mr. Richards^ Page the 
 83d, ^c. He there fuppofes a Man to give his 
 three Daughters, whom I will call J. B. and C, 
 . an Annuity of ao /. per Jfwnm each during their 
 Lives 5 and to order, that when any of them 
 
 dies,
 
 [ 403 ] 
 
 dies, the Annuity of the Siller dying Ihould 
 fall to the other two in equal Shares ; and at the 
 Death of another, that the whole Ihould belong 
 to the Survivor for her Life. He goes on thence 
 to compute the Value of this Gift to each of 
 thefe Women; and gives us the Value of the 
 Share of the eldefl Siller thus. Firll ao /. per 
 Jmuim during her own Lile. Secondly, lo/- 
 per Annum in Reverfion of the youngell, to con- 
 tinue from thence during the ]oint-Lives of the 
 other two. Thirdly, lo /. per Annum in Rever- 
 fion of the Second, to continue from thence du- 
 ring the Joint-Lives of the Firll and Third. 
 Fourthly, 40 /. per Annum in Reverlion of the 
 two youngell Lives during her own Life. I have 
 here omitted the Values, becaufe I do not intend 
 to enter into that Matter, whether the Values 
 there given, or the Method by which they are 
 computed, be right or no, though that too may 
 very well be quell ion*d ; but the fingle Point I 
 infill on here, is that more Articles are inferted 
 than do belong to, or can polTibly ever be en- 
 joy 'd by this Siller. 
 
 I will not charge the Author with ufing fome 
 Art in the ExprclFions here \ but he mull give 
 me Leave to vary them a little, and I would ex- 
 prefs the Thing thus. Firll, 20 /. per Annum to 
 A* during her own Life. Secondly, 10/. per 
 Annum in Reverfion after the Death of C, to 
 continue during the Joint-Lives of A, and 5. 
 Thirdly, 10/. per Annum in Reverfion after the 
 Death of 5., to continue during the Joint-Lives 
 of A. and C If we read the Account thus, it 
 G c 2 appears
 
 [ 404 ] 
 
 appears evidently that there are two Articles or 
 two Annuities of i O /. per Ann. each, on the feve- 
 ral Joint-Lives of A. and B., and of J. and C. .* 
 And a Man may read as he runs, that this ne- 
 ceffarily fuppofes that E. furvivcs C, and at the 
 fame Time that C. furvives B. : But fince *tis 
 impolTible in Nature that both Ihould happen, 
 'tis impolTible in Fad that A. Ihould enjoy both 
 thofe Annuities. She has undoubtedly a Chance 
 to one or other of thofe two Annuities, and to 
 determine to which of the two (he has a Chance, 
 and to compute the Value of it, may have fome 
 Difficulty in it, but there is none at all to fee 
 ilie cannot poifibly have a Chance to both. Her 
 firft reverlionary Interefl commences on the Death 
 of C, and not before, and from thence during 
 the Joint-Lives of herfelf and B. fhe will receive 
 lo/. per Annum: But when, or which Way, 
 v/ill the fecond reverfionary Interefl arife and 
 commence, or how can Ihe ever receive the lo/. 
 pr Annum on the Joint-Lives of herfelf and C, 
 when C. is already dead and gone ? We fee then 
 in this Cafe, juft in the fame Manner as in the 
 Cafe of Annuities for three Lives, that in both 
 Accounts the Annuity during the Joint-Lives 
 of the two Survivors is twice inferted, when it 
 ought to ftand there once only. From hence 
 therefore, as Mr. Richards concludes that his 
 Computation of each Sifter's Portion is a right 
 one, from its Agreement vvith the Value of the 
 whole Eftate of all the three Sifters on the Sur- 
 vivor of the three Lives ; fo from the fame 
 Agreement I conclude, fince his Computation 
 
 of
 
 ■ [ +05 ] 
 
 of the Value of each Sifter's Portion is a wrong 
 one, that the Computation of the Value of the 
 whole Eftate on the Survivor of the three Lives 
 is a wrong one likewife. 
 
 If we carry our Enquiries further, and exa- 
 mine whence this Excefs in the Valuation of thefe 
 Annuities arifcs, we fhall fee plainly that it arifes 
 from the original Hypothefis, either in part or in 
 the whole, for moft certainly it alligns more 
 Chances of Vitality on the Lives of three Per- 
 fons and the Survivor of them than do or can 
 pollibly belong to them. The Hypothefis af« 
 ferts, that an Annuitant on the Lives, fuppofe of 
 j^. B. and C. and the Survivor of them, has kvcn 
 Chances of Vitality, or feven Chances that fome 
 or one of them is living ; and I affirm on the 
 contrary, that he has fix only, and that he can- 
 not poflibly have more than i\x^ either at the 
 Commencement, or at any Time during the Con- 
 tinuance, of the Intereft in their Lives, The 
 Seven feveral Chances of Vitality aifign'd to the 
 Annuitant upon thefe three Lives and the Survi- 
 vor by the Hypothefis ftands thus ; and may be 
 rank*d in three Clafl^es, as follows. 
 
 The firft on the Lives of A. B. and C. all living. 
 
 The fecond on the Lives of J. and B, C. dead. 
 The third on the Lives of J. and C JB. dead. 
 The fourth on the Lives of B. and C, J, dead. 
 
 The fifth on the Life of A B. and C. dead. 
 
 The fixth on the Life of B. A and C. dead. 
 
 The fcventh on the Life of C J, and B. dead. 
 
 Cc 3 In
 
 [ 4o6 ] 
 
 In the Order and Manner in which I have 
 placed thefe Chances, which does not in Sub- 
 llance vary from that in which the Hypothefis 
 places them, I think it appears on the View, 
 that in the third or lalt Glafs there is one Chance 
 inferted which the Annuitant has no Right or 
 Pretence to, there being in Fad: no fuch Chance. 
 At the Commencement of the Interell in the 
 Lives of thefe three Perfons, and whilft they are 
 all living, there is undoubtedly a Chance that A. 
 may become the lafl: Survivor, that jB. may be- 
 come fuch, and that C. may become fuch : But 
 the Hypotheiis does not ftate it in this Manner ; 
 and if it did fo ftate it, then there muft be three 
 fuch Chances reckon 'd, for fo many are then fub- 
 ilfting, or there are none, and thus we Ihall have 
 nine Chances inftead of Seven. But the Truth 
 is, and the Hypotheiis ftates it in this Manner, 
 VIZ. when, and if A. is dead, that either B. or 
 C. is the laft Survivor ; when, and if E. is dead, 
 that cither A or C is the laft Survivor; and 
 when, and if C. is dead, that either A' or B. is 
 the laft Survivor. Now thefe Chances, or any 
 of them, do not commence and accrue to the 
 Annuitant, unlcfs and until either A^ or B. or 
 C is dead ; and when any one of them is dead, 
 fuppofe A.^ *tis as plain as Words and Figures 
 can r-iake it, that between the two Survivors JB. 
 and C. there remain to the Annuitant no more 
 than two Chances of Survivorfliip, 'oiz, either 
 that B. or that C. will furvive ; for after A* is 
 dead there can be no fuch Chance that A^ will 
 
 furVive j
 
 [ 407 ] 
 
 fiirvive ; it being now, to this Purpofe, the famp 
 Thing as if A. never had any Being at all. 
 
 And that the Chances of the third Clafs, that 
 is the Chances of the laft Survivor, do not com^ 
 mence 'till one of the three is dead, is as evident 
 aPofition as the former; becaufc whilft all three 
 are living, the Annuitant's Chances arc thofe of 
 the fecond Clafs, and no other ; and the Hypo- 
 thefis ftates it in this Manner. But whether it 
 does fo Hate it, or does not fo ftate it, in the 
 third Clafs, and after one of the three is dead, 
 there are then but two Chances left \ and 'tis 
 impollible to make three Chances of them, but 
 by fuppofing that they are all three living. Now 
 if we make this Suppolition, that all three of 
 them are a fecond Time living, as in Truth the 
 Hypothefis does ; the Confequence mull be, and 
 we fee the Confequence in the Computation is, 
 that the Chances on the Survivorfhip of two are 
 a fecond Time brought to Account ; for fo long 
 and as often as all three are living, or fuppofed 
 to be living, fo long and as often the Chances 
 of the Survivorfhip of two muft continue and be 
 accounted. 
 
 To fhew more apparently, that this Hypo- 
 thefis for calculating the Chance of Vitality oa 
 thefe Lives, does certainly infert more Chances 
 than belong to them, if not the individual one 
 juft mentioned ; I crave leave to add one Argu-» 
 ment more, which in Subflance is the fame, as I 
 urged under the Head of the Chances of Survi- 
 vorfhip on two Lives, and which perhaps may 
 appear more obvious and open than the former. 
 Gc 4 It
 
 [ 4o8 ] 
 
 If we make a Calculation of the Chance of the 
 Buratian of three Lives or one of them, all 
 fuppofed to be often Years of Age, according 
 to the Method prcfcribed by the prefent Hypo- 
 thelis ; fuch Duration comes out to be equal to 
 a Term of 6i Years or near it. If we make a 
 Calculation of fuch Chance and on the fame 
 Lives by the Bre/Iaw Table, and that Table is 
 by the Dodor recommended, and molt certainly 
 is a good Rule, fo far as the Table itfclf is a 
 jull one, to calculate the Chance of the Duration 
 of a iingle Life, and the Reafon which proves 
 fuch Table to be a good Rule, for calculating 
 the Chance of the Duration of a fingle Life, is 
 the very fame and equally good to prove, that 
 the Table is a good Rule to calculate the Chance 
 of the Duration of two, three, or more Lives, 
 and the Survivor of them, at Icail where the 
 Lives are of equal Age: Now by this Table, 
 the Chapce of the Duration of thefe three Lives 
 or any one of them, is equal only to a Term of 
 j8 Years \ the Method of making which Calcula- 
 tion, and the Reafons confirming the Jullice of 
 it, may be feen in page 346 of this Treatife, 
 to which I crave leave to refer myfelf. 
 
 Since then the Chance of the Duration of 
 thefe Lives, in one Way of computing, is equal 
 to a Term fhorter by three Years, than in the 
 other Way of computing the fame, and fines 
 it is impoffible both fhould be right, I mull leave 
 it with the Favourers of this Hypothefis to make 
 out, that the Error does not lie on the fame fide 
 
 on '
 
 [ 409 ] 
 
 on which the Excefs lies ; and an Error on one 
 lide or the other, there niuft certainly be. 
 
 This being the Cafe, if Mr. Motvre has de- 
 monftrated that his Rule for calculating the Va- 
 lue of Annuities for three Lives, is confonant 
 to the Dodor's Hypothefis, for calculating the 
 Chance of the Duration of three Lives, which I 
 have Reafon to believe he has not done, though 
 I never faw his Treatife ; yet, admitting it to 
 be fo, I do not fee any Thing is gained by his 
 Demonftrations, unlefs the Satisfidion of feeing 
 that fuch his Demonftrations have deftroyed that 
 very Thing, which they were calculated to 
 ftabliih; for if the Hypothefis is falfe, the Rule 
 too, if it be mathematically confonant to it, and 
 for that very Reafon, muft be falfe too. But I 
 will venture to fay, whatever Mr. Motvre has 
 demonftrated, or fuppofed himfelf to have de- 
 monftrated, that his Calculations have not prov'd 
 that his Rule is in all Inftances confonant to the 
 Hypothefis, unlefs they have prov'd, which I 
 may be very confident they have not done or 
 ever can do, that a Term for 59 Years is equal 
 to and one and the fame Thing, as a Term for 
 6^ Years. I have made a Calculation, and fet 
 it forth in page 389 above^ of the Term to 
 which three Lives of 12 Years of Age or one of 
 them, has an even Chance of Duration, where 
 fuch Chance is calculated by the Dodbor's Hy- 
 pothefis, and the Term is barely ^g Years : But 
 an Annuity for three fuch Lives, computation 
 being made at 4/. per Cent. Intereft, at which 
 Rate of Intereft Mr. Richards tells us he ftridly 
 
 purfued
 
 [ 4IO ] 
 
 purfucd Mr. Moivre's Rule, ii^ forming his 
 Tables, is in thofe Tables Itated at 23-01 \ and 
 the Value 23-01 is equal to a Term ot 65 Years. 
 Now this Term of 6^ Years, for which the 
 Annuity muft continue, or the Value of it can- 
 not be 23-01, is not only monltroufly abfurd, as 
 making the Annuity to have a Chance of Conti-^ 
 nuance, longer than anyone of theLiveson which 
 it depends has any Chance of Continuance, 
 but it demonftrates that Mr. Momreh Rule for 
 calculating the Value of Annuities on fuch Lives, 
 is not agreeable to the Dodor's Hypothelis for 
 calculating the Chance of the Duration of thefc 
 Lives. 
 
 And as the Rule for calculating the Value of 
 Annuities on thefe Lives, does not in all In- 
 ftances agree with the Hypothefis for calculating 
 the Chance of the Duration of fuch Lives \ for 
 we fee here it varies in the moft material Point ; 
 fo neither does this Rule in any Sort agree with 
 the Rule which the Doftor prcfcribes, for cal- 
 culating the Value of Annuities on thofe Lives, 
 The Dodor's Rule for calculating the Value of 
 Annuities on two, three, or more Lives, is the 
 fame in Subftance, with that which he ufes for 
 calculating the Value of an Annuity on a fingle 
 Life, iiiz^ by eftimating the Annuity on a Life 
 or Lives, as an abfolute one for fo many Years as 
 fuch lingle Life, and where more Lives than one, 
 for fo many Years as any one of the given Lives, 
 can poifibly have a Continuance ; after having 
 dcduded thereout a Value corrcfponding to the 
 Chances of Mortality, which in each Year of 
 
 fuch
 
 [4X1 ] 
 
 fuch Life or Lives may have arifen, or incur'd 
 during fuch poflible Continuance. By this 
 Means, as well in the Cafe of a fingle Life, as 
 of two or more Lives, the Chance of the Dura- 
 tion of the Annuity in the Lives, is not ade- 
 quate to, but is always lefs, or ihould be Icfs, 
 than the Chance of the Duration of the Lives 
 themfelves ; and the Dodor does not alledge or 
 fuppofe, tbat his two Rules would produce one 
 and the fame Term for the x\nnuities and for 
 the Lives, though if they had concurred in fuch 
 Produce, I do not think fuch Concurrence would 
 have been any Objedion to the Rules ; but Mr. 
 Moivre's Rule, whatever he may alledge or 
 might mean, does not agree with either of them. 
 The Do6tor*s Method for calculating the 
 Value of Annuities on two or more Lives, as 
 he himfelf very jullly complains, requiring a 
 very long and tedious Calculation, and being at- 
 tended likewife with foine Intricacies, Mr. Moivre 
 has found out, or imagines he has found out, 
 another Rule for this Purpofe, which is lefs dif^ 
 ficult and more concife, and which is furmifed 
 to be built on the Doctor's Foundations, and 
 agreeable thereto. Whether there be any and 
 what Agreement between thefe two Rules, we 
 ^all fee prefcntly j but in the Values which 
 they feverally produce, as the Values of Annui- 
 ties on the fame Lives, and in the Terms to 
 which thofe Values correfpond, fure I am that 
 they differ very widely, and that Mr. Moivre's is 
 by much the worft. I have by the Doctor's 
 Rule made a Calculation^ and a tedious Piece of 
 
 Work
 
 [ 412 ] 
 
 Work it was, of the Value of an Annuity for 
 two Lives, and for three Lives, and the Survi- 
 vor, all fuppofed to be of ten Years of Age, 
 and at 61, per Cent. Intereft; and the Value of 
 the firll Annuity is 15-41, which is equal to a 
 Term of 45-0-00, and the Value of the fecond 
 is 1 5-9 1 5 vvhich is equal to a Term of 55-0-00. 
 If we confult Mr. Kichards*s Tables for the 
 Value of Annuities on two Lives, and three 
 Lives of 12 Years of Age, which is the Age at 
 which his Computations are made ; and he alTures 
 us, thofe Tables were framed by Mr. Moivre's 
 Rule, and upon Tryal, I am fatisfied that they 
 are rightly and juftly fo con{lru6ted ^ we find 
 the Value of the fiill: of thefe Annuities, <viz. 
 that for two Lives to be 15-59, which is equal 
 to a Term of 48-0-00, and the Value of the 
 iecond, dIz. that for three Lives to be 16-19, 
 which is equal to a Term of 61-0-00. Now 
 the Value of an Annuity muft be greater, and 
 the Term correfponding to it longer, where the 
 Annuity depends on two, or on three Lives of 
 10 Years of Age, than where it depends on 
 the like Lives of 12 Years of Age, and if the 
 Rules which produced thefe Values and Terms 
 were confonant to each other, would certainly 
 be fo ; whereas we fee here that the Annuities 
 depending on the elder, and confequently the 
 ihorter Lives are of a greater Value, and equal 
 to a longer Term, than thofe which have their 
 Dependance on younger, and confequently longer 
 Lives; the one an even three Years, and the 
 other fix Years ; which is not only very abfurd, 
 
 but
 
 [413] 
 
 but evidently demonllrates that the two Rules 
 are not the fame in Subltance; but that Mr. 
 Moivre's of the two, is the moll faulty, as mak- 
 ing an Addition to a Value already too great, 
 and to a Term already too long. 
 
 Having fufficiently proved that Mr. Mom-e's 
 Rule, for calculating the Value of Annuities on 
 three Lives, does not, in all Circumftances, 
 and in thofe which are moft Material, agree, 
 either with Dr. Halle/s Hypothecs for calculat- 
 ing the Chances of Vitality on fuch Lives, or 
 his Rule for computing the Value of Annuities 
 on fuch Lives, lince his Rule produces fuch a Value 
 and Term for thefe Annuities, as varies greatly 
 from the Value and Term which the Do6bor*s 
 Hypothelis or Rule produces : Let us examine 
 now wherein thefe two great Authors agree, for 
 in one Particular at leaft, they certainly do agree; 
 but it is in a very unhappy Particular, 'viz. the 
 moft weak Part of the Dodor's Hypothelis. 
 This, as I have before obferved upon it, in ftat- 
 ing the Chances of Vitality on three Lives, has 
 Hated them at feven, whereas there are no more, 
 and I think I have prov'd beyond all Contradic- 
 tion, that there can be no more than fix ; and 
 herein the Rule of the one, is confonant to the 
 Hypothelis of the other ; for it makes Account 
 of fuch fcventh Chance, and inferts a Value in- 
 Head of, and as an Equivalent for, fuch Chance. 
 
 There being then this Confonance between 
 them, it may be ask'd. What gives occalion 
 that they produce fuch a Variety in the Values, 
 and in the Terms of the Annuities on thefe 
 
 Lives
 
 [ 414 ] 
 
 Lives ? The Anfwer to this Quellion will plain- 
 ly fhew, there is a Fault in Mr. Moivre*s Rule, 
 conlider'd feparately and dillindly by itfelf, over 
 and befides the Defed which is in it, in that 
 Inftance wherein it agrees with the Dodtor's 
 Hypothefis. The fupernumerary Chance of 
 Vitality, which the Hypothelis brings to Ac- 
 count on three Lives, is a Chance arifing be- 
 tween^e Determination of the Intereft on the 
 three Lives, and the Commencement of the In- 
 tereft on the fingle Life, that is during the In- 
 tereft of the two furviving Lives ; for it gives 
 three Chances of Survivance between thofe two 
 Nominees, when there can be no more than two; 
 by which means the Chance of Duration of one 
 or other of the three Lives, is extended beyond 
 its due Limits, for a certain Number of Years 
 equal to fuch extraordinary Chance. Mr. Moivre 
 enters into the fame Notion, fuppofes the fame 
 fupernumerary Chance, and in his Rule inferts 
 a Value correfponding to fuch Chance, and in 
 the fame Place : And very likely he may have 
 demonltrated, for in Fa(5t the Cafe is fuch, that 
 his Rule does infert fuch and no other, or more 
 Values on fuch Lives, than correfpond to the 
 Chances on fuch Lives, which the Hypothefis 
 has inferted. But the Chance extraordinary in- 
 duced in one, is a Thing {o different from the 
 Value extraordinary induced in the other, that 
 thefe Things, fuppofcd and fccming to be the 
 fame, create the Difference of the Values and 
 Terms on thofe Lives, according as they arc 
 computed by the one or the other of the Rules. 
 
 That
 
 [ 4^5 ] 
 
 That this Chance in the Hypothefis, is of 
 a different Nature and Account from the 
 Value in the Rule, will appear from hence. 
 The Hypothecs, by inducing this Chance too 
 much, inlarges the Term, to which all three 
 Lives arc equal for a Term, fuppofc of feven 
 Years, and whether this be the exadt Number 
 of Years, by which fuch Term is inlarged, 
 is not here material ; but a Term of feven 
 Years, or any other Number of Years, whe- 
 ther inferted during the three Joint-Lives, or 
 during the two furviving Lives, or during the 
 Life of the fingle Survivor, or wherever it 
 be inferted, does no more and goes no far- 
 ther, than to add feven Years too much to 
 the Term to which thof» Lives are equal. But 
 where a Value correfponding to fuch fupernu- 
 merary Chance, or to fuch additional Term, is 
 to be added to fuch Account, or is inferted 
 more than belongs to fuch Account, there is 
 great Difference, whether fuch a Value, as corre- 
 fponds to the like Term, be taken during the 
 three Joint- Lives, or during the two furviving 
 Lives, or during the fingle furviving Life, or in 
 Reverlion after them all. A Term of feven 
 Years added, let us fuppofe it inferted where 
 we will, is only an additional Term in Rever- 
 flon after the End of a prior Term, and inlarges 
 fuch Term no farther than the feven Years fo 
 added j but a Value equal to feven Years, if 
 taken and inferted in the middle of a longer 
 Term, is not a Value wholly in Reverfion of 
 the former Value, but ihall increafe the original 
 
 Term,
 
 [ 4i6 ] 
 
 Term, perhaps twice feven Years, and of 
 necelTity mufl increafe fuch Term more than 
 fevcn Years ; becaufe the Value of a Term in 
 Reverlion, fuppufc after twenty Years, is much 
 greater, and theTermcorrefponding much longer, 
 than the \ al ue of a Reverlion, and the Term cor- 
 refpunding, iuppofe after a Term of forty Years. 
 Now lince Mr. Moivre's Rule infer ts, as it ma- 
 nifeftly does, a Value correfponding to the Term^ 
 fuppofcd to arife immediately after the Deter- 
 mination of the Intereffc on the three Joint-Lives, 
 and during the Lives of the two furviving No- 
 minees ; the Value of a Term for leven Years in 
 Reverlion, then and there taken and added to 
 the Account, will add fuch a Value 'to the 
 V/hole, as ihall greatly exceed a Term of feven 
 Years. 
 
 For Inftance, and not that this is exadly the 
 Cafe • if the true and proper Term of thefe 
 three Lives was a Term of forty Years, the 
 Value of an Annuity for thofe Lives at 4 /. per 
 CefJt, would be 19-78 ; if to this Term we add 
 leven Years, as we fuppofe the Hypothefis to 
 do, the Term would then be 47 Years and no 
 more, and the Value of that Term is 21-05: 
 But if to the Value 19-78, we add the Value of 
 a Term for feven Years in Reverfion alter 20 
 Years, as we fuppofe Mr. Moivre's Rule to do, 
 and as in Fad it evidently does, the Value of 
 the whole Annuity will iland thus : The Value 
 of thofe feven Years in Reverlion after 20 Years 
 is 2-74, and if this Sum be added to 19-78, the 
 total Value will be 22-52, which is equal to a 
 
 Term
 
 [ 417 ] 
 
 Term of 59 Years. From hence 'tis apparent 
 that Dr. Halleys Hypothelis and Mr. Moivres 
 Rule do not agree in all Inftances ; and it fol- 
 lows likewife, though the Hypothefis were ad- 
 mitted to be right, yet that the Rule mult be 
 falfe throughout, for that one and the fame Er- 
 ror runs through the whole Scheme of it. The 
 Error I mean is, that it fuppofes that to add to 
 or fubftrad from a given Value a proportional 
 Part of it is one and the fame Thing, as to add 
 to or fubftrad from a given Term a like propor- 
 tional Part, which we have before proved, and 
 which appears from the Inftancc juft now given 
 to be entirely wrong. 
 
 Now this Defeat in the Rule has fo ftrong 
 an Operation in all Computations of this Kind, 
 that if we could fuppofe an Annuity on a fingle 
 Life, Intereft computed at 4/. per Cent.^ to be in 
 Value 19-78, and to be equal to a Term of 40 
 Years ; and if the Bre/Jaw Table be a good Rule 
 to eftimate the Duration of a Life, a Perfon of 
 10 or 12 Years of Age has an even Chance to 
 live 40 Years ; and if the Life be equal to that 
 Term, I make no doubt but the Annuity de- 
 pending on that Life will be equal to the fame 
 Term: Upon thefe Suppolitions, and if we 
 make ufe of this Rule to calculate the Value of 
 an x\nnuity for three fuch Lives, the Value pro- 
 duced will exceed the Value of the Inheritance 
 confiderably. But in Truth, to difcover the 
 Extravagance of the Eltimates of the Value of 
 Annuities on thefe Lives, when made according 
 to this Rule, and that fuch Extravagance does 
 * D d not
 
 [ 4i8 ] 
 
 tiot proceed Intirely from Dr. Halley*s Hypothe* 
 lis, but from theRuleitfelf as introducing a Value 
 inltead of a Term, we need not to go out of Mr. 
 Kichards's Tables which were conrtru6ted by the 
 Rule. If we confuk them, the Value of an Annui- 
 ty for three Lives, fuppofe of the Age of 15, as a 
 Medium between la and 17, is 23-01, Intereft 
 flated at 4/. per Cent.^ which is equal to a Term 
 of 65 Years; but if we make Ufc of the BreJIaw 
 Table to compute the Duration of thefe Lives 
 or any one of them, and I am fure thofe Tables 
 extend the Chance of fuch Duration beyond 
 the Term which can well be juftified, yet the 
 Term of thofe Lives is no more than 53 Years 
 and an half; and if we follow Y>r, Hal/ey's Hy- 
 pothefis in making fuch Computation, the Term 
 is only S^ Years and a Quarter. If we put 
 this Matter into another Light, and compute 
 upan the BreJJaw Tables how many Perfons of 
 the Age of 15 live to be of the Age of 80, and 
 here again the Allowance is fufficiently large, it 
 appears manifcrtly that not one in 15 arrives 
 at that Age ; the Number of Perfons living of 
 the Age of 15 being 628, and thofe of the Age 
 of 80 no more than 41 \ and if we divide 628 
 by 41, the Number arifing is 13 and fomething 
 over; which fhews, that to have an even Chance 
 that any one of our Nominees Ihall live to be 
 of the Age of 80, we muft nominate more 
 than 15 Perfons. But MY.Rkhards*sTdb\cs for 
 the Value of an Annuity for three fuch Lives, 
 which were formed by Mr. Moivre's Rule, fup- 
 pofe there is an even Chance that one Perfon in 
 
 three
 
 [ 419 ] 
 
 three of fuch Age will live to be So Years old, 
 for they compute the Annuity to continue for 
 6^ Years from the Time of the Grant made, at 
 which Time all the three Nominees in the Grant 
 are of the Age of 15. This, I think, is fo plain 
 and clear a Gafe, that I need not to add any 
 Thing farther to prove, what I have undertaken to 
 prove, that in this Value of an Annuity for three 
 fuch Lives, which makes it an even Chance that 
 one of the Lives continues in Being for 6^ 
 Years, more or greater Values, though it does 
 not difcover in particular what thofc more or 
 thofe greater Values are, have been carried to 
 the Account, than do or can belong to the 
 Proprietor of an Annuity for three fuch Lives. 
 
 Since then, neither Dr. Halky's nor Mr. — 
 Moivre's Rule, for calculating the Value of An- 
 nuities for two, three, or more Lives and the 
 Survivor of them, nor any Tables which have 
 been conllruded by thofe or by any other Rules, 
 do give us the right Value of them; I fhall pro- 
 ceed now to propofe my own for that Purpofe. 
 My Method, and which I believe is the only 
 one to do this within any Degree of Juflnels, is 
 to compute here, as we d^d in the Cafe of An- 
 nuities on a fingle Life, to what Term the given 
 Lives or one of them has an even Chance of Du- 
 ration ; for the Annuity moll certainly depends 
 on and mull continue as long as any of the Lives 
 continues ; and when we know for what Term 
 the even Chance is that any one of the Lives 
 and the Annuity attendant on it will continue, 
 we know on the fame even Chance what the Value 
 D d 2 ' cf
 
 [ 420 ] 
 
 of the Annuity is, for 'tis equal to that Term 
 to which the Lives are equal. 
 
 To compute the Term to which two, three, 
 or more Lives, meaning concurrent Lives and 
 the Survivor of them, are equal ; I propofe this 
 ]^vIcthod. Where two Lives are of equal Age 
 ' apd Goodncfs there can be no Difference which 
 of the two is fuppofed the Survivor, the Chances 
 of Survivorfhip between them being equal ; here 
 then we may take the Term to which either of 
 the Lives is equal, limply as it is, as Part of fuch 
 Term. The fecond or other Life we muft look 
 'upon as a Life or Term in Reverlion, as in rea- 
 lity 'it is; and when it falls into PoirefTion, we 
 muft not account of it as a Life equal to the 
 fame Term, as it might be eflimated at the Time 
 ivhcn the Interell in the two Lives firll com- 
 mences ; becaufe fo many Years are elapfed be- 
 fore it commences, as incurred during the conti- 
 nuance of the firft Life ; but we muft ftate fuch 
 fecond Life as cqual'^to fuch a Term only, as fuch 
 Life will be equal to at the Time of its fuppos'd 
 Commencement. This then would be the Term 
 to which fuch fecond Life would be equal if it 
 were a Thing certain that it did ever take Place; 
 "but there being a Contingency,- by the Death of 
 fuch fecond Perfon during the Life of the firft, 
 that it may never come into Poifcffion at all, a 
 ■Deduction is to be made out of fuch reduced 
 'Terfn' equal' to that Contingency : And thefe 
 two Terms, viz. the Term to which the firft 
 Life is equal taken fimply as it is, and the 
 .^Tenn to which the fecond Life is equal when 
 
 compute cl
 
 [431] 
 
 computed in this Manner on thefe two Reduc- 
 tions • thefe two Terms put together are the 
 Term to which the two given Lives and the 
 Survivor of them are equal. 
 
 This Method purfued will give us the Term 
 for any three or more given Lives. The Term 
 for two firft Lives being known • the third Life 
 as well as the fecond being only a reverfionary 
 Intereft, and to have its Continuance from the 
 Time when it commences, and fuch Commence- 
 ment depending likewife on a Contingency : 
 Such reverfionary Life muft be computed to 
 commence from the Expiration ot both the 
 precedent Lives, and for the Contingency of 
 Non-commencement a Dedu6tion mull be made 
 from the reduced Term, in Proportion to the 
 Number of Years by which the Lives ol both 
 the precedent Perfons, or the Terms to which 
 their Lives are equal, exceed the Term to which 
 the third or laft Life is equal. And thefe three 
 Terms, 'uiz. the Tern^oi' the firit Life, taken 
 fimply as it ilands, and the Term in both the 
 other Lives, fuch double Reductions being firft 
 made, give us the Term for the three giver> 
 Lives and the Survivor. 
 
 I do not pretend to prove this Rule right by 
 a Mathematical Demonftration, for I am not 
 Mafter enough of the common Procefles in Al- 
 gebra to enter into that Sort of Proof, and pof^ 
 libly the Thing may not be capable of it* and 
 even that Proot fometimes fails us for Want of a 
 proper Application j fo we muft content ouc 
 Iblves with an Inftance or two, which may explain 
 D d 3 my
 
 [ 422 ] 
 
 my Meaning and Method ; and at the fame Time 
 ihow the Grounds and Foundation of the Rule. 
 
 We will fuppofe then the Eftate, whofe 
 Value for two or for three Lives we are enquir- 
 ing after, to be lOO L per Jmjtim-^ the Rate of 
 Interefl to be 6 /. per Ceut. ; and the Term to 
 •u'hich a lingle Life for the different Stages of 
 of Life is equal, to be according to the Efti- 
 mate exhibited in my Table : Though at every 
 other Rate of Intereft, and upon any other fix'd 
 Eftimate of the Term to which any fingle Lives 
 are equal, the Method will be the fame, and 
 will produce the Term, to which two or three 
 Lives are equal. We will put it then that N*^. i. 
 has an Intereft in fuch an Eftate for a Term of 
 a8 Years in immediate PofTeflion; N^. 2. for a 
 fecond Term of 28 Years to commence from 
 the Determination of the firft; and N*'. 5. lor 
 the like Term to commence from the Determi- 
 nation of the fecond ; that N®. 4. has an Interefl: 
 in the like annual Eftate for the Lives o^ J. and 
 S. and the Survivor; and N^. 5. for the Lives 
 of J. B. and C. and the Survivor, all of 1 2 Years of 
 Age. Now let us examine what is the Value of 
 thefe feveral Intcreits belonging to thefe diffe- 
 rent Perfons, and compare them together; and to 
 avoid Perplexity, we will confider them feparately. 
 
 The Value of the Term belonging to N''. i., 
 being a Term for 28 Years in immediate Polfef- 
 fion, is 1340/. : the Value of that belonging to 
 N^. 2., being a Term for 28 Years in Rever- 
 fion after 28 Years, is 260/.; and the Value 
 ♦if that of N°. 3., being a farther like reverfion- 
 
 ary
 
 [ 423 ] 
 
 ary Term, is 54/.; and the Value of all the 
 three Terms, being all together a Term of 84 
 Years, is in all 1654. The Intereft of N°. 4. 
 in the Life of A, is equal to the Intereft of 
 N^. I., for they are bothfuppofed to be Eftates 
 in Poflcffion, and by my Table the Life of X is 
 ellimated to be equal to a Term of 28 Years; 
 the Value of that Intereft therefore is 1340/, 
 The Intereft of N^. 4. in the Life of B. fup- 
 pofed to be the Survivor is not equal to the 
 Intereft of N^. 2., becaufe the Intereft of N^. 2., 
 though in Reverlion, is an abfolute Term, and 
 certain both as to its Commencement and 
 Continuance ; whereas the Intereft of N^. 4. in 
 the Life of J5., if ever it takes Place, cannot be 
 computed to have a Continuance for a Term of 
 28 Years, but for fuch a Term only as the 
 Life of B. is equal to at the Death oi J,\ and 
 befides fuch Intereft may never take Place at 
 all, becaufe of the Contingency that B. may die 
 before A* At the Death of ^., that is at the End 
 of 28 Years, the Age of £. will be 40, a Life of 40 
 by my eftimatc is ftated to be equal to a Term of 
 21 Years, omitting the Fraction, and the Value 
 of a Term for 21 Years, in Reverlion after 28 
 Years, is 228 : And this would be the Value of 
 the Intereft of N^. 4. in the Life of 5., if the 
 Intereft were fure to take Place. But, there bet- 
 ing a Contingency that B. may die before A.^ a 
 farther Dedudion is to be made out of this 
 Term in Proportion to fuch Contingency; and 
 this Proportion is thus to be adjufted, 'u'tz. A, 
 and E, being of the fame Age, the Chance is 
 D d 4 juft
 
 [ 424 ] 
 
 jufl an equal one which of the two becomes the 
 Survivor ; and an equal Chance to the Whole 
 being the fame Thing as an even, Chance to a 
 Moiety, one half of this Term of ai Years 
 muft be (iedudled or withdrawn, and then lo 
 Years and an half will be left, as the Term to 
 which the Life of B. is equal ; and the Intereft of 
 ]S°. 4, in the Life of JB., will be in Value J4p/.; 
 and his Intereft in the two Lives of J. and B, 
 will be equal to a Term of 38 Years and an half, 
 in Value 1489/. 
 
 If we proceed in this Method and compare 
 the Intereft of N^. 5. in the Life of C with the 
 Intereft of N^. 3. in his reverfionary Term of 
 28 Years, it will appear on the Comparifon that 
 the Intereft of the former falls much fliort of 
 the Intereft of the latter, for the fame Reafons 
 and under the fame Heads as in the Cafe pre- 
 cedent. The Intereft of N^. 5- in the Life of 
 C. cannot be computed equal to 28 Years, be- 
 caufe at the Commencement of the Intereft of 
 N^. 5. in the Life of C, which is after the De- 
 termination of the two precedent Interefts in the 
 Lives of J. and 5., there can be no Pretence 
 that C. ftiould have an even Chance to live 28 
 Years ; but his Life moft be computed equal to 
 a Term anfwerable to the Age he will be of at 
 the fuppos*d and computed Time of the Ex- 
 piration of the former Lives j and it being fup- 
 pofed that A lives 28 Years, and computed 
 that B. will live 10 Years and an half, in all 38 
 Years and an half; the Age of C at the Com- 
 mencement of the Intereft in his Life will be 
 
 50
 
 [ 425 ] 
 
 50 and above, and a Life of ^o by the Eftimate 
 in my Table is ftated at 17 Years, and a Term 
 of 17 Years, in Reverlion after 38 Years, is in 
 Value 107/.: This therefore would be the Va- 
 lue of the Interell in his Life, were it fure to 
 take Place. But this being uncertain, becaufe of 
 the Contingency that C. may die before X, or 
 B., or before either of them, this Term muft be 
 reduced as before ; and the Chance with him 
 that he outlives them both being one in three, 
 and the Chance againll him that one or other 
 of the two others outlive him being two in three, 
 and a Chance of one in three to the whole Term 
 of 17 Years being the fame Thing, as an even 
 Chance to a third Part of the Term, the In- 
 terell of N°. 5. in the Life of C will be equal to 
 a Term of five Years and two thirds of a Year 
 in Reverlion after 38 Years, which is in Value 
 about 42/. : The Intereft therefore of N°. 5, in 
 the three Lives all together, will be equal to an 
 abfolute Term of 44 Years, and in Value 1531/. 
 or thereabouts. 
 
 To thefe Computations of the Term, and 
 Value of the feveral Annuities of N*^. 4. and 
 N°. 5. refpedlivcly, I fee no Objedion which 
 can be made ; and on the contrary, I apprehend 
 they are confirmed by Dr. HaUefs Maxim, 
 which with a very little Variation may be ap- 
 plied, and will be rightly applied here. The 
 Purchafor of an Annuity for the Term of a 
 given Life, muft pay for fuch Parts, and only 
 fuch Parts, of the Term and the Annuity at- 
 tending on it, as he has Chances, or the even 
 
 Chance
 
 [ 426 ] 
 
 Chance, that his Nominee is living, and that he 
 enjoys the Annuity. That the Interelt of N-:-. 4. 
 in the Life of A. is equal to a Term of 28 
 Years will admit no Difpute; becaufe it is fup- 
 pofed in the Cafe that the Life of ^. is equal to 
 a Term of 28 Years : and that his Intcreft in the 
 Life of B. is equal to the very Term which I 
 have afligned, neither more nor lefs, will I think 
 almolt as little admit of a Difpute. Upon the 
 Interelt of N^ 4. in the Life of B. there can be 
 no Sort of Pretence that it isaf any Value during 
 the Life of ^. ; or when it commences, that it 
 can, be computed to continue any longer than 
 for fuch a Term as the Life of B. is equal to, 
 or has a Chance to continue, from the Time of 
 fuch Commencement : And here again 'tis fup- 
 pos'd in the Eftimate, that the Life of 5., when 
 the Intereft in his Life commences, is equal to a 
 Term of 21 Years, he being then of the Age of 
 40. As to the Deduction made out of this 
 Term of 21 Years, by which it is reduced to 10 
 Years and an half; it muft be admitted, that 
 there i.': a Contingency that B. does not furvive 
 A-i and they being of the fame Age, *tis plain 
 that the Chance of Survivorlhip is juft equal ; 
 and I fuppofe it will be admitted without an 
 Algebraical Calculation, though it is capable of 
 bei:)g fo prov'd, that he who has an equal 
 Chance or one Chance in two to the whole 
 Term of 21 Years, has jufl an even Chance to a 
 Moiety of it ; and iince the Intereft of N". 5. 
 in three Lives, in all refpedts ftands on the fame 
 Foot; I conclude that the Terms and the Value 
 
 of
 
 [ 427 ] 
 
 of the Annuities attending on them, which I 
 have affigned to the refpedive Annuitants, arc 
 the very Terms and Values which they have 
 rcfpedively the Chance to enjoy. 
 
 If then thefc my Computations of the Term 
 and Value of thefe feveral Interefls are right, 
 the Interert of N°. 4., which is for the Lives of 
 j^. and B. and the Survivor, is equal to a Term 
 of 38 Years and an half, and is in Value 14S9: 
 And the Intereft of N^. 5., which is in the Lives 
 of A. B' and C and the Survivor of them, is 
 equal to a Term of 44 Years, and in Value 
 1 53 1, or thereabouts. To which I add by way 
 of Corollary, that the Term to which fuch two 
 or fuch three Lives and the Survivor are equal, 
 at every Rate of Intereft, whether it be higher 
 or whether it be lower, is and muft be the fame; 
 for that the Rate of Intereft has no Influence, 
 and it is impofTible it ftiould have any, on the 
 Chance of the Duration of the Lives. 
 
 I cannot forbear obferving here the great 
 Difference which there will be in the Value of 
 the Intereft of N^ 4. in the Life of 5., if fuch 
 Value be computed by taking a Moiety of the 
 Term, and what it will be if we take a Moiety 
 of the Value of fuch Term. Here, where we 
 take a Moiety of the Term of 21 which is 10 
 Years and an half, the Value correfponding to 
 fuch 10 Years and an half, which is undoubtedly 
 the Value of the Intereft in the Life of 5, is 
 144: But a Term of 21 Years, in Reverfioii 
 after a 8 Years, is in Value 2-a8, and if we take 
 a Moiety of fuch Value, we fhall have no more 
 
 than
 
 [ 428 ] 
 
 than 1-14, as the Value of the Intereft in the 
 Life of 5 ; and the Value of the Annuity for the 
 two Lives and the Survivor will be no more than 
 ^4-54) equal to a Term of 33 Years and fomething 
 over. And if we compute the Intereft of N°. 5. 
 in the Life of C on the Chance of Survivorfhip in 
 the fame Manner, by taking one third of the 
 Value of the Term when we fhould have one 
 third of the Term itfelf, we Ihall find the like 
 Deficiency ; and inltead of the Value 42, which 
 is the Value of the third Part of the Term, and 
 which on my Computation is the Value of the 
 Intereft in the Life of C, we fliall have no more 
 than ..38, as being a third Part of the Value 
 1-14: So that, in this Way of computing, the 
 Value of the Annuity for the three Lives would 
 be no more than 14-92, equal only to a Term 
 of 39 Years or Icfs. 
 
 To make this Obfervation applicable to the 
 prefent Purpofc, I muft take Notice that Mr. 
 Richards, in his Table for the Value of Annui- 
 ties on a jingle Life, makes the Term, to which 
 a lingle Life of 12 Years of Age is equal, to be 
 a Term of 28 Years or near it, Litereft com- 
 puted at 6/. per Cetit.-^ and that an even 28 
 Years is the Term to which I eftimate a fingle 
 Life of that Age to be equal, at that and at every 
 other Rate of Intereft : But in his Table for the 
 Value of Annuities for two fuch and three fuch 
 Lives, he makes the one equal to a Term of 48 
 Years and above, which I make no more than 
 38 Years and an -half ; and the other 60 Years 
 and above, which I make only 4y^Years. Now^ 
 
 fmce
 
 [ 429 ] 
 
 fincc in forming thefe Tables he made Ufe of 
 Mr. Moivre's Rule, and that Rule direds the 
 Ufe of the Value of the Terms ana not the 
 Terms themfelvcs, in making our Computations 
 of the Value of thefe Annuities; and lince, as I 
 obferved before, the Way of computing by the 
 Values produces a fhorter Term, as the Term 
 to which two and three fuch Lives fhall be equal, 
 than the Way of computing by the Term pro- 
 duces : From hence I infer, if all the Chances 
 of Vitality and the Terms correfponding to 
 them are brought to Account in my Calculati- 
 ons, as I am almoft confident they are, that in 
 thefe Calculations there are inferred, where or 
 by whofe Hands I lliall not here diiputc, more 
 Chances of Vitality, as belonging to two and 
 to three fuch Lives, than do or can belong to 
 them ; otherwife the Term to which two fuch 
 Lives would be equal, in that Way of Compu- 
 tation, would be little more than a Term of 35 
 Years ; and the Term to which three fuch Lives 
 would be equal would not only fail fhort of 
 being a Term of 60 Years, but even of a Term 
 of 39 Years. 
 
 What Values too great or too many are hy 
 Mr. Moivre's Rule here inferted fo as to fvvell 
 the Values and the Terms in thefe Annuities to 
 fo nfonllrous a Degree, we have already conli- 
 dered ; and have ihown, that upon a fiditious 
 Suppofition that A- has a Chance to furvive B.^ 
 and 5. al fb a Chance to furvive ^., and fuch a 
 Chance that both may happen and take Effect, 
 a Value anfwcring to fuch double Chance is 
 brought to the Account ; and this appears evi- 
 dently
 
 [ 430 ] 
 
 dently to be the Cafe here on thefe two Lives. 
 Here where the Value of the Chance of Surv-i- 
 vorlhip on the Life of B. only is inferted, the 
 Value is no more than 14-54; but if we add to 
 this Sum 1-14, which is ths imaginary Value of 
 the Chance of Survivorlhip on the Life of A.y 
 the total Value will be 15-68; which is the 
 Value, within few Decimals, given in Mr. Rich- 
 ards's Table as the Value of an Annuity for two 
 fuch Lives and the Survivor. Now 'tis not only 
 impolfible that A fhould in Fad furvive B. and 
 B. alfo furvive A^ and an Annuity cannot fubfiH 
 unlefs the Life on which it attends fublills ; but 
 there is this farther Error, that we have already 
 inferted in our Account a Term and a Value an- 
 fwerable to the whole Life of A. ; therefore no 
 Value or Term or any Chance of either can be 
 now remaining. And if on the Life of J, there 
 is no Chance left, and when he is dead fure 
 enough there is none, and if the Chance on the 
 Life of B' furviving is only a Chance to a Moi- 
 ety of the Term 10 which his Life is equal at 
 the Time he becomes the Survivor, which I 
 think is very evident ; it follows that my Rule 
 inferts all and the Whole, and Mr. Moivre's Rule 
 more than all and the Whole, of the Term be- 
 longing to fuch Lives. 
 
 In the Inftance already given, the Lives are 
 fuppofed to be o^ the fame Age, but the more 
 common Cafes are that they are of different 
 Ages, but there the fame Rule will give us the 
 Term to which two or three Lives are equal. 
 The Calculation is indeed fomewhat longer, but 
 {lands on the fame Reafon as in the former Cafe. 
 
 Here
 
 [431 ] 
 
 Here the youngcft of the Lives, whether they 
 be two, three, or more Lives, lliould be placed 
 firit; and the Term to which fuch firft Life is 
 equal eftimated as it itands by itfclf, muft be 
 taken as Part of the Term. As for the fecond and 
 third Life, they muft be placed in Order as they 
 are in Juniority of Age ; and out of the Term 
 to which they are rcfpcdiveiy equal at the Time 
 of their Nomination, a double Dedu6lion muit 
 be made here as before : One muft be made in 
 Regard that the Intereft in their Lives is a rcver- 
 lionary one, and commences only from the Ex- 
 piration of the precedent Kftates ; and this is 
 regulated by computing thofe Lives as equal to 
 fuch a Term as Lives are computed at which 
 arc of that Age of which the Nominees are when 
 the Intereft in their Lives refpedively commences. 
 Another Dedu6lion is to be made on the Ac- 
 count of the Contingency, that by Death inter- 
 vening fuch their Intereft may be prevented from 
 ever taking Place : And this is adjufted by fub- 
 ftrading out of the Term, when reduced in the 
 Manner before direded, fo many Years as are 
 equal to the Chance that there is againft fuch le- 
 cond Perfon that he does not furvive the firft, in 
 Cafe of two Lives ; and in Cafe of three Lives, 
 fo many Years as are equal to the Chance that 
 the third Perfon does not furvive the two prece- 
 dent ones. 
 
 In the former Tnftancc, where the Lives were 
 all of the fame Age, and where the Chance of 
 Survivorlhip againft the fecond Life was an equal 
 Chance or one in two, and the Chance againft 
 the third Life was two in three • there the Term 
 
 to
 
 [ 432 ] 
 
 to be deduced appear'd readily to be a Moiety 
 of the reduced Term in one Cafe, and in the 
 other two Thirds of the reduced Term. Here, 
 where we fuppofe the Lives to b^ of different 
 Ages, if we take three Lives whofe Term is, 
 fuppofe 28, ai, and ii Years; the Manner of 
 making the Computation is this. On the two 
 firil Lives, all the Chances put together are 49, 
 'viz. on the firll: Life 28, and on the fecond 2 1 ; 
 -and therefore the Chances of Survivorlhip with 
 the fecond Life will be ^| Parts of the reduced 
 Term of fuch fecond Life, and the Chances 
 againft him ^f Parts of the fame Term : So that 
 if we dedud thefe ^^ Parts out of the reduced 
 Term of the fecond Life, the Term remaining 
 will be the Term to which fuch fecond Life is 
 equal ; and then the Term of the firll Life taken 
 limply as it ftands, and the Term on the fecond 
 Life thus doubly reduced, when both put toge- 
 ther, give us the Term to which the two Lives 
 and the Survivor of them is equal. On the three 
 Lives, all the Chances put together are 60, 'viz^ 
 on the two firft Lives 49, and on the third Life 
 I j; and therefore the Chances of Survivorship 
 with the third Life will be \~ Parts of the reduced 
 Term on fuch third Life, and the Chances againft 
 him f J Parts of the fame Term : So that if we 
 dedud thefe ^^ Parts out of the reduced Term 
 of fuch third Life, the Term remaining will be 
 the Term to which fuch third Life is equal ; and 
 then this remaining Term, added to the Term 
 for the two Lives, gives us the Term to which 
 all three Lives and the Survivor of them is equal. 
 We may calculate the Term to which this fecond 
 
 and
 
 [ 433 ] 
 
 ahd tiiird Life are equal, on the Chance of Siir^ 
 vivorihip, in another and more eafy Method^ 
 which will produce one and the fame Term, viz, 
 by taking f ^ Parts of the reduced Term on the 
 fecond Life as the Term to which fuch fecond 
 Life is equal, and ^t Parts of the reduced Term 
 on the third Life as the Term to which fuch third 
 Life is equal. 
 
 To make my Rule and the Manner of ufing 
 it more intelligible, we Will calculate the Chance 
 of the Duration of two and of three Lives of 
 different Ages. We will fuppofe then J. to be 
 12 Years old, and his Life to be equal to a Term 
 of 28 Years: B, to be 40 Years old, and his 
 Life to be equal to a Term of ai Years : And 
 C. to be 60 Years old, and his Life to be equal 
 to a Term of 11 Years, omitting Fractions, 
 The Life of A.y taking that Life firfl, is equal 
 to is Years ; and this itands fimply as it is, and 
 is the firft Term. At the Knd of 28 Years, 
 when the Life of J. is fuppofed to determine, 
 B. will be 68 Years old, and his Life will be 
 then equal to a Term of 8 Years, and this would 
 be the Intereft in his Life, if it were fure to take 
 Place. The Chance of its taking Place or notx 
 taking Place is ^| againft him, and %^ with him ^ 
 therefore his Intereft is equal to fj Parts of a 
 Term of eight Years. The Term to which 
 fuch Intereft is equal is adjufted by multiplying 
 the firft and laft Terms, that is 21 by 8, and 
 dividing by the middle Term, that is by 49 ; and 
 it comes out to be a Term of 3-1-63. This 
 th?n is the Term to which the Life of B, is 
 E e ?qual .
 
 C 434 ] 
 
 equal ; and if this be added to the former Term 
 on the Life of J., the Whole will be 31-1-63 ^ 
 and the Intereft in the Lives of J. and B. and 
 the Survivor of them, will be equal to that 
 Term. After the End of this Term of 31-1-63, 
 C. the third Life will be 90 Years old; the 
 Term in his Life will be equal to one Year only, 
 and is the Intereil in his Lite, if it were fure to 
 take Place. The Chance of its taking Place is 
 49 in 60 againlt him, and 1 1 in 60 with him ; 
 his Interelt therefore is equal to Is Parts of one 
 'i^: //::^/^;r Year, or 365 Days, which comes out to be 67 
 Days: And if all thefe Terms, wa. 28-0-00, 
 3-1-63, and 0-0-67, ^'^ put together, the total 
 Term will be 3 1-2-39 ; and is the Term to which 
 the Lives of A. B. and C and the Survivor is 
 equal. 
 
 That the Annuitant who has an Eflate for 
 thcfc two or for thefe three Lives and the Sur- 
 vivor, in this Way of computing his, Intereft, 
 has all the Parts of the Terms of thefe two or 
 thefe three Lives for which he has a Chance 
 that his Nominees, or any one of them, are living^ 
 and if he has all the Terms he will enjoy all the 
 Parts of the i\nnuity which he has a Chance to 
 enjoy, lince the Annuity vi'ill attend on the 
 Term, is proved, if not mathematically, yet 
 by a little common Reafon. The Life of J, is 
 fuppofed to be equal to a Term of 28 Years ; 
 and a Term of 28 Years is accordingly aflign'd 
 to the Annuitant as Part of his Term. During 
 the Life of ^. the Interelt of the Annuitant in 
 the Lives of B. and C. is, in fome Senfe, no 
 
 other
 
 [ 435 ] 
 
 Other than a dormant one, for as to the Receipt 
 and Payment of the Annuity, the Being of one 
 Life avails as much as the Being of all three 
 Lives : But the Perfons themfelves, during this 
 Time, are living on and advance in Age, and 
 their Lives decay and are impair'd in Proportion 
 to the Duration of the firit Life ; fo that their 
 Lives, at the Commencement of the valuable 
 Intereft in them, can be reputed equal to fuch a 
 Term only as their Lives are refpediively equal 
 to at the Commencement of fuch Intereft. In 
 Purfuance of this we have reduced the Term to 
 which the Life of B. is equal to a Term of eight 
 Years, and the Life of C. to a Term of one 
 Year; thofe being the Terms refpedively to 
 which the Lives of B. and C, rcfpcdively arc 
 equal at the Commencement of the profitable In- 
 terefl in their Lives. 
 
 As to the Contingency of Survivorfhip, I 
 have inferted all the Chances that can poiTibly 
 arife on each of them, all I mean from which the 
 Annuitant can receive any Benefit ; and I may 
 be confident that my Calculation of the Terms 
 anfvvering to fuch Chances is right, fince 'tis in- 
 flru(5ted, as I apprehend, by the common Rules 
 of Chances, and is the fame which Dr. BalJey 
 makes Ufe of, with this Difference, that I apply 
 the Rule to afcertain a Term, which he applies 
 to afcertain a Value. In my Way I argue thus. 
 The Number of Years to which the Life of B. 
 is equal, when the Chance of Survivorfhip is to 
 be calculated, is no more than eight Years in 
 the whole, and that is all the Term he can have 
 E e 2 if
 
 [ 436 ] 
 
 ff all the Chances of Survivorihip were with hiifij 
 and he were in Fa6t become the Survivor. The 
 Life of J. is fuppofed to be equal to 28 Years, 
 and the Life of B. to be 21 Year.?; fo that the 
 whole Number of Years or Chances between 
 them both is 49; of thefe 49 21 are with Bi 
 that he becomes the Survivor, and 28 are againf! 
 him that he does not: Therefore the Chance of 
 B. is a Chance to ^^ Parts of a Term of eight 
 Years, which on a Computation proves to be a 
 Term of 3-1-63. Dr. Halley reafons thus. The 
 total prefcnt Value of an Annuity, fuppofe o^ 
 ioooo/. for one Year is 9434 /. ; and no more 
 than that Sum is fuch an Annuity worth, Inte* 
 reft competed at 6 /. per Cent.^ where the Annui- 
 tant has all the Chances there are to receive it, 
 iind the Annuity were an abfolute one. The 
 whole Number of Chances, fuppofe in the firft 
 Ycrtr of the Life of a Pcrfon of 10 Years old, 
 is 661 ; of thefe Chances S^'^^ are with the No- 
 minee that he does forvive the Year, and % are 
 againll him that he does not furvive the Year : 
 Therefore the Chance of the Annuitant to the 
 Annuity for the flrfr Year is f ^f Parts of the 
 Sum of 9434, which is ^^ 1 9. This is my Way 
 of Px.ealcning cxadly, only differently applied :' 
 From whence I conclude, that the Annuitant 
 has all the Terms, with the Annuity attending 
 on them, which he has the Chance that any one 
 cf his Nominees is living. 
 
 To prevent any Surmifc which may be itiadc,. 
 that I have myfclf, ' under a former Head, dif- 
 approved this Way of Reafcning ifvDr. Halley i 
 
 as
 
 [ +37 ] 
 
 IS I may feem to fome to have done, I obfcrve, 
 that I do admit the Way of Reafoning to be 
 right in general, but I urge, that 'tis wrongly 
 applied in the Manner and in the Inllance in 
 ■which 'tis there applied, I have aflerted, in 
 Oppofition to the Do6tor, that 93 19 /. is not the 
 Value of the Annuity for the firll Year of the 
 Life of a Perfon of 10 Years old, where the 
 Annuity is for his Life abfolutely; but where 
 the Annuity is tor one Year, and the firft Year 
 only of the Life of a Perfon of fuph an Age, if 
 he Ihall fo long live, and then to determine ; 
 there I admit that 9319/. is the true Value of an 
 Annuity for fuch one Year, and the Annuity 
 fo determinable, The Difference of thefe two 
 Cafes is this : In the firft Cafe I contend that 
 there are no Chances of Mortality arife which 
 are to be dedu6ted in the firft Year of the Life 
 of the Nominee, but that fuch Chances as arc 
 computed to arife in fuch firft Year arc to be 
 difcounted out of the whole Life at large ; but 
 in the latter Cafe, the Chances of Mortality 
 arifing in the firft Year muft be then allowed and 
 dedudted, or they cannot be allow'd and deduc- 
 ted at all, the Intercft expiring with the Year. 
 Therefore, where there are Chances of Morta* 
 lity againft an Annuitant, and an Allowance and 
 Dedu(5lion ought to be made for them, there 
 the Reafoning is rightly applied ; but where 
 there are no Chances againft him, or none that 
 are ro be then and there deducted, here it will 
 be wrongly applied. But in the Cafe on the 
 Life of B.y and when his Chance of furviving A^ 
 E e ^ conges
 
 [ +38 ] 
 
 comes under Confideration, 'tis obvious, and 
 very certain, that there are many Chances againft 
 him that he does not out-live a8 Years and be- 
 come the Survivor, and that thofe Chances muft 
 then and there be dedu6ted, or there is no Room 
 to dedud them at all, and therefore the Reafon- 
 ing is juftly applied. 
 
 I am fenfible, that by this Method, fo far as 
 we have hitherto gone into it, no Provifion is 
 made for the pofTible Contingency that a diife- 
 rent Perfon or Perfons, where they are of ufi- 
 cqual x^ges, may become the Survivors or Sur- 
 \^ivor y which yet may produce a Variance in the 
 Term for the two or the three Lives refpedive- 
 ly. Now the great Difficulty in adjufting the 
 Value of Annuities for two or for three Lives 
 and the Survivor, is to afcertain what is the 
 Chance, or the Value of the Chance, on the pof- 
 fible Contingency of Survivorfliip, when two of 
 three Nominees of unequal Ages are to become 
 the Survivors, and when one of two is to be 
 left the fole Survivor. Upon this I obferve, 
 that fome Perfons have been fo careful to make 
 a Provifion fufficient for this polBble Contingen- 
 cy, that in the Cafe of two Lives they have fu{ - 
 pofed both the Nominees to be the Survivor ; 
 and in the Cak of three Lives, all the three to 
 be furviving when there are two only. Now 
 this muft certainly be wrong ; for though on 
 two Lives there is a Contingency that j4. may 
 furvive B., and a Contingency that B. may fur- 
 vive A.J yet 'tis impoifible it Ihould happen that 
 J. does ia Fac^ furvive B., and B, alfo furvive 
 
 J.'
 
 [ 439 ] 
 
 J. : And on three Lives, though there is a 
 Chance that A.^ and that £., and thatC, whilft 
 all three are living, may be the lull Survivor, 
 yet when one of the three is dead, there is no 
 Chance that fuch one who is dead can be thelall 
 Survivor. Notwithftanding which, in fome Cal- 
 culations, and from the bell of Hands too, we 
 have feen that on two Lives the Intereft in the 
 Lives of both is computed as in Being, when 
 the Intereft of the one or the other is extinguifti'd 
 by Death ; and that on three Lives, a Chance 
 of all three becoming the laft Survivor is infert-- 
 ed at a Time when one of the three is fuppofed 
 to be dead and gone. 
 
 On my Method I obferve ; whether the Cal- 
 culation be made on two, on three, or any other 
 Number oi Lives propofed, if the Perfons are 
 all in the fame Period of Life, and their Lives of 
 equal Goodnefs, the Contingency of Survivor- 
 fiiip amongft them is entirely out of the Quefti- 
 on, becaufe there can be no Difference in the 
 Chance : And with Regard to fuch Lives, I 
 may venture to fay, though I will not undertake 
 to prove by Algebra, that my Method is ma- 
 themadcally right. Where Perfons are in diffe- 
 rent Periods of Life, this Contingency of Sur- 
 vivorfhip muft undoubtedly have fome Influence 
 on the Term to which the Lives are equal, or 
 on the Chance of the Term to which they may 
 be computed equal : And yet it will not make 
 fuch a Variance in the Term, or in the Chance 
 of the Term, as may commonly be apprehend* 
 E e 4 ed.
 
 [ 440 ] 
 
 cc^, or fuch 4s perhaps would be thought of 
 Moment. 
 
 But be the Influence more or lefs ; the Rule, 
 fo far as we have already confider'd it, dire(5ts 
 that the yoUngcfl oC the given Lives be taken 
 as the firft, that the reft be placed as they arc 
 in Juniority of Age, that we compute the Term 
 to which thofe Lives arc feparately and fpccef- 
 fivcly equal, that we make this Computation 
 once only, and that we take the feveral Terms 
 fo arifing, when all put together, as the Term 
 to which the two Lives or the three Lives and 
 the Survivor are equal. Now 'tis obvious that 
 this Order of placing and computing the Chance 
 of the Term On unequal Lives does not provide 
 for all the feveral Contingencies of Survivorlhip 
 which there are on thofe Lives ; but yet I fee 
 Reafon to think that, in general, it muft pro- 
 duce a longer Term, as the Term to which thefe 
 Lives are equal, than any other Order of placing 
 them will produce. Upon the Calculations I 
 have made for this Purpofe, and I have made 
 great Variety of them, I have found that it does 
 produce either one and the fame Term as any 
 other Order of placing them produces, in fome 
 Inftances to the Exa(5lnefs of a finglc Day even 
 in the Cnfc of a large Number of Years, but 
 moft commonly produces a longer Term that! 
 any other Order of placing and computing does. 
 
 However, fincc there are Inftanccs, for fomd 
 few have f^l'en under mv Obfervation, where 
 this Method of proceeding will not give us the 
 Ipngeft Term as the Term for fuch Lives j we 
 
 nauft
 
 [ 441 ] 
 
 muft vary the Pofition of the Lives given as oft 
 as thofe polTible Contingencies on the Lives ad- 
 mit of; and then proceed, in the fame Method 
 as before, to calculate to what Term each of 
 the given Lives will be pqual in all the poflible 
 Shapes of Sprvivorfhip : And that Term, which, 
 lipon repeating the Calci^lation fo often*, and 
 which, upop putting together the feveral Terms 
 fo collected, comes out to be the longeft Term 
 foT the Lives nominated, is the Term to which 
 thofe Lives are equal : And this, I think, allots 
 to the Annuitant juft the Term which he has a 
 Right to, viz. the beft of the Chances in the 
 Term of the giycn Lives. 
 
 Mr. R.icbards^ in his Tables for the Value of 
 Annuities on two and on three Lives, which he 
 form'd by Mr. Mohres Rule, has allign'd fuch 
 a Value of Annuities for fuch Lives as 1 have 
 juft mention'd, 'viz. for two Lives of ii and 42, 
 and for three Lives of 12, 42, and 62, as that 
 the Terms correfponding to thofe Values, Inte- 
 ^eft computed at the Rate of 6 /. per Cent, do 
 much exceed the Terms to which I have made 
 fuch Lives, or Lives rather better, to be equal : 
 For the Value of the Annuity for the two Lives 
 given in the Tables is equal to a Term of 35 
 Years and above, which in my Way and on my 
 Lives I make no more than 31- 1-63; and the 
 Value on the three Lives there given is equal to 
 a Term of 40 Years, or near it, which in my 
 Lives I make no more than 31-2-39. And yet 
 I obferve, on his Tables for the Value of Annu- 
 ities on a fmgle Life, computed at this Rate of 
 
 Interelt,
 
 [ 442 ] 
 
 Intercft, and theObfervation holds good on Dr. 
 Halleys Table for the V'alue of the like Annui- 
 ties, and at the fame Rate of Intereft, that the 
 Term correfponding to the Value, given in thefe 
 Tables, as the Value of the Annuity for every 
 one of thefe Lives, when fingly taken, is a Term 
 Ihorter than I affign to thofe Lives, and the Life 
 of 42 in particular falls Ihort of the Term to 
 which I compute fuch Life to be equal by five 
 Years, or near it : The Confequence of which 
 muil be, that the Term correfponding to the 
 Value of all the three Lives fingly taken and 
 put together, in their Way of computing, muft 
 fall fhort likewife o^ the Term which I alfign to 
 the fame three Lives fingly taken and put toge- 
 ther. Therefore, fince the Term, to which thefe 
 three Lives fingly taken and put together is equal, 
 in their Way o( Eftimate, falls fo much fhort 
 of the Term, to which thofe Lives fingly taken 
 and put together is equal, in my Way of Efti- 
 mate : And fince the Term to which thefe three 
 Lives taken as combined and furviving Lives, in 
 their Way of computing is equal, fo greatly ex- 
 ceeds the Term to which thefe Lives taken as 
 combined and furviving Lives, in my Way of 
 computing is equal : From hence it neceffarily 
 follows, either that in my Account of the Term 
 tor the combined and furviving Lives I have not 
 inferred all the Parts of the Terms in thefe Lives 
 which the Annuitant has a Chance to enjoy, or 
 that Mr. Richards has inferted more than all the 
 Parts of the Value of the Annuity which he has 
 a Chance to receive. 
 
 Upon
 
 C 4+3 ] 
 
 Upon either of our Computarlons 1 will make 
 no farther Refledions, having infifted very large- 
 ly on them before * and Ihall add only this. 
 Calculations in Algebra may be right, infallibly 
 right ; but then they mull be grounded on Prin* 
 ciples or Suppofitions ; and thofe Principles, un- 
 lefs they arc feif evident, and the Suppofitions, 
 unlefs they are fuch as all Mankind will concur 
 in, muft be proved ; and by a Proof which per- 
 haps may arife from fomething which has no Rc"* 
 lation to nor can be meafured by a mathematical 
 Rule: And therefore I leave it with the Authors 
 to make good the Suppofitions on which their 
 algebraical Calculations are founded. 
 
 I fhould here put an End to this Difcourfc ; 
 but I cannot forbear firil taking Notice of ano- 
 ther algebraical Calculation, made by a different 
 Hand, to adjuft the Credibility of human Tef- 
 timony; where we meet with a moil wicked 
 Application of the Mathematicks. It is not in- 
 deed diredlly and immediately to the prefent 
 Purpofe, but it will ferve to Ihovv us that Ma- 
 thematicians fometimes make very ilrange Sup- 
 pofitions, and then build thereon very good 
 Calculations, and from right Calculations fome- 
 times deduce very wrong Confequences : And 
 thus far the Calculations have an Affinity the one 
 to the other, that this Writer makes ufe of the 
 Value of Money as a Meafure to afcertain the 
 Duration of the Credibility of human Tellimo- 
 ny much in the fame Manner, which Dr. Halley 
 and Mr. Moivre make Ufe of it to determine the 
 
 Chance
 
 [ 444 ] 
 
 Ghancc of the Duration of a Life or Lives ♦ 
 and I think with equal Succefs. 
 
 I have the rather chofcn to make fome Re-» 
 marks on this Calculation, becaufe it has been 
 taken Notice of by the Author of that celebrated 
 Piece, Chnjltamty as old as the Creation^ and I 
 fuppofe he would be underftood that it was ap-^ 
 plicable and of Moment to his Purpofe. He 
 does not indeed direclly make an Application of 
 it \ but whether he did not know how to make 
 it, or meant to leave it to his willing Reader to 
 do it, I can't fay; but I believe no Man makes 
 any Doubt of his Good-Will. 
 
 The Propofition is this ; which I Ihall give 
 you in the Author's own Words, becaufe I think 
 it to be a very extraordinary one, ilnce 'tis not 
 only founded on chimerical Suppolitions, but 
 the Conclulion is diredtly wrong. 
 
 Jt the Rate of 6 1. per Cent, hitere/i^ the pre- 
 feiit Vahie of any Sum payable after twelve Tears is 
 but half the Sujn ; fo if the Proportion of Certitude 
 tranftnittcd by each Reporter be ^-f^ the Propor- 
 tion of Certainty after tv-nlve 'Tranfmifftons will he 
 but as half \ and it i£ill grow by that Ti?ne an 
 equal Lay whether the Report be true or no. 
 
 If we admit his firft Pofition, on the preferit 
 Value of a Sum of Money payable in future^ to 
 be true, as moft certainly it is ; yet it is 
 grounded on a Suppofition of his own making, 
 to ferve a Turn, for we may fuppofe Money at 
 any other Rate of Interefl as well as at 6 /. per 
 Cetit. : And if we admit the fccond Pafition, on 
 the Proportion of Certainty or Credibility, to 
 
 be
 
 [ 44S ] 
 
 be true likewife, and here the Suppofitlons ar^ 
 fuch that no Man furely but bimfelf co^ld make : 
 Yet the Conclufion, 'viz. it will grow an equal 
 Lay whether the Report be true, has neither 
 Senfe nor Truth in it. Wc do indeed, in a 
 common Way of fpeaking, fay that fuch a Re- 
 port is true, or fuch a Report is falfe ; but the 
 Meaning of that Expreffion is, that the Fa6t 
 reported is true or is falfe : And furely here, 
 vvhere the Propofition is to be demonftrated by 
 a mathematical Proof, it would have been much 
 more proper, at leaft, to have concluded that it 
 would be an even Lay whether the Fait report- 
 ed were true, and not whether the Report were 
 true ; for moft certainly there may be a true 
 Report of a Fad not true. 
 
 If then we underltand this Propofition in fuch 
 a Manner as to make Senfe of it, yet *tis neither 
 true nor juft, becaufe there is plainly more in the 
 Ccnclufion than in the Premifes; for the Premi- 
 fes and the fubfequent Calculations do not Ihow, 
 nor are they intended to fhow the Duration of 
 the Truth of a Fa6l reported, for that is per- 
 petual, but the Duration of its Credibility j and 
 the Truth of a F'adt and the Credibility of it arc 
 very different Things. Matters of Fa6l may be 
 true, and many are fo, which yet are not, nor 
 ever were credible, to fome Perfons at leaft, for 
 want of proper Evidence to prove them j and 
 if once they were credible they may ceafe to 
 be fo, and will ceafe to be fo, if and when the 
 Evidence is loft or funk ; but if once they were, 
 they will always eontinuc to be true. Here then 
 
 Tt)ie
 
 [ 446 ] 
 
 the Author manifcftly over-docs the Matter, 
 when he concludes to the Verity or Reality of 
 a Fad:, whereas his Undertaking was, and he 
 could undertake, only to compute its Credibility. 
 Therefore the Conclulion fliould have been this, 
 and no more than this : Alter twelve Tranfmif- 
 iions it will grow an equal Lay whether the f adt 
 reported be credible or no. 
 
 But when the Propolition is Hated, as it ought 
 to have been, in this plain Language, and is un- 
 derflood in this Senfe, it manifeflly appears to 
 be falfe : And I cannot but think that the Author, 
 if he would have given himfelf Leave to have 
 exprefs'd his Thoughts in this open Manner, 
 muil have feen the Falfity of it. Admitting 
 that by twelve Tranfmiffions of a Fad reported 
 one Half of the Degrees of Certainty or Credi- 
 bility of the Fad are funk, does it follow that 
 the other Half are funk too, and that there are 
 none left ; or that a Fad: reported ceafes wholly 
 to be credible, becaufe or when 'tis but halt as 
 credible as it was at firft ? 'Tis downright pal- 
 pable Nonfenfe; and it would be as good Rea-. 
 fpning and as much Senfe to fay of a Sum of 
 Money, fuppofe a looo/. that when 'tis funk to 
 500 /. it ceafes to be any Money at all, or that 
 the Sura remaining, il there could be faid to be 
 any, was of no Value. 
 
 For my own Part, I mull confefs, I cannot 
 fname any Idea of the Relation between the In- 
 crcafe and Decreaie of Money, or of any other 
 mathematically mcafurable Quantity, and the 
 Increafe or Decrcafe of the Credibility of a Fad 
 
 reported ;
 
 [ 447 ] 
 
 reported ; and molt People, I believe, would 
 look upon a Pound or an Ell of Credibility a,s a 
 very great Rarity : But if any one can conceive 
 a Similitude between them, they mult admit this 
 farther Similitude in the Cafe. According to 
 my little Mathematicks, if out of a Pound of 
 Money in a Purfe we take at Itated Times, let 
 thofe Times return never fo often, a Part in Pro- 
 portion to what we find in the Purle ; or if from 
 a Line of an Ell long we Ihould cut off once a 
 Year, or once a Day if you will, a Part of fuch 
 Line in Proportion to what is remaining ; I think 
 we Ihall never, as long as the World lafts, get 
 to the Bottom of the Purfe, or to the End of 
 the Line, fo as that the one or the other fhall 
 be reduced to nothing; and therefore if the 
 Credibility of a Fad: is compared to a Pound o^ 
 Money or to an Ell of Line, fo proportionably 
 decreafmg, it can never be totally annihilated, or 
 a Fa(5l reported , that was once credible, ever 
 lofe the Whole and all its Degrees of Credibi- 
 lity. 
 
 Or, if the Credibility of a Fad reported may 
 be refemblcd to Money, take it in this Light. 
 If a Bag of Money is tranfmitted through feve- 
 ral Hands, on thefe Terms, that every one into 
 whofe Hands it comes is at Liberty to take out 
 a certain proportionable Part of what he findg 
 in it, the Cafe never will be fuch that there Ihall 
 be no Money in the Bag, though it goes through 
 a Million of Hands, unlefs and until it falls into 
 the Hands of fome one who is Knave enough to 
 take more than his Proportion, and take all. Juft 
 
 in
 
 [ 448 ] 
 
 in the fai^c Manner will it fare with a Fa6t tranA 
 mittcd by Report through feveral Hands, fup- 
 poling it to lofe a proportionable Part of its 
 Credibility by every Hand through which it 
 gocS ; if the Fa6t was once credible, it will 
 continue eternally credible, unlefs and until the 
 Report falls into the Hands of a Perfon whofe 
 Character is fuch that from his Hands, and his 
 Hands are the fole Hands through which it can 
 any farther be tranftnitted, it will entirely ceafc 
 to be credible. 
 
 From thefe Premifes then, and if we can make 
 the Suppofitions and Refemblances here made, 
 the Conclulion will not be, that after twelve 
 TranfmilFions of a Fa6t reported it will become 
 an even Lay whether it be true or be credible or 
 not : On the contrary, the Conclulion will be,' 
 and it will be a mathematically neceflary one, 
 thoup^h the Tranfmillions fhould be fo many as 
 not to be knov^able and numerable, which is al-^ 
 moll always the Cafe, yet that it can never be* 
 come an equal Lay that 'tis not credible. Nay^ 
 if we make fuch Suppofitions here as may be 
 made and mull be made on a Report which has 
 gone through twelve TranfmilTions, "viz. that 'tis 
 a Fa6t worthy of Notice and of fome Confe- 
 quence, and that 'tis reported as fuch Fads ufu-^ 
 ally and commonly are reported ; the Inference 
 will be, and the Inference here too will be ma* 
 thematically ncceffary, that the Fad, whether 
 it "be a recent one, or one of former Ages, by 
 this Multiplicity of Tranfmiifions will gather 
 
 I>cgrc$*
 
 [ 449 ] 
 
 Degrees of Credibility, and as it grows older in 
 Age will grow llronger in Credibility. 
 
 This Author's Suppofitions in his fecond Po- 
 fition are, that a certain and ftated Proportion 
 of Credibility, may be afligned to twelve Re- 
 porters of a Fa6t, and that every one of fuch 
 twelve Tranfmiflions, is from a fingle Perfon to 
 a linglc Perfon only ; which I fay are fuch Sup- 
 pofitions as no one could make but himfelf, 
 and moll certainly they fubfifl no where but in 
 the Imagination of a moft fruitful Brain. When 
 and where is that Report to be met with, that 
 every Report-Maker has jufl: the very fame De- 
 grees and Parts of Degrees of Credibility apper- 
 taining to him, which this Calculator has fcve- 
 rally carved out for him ? What is that Fadt, or 
 how can we conceive a Fadt to be fuch, that 
 there fhall be twelve Reporters of it who Ihall 
 fucceffively tranfmit it from one to the other, 
 and no one of them Ihall make a Report of the 
 fame Fa(5t to more than a fingle Perfon ? Now 
 if any one of thefe Tranfmitters of this Report 
 fhould happen to have more Degrees of Credi- 
 bility inherent in him than this Author has 
 been willing to allow him, the Auditor will be 
 cruelly at a Lofs, notwithftanding this Mathe- 
 matical Rule which he may have in his Pocket, 
 when and where he is, and when and where he 
 is not, to give Credit to a Report ; and if any 
 one of theie Tranfmitters fhould commmcate the 
 Fa(^ to more than one Perfon, and if it be not 
 worth relating to more than one Perfon, he may 
 e'en as well keep it whoUv to hinifelf, bat if he 
 F f ' io
 
 [ +5° ] 
 
 fo divulges it there's an End of our fuppofed fin- 
 gle Tranfraiflions, and the Report may come to 
 the laft Hand by a double, treble, or fextuplc 
 Tranfmiflion, and then the Evidence is enlarged, 
 as this Author himfelf admits, in Proportion to 
 the Degrees of Credibility inherent in each 
 Reporter, > and to the Number of Perfons re- 
 porting. 
 
 And after all thefe wild Suppofitions, and the 
 Calculations grounded upon them, wc remain 
 ftill in the fame State of Uncertainty as to the 
 Credibility of a Fa6t reported ; for all the Dis- 
 putes that ever were or poflibly can be on a 
 Matter of Fa6t, if Perfons know or mind what 
 they are difputing about, though I doubt that 
 is not always the Cafe, are on that Point folely 
 how far and to what Degree the Pcrfon firft re- 
 lating a Fadt, and the Perfons fuccellively tranf- 
 mitting the Report of it, are credible Perfons ; 
 for the Credibility of a Fa6t depends folejy and 
 wholly on the Credibility of the Perfon teltifying 
 and reporting it. Now the Credibility of the 
 Perfon depends on his Character, that is on his 
 Abilities, and the Opportunities which he had 
 to be rightly informed in the Things he faw, or 
 were related to him, and his Honefly and Inte- 
 grity in faithfully tranfmitting what he fo faw, 
 or was related to him ; and his Character in 
 thefe Tnftances will be the fame, whether he be 
 the firft, or the twelfth, or the thoufandth 
 Hand : So that the fole Difficulty is, to difcover 
 and form a right Judgment what Degrees of 
 Ocdibility may juftly be alTigned to each Perfon. 
 
 Now
 
 [ 451 ] 
 
 Now this Gentleman takes it for granted, that 
 this Matter, which is the very Thing and the 
 jonly Thing, which is or can be in Difputc, is a 
 -iettled and flablifh'd Point ; and very gravely 
 ftates the Degrees of Credibility of Perfons, at 2 
 Rate and Proportion certain, for every one the 
 -fame: And then with great Sagacity, and upon 
 a juft Calculation, finds out that if 50 Degrees 
 of an 100 are gone, there is no more than one 
 Half left, which I fuppofe he might have done 
 without the Aid of a Mathematical Rule : But 
 the Difcovery which he makes in his Conclufion, 
 that when one Half is left they are all gone, 
 that there is fo new and peculiar to himfeli^ that 
 I doubt, even with the Aid of his Mathematical 
 Rule, he will not be able to make it good. 
 
 I have aflerted before that this Author's firft 
 Polition on the prefent Value of a Sum of 
 Money payable at the End of twelve Years, 
 depends on a Suppofition of his own making, 
 and one made on purpofe to ferve a Turn ; and 
 molt certainly it does depend on this, that the 
 Sum of Money, calculated to be the prefent 
 Value of a Sum payable at the future Time, is 
 jfuppofed to be put out at 6 /. per Cent. Intercfl", 
 that it is computed at Compound Interefl from 
 the End ©f each Year and no oftner, and thaC 
 at the Hated future Time it will become juift 
 double the Sum, and be equal to the Sum total 
 .then payable. But what if Perfons Ihould diifet 
 in their Opinion about the Degrees of Credibility, 
 which^nay beaffigned to thefevcral, or any of the 
 Reporters of a Fad, and one Ihould think that 
 F f 2 the
 
 [ '452 ] 
 
 the PibpoTtion here ftated is too great, and 
 another ihould think it too fmall ? Why, truly 
 for this pofTible Cafe the Author has made a 
 Provition tolerable good ; for as there are diffe- 
 rent Degrees of Credibility in the Parties teilify- 
 ing or tranfmitting the Report of a Fa6t, fo 
 there are different Rates of Interefl of Moneys 
 :and ai*the Proportion of Credibility runs higher 
 or lower, we mull: take a higher or lower Rate 
 of Interell: ; and the Receipe will certainly keep 
 either at Land or at Sea : So that a Perfon need 
 only to have a common Table of Interelt, com- 
 puted at Variety of Interefls, and by a proper 
 Application to the particular Cafe, may readily 
 calculate how many Degrees of Credibility there 
 muft be left before they are all gone, and he 
 may jultly withhold his Affent to the Truth of 
 a Fa(^ reported. 
 
 But, lince this Gentleman fuppofes that the 
 Interell of Money is computed as due once a 
 Year only, and once a Year only fuchlntereft 
 is accumulated, and fuppofes that the Tranfmif- 
 lion of the Report of a Fad from one Hand to 
 another in one Cafe, mull be jufl as often and 
 correfpond to the Times of computing, and 
 compounding Interefl in the other; what if a 
 Perfon, who knows Res agere^ and underflands 
 the Difference of compound Interell, where it is 
 computed once a Year, and where it is comput- 
 ed every half Year, or every Quarter, or every 
 Dav, Ihould have a Mind to compute and accu- 
 mulate his Interell every Dav, and if one, who is 
 as credulous as the other is covetous, Ihould 
 
 claim
 
 [453 ] 
 
 claim as many Traofmiflions of a Report as the 
 moneyed Man takes Times of accumulating his 
 Intereft; What fhall .yve do here ? This feems to 
 be. a Cafe omitted : but, where Interefl is com- 
 puted and in Fa<5t accumulated every Day, I 
 iuppofe the moneyed jMan in ten Years Time per- 
 haps, or thereabouts, may fee his original Sum 
 doubled, or, which is the fame Thing, the pre- 
 fent Value of the Sum payable at the End often 
 Years, Intereft fo often compounded, will be no 
 more than half the Sum ; and then the Perfon 
 who is meafuring the Degrees of Credibility, and 
 has a Title to as many Tranfmiffions as the other 
 takes Times of accumulating Intereft, inftead of 
 twelve Tranfmilfions will be entitled to ten 
 Times ^6^^ that is 3650 Tranfmiffions, there 
 being fo many Days of accumulating Intereft in 
 the Space of ten Years, before the Degrees of 
 Credibility will be funk to one Half. And the 
 latter will have greatly the Advantage too, be- 
 caufe the moneyed'Man will not fee his Principal 
 doubled in ten Years Time, unlefs in Fact as 
 well as in his Computations he accumulates In- 
 tereft daily, whereas the Dealer in Credibilities 
 can never meet with a Difappointment; for if 
 his Computation is right, and after fuch plain 
 Inftru(flions there is no great Difficulty in making 
 it right, the reft is ready cut and dried to his 
 Hands. 
 
 Upon the Whole ; I think the Reprefentation 
 
 here given, of the Duration of the Credibility 
 
 of a Fa6t reported on human Evidence, is a 
 
 Piece of meer Pageantry, and that not well 
 
 F f q drawn
 
 [ 454 ] 
 
 drawn neither : And the Ufe I would make of 
 it is only this, that where vain or wrong Suppo- 
 fitions are made, and on them juft Calculations 
 are founded; and where juft Calculations aire 
 made, and from them wrong Inferences arc de- 
 duced, we can make no fafe or juit Conclufions: 
 and there I reft the Matter. 
 
 TABLE
 
 f 455 ] 
 
 T A B L E I. 
 
 Showing the Value of Annuities, for theTerm, 
 and at the Intereft, in the Table \ in Years, 
 Quarters, Days ; and the centefirnal Parts 
 of a Year, correlponding to the Quarters and 
 JJays. 
 
 i "S 
 
 ^l.perCt. 
 
 
 ^l.perO: 
 
 
 5 I. per C 
 
 
 li 
 
 
 C.P. 
 
 
 C.P. 
 
 
 C.P. 
 
 
 Y.Q^D. 
 
 
 Y. Q. D. 
 
 
 Y. Q. D. 
 
 
 I 
 
 00 3 80 
 
 91 
 
 00 3 77 
 
 96 
 
 00 3 ^9 
 
 95 
 
 2 
 
 01 3 58 
 
 91 
 
 ®i 3 47 
 
 88 
 
 01 3 40 
 
 85 
 
 3 
 
 02 3 2p 
 
 83 
 
 02 3 11 
 
 78 
 
 02 2 80 
 
 72 
 
 4 
 
 03 2 77 
 
 71 
 
 03 2 44 
 
 d2 
 
 03 2 14 
 
 54 
 
 5 
 
 04 2 29 
 
 5j 
 
 04 I 6p 
 
 44 
 
 04 I 2p 
 
 M 
 
 6 
 
 05 I 62 
 
 42 
 
 05 84 
 
 23 
 
 05 25 
 
 07 
 
 1 
 
 q6 87 
 
 24 
 
 05 3 87 
 
 99 
 
 05 3 14 
 
 7P 
 
 8 
 
 07 II 
 
 03 
 
 o5 2 80 
 
 72 
 
 o5 I 77 
 
 45 
 
 9 
 
 07 3 22 
 
 81 
 
 07 I 62 
 
 42 
 
 07 40 
 
 II 
 
 lO 
 
 08 2 22 
 
 ^ 
 
 08 3d 
 
 10 
 
 07 2 89 
 
 71 
 
 II 
 
 op I 14 
 
 2p 
 
 08 3 00 
 
 75 
 
 08 I 22 
 
 31 
 
 12 
 
 op 3 87 
 
 99 
 
 op I 44 
 
 37 
 
 08 3 44 
 
 87 
 
 13 
 
 10 2 62 
 
 ^7 
 
 cp 3 80 
 
 97 
 
 09 I 51 
 
 39 
 
 14 
 
 II I 2p 
 
 33 
 
 10 2 18 
 
 55 
 
 op 3 51 
 
 89 
 
 1(5 
 
 II 3 80 
 
 59 
 
 II 3^5 
 II 2 51 
 
 10 
 6i 
 
 lo I 47 
 10 3 33 
 
 38 
 84 
 
 12 2 33 
 
 I? 
 
 13 6p 
 
 19 
 
 12 55 
 
 15 
 
 II I 07 
 
 27 
 
 i8 
 
 13 3 II 
 
 78 
 
 12 2 51 
 
 ^4 
 
 11 2 66 
 
 68 
 
 ip 
 
 14 I 36 
 
 35 
 
 13 44 
 
 12 
 
 12 2p 
 
 08 
 
 20 
 
 21 
 
 14 3 51 
 
 8p 
 42 
 
 13 2 33 
 
 14 03 
 
 59 
 01 
 
 12 I 77 
 
 12 3 2)- 
 
 4^i 
 82; 
 
 15 I 6i 
 
 22 
 
 15 3 <5p 
 
 94 
 
 14 I 6p 
 
 45 
 
 13 58 
 
 16 
 
 23 
 
 i5 I 6p 
 
 441 
 
 143 33 
 
 84 
 
 13 I 84 
 
 48 
 
 24 
 
 16 3 62 p2 
 
 17 I 55] 40 
 
 15 I 00 
 
 25 
 
 13 3 14 
 
 79; 
 
 25 
 
 15 2 40 
 
 di 
 
 14 33 
 
 op
 
 [456] 
 
 TABLE I. continued. 
 
 o 
 
 I L per a 
 
 
 4/. per C. 
 
 
 5 /. /»« 
 
 ' C. 
 
 
 
 
 C.P. 
 
 
 C.P. 
 
 
 
 C.P. 
 
 
 y^D. 
 
 
 Y. Q.D. 
 
 
 Y.Q, 
 
 D 
 
 
 25 
 
 17 3 40 
 
 86 
 
 15 3 80 
 
 97 
 
 14 I 
 
 44 
 
 37 
 
 27 
 
 18 I 22 
 
 31 
 
 16 I 22 
 
 31 
 
 H 2 
 
 51 
 
 64 
 
 28 
 
 18 3 00 
 
 75 
 
 16 2 51 
 
 64 
 
 14 3 
 
 51 
 
 89 
 
 29 
 
 19 (52 
 
 17 
 
 16 3 80 
 
 97 
 
 15 6 
 
 47 
 
 13 
 
 31 
 
 19 2 29 
 
 19 3 84 
 
 58 
 98 
 
 17 I 14 
 17 2 25 
 
 57 
 
 15 I 
 
 45 
 
 i5 
 59 
 
 15 2 
 
 33 
 
 32 
 33 
 
 20 I 44 
 20 3 00 
 
 37 
 75 
 
 17 3 3^ 
 
 18 40 
 
 85 
 II 
 
 15 3 
 
 18 
 
 80 
 
 16 
 
 00 
 
 34 
 
 21 44 
 
 12 
 
 18 I ,44 
 
 37 
 
 16 
 
 69 
 
 19 
 
 35 
 3^ 
 
 21 I 84 
 21 3 29 
 
 48 
 83 
 
 18 2 44 
 
 1 8 3 40 
 
 62 
 
 86 
 
 i5 I 
 
 _44 
 
 12 
 54 
 
 16 2 
 
 14 
 
 37 
 
 220 6i 
 
 17 
 
 19 36 
 
 lo 
 
 16 '2 
 
 73: 
 
 70 
 
 38: 
 
 22 I 87 
 
 49 
 
 19 I 29 
 
 33 
 
 16 3 
 
 40, 
 
 86 
 
 19 
 
 22 3 14 
 
 79 
 
 19 2 22 
 
 56 
 
 17 
 
 OS 
 
 PI 
 
 d2 
 
 23 40 
 
 II 
 
 19 3 li 
 
 Z? 
 
 17 
 
 5 5 
 
 11 
 
 41 
 
 23 I 61 
 
 42 
 
 20 00 
 
 — 
 
 ^71 
 
 18 
 
 30 
 
 42 
 
 23 2 77 
 
 71 
 
 20 73 
 
 20 
 
 17 I 
 
 66 
 
 43 
 
 4? 
 
 24 00 
 
 "~~ 
 
 20 I 51 
 
 39 
 
 17 .2 
 
 II 
 
 53 
 
 44 
 
 24 I 07 
 
 27 
 
 20 ,2 25 
 
 57 
 
 17^ 
 
 •5-5; 
 
 ^5 
 
 45 
 4d 
 
 24 2 II 
 
 53 
 78 
 
 20 2 84 
 20 3 55 
 
 73 
 90 
 
 17 3 
 
 07 
 
 77 
 
 88 
 
 24 3 II 
 
 17 3 
 
 47 
 
 47 
 
 25 07 
 
 02 
 
 21 18 
 
 05 
 
 17 3 
 
 84 
 
 98 
 
 48 
 
 251 00 
 
 25 
 
 21 73 
 
 20 
 
 18 
 
 29 
 
 08 
 
 4P, 
 
 25';i 80 
 
 47 
 
 21 I 33 
 
 34 
 
 18 
 
 62 
 
 17 
 
 :5oi 
 
 ■5-1 ' 
 
 .25'.2 ,<52 
 
 67 
 86 
 
 21 I 80 
 
 47 
 60 
 
 18 I 
 
 00. 
 
 15 
 33 
 
 25, 3 40 
 
 21 2 36 
 
 iS I 
 
 29= 
 
 !69 
 
 27 2 66 
 
 68 
 
 22 2 -40 
 
 61 
 
 x8 3 
 
 66 
 
 93 
 
 ; 79' 
 
 29 44 
 
 12 
 
 23 I 51 
 
 39 
 
 19 I 
 
 3 5 
 
 34 
 
 ^0 
 
 3;o 77 
 
 21 
 
 23 3 55 
 
 90 
 
 19 2 
 
 3 3 
 
 59 
 
 90 
 
 310 00 
 
 — 
 
 24 .1 oy 
 
 26 
 
 19 3 
 
 00 
 
 75 
 
 roo 
 
 31 2 36 
 
 60 
 
 24 2 00 
 
 50 
 
 19 3 
 
 3^ 
 
 85 
 
 Fee 
 
 33 I 29 
 
 33 
 
 25 
 
 
 20 — 
 
 ■ »■■■■ 
 
 — •
 
 [ 457 ] 
 
 TABLE I. continued. 
 
 o 
 
 6 1 per C. 
 
 
 7 /. per C. 
 
 
 8 /. per C. 
 
 
 I 
 
 Y.Q.D. 
 
 C.P, 
 
 94 
 
 Y. Q. D. 
 
 00 3 (56 
 
 a P. 
 
 93 
 
 Y. Q. D. 
 
 C.P. 
 
 
 00 3 
 
 <^9 
 
 00 3 62 
 
 92 
 
 
 2 
 
 01 3 
 
 29 
 
 B3 
 
 01 3 22 
 
 81 
 
 01 3 II 
 
 7« 
 
 
 3 
 
 02 2 
 
 61 
 
 ^7 
 
 02 2 44 
 
 62 
 
 02 2 29 
 
 58 
 
 
 4 
 
 03 I 
 
 11 
 
 46 
 
 03 I 51 
 
 39 
 
 03 I 22 
 
 31 
 
 ', 
 
 2 
 6 
 
 04 
 04 3 
 
 11 
 62 
 
 21 
 92 
 
 04 36 
 04 3 07 
 
 10 
 
 77 
 
 03 3 87 
 
 99 
 6z 
 
 s 
 
 04 2 44 
 
 r 
 
 7 
 
 05 2 
 
 29 
 
 5« 
 
 05 I 51 
 
 39 
 
 05 77 
 
 21 
 
 s 
 
 8 
 
 o5 
 
 77 
 
 21 
 
 05 3 80 
 
 97 
 
 05 3 00 
 
 75 
 
 J. 
 
 9 
 
 06 3 
 
 18 
 
 80 
 
 0(5 2 03 
 
 51 
 
 o5 I 00 
 
 25 
 
 , 
 
 lo 
 
 07 I 
 
 40 
 
 3^ 
 
 07 07 
 
 02 
 
 o5 2 77 
 
 Zi 
 
 \ 
 
 II 
 
 07 3 
 
 47 
 
 88 
 
 07 2 00 
 
 50 
 
 07 51 
 
 14 
 
 \ 
 
 12 
 
 08 I 
 
 47 
 
 3^ 
 
 07 3 69 
 
 94 
 
 07 2 14 
 
 54 
 
 
 IS 
 
 08 3 
 
 3^ 
 
 «5 
 
 08 I 40 
 
 3^ 
 
 07 3 55 
 
 90 
 
 
 14 
 
 09 I 
 
 14 
 
 29 
 
 08 2 87 
 
 74 
 
 08 87 
 
 24 
 
 I 
 
 LI 
 I5 
 
 09 2 
 
 77 
 
 71 
 10 
 
 09 40 
 
 09 I 73 
 
 1 1 
 
 4) 
 
 08 2 22 
 08 3 36 
 
 56 
 85 
 
 
 10 
 
 36 
 
 
 17 
 
 10 I 
 
 80 
 
 47 
 
 09 3 03 
 
 16 
 
 09 44 
 
 12 
 
 
 i8 
 
 10 3 
 
 29 
 
 «3 
 
 10 22 
 
 06 
 
 09 I 44 
 
 37 
 
 
 19 
 
 II 
 
 5« 
 
 16 
 
 10 I 29 
 
 33 
 
 09 2 36 
 
 60 
 
 
 20 
 
 II I 
 
 80 
 
 47 
 
 10 2 33 
 
 59 
 
 09 3 25 
 
 82 
 
 
 21 
 
 II 3 
 
 03 
 
 7'5 
 
 10 3 29 
 
 83 
 
 10 07 
 
 02 
 
 
 22 
 
 12 
 
 II 
 
 03 
 
 II 22 
 
 06 
 
 10 73 
 
 20 
 
 
 23 
 
 12 I 
 
 18 
 
 30 
 
 II I 07 
 
 27 
 
 10 I 44 
 
 37 
 
 
 24 
 
 12 2 
 
 i8 
 
 55 
 
 II I 80 
 
 47 
 
 10 2 II 
 
 03 
 
 
 25. 
 26 
 
 12 3 
 
 II 
 
 73. 
 
 II 2 S5 
 
 65 
 
 83 
 
 10 2 62 
 10 3 22 
 
 67 
 
 81 
 
 
 13 
 
 00 
 
 II 3 29 
 
 
 27 
 
 13 
 
 77 
 
 21 
 
 II 3 87 
 
 99 
 
 10 3 (56 
 
 93 
 
 
 28 
 
 13 ^ 
 
 55 
 
 40 
 
 12 51 
 
 14 
 
 II 18 
 
 05 
 
 
 29 
 
 13 2 
 
 29 
 
 5B 
 
 12 I II 
 
 28 
 
 II 58 
 
 16 
 
 
 30 
 
 
 
 13 3 
 
 03 
 
 7^ 
 
 12 I 58 
 
 51 
 
 1 1 I 03 
 
 26 

 
 [ 458 ] 
 TABLE I. continued. 
 
 o . 
 
 61. 
 
 per a 
 
 
 7 /. /Jgy C. 
 
 
 8/. 
 
 perC. 
 
 
 
 Y. 
 
 C^D. 
 
 CP. 
 
 y.Q,D. 
 
 a P. 
 
 Y. 
 
 Q,D. 
 
 c.p. 
 
 .91 
 
 13 
 
 3 ^6 
 
 93 
 
 12 2 18 
 
 55 
 
 II 
 
 I 36 
 
 35 
 
 3i 
 
 14 
 
 33 
 
 09 
 
 12 2 66 
 
 68 
 
 II 
 
 I 66 
 
 43 
 
 33 
 
 14 
 
 84 
 
 23 
 
 12 3 18 
 
 80 
 
 II 
 
 2 00 
 
 50 
 
 34 
 
 14 
 
 I 47 
 
 38 
 
 12 3 58 
 
 91 
 
 II 
 
 2 22 
 
 5<5 
 
 35 
 ^6 
 
 14 
 
 2 03 
 
 51 
 63 
 
 13 03 
 
 01 
 
 10 
 
 II 
 
 2 44 
 
 62 
 
 14 
 
 2 47 
 
 13 36 
 
 II 
 
 2 66 
 
 68 
 
 37 
 
 14 
 
 2 87 
 
 74 
 
 13 66 
 
 18 
 
 IX 
 
 2 87 
 
 74 
 
 38 
 
 H 
 
 3 33 
 
 84 
 
 13 I 00 
 
 25 
 
 II 
 
 3 18 
 
 80 
 
 39 
 
 14 
 
 3 ^9 
 
 94 
 
 13 I 25 
 
 32 
 
 II 
 
 3 40 
 
 86 
 
 42 
 41 
 
 15 
 15 
 
 11 
 
 05 
 12 
 
 13 I 47 
 
 38 
 43 
 
 II 
 
 3 62 
 
 92 
 
 44 
 
 13 I 66 
 
 II 
 
 3 80 
 
 97 
 
 42 
 
 15 
 
 73 
 
 20 
 
 13 I 84 
 
 48 
 
 12 
 
 03 
 
 01 
 
 43 
 
 15 
 
 I II 
 
 28 
 
 13 2 II 
 
 53 
 
 12 
 
 18 
 
 05 
 
 44 
 
 15 
 
 I 40 
 
 3<5 
 
 13 2 29 
 
 58 
 
 12 
 
 29 
 
 08 
 
 45 
 45 
 
 15 
 
 I 66 
 
 43 
 50 
 
 13 2 44 
 
 62 
 66 
 
 12 
 
 40 
 
 II 
 
 15 
 
 2 00 
 
 13 2 58 
 
 12 
 
 47 
 
 13 
 
 47 
 
 15 
 
 2 22 
 
 56 
 
 13 2 73 
 
 70 
 
 12 
 
 55 
 
 IJ 
 
 48 
 
 15 
 
 2 44 
 
 62 
 
 13 2 87 
 
 74 
 
 12 
 
 62 
 
 17 
 
 49 
 
 15 
 
 2 66 
 
 68 
 
 13 3 07 
 
 77 
 
 12 
 
 69 
 
 19 
 
 50 
 51 
 
 15 
 
 2 87 
 
 24 
 79 
 
 n 3 18 
 
 80 
 83 
 
 12 
 
 77 
 
 21 
 
 15 
 
 3 14 
 
 13 3 25> 
 
 12 
 
 84 
 
 23 
 
 60 
 
 16 
 
 62 
 
 17 
 
 14 00 
 
 — 
 
 12 
 
 I 40 
 
 3<^ 
 
 70 
 
 16 
 
 I 51 
 
 39 
 
 14 40 
 
 II 
 
 12 
 
 1 69 
 
 44 
 
 80 
 
 16 
 
 2 00 
 
 50 
 
 14 55 
 
 15 
 
 12 
 
 I 80 
 
 47 
 
 90 
 
 16 
 
 2 29 
 
 58 
 
 14 66 
 
 18 
 
 12 
 
 I 84 
 
 48 
 
 100 
 Fee 
 
 16 
 
 2 40 
 
 61 
 67 
 
 14 73 
 
 20 
 29 
 
 12 
 
 I 87 
 
 49 
 
 16 
 
 2 61 
 
 14 I 14 
 
 12 
 
 2 — • 
 
 50
 
 [ 459 ] 
 
 TABLE ir. 
 
 Showing in the firft Column the Age of Pe^- 
 fons, in the fecond the Term of Years and 
 Quarters to which any given Life, in thofe 
 Periods, is computed to be equal, accord- 
 ing to my Way of making the Eftimate; 
 to which I have added, in the third Column, 
 the Difference or Decreafe in the Term, 
 from one Period to another. 
 
 Ages of Pcrloha 
 
 Term to which 
 
 Decreafe in 
 
 from 
 
 fv.chLives equal. 
 28 
 
 the Term. 
 
 lo to 15 — 
 
 15 20 — 
 
 27 
 
 2 
 
 6 
 
 i 
 
 20 — 25-^ 
 
 26 
 
 3 
 
 
 
 3 
 
 25 — 30 — 
 
 25 
 
 3 
 
 I 
 
 
 
 30 — 35 — 
 
 24 
 
 2 
 
 I 
 
 I 
 
 35—40 — 
 
 23 
 
 
 
 I 
 
 2 
 
 40—45 — 
 
 21 
 
 1 
 
 I 
 
 3 
 
 45 — 50-- 
 
 19 
 
 I 
 
 2 
 
 
 
 50 — 55 — 
 
 17 
 
 
 
 2 
 
 I 
 
 55— (?o— 
 
 14 
 
 2 
 
 2 
 
 ■2 
 
 6o-^d5^^ 
 
 II 
 
 2 
 
 3 
 
 
 
 ^5— 70— j 
 
 8 
 
 D 
 
 I 
 
 2- 
 
 70— 7;— ■ 
 
 4 
 
 
 
 4 
 
 
 
 75 — 80-- 
 
 2 
 
 ~— 
 
 *"■ 
 
 "■■" 
 
 80 &c. 
 
 I 
 
 — 
 
 — 
 
 "■" 
 
 From this and the precedent Table is rea* 
 aily feen the Value of an Annuity for any 
 Life given at any Rate of Intereft: For this 
 Table gives the Term to which the Life is 
 equal, and the precedent Table gives the 
 Value, or Year's Purchafe, to which fuch 
 Term is equal j that is the Value of the 
 Annuity.
 
 [ 46o ] 
 
 T ABLE III. 
 
 Shewing, in the firft Column, the Age of 
 •. ''Terfbns, in the lecondthe Value of Annui- 
 ties for every fifth Year of Age, from the 
 loth to the^7oth Year of Man's Life, as 
 , ; ilated by Dr. Halley^ Intereft computed at 
 ■■dLfer Cent : To which I have added,- in 
 the third Column the Years, Quarters, and 
 Days, to which fuch Value correfponds, 
 in the fourth Column the Difference or 
 Decreafe in the Term, from one Period to 
 another ; and in the fifth, the Difference or 
 Decreafe in the Value, in the fame Manner. 
 
 Ages of 
 Pet ions 
 
 lO 
 
 Years Value 
 or Purdiale. 
 
 IS 44 
 
 Term to which 
 fuchVahieeqval 
 
 Dec'catc in 
 the Term. 
 
 Dccreale in 
 the Value. 
 
 28 1 
 
 00 
 
 
 ., .- . . 
 
 
 
 15 
 
 13 33 
 
 27 2 
 
 6i 
 
 2 30 
 
 00 
 
 II 
 
 20 
 
 i.z 78 
 
 25 
 
 00 
 
 2 2 61 
 
 00 
 
 55 
 
 25 
 
 12 27 
 
 22 3 
 
 45 
 
 2 45 
 
 00 
 
 51 
 
 30 
 
 II 72 
 
 2^ 3 
 
 45 
 
 2 00 
 
 CO 
 
 55 
 
 35 
 
 II 12 
 
 18 3 
 
 45 
 
 2 00 
 
 00 
 
 60 
 
 40 
 
 10 57 
 
 17 I 
 
 10 
 
 I 2 35 
 
 00 
 
 55 
 
 45 
 
 op 91 
 
 15 2 
 
 00 
 
 _i 3 10 
 
 00 
 
 66 
 
 50 
 
 op 2 I 
 
 13 3 
 
 30 
 
 I 2 61 
 
 00 
 
 70 
 
 55 
 
 08 51 
 
 12 I 
 
 12 
 
 I 2 18 
 
 00 
 
 70 
 
 60 
 
 07 60 
 
 10 2 
 
 00 
 
 I 3 12 
 
 00 
 
 9^ 
 
 ^5 
 
 06 54 
 
 08 2 
 
 CO 
 
 2 00 
 
 01 
 
 06 
 
 70 
 
 05 JO 
 
 06 2 
 
 30 
 
 13 61 
 
 01 
 
 24 
 
 oi
 
 [ 46i ] 
 
 TABLE IV. 
 
 Shewing, the Value of Annuities for a Life, 
 
 ■ at the Ages jnentioned, Intereft computed 
 
 • at ^L fer Cent., as ftated in Mr. Hayes's 
 
 Tables ; with the fame Additions as in the 
 
 precedent Tables. 
 
 AgC.^ of 
 
 Years Value 
 
 Term to which 
 
 Dfccreafe in 
 
 Decreale in 
 
 Per ions 
 
 or Pnrchafe. 
 
 15 54 
 
 fuch Value equal 
 
 the Term. 
 
 the Value. 
 
 50 
 
 24 3 23 
 
 
 
 
 
 35 
 
 14 51 
 
 22 55 
 
 2 2 68 
 
 1 03 
 
 40 
 
 13 49 
 
 19 3 38 
 
 2 I 27 
 
 I 02 
 
 45 
 
 12 41 
 
 . 17 2 22 
 
 2 I Od 
 
 I 08 
 
 5c 
 
 II 28 
 
 15 I 45 
 
 2 77 
 
 I 13 
 
 55 
 
 10 10 
 
 13 84 
 
 2 61 
 
 I 18 
 
 60 
 
 c8 80 
 
 II 33 
 
 2 51 
 
 I 30 
 
 65 
 
 07 26 
 
 08 3 07 
 
 2 26 
 
 I 54 
 
 70 
 
 05 54 
 
 06 I 84 
 
 2 I 13 
 
 I 72
 
 [ 462 ] 
 
 T A B L E V. 
 
 Showing, the Falue of Annuities for a Life, 
 at the Ages mentioned, Interefl: computed 
 at 6 1. ferCmt,^ as ftated by Mr. Richards : 
 with the {ame Additions as in the precedent 
 Tables. 
 
 Ages of 
 
 Years Value 
 
 PcrLbns 
 
 or Purchafe. 
 
 7 
 
 13 42 ■ 
 
 12 
 
 13 3^ 
 
 I? 
 
 12 pp 
 
 22 
 
 12 54 
 
 27 
 
 II 5)7 
 
 32 
 
 II 47 
 
 37 
 
 10 po 
 
 42 
 
 10 24 
 
 47 
 
 09 58 
 
 52 
 
 08 91 
 
 57 
 
 08 II 
 
 52 
 
 07 08 
 
 <^7 
 
 05 90 
 
 72 
 
 04 78 
 
 77 
 
 03 29 
 
 82 
 
 01 35 
 
 Term to 
 
 which 
 
 Dccrealc in 
 
 fuch Value eqiial 
 
 the Term. 
 
 28 
 
 z^ 
 
 ' 
 
 27 3 
 
 00 
 
 I 36 
 
 25 3 
 
 74 
 
 I 3 17 
 
 23 3 
 
 7<^ 
 
 I 3 Sp 
 
 21 3 
 
 00 
 
 2 75 
 
 20 
 
 00 
 
 I 3 00 
 
 18 
 
 77 
 
 I 3 14 
 
 16 I 
 
 10 
 
 1 3 67 
 
 14 2 
 
 74 
 
 -I J2 27 
 
 13 
 
 50 
 
 I 2 24 
 
 II I 
 
 1^ 
 
 I 2 6^ 
 
 cp 1 
 
 00 
 
 I 3 75 
 
 07 2 
 
 00 
 
 2 00 
 
 05 3 
 
 II 
 
 I 2 80 
 
 03 3 
 
 II 
 
 2 00 
 
 01 I 
 
 80 
 
 2 122 
 
 Decreale in 
 the Value. 
 
 06 
 
 57 
 45 
 57 
 50 
 57 
 66 
 66 
 
 61 
 80 
 
 03 
 18 
 12 
 
 49 
 93
 
 [ 463 ] 
 
 TABLE VI. 
 
 Showing the Value of Annuities for a Life, at 
 the Ages mentionM, Intereft computed at 
 ^L ferCent.^ as ftated by Mr. Morris: 
 with the fame Additions as in the precedent 
 Tables. 
 
 Ages of j Years Value, 
 Perfons. I or Purchaft. 
 
 Term to which 
 fuch Value equal. 
 
 Decreafe infDecrcafe in 
 
 the Term. | the Value. 
 
 lO 
 
 16 72 
 
 28 68 
 
 
 
 M 
 
 16 53 
 
 27 2 68 
 
 2 GO 
 
 19 
 
 20 
 
 15 6a 
 
 25 22 
 
 2 14^ 
 
 91 
 
 25 
 30 
 
 14 80 
 
 22 3 68 
 
 2 45 
 2 »} 
 
 82 
 84 
 
 13 96 
 
 20 3 45 
 
 35 
 
 13 08 
 
 18 3 60 
 
 I 3 75 
 
 88 
 
 40 
 
 12 2p 
 
 17 I 15 
 
 I 2 45 
 
 79 
 
 45 
 
 11 3P 
 
 15 2 30 
 
 I 2 75 
 
 po 
 
 50 
 
 10 45 
 
 13 3 15 
 
 I 3 15 
 
 94 
 
 55 
 
 09 55 
 
 12 I 18 
 
 1 I 88 
 
 po 
 
 60 
 
 08 40 
 
 10 2 00 
 
 I 3 18 
 
 I 15 
 
 65 
 
 07 12 
 
 08 2 50 
 
 I 3 31 
 
 I 28 
 
 70 _ 
 
 05 70 
 
 06 2 45 
 
 2 05 
 
 1 42 
 
 TABLE Vn. 
 
 Showing, in the firft Column, the Rate of In- 
 tereft, in the lecond the Value of an Annu- 
 ity for a Life of 1 2 Years Age, tranfcribed 
 from Mr. Richards's Tables : To which I 
 have added, in the third Column, the Termi 
 of Years to which fuch Value correfponds, 
 as I calculate the fame. 
 
 Intereft. 
 
 Value. 
 
 Term of Years. 
 
 4 
 
 17 20 
 
 2p 3 00 
 
 5 
 
 15 23 
 
 2p I 44 
 
 6 
 
 13 36 
 
 27 3 00 
 
 7 
 
 1 1 72 
 
 25 I 23 
 
 8 
 
 10 67 
 
 25 00
 
 [ 464 ] 
 
 TABLE Viri. 
 
 Showing, in the firft Column, the Rate of In- 
 tereft, in the fecond, the Value of an Annu- 
 ity for a Life of 30 Years Age, extraded 
 from Mr. Hayes's Tables : To which I have 
 added, in the third Column, the Term of 
 Years to which fuch Value correlponds, as 
 Hated by Mr. Hayes himlelf 
 
 Intereft. I Value. | Term of Years. 
 
 4 
 
 15 54 
 
 25 
 
 5 
 
 13 12 
 
 22 
 
 6 
 
 II 27 
 
 20 
 
 7 
 
 09 8j 
 
 18 
 
 8 
 
 08 68 
 
 16 
 
 Mr. Hayes feems to me, in this Table, lirft ^ 
 to have ftated to what Term a Life of 30, or 
 any other Age, is equal, and from thence to 
 have computed the Values : Now, in what 
 Way of Thinking, or on what Grounds, he 
 could in the firft Inftant fuppofe that one and 
 the fame Life could be equal only to a Term 
 of 16 Years, and at the fame Time be equal 
 to a Term of 25 Years, I confefs exceeds my 
 Comprehenfion. 
 
 TABLE
 
 [ 465 ] 
 TABLE IX. 
 
 Exhibiting a Computus of the Value of an An- 
 nuity of looco /. per Anvtim^ for a Life of lO 
 Years Age, made according to Dr. Halleyh 
 Rule, for every Year of Life to an lOO Years 
 of Age. 
 
 
 I 
 
 multiply'c 
 
 1 by 
 
 ^53 
 
 proiluccs 
 
 
 :nrtgrali.--Kr;iCtions. 
 
 // 
 
 9 434 
 
 6 160 402 
 
 
 9 319—543 
 
 
 2 
 
 8 5)00 
 
 646 
 
 5 749 400 
 
 
 8 698 — 022 
 
 
 3 
 
 8 S96 
 
 640 
 
 5 373 440 
 
 
 8 129 — 171 
 
 
 4 
 
 7 ^21 
 
 634 
 
 5 021 914 
 
 rt 
 
 7 597—297 
 
 
 6 
 
 7 473 
 
 628 
 622 
 
 4 ^P3 044 
 
 t-, 
 c 
 
 7 099 — 605 
 
 
 7 050 
 
 4 385 I0O 
 
 6 634 — 026 
 
 
 7 
 
 5 650 
 
 6i6 
 
 4 095 400 
 
 S 
 
 6 197 — 183 
 
 
 8 
 
 6 274 
 
 610 
 
 3 827 140 
 
 <L) 
 
 5 789 — 611 
 
 
 9 
 
 5 pip 
 
 (5o4 
 
 3 575 075 
 
 d 
 
 5 408 — 388 
 
 70 
 
 JO 
 
 II 
 
 5 584 
 
 598 
 592 
 
 3 339 232 
 
 CO 
 
 5 051 — 521 
 
 
 5 2d8 
 
 3 118 655 
 
 4 718 — 058 
 
 
 12 
 
 4 P70 
 
 585 
 
 2 912 420 
 
 
 4 406 — 054 
 
 
 13 
 
 4 688 
 
 57P 
 
 2 714 35 2 
 
 VO 
 
 4 i®6 — 286 
 
 
 14 
 
 4 423 
 
 573 
 
 2 534 379 
 
 vo 
 
 3 834—105 
 
 
 15 
 16 
 
 4 173 
 
 557 
 560 
 
 2 36(5 cpr 
 
 
 3 579—372 
 
 
 3 P3^ 
 
 2 204 160 
 
 3 334—386 
 
 
 17 
 
 3 714 
 
 553 
 
 2 053 842 
 
 "O 
 
 3 107—115 
 
 
 18 
 
 3 503 
 3 305 
 
 546 
 
 53P 
 
 912 638 
 I 781 395 
 
 -5 
 
 2 893 — 365 
 
 
 •\ 
 
 2 t»95 ■ ■ 
 
 30 
 
 20 
 21 
 
 3 118 
 
 531 
 523 
 
 I 65^ 658 
 
 E 
 
 CO 
 
 2 504 — 514 ■ 
 
 
 2 941 
 
 I 538 143 
 
 2 326 — 657 
 
 
 22 
 
 2 775 
 
 515 
 
 I 429 125 
 
 C3 
 
 2 162 — 043 
 
 
 23 
 
 2 (5i8 
 
 507 
 
 I 327 326 
 
 <« 
 
 2 008 — 038 
 
 
 24 
 25 
 
 2 470 
 
 49P 
 490 
 
 I 2g2 53=^ 
 
 
 I 864 — 426 
 
 
 2 330 
 
 I 141 700 
 
 I 727—153 
 
 
 25 
 
 2 ip8 
 
 481 
 
 I 057 238 
 
 
 1 5P5,— ,99 
 
 
 27 
 
 2 074 
 
 472 
 
 978 928 
 
 
 1 480 — 648 
 
 - 
 
 28 
 
 I P56 
 
 4^3 
 
 905 628 
 
 
 r :?7o — <^'^^-'
 
 L 466 ] 
 TABLE IX. continued. 
 
 29 
 
 I 845 
 
 454 
 
 837 630 
 
 
 I 267.— 143 
 
 30 
 
 I 741 
 
 445 
 
 774- 745 
 
 
 I 017 — 053 
 
 31 
 
 I 643 
 
 436 
 
 ni6 348 
 
 
 I 083—485 
 
 32 
 
 I 550 
 
 427 
 
 661 850 
 
 
 I 001 — 189 
 
 33 
 
 I 462 
 
 4'7 
 
 609 654 
 
 
 922 — 212 
 
 id 
 35 
 
 ^379 
 I 301 
 
 407 
 397 
 
 561 233 
 
 
 849 — 064 
 
 516 497 
 
 781 — 256 
 
 3^ 
 
 I 227 
 
 387 
 
 474 8^9 
 
 'rt 
 
 718 — 251 
 
 37 
 
 I 15^ 
 
 377 
 
 436 566 
 
 S 
 
 660 — 306; 
 
 38 
 
 I 05)2 
 
 367 
 
 400 764 
 
 ^ 
 
 606 -r 198: 
 
 39 
 
 I 031 
 
 357 
 
 3<58 0^7 
 
 rt 
 
 w) 
 
 55^-"55i 
 
 40 
 41 
 
 ■9-1'^ 
 
 34^ 
 
 336-312 
 
 
 508—524 
 
 972 
 
 335 
 
 525 620 
 
 492 — 408 
 
 43 
 
 5?72 
 
 324 
 
 314 928 
 
 
 476 — 292 
 
 43 
 
 91"- 
 
 313 
 
 304 236 
 
 460 — 176 
 
 :44 
 
 972 
 
 302 
 
 293 544 
 
 
 444,— 060 
 
 45 
 45 
 
 726" 
 726 
 
 292 
 282 
 
 211 992 
 
 72 
 
 320 — 472 
 
 204 732 
 
 309 — 483 
 
 47 
 
 72^ 
 
 272 
 
 197 472 
 
 t* 
 
 298 — 494 
 
 48 
 
 725 
 
 262 
 
 190 212 
 
 -13 
 
 287 — 505 
 
 4P 
 
 725 
 
 252 
 
 182 952 
 
 b" 
 
 276 — 516 
 
 IP 
 51 
 
 543 
 543 
 
 242 
 232 
 
 131 406 
 
 3 
 
 CO 
 
 198 — 528 
 
 125 976 
 
 190 — 386 
 
 52 
 
 543 
 
 222 
 
 120 546 
 
 CD 
 
 182 — 244 
 
 53 
 
 543 
 
 212 
 
 115 116 
 
 H 
 
 174 I02 
 
 55 
 
 543 
 406 
 
 202 
 192 
 
 109 686 
 77 952 
 
 165 621 
 
 117 — 615 
 
 5^ 
 
 406 
 
 182 
 
 73 892 
 
 
 III 521 
 
 57 
 
 40^^ 
 
 172 
 
 69 832 
 
 
 105 — 427 
 
 58 
 
 40(5 
 
 162 
 
 65 772 
 
 
 99—333.
 
 1 467 ] 
 
 TABLE IX. continued. 
 
 59 
 
 405 
 
 152 
 
 61 712 
 
 
 93 — 239 
 
 60 
 
 303 
 
 142 
 
 43 026 
 
 
 6) — : 061 
 
 6\ 
 
 303 
 
 131 
 
 39 693 
 
 
 , 60 —033 
 
 6i 
 
 -303 
 
 120 
 
 3;6 360 
 
 
 5 5 —005 
 
 6Z 
 
 ■?o3 
 
 ■ 109 
 
 3S ^'^7 
 
 
 49 — 638 
 
 64 
 
 303 
 
 98 
 
 29 594 
 
 1 1 
 
 44 — 61Q 
 
 ^5 
 
 227 
 
 88 
 
 19 976 
 
 .-a 
 
 30 — 146 
 
 66 
 
 227 
 
 78 
 
 17 70 (^ 
 
 c3 
 
 26— ^' 520 
 
 61 
 
 227 
 
 68 
 
 15 .43^ 
 
 6 ■ 
 
 23'>>H 23:5 
 
 68 
 
 227 
 
 58 
 
 13 1 56 
 
 
 19 ^— 607 
 
 69 
 
 227 
 
 49 
 
 II 123 
 
 CJ 
 
 16—547 
 
 70 
 
 169 
 
 41 
 
 6 929 
 
 to 
 
 10,— 319 
 
 71 
 
 169 
 
 34 
 
 5 74^ 
 
 
 8 - 45S 
 
 72 
 
 1.69 
 
 28 
 
 4 732 
 
 «^ 
 
 7 "— 105 
 
 73 
 
 169 
 
 23 
 
 3 887 
 
 1-1 
 
 5 - 58^ 
 
 74 
 
 1(59 
 
 20 
 
 3 3S0 
 
 ^O 
 
 5 -^ 075 - 
 
 75 
 
 126 
 
 18 
 
 2 268 
 
 >> 
 
 3 - 285 
 
 7^ 
 
 126 
 
 16 
 
 2 016 
 
 -T3 
 
 r— 033 
 
 77 
 
 126 
 
 14 
 
 I 764 
 
 '72 
 
 2 — 442 
 
 78 
 
 12^ 
 
 12 
 
 I 512 
 
 
 2 — 190 
 
 19 
 
 125 
 
 10 
 
 I 266 
 
 
 r — 599 
 
 80 
 
 94 
 
 8 
 
 ' 752 
 
 s" 
 
 3 
 
 I — 091 
 
 81 
 
 P4 
 
 7 
 
 658 
 
 CO 
 
 — 658 
 
 82 
 
 94 
 
 6 
 
 564 
 
 -.; ■.;-^ 564 
 
 83 
 
 94 
 
 5 
 
 470 
 
 .— 
 
 — 470 
 
 84 
 
 94 
 
 4 
 
 37^ 
 
 H 
 
 - 37<^ 
 
 85 
 
 71 
 
 3 
 
 213 
 
 
 — 213 
 
 86 
 
 71 
 
 2 
 
 142 
 
 
 — 142 
 
 87 
 
 71 
 
 I 
 
 71 
 
 
 — 071 
 
 88 
 
 71 
 
 1 
 
 35 
 
 
 — 035 
 
 89 
 
 71 
 
 I 
 4 
 
 17 
 
 
 — C17 
 
 90 
 
 53 
 
 4 
 
 13 
 
 
 - 013 
 
 Gg a
 
 [ 468 ] 
 TABLE IX. continued. 
 
 Totals. Col. 2. Column 4 
 
 Page I. 
 Page 2. 
 Page 3. 
 
 Totals. 
 
 134 
 
 061 jp 
 
 C85 
 
 357 
 
 28 
 
 258 
 
 9 
 
 8io 
 
 401 
 
 5 
 
 409 
 
 
 357 
 
 724 
 
 157 72889 253 482 
 
 Column 5. 
 
 119 (545 — 012 
 
 14 841 — 500 
 
 541—12? 
 
 135 027—535 
 
 130-265-347 
 
 134-486-512 
 
 135-027-635 
 
 N. B, Where this Computation 
 
 is carried on for 41 Years 
 
 only, the Produce is 
 Where carried on for 58 Years, 
 
 the Produce is 
 Where carried on for po Years, 
 
 the Produce, as above, is 
 
 That is, in the firfl Cafe, the Value is 1 3-02 
 In the fecond Cafe, the Value is 13-44 
 In the laft Cafe, the Value is 13-50 
 
 From whence *tis apparent, where the Value 
 of fuch an Annuity is Hated at 13-44 only, as 
 in Dr. Halleys Table, that there is an Omilfion 
 of 32 Years in making the Calculation.
 
 [ 469 ] 
 
 T A B L E X. 
 
 Showing the Number of Perfons living in the 
 City of BreJIa^isu in Silefia^ and their Ages 
 current refpedively, from i to 84 Years 
 Age. 
 
 Age current.|Perfons living. 
 
 A. 
 
 P. 
 
 I — 
 
 1000 
 
 6 — 
 
 710 
 
 2 — 
 
 855 
 
 7 — 
 
 692 
 
 3 — 
 
 798 
 
 8 — 680 
 
 4 — 
 
 760 
 
 9 — 570 
 
 5 — 
 
 732 
 
 10 — 661 
 
 The reft of this Table is inferted in the third 
 Column of the precedent Table, and need not 
 be here repeated : And I have there added the 
 Age current and Perfons living from 84, where 
 Dr. Halleys Table breaks off, to Age an 100 ; 
 And the Table, with thefc Additions, makes 
 the total Number of Perfons there living to bo 
 the fame as fuppofed by the Dodor, viz* 3400P, 
 
 ^g 3
 
 [ 47° ] 
 
 o 
 
 O 
 
 s 
 
 3 
 
 X 
 
 ><i 
 
 o 
 
 co! 
 
 00 
 
 <*- 
 
 o 
 
 (J 
 
 
 > 
 
 v> 
 
 
 ra 
 
 n* 
 
 m 
 
 e 
 
 
 « ^^ ! — CO 
 
 O ;00 
 O I rJ 
 
 '^1 CN 
 
 o 
 
 
 ml ?^ 1 
 O CO 
 
 SJ 
 
 C 
 
 9.V. 
 
 O 
 
 
 CO 
 
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 "^ t-^ ■* 'n 
 
 t^ Oj '^1 O 
 0\ CO 
 
 
 ■^iCQ. 
 
 O 0\ 
 
 160 - 
 
 Ox ho 
 
 -S; 
 
 ^ 
 
 cc 
 
 _ 
 
 rv^ 
 
 r<~> 
 
 n 
 
 ^f- 
 
 r-- 
 
 «-- 
 
 ,»i 
 
 0\ 
 
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 ':h 
 
 rr, 
 
 r^ 
 
 
 n 
 
 N 
 
 M 
 
 
 
 >> 
 
 
 
 
 
 ^^ 
 
 
 
 
 >, 
 
 .ii: 
 
 >^ 
 
 
 
 c 
 
 H 
 
 *-> 
 
 o 
 
 ,&i>-! 
 
 & 
 
 -o 
 
 Uh 
 
 f.T. 
 
 >< 
 
 t- 
 
 
 ■o 
 
 -n 
 
 '-f; 
 
 ■c 
 
 »-N 
 
 CO 
 
 c 
 
 r 
 
 
 
 
 
 
 
 
 
 >, 
 
 
 
 
 t4 
 
 
 >s 
 
 t-i 
 
 •g 
 
 -= 
 
 o 
 
 
 H 
 
 r" 
 
 r* 
 
 Xl 
 
 Lu 
 
 «- ■- <~l 
 
 -6 
 
 
 
 -a 
 
 "? 
 
 
 
 
 
 3 
 
 a: 
 
 o. 
 
 
 c 
 
 3 
 
 
 C3 
 
 
 
 
 nl 
 
 
 
 
 
 rt 
 
 -o 
 
 i 
 
 
 
 
 
 c 
 
 3 
 
 
 Z 
 
 a: 
 
 
 -^ 
 
 r<^ 
 
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 «*N 
 
 M 
 
 oc 
 
 rr^ 
 
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 tuu 
 
 
 < 
 
 
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 k. 
 
 
 QJ 
 
 
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 OJ 
 
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 T3 
 
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 <
 
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 By Robert Jenkin, D. D. late Lady Margaret's ProfeiTor of 
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