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:» 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 te 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 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 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^., 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 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 ^"'^ ^^ '^^^ 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 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 — — 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, 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 >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 > v> ra n* m e « ^^ ! — CO O ;00 O I rJ '^1 CN o ml ?^ 1 O CO SJ C 9.V. O CO - NO "^ t-^ ■* 'n t^ Oj '^1 O 0\ CO ■^iCQ. O 0\ 160 - Ox ho -S; ^ cc _ rv^ r<~> n ^f- r-- «-- ,»i 0\ () ':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<^ »--. «*N M oc rr^ «^ •^ •" «") tuu < t/> k. QJ ^ c c >u u H H k. u OJ > T3 o C x> D < BOOKS printed for S. Bi rt, at the Bible and Ball in Ave-Mary-Lane. 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