rules for correcting the usual methods of computing amounts and present values, by compound as well...

Rules for Correcting the Usual Methods of Computing Amounts and Present Values, byCompound as well as Simple Interest; And of Stating Interest Accounts. Offer'd toConsideration, by Thomas Watkins, Gent. F. R. S.Author(s): Thomas WatkinsSource: Philosophical Transactions (1683-1775), Vol. 29 (1714 - 1716), pp. 111-127Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/103043 .

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( ltl )

ris percurrit lineolam tht,corpus aliud in circulo in-eadetn di^.

Rantia motum pexurret arcum-x KN; & Area fed}oris b

Circuli & Trajedroriz fimul deScripe:t erunt-x R N x CN

a- & gyx t CNquz dllzAre:r funt in--ratIone daeas fcil. ut b adt .- Adeoque ubi eR a=b, uei Xt in Spirali Hyperbolica, Area fic detcriptaerit femper xqualis Arez feiN;torisarculatis in zquali-tempore defcriptse.

NoucOr ^4. I 7 I 3 , I , . , l, . ;.. _ .

ItI. Rigtes for aorreSxg thc U0XAI Methods of {omp£ng Amogtts and trc%ent telxes, b Compognd Js well as Simple Axterc#; nd of fatl}ng lxteref A;;ognts. Of;cGd to Cvn#rderatfon} by Thotnas Watkins, &&4t. F. R. Sr

1.* Qf Compound Inecre0X-

E! E Suppofitiorr whereon the Metho8- of computing 1 IzY Compound IntereR is founded ; sir. That all In

t-ereR Money, Rents, cFc. are orSmay be conRatxlyreceivid, and pUt OUt again arlntere0, th; Mement they become du¢, W2hOUE any Cl?arge; or-Trouble, being i-mpradicable; ther>: fore all Computations ly- this Method- (-¢xcept of FeefSimples or oeher Perpetuities) muA meds be errogeow Thuv bE lnSance, the Amount of a Sum of Money, or Aniluity, for want of Dedildions OUt Of tlle ProfitsX for the "navokable TrQUSIeF Cliarge, atid Delay- in the M^nafgementfi will be tOQ great and fort ttze fame reaGon, the prefent Avatti¢^- of a Sum of Money payable in any time to come, will bc tOQe}-Stle; alSo the prefent*value of an: Annuaty (being only the-Amou<oFa the diSerence between tl e Annuity, and tntere&^ of the fad prefent value) will be toot much. But ill long terms of \eart,^ as vllat diSercace becomes IeSsSo does tlle ErroB ai tl^ Etlexm is

greav

( ltl )

ris percurrit lineolam tht,corpus aliud in circulo in-eadetn di^.

Rantia motum pexurret arcum-x KN; & Area fed}oris b

Circuli & Trajedroriz fimul deScripe:t erunt-x R N x CN

a- & gyx t CNquz dllzAre:r funt in--ratIone daeas fcil. ut b adt .- Adeoque ubi eR a=b, uei Xt in Spirali Hyperbolica, Area fic detcriptaerit femper xqualis Arez feiN;torisarculatis in zquali-tempore defcriptse.

NoucOr ^4. I 7 I 3 , I , . , l, . ;.. _ .

ItI. Rigtes for aorreSxg thc U0XAI Methods of {omp£ng Amogtts and trc%ent telxes, b Compognd Js well as Simple Axterc#; nd of fatl}ng lxteref A;;ognts. Of;cGd to Cvn#rderatfon} by Thotnas Watkins, &&4t. F. R. Sr

1.* Qf Compound Inecre0X-

E! E Suppofitiorr whereon the Metho8- of computing 1 IzY Compound IntereR is founded ; sir. That all In

t-ereR Money, Rents, cFc. are orSmay be conRatxlyreceivid, and pUt OUt again arlntere0, th; Mement they become du¢, W2hOUE any Cl?arge; or-Trouble, being i-mpradicable; ther>: fore all Computations ly- this Method- (-¢xcept of FeefSimples or oeher Perpetuities) muA meds be errogeow Thuv bE lnSance, the Amount of a Sum of Money, or Aniluity, for want of Dedildions OUt Of tlle ProfitsX for the "navokable TrQUSIeF Cliarge, atid Delay- in the M^nafgementfi will be tOQ great and fort ttze fame reaGon, the prefent Avatti¢^- of a Sum of Money payable in any time to come, will bc tOQe}-Stle; alSo the prefent*value of an: Annuaty (being only the-Amou<oFa the diSerence between tl e Annuity, and tntere&^ of the fad prefent value) will be toot much. But ill long terms of \eart,^ as vllat diSercace becomes IeSsSo does tlle ErroB ai tl^ Etlexm is

greav



( t 12 9 gr^ater; at1d ln Fee SimpIvs st van8flles s tlle contrary to 5rlliclE llappens ill .4Xnounts of itms of Molzcy, and Annuitic;

AJ1 wbich is propos'd to- le reEtifyvd, ouly by a juR- ro dbuAion of thc R-ate, and Atllzuity; (wlwid -is dme by de dudsng fo 1nuc;z per Cent. tllereout, astlle sv-hole Trouble, and Cllarge of Manwement is fi1ppos'd to amount to, Mand redtl- cxng the Remaind-er, by a DiScQunt cquivalent tO tl]D uppos'd 10fi5 of tinle) and tlzen bWy -Nm wirll dze Rve^ So; reduc'd brSums of Maney, and \Vitll tlle ;Rae and Annuity reduc?d -;n the likeproportionfor A-nnuitys, accosrdingtothecommm Method of Compwnd lnterelt; as follows. Put r for tlle rateofIntereSof I 1.[ for theCirge a-ndTroubleeftheMae

nagement of X 1. IllenAis r {r the Rate after dedud;ting

tllc Eaid Charge, = (puttong d- :for I - Ct)- dr, AxdbSor tlle IDiScount pt t for the time lo-0, vllat is for- f-ubh part of ehe P=iod of time ttn xvllich ife Payments are made (wllether Yearly, X Yearly, QMarterly,+er otllerwiS4-as is fiuppos'd tO bW fpent in receiving and put:tsng tllem Qur again at Inte refi lXen, dtrw b¢ng =>.t-lle lntereA of litl. for tbat time;

- -dr fay, a ^ f iso: x *: dr^* d - (put-ting c-for 1 idtr)

wh-ich is ekqual to the reduc'd R-ate, near enough for

rad:tice, fors which put t. ButK if theutmoft- accll.ragy be rev quir'd, the IXi{Eount-itWelf muR be madetwitll regard tO tbe like lofs of gime, which is done-by a Series of lDifGounts rais'd

thus;c(xit):tir do Mr tdzr tdr grc e u c

td2t2 t d3r3 t3 tr

Whence Gr _ + X - a grv. (putt}ng f for {td-r ._ -,1 ............. l dr s , _ , .

c X _q ga FX6. x ,6r,,--(-X-f xdr)- q$_qe > Wr, is = ,-the true Ratecreduced. Put s _ X + 17 n. the:;omes s = tht p.tefcat Sin or Value, a9-the Amount

Thea



C 11a,) -- e fl -

Thenwillt d 4 r Jra^t f3&a.-X d-r - xpI-+g xp psr

.., ,, ,,

lbb exadly = w : But t de > x p m isfuflicient for pradticeP

And; br -tlle Amoo3nts a^nd prefent values of Annuitys.

Put. ^4 Annuity per anaxm.

aAnnuity 2yearlyf quarter}ys(a

R = Yearly rate >of:Jnterefi of I t r-Rate 2 yearly, quarterSy, <Fa

ol Reducsd rate yearly, 2 yearly, quarterly, &>

>= Number of Years, 2 years, quartersa eEv

Then will a R = a-be-reduc'-d Annuity taken yearly w yearly, quarterly; or otherwi-le; and by Compound IntereR Ewill be { tl- s a_ _ ' +t1 I w 4 1 AL t, _t s

g 4 r R 3"_ t Sr--I- m

* 4-,- and R a ( = 8 ) = P° Whence tlle

theorems for-folvinSg a-ll eho other--Caes are eafily deducd And if the Rate be requit?d, when 'tiS- for a OSasm of Money the Solutioz {s ob-vious: xvhen for the Amount or Val-ue oiF-

mrAJ a an Annuity, fiinn I irtF =--- are the Equaw

A -a-pt tions whence-Theorems-for thc Rate are ul'ually;denv'd, whicll by thws Corredion -bEcome X + rIn =--- That ot a-pr vlse Eatne 1: may be had on both fildes the- EqRationf pue st0* forr, and 'twill be -I+t"= --0___ a : tllen,by

4. C - 4_p#r,

the Rate aGumd as near the trutll as may be,- Endi the va:!ue

<>f;W d + tr=d -d-g) and in any- Thcorem for tlle

7



( s14) URsee, puetxng#for", andp«¢forp, therefiute wlll be th-e Rate redtlced nearly * and bytepeated Operations correfting g a*toi19 tit>rcSy , tLle true , alld tllC"Ct t til: -Nvl ole Rates may be fioundZ

oThe onlv diXiculty that remains} is the rigllt afruming vle Quantitits C a1ld t 5 th e impoSlbility of doing xvhicll ̂ -virh perteS LsadctCis) I Suppofe to be tll-e reabn why 1lelter Clliss

ntr ay Mesbod of GereEtion to tlle like purpoWe, llas yet been taken notice of by tIle Writers QN thiS Si: jed; and wllat may theretore be very likely to be objeEted to tllisw But tlle fame Objed;tion I take to be of equal force again& ttle E(timates -of any:otler jUncertainties whatfoever, as Eftates for Lives, -Infurances eFc

Fir then} for the Qumtity , which ss put for t&e Trouw ble and Charge of Managtlent, viz. of colleEting and plaw Acingout the-Money on good Security, together witll all Contingencies attenditxg s}ze itne, as travelling Cltarges, Ex pes, Actorneg's Bills, cEc. of all whiclz, tlle principal Arti sle is tlle Charge of Colledrion or Receiver's Fees} wllich is gammonly a fix'd Rate, cuRomarilyallonr'd in the Place, or upon tlles E0ate it rClf OUt 0 wffiich the Purcllate is made, if it be a Rent; and for IntereR Money; or any other Annuitys the like Ellimate is .to be made, whether the Proprietor adrs fiar llimSelf, or by another.- Then for the Charge of placit - OUt thc Mony at Intereft when receiv'd ; though thisi be for x;hv mo part defray'd by the Borrower, yer becauSe it hlglz- sy codcerns the Lender tO lie It be fecurely done, tllere are ufual Allosrances made to Agentssnd Scriveners, to encourage theirCare andfidelity therein; befldes theTime, Expence and Tsouble of Elle Proprietor hi-mfelf, in finifhing Contrads, intpeAing Securities ;. and wllatever is fav d in tllis ArtF

clew we muft fuppofe to be fully made up by an Equivulent Degree of RiSque in the Security.

In.the!next-place, zfbr the lofs of tim-e; tllougll iselfo im- eettlible for this to be exadtly atceriain'd, nor perhaps fo aoar!y as theffiormer, fillce it depends very much on the Dili- venee of the NAanager : \tet if the u(ual times of Payment

f



( 115 ) oftlleparti¢ularRent orAtlnuity to be purctlafed,5ritilanzoderate degree of dili ence in tlle Manager, and tlle uSilal indulgence pradris'd by Men of Bufinefs in tllis CaSe toXvards onv allortler be tSfers'd, a reafonable E0imate may lve alSo ulade Qf t[lfv 1012

of time; In wllicl] -'tiS to be noted, trllat [ntereR Money-lJei,lg ufually paid sn {;inall Sums, wllen any Sum of Moaley to lze made up"of Weveral fuch Payrnents, is intended to be put Out

at Int@rtS, the wllole muft lie dead till the 1al Paymetzt be mai; alSo that tlle Principal lies dead tometitnes as well a-s tlle IntereR; --and tnat on tlle other llafnd, to fave time, Bor- rowers may be found OUt, and treated Witlt during tlle rime 0

ColleEtion; it this Advantage is in a great Meafure loR by tl!e difficulty of fixing the time orQgactum of a Loan, till the whole be patd in ENote -alSo, that if tlse Charge of C'olledi- on, -or-Lefs of timef on the Rene, or Annuity, of any !particular Eftate or Place, lze found tO difEer from rhat of the tn- tereR of the PurchaSe Mony, and fQ XUCil exadnefs be required. as that the Compuration be made witll regard to fuch difference: 1t muR be dene, eisher by aXuming a proper Medium for totll, or more acruratelg tllus: For the reducd

Xat: f the ilntereMonyt put ) } Tllen r : X : A : aiX,

and t : X + t1 -I :: 4 .- t + t1 - t a x } -m = r r g

I ttl X p, TQ g-ive a Specimen of this Method in Numberss firflc the

Quantities c and t are tO -be afEum'd-, which are not hereetobe accomodated to any particular Place or Ellates but tO be taken in general: And lSrS for c tbe Charge of Management; the ufiualRates of Colledors Fees in thetzKingdoms, asl am inform'd, are Sd. x^J and lGd.per Pound, but tlle moft ufiual lxd. wllichas 5 per Ctt. However tO be withincompaSs, 1 {llall take 4 per Cent. for the Medium, including wluat further

R trou-



( 1169 tr4uble and cljarge may attend th-e }teceipt of thse-Mony, be61desReceivers Fees; - and X per Cent fo£all the otherwCharges before nzeneionad, in placing it out at - Interell, both which

make 6 per Ceat. fo that a is-oy 06, andKd ( I {)-o, 94.

Next for-t elle Lofs of Time t f1nce few Annuities aw paid yearly, and a E)itcount being giveni for the Lots of Time, we are to lofe no more ehan xs diScounted br ; therefote l choofe :Half-yearly Paymenes for ti Examples, being t11e moR llfualS which with little Alteration may (erve for quarterly; and con Ildering the before meneioa'd CircumRances relating £0 thew Time l look upon two Months the leaA, and kvens or eight :hIonths tlle ma*, that can well be Suppostd- to be fpent,- one Time with another, in recaving and ptstting out the Monys upon a moderate Management , between whish theMedium is abolltfour Months and halzf; wfhichbexng X of 2 aXtSgi6

+3-t: and xf-be - * , the yearly Ratesof 4-, 5, 6, 8 and e

optr Cext. will pfoduce-for HalEyearly llates reducSd of X 1. o.OtEy39, o.ox30g3, 0.0X76J6, o.o36S6g andoo4g4 each-

t; But if t be--q-q5 &c- xdr>2twill bes o^Ox8s4X1g

o ot30ggg X 55, o. ow76X777ss o.o365g6z89 a2 ° °4S45t 3 59

eacll ty (fo that eacll Rate-loWes by this -Effimateabout 112 Part.3

Whence the following Amounts, and prefent;values of I

per Axnxns computed Eialf-year3g, are produc , -and compared w¢Xtll thoSe o£ the uwual Method xmpud yeady? eo agrec- w.ith'the..common Tables.

j£gres-s



- *g * # - -

I Pref V4lges-of I 1. per Prey^. FEales of I 1. per r An. at 5 per Cent by DifferenceJ An. at 6 per -Cent 67 Di ferences c Conzp. Int. I Co.lSt.cor. Comp.;:lnt. I Gn.- lnt. cor.

* -- ; . t 4. i

5 At32948 | A>Q8343 ,;4605 4,2Ie 36 3}97582 ,23654 I0 7t72I 74 1 7t333IX ,38860 7,36009 7,oo325 35687 15 Iox37966 9,9I935 ,4603I 9,7122S 9,go84) ,40382 2O I 2 ,462 WI I I ,9 7 7 5 3 1 48468 I 1 a46992 g I t ,o6 3 7 3 ,+o6 I 9

3c I5237;45 I4,9I902 45343 I3,76483 I3,4t805 ,34678 1 t7tI5909 gI6,78to0 - ,37709 Ysno46go I4,7-8309 263tI

5< i8j2S593+ 17i96l9 *- |949 15,76I86 I;i-i574$6 ,18730 7C 19}34%68 I 9,18247 ,I 602 l I 6s 38454 I6t2995 3 ,0850I

I oC I ,8479 I I 9, 7 92 3 t ,° 5 5 60 1 6,6 t 7 5 5 I 6, 59 5 l o - | ,0: 245

F.s. 20,Q0000: Aopoooo ,00000 I6,66666 I6,66666 t ,°°°

_,j X sE* *-.t-

amoxats of I 1. per Arle I D 6 per Ccnt. by DzffG^.erces >

. . .

Gompo 1rt. | Co. lnt. cor. F

.

l 5X63 jog 5 24 I 38 1zet 57 I 0 I3018C:79 I4,07854 1}l0t25 i

36,785 59 3t,9 1o54 s8 7 SoS 79}058Ig 6S,83976 wO>2I843;

I 54,76 [ 9 7 T 3oSO 7 3 3 2 3 *9 5464 | S;XloSt8 4_2)0I4>9, 1 l 1 tl I 3s9 i

I ?46, 59991 r 28S, 4 4841 458 r 5 I );

5638s 368c13864, 9 J j3 I 7 73,39z X

j The fame at IOper Ce72t. }

z 6,ioyio iip7051 eS°8o-5 ! I 5,9s743 I4,32680| 1 .-6to63 57>2 /50o 49I7930 8>ogi7<B

t64,494OfL I33>-96428 30X54974 44o-Z59257 340}2Igoo lo2^,373s7

3s3g,8I 648 2062Z5 7 161 ' 9721244SI

2Q474,002 7 I 2 2 5 5, 3 448 8 2 I 8} 6679 I 37796,I27Z7257),@926,6522os7344

w394591 *99943

3,2oS9g

7,713>9 146z49898

63 7e,4OQ 2 I 8,69446

7°4,°9845 IOt

3

lob

10

o

3-° 4o 6v

I 5,5 j63 S)t 3 1o4| :,57789 ,57846;

33,6595 :9,85996 6d,438S5 58,74 546

X xto,7997 7 I04,30°79 35t,5837t zg9,8S,64,

Yo 9 7 I1e L88t 7 sn ,5 34 16IN,O: 5 I 6 tpo5)9267 t

}0

'4o 6..Q

| The fatne at 8 per Cent.

i , s 666 3 5 Z 4 3 3 , 44 o 2 I4-148656 1 3sI S092 I 8 33564 45,*16I96 AO>I 3759 5,624}

. I1 3,:832l 95,S9I623 t 7,76699 2 59,o5 6 5; zog, x S 7 3 7 4R,899 I 5

I ^5 3,2 I 3 3| 92Q,9QO40 34 ;, 3 1 29 845886,9354

3918?Q578 I968,.8776

2 748X) 5 t 5 j I 65 39,o9S9 I og45;41+68 ..... .

Nores That thxs Corrdion is alfio applicable to tbeValuati ons of s ERates for Lssts, in which the firfl: Step being to find an Eai?alene iis Years- of Continuance, brings tbem to the Cateaof ERates for Years.

R X Of

( 117

__ J _- s __

< XmotznDs of I' 1. per An. sk At 5 per Cent. by Dzierenccs.

- ComJp. lat. l Co. [nt; car. ;



1I. Q£Simple-TntereS.

Tlle abltirdity ofh the Suppofition, otx wilich the ufual Nleth-od of computing. pvrefent t.alues byt Simple^lt}tereRs is bunded, w. l liat tlze -Rent or- Annutty is;conttantly- recei- ved, and put out again at [ntereR, as it bEcOs due; >b-Ut tlSt tIze Interek of the Purcllae Mony lies dead during thev wlnle Term, is fo appa£ent, atad the Errors arifing from it [0 grOrS, tllat the Wrieers wllo IzzYe laid dosvn tllts MietilOd llaVe at tllt lame time caution'd agaitlR the Ufe of it for any more than 6 Otr 7 Years, the Error for that time being nor xnfiderable.

The fame Suppofstondoes-alfo occahe-n-tile miScomputsa- tion of Amounts, or rather the miEapplicationoof them to their proper Cates. Wherefores fince tlle Simple-<Intere{t of Mony xs of equal -Valut pro rats, and of tlze fame regard with Rents orAnnuitysy being eacll tlle Origin-alProSts ifluing alike from a principal Stock, Eflate or Value, and equally improveable; This generalPRule may [erve fUr a juR Corredtion of tllis Meehod, vi. Ti;zat fuppofinginoanyGaGesanlatereCought to be, or not D bc allovred to eitlier of tllofe Protits, the fame be done in the like nCafe tO the other. Thus, in ehe CaGe of DeAs, or Amounts of Sums of Mony, ReNts, or Ahnuitys for tlle tsme pa(}> itS utual in pradtiSe tO allow- no IntereA to eiZ ther: For though tlle Law, to curb the exorbitant Avarice of Ufurers,and fbr otller ReaSons,does more expreny diraIlow Intee rell UpOIl latere& for aDebt; our Courts bothefLawandEquffly, as I am inform'd, will bewas far f£om allowing the Charge of 1n terell againA atTenant for Rent in Arrear; exceB on a nome ptnx (uthicll isrow becomealmollobSolete)(otllatin tllis Cafe (putring avand r forthe i?9tinllityandRateyearly Half-yearly, or otlletwsSe) as pr r 4r p is the Amount d a Sum of A4o¢y, fo iS. a n tlle Atnoun; of an Annuity or R¢nt in Axreart and nor

...r

- I r S I X Q>as Arithmetivans commonly makeit. But in

the 67omputation of prefent Valuesfor Amounts for the time to come,

( 118)



(. Il.0g y come tile farne liEitAg made on the Expedation of a onilanf regwlar Income of tlle ProSts, wiebout any extrtordinary In- terruption; an Interell ougllt : to be allott- d tO botb eEpecially in prefent Values 5v11ich are follnd loy Setting the Amounes of botll againtt eaah aher: fo tilat in tlleSe Cafes, puttXng o8^^ =

S _ I, if -xr + X s d n, be made the Amount of a Ptent or A¢ nuicy ; then 1^ x rSr I X pt-r n + p, will be ehe proper Amount of a Sum of Mency, and not p ran p : and conSequently

c r + + X a n * = p will be the prefent Value of a Rent or z xr +X xrnJr I Annuity, tlle fub:fi£quentsIntereEt being remittedzon both-fides in lieuS of the Lofs of Time and Charge of Management; which fuicll as ar-e apt eo depreciate long Futurities, may tl ink the propereR Method of approacllmgx the true Value; bur 1 rather look upon tlle fo-rmer Metllod of! Compound IntereR correded as more exad-, as well as more general, the Interelt remitted in tlliSs being in ffiort Terms lisX and in long Terms more than an Equivalent for t11¢ Troublew C:harge and Delay ilt the Management. But Xt iS 11oweva the moR exad of any of tlle--Methods that have yet been deduc'd from Simple ln tere0. Tlle xedudd Rate may alfo>i¢fome Cafes be properly made uSetof fbr Amountsa but not for preSent hlues, except for lhort ̂ £erm$; and then, f1nce t : a : : s x t Lt I X t n ww

^ Xt + IX n w twillbe sst+t xan||._aI wrsw r r 1 x+ tfxan s- oC

x pt n + px and>X! X t +: X X g a +;l - r P

-

aano



( l2O )

E:samtles -of this Mer-llod comparsd wisth thofe of ;tile for aler, wrill ftand as followrs; in whiciltall is computed Haif^ yearly cs;cepr the laR Cotumn of Con1pound Inte--refi.

Amollnts of I I. at -5 per Cent. -compured 6 Seteral Ways. -,_ . , _

| I 5 + 3 4 5 6 . - ,. . ..

> StxnSle [7f. Ic{.forBonh Sim.lat.cor. 1a.&7tbereirst - Co. Irzt. cor. Cor^p lnt.

k _ I " .. .

wrn + p= w ldtillptn-pl X Sr + l_X sX tElxPt tl + n X 5-RI XP-"

_ , . . . -- -- ,

5 tx25 I,!5 I,z78I3 ,25S°t I 55655 I- ;76^S

I0 I,5 I,5 I,61875 Is56338 I,5789t I,62889

50 2, & : 1 2s487.5o 2>340t 1 2>493 2,6S3 >°

30 2,5 2, | ,60625 3933QA8 3,936t.5 4,32I94

4° 3, 6., - 4i9750° S 4,53419 6,2I504 7,4O00 60 4, . 2, 8,462Fo 7,58Ip4 I 5,49408 18-,679I9

8o S} 2g I2s9S000 I I,48345 3S,6z6X2 49s56X44 I00 6> - 2, Y sk43 7 SO I6,2 38 74 96,29634 t 3 I ,5QI-26

Amounts of I 1. at 26 p-er- Cent. by the Jame tEreoreau.

5 I3 | I ,3 I>34Q5 I3ro63 1 Is31338 ;ts33gn 2 Et I-,6 | I,6 Is7 710 t ,69758 I,72471 I}79°84 20 2,: ;t 5nvomo " ,7094S 2,97463 ,:07l; 30 2,8 2, 4,39)0 4,°O87Q 5>I 039 5,74349

4d g ", 6}2440 5>626 2 3 8,84844 10O:8S7:

6e 4,6 29, I I,026O 9,76y 2S 26,320S6 3^ 98769

8e 5,8 *, 17,24S0 I55l^954 78,294S8 l°Sz79599

100 7, , 2, :4,pI0O - 2I>715I0 232>89856 33gsvo20S

SmounXs of t 1. gZt I0 pes Ce-xlt. by tbe fiase theQrenzs.

S tZ5 }^5 | 1,6125 I^54749 I5597° I,6I0iI 10 2s ^ ̂ tt750 2s30t 5 7 4 43 6S 2}593 7-4 zo 3, : 4,95°s 4,45951 5}91 793 6,7275 sQ 4v 49s 1 8,42So 7,3838t I4}39643 I 7,44g4 40 5, 29 I2}9000 ItI6449 35}02I90 45,z5925 60 i 7, 2, -2,8Soo :t,4°493 :07,:57If 304,48I65 80 | 9, 2, 4o,Sooo 34wyso84 1nz5,5335 204S,4oo3

I00 IIs 4 60X/5Qo 5I2t0o22 7^? 5Ss5t9s I3730)6It7 . _

- n

l ears



-w / a *

*UX

- . . !

ES.

* , X X s rw

Pre/: Falwes nf I 1. yer An. 4t 6 per Cent. by

_ 4

Sz.ln.cor Co.Ent. cot. Comp. Int -

4,2334-9 3s97582 +>4 I236 Eit 5 5 79 7,°322, 7,36009 9i39 341 9, 3O843 9,1 I.22 5 <-39t 35° I 1,06373 I I ,4699t

It,87575 I 3,418OS I 3;,76483 I 3s99 74{ I 4, 7 8 309 I 5 ,e463(> I 4y6g544 I5}5z7456 I5,76I86 t5,47252 I6,:9953 I6 38454 I5,99759 16}595IO I6,6t 755 16,-66686 I6Ogd66 | I6,66666

120

k S E

s It

I S

20

3o 4t Se-

PreC. V4lges of 11. per An. at 5 per- I Cent. by

Sin. j.con Co.lnt. cor. I CornpW Int.

4,35it9: 4s:°8s43 4:,3tR48 7,64+73 7s333I4 7s7:I 7*-

10,1082I 9Z9I935 1o,379b6 II ,95980 IIn97 753 I2,46221 I 4,4,5407 t 4s9I902 I 5,37445 I 5^9 799° r I6}78200 I 7,1 59@9 I606682 I 7,96I90 I8,Z55.93

7U I8*lOp86 19,I 8z^7 t9,34568; . x8,9 I525 l9,792 3t I 9,84791

26,-0000 l o,OtO00 b 20X00000

The Theoremsv tO tlle - preceding Columns of Amounrs (of wthich the fourth audtlSfth are infinitely vatiable sn tlle ReF fiult, by iOuming a and t ln the rEdus'd Rates at Pleafure may ferve:to anEwer all the fimple Cafes of Amounts that oc cur in Buf1nefs: To inRance in fome,

x. The firS Column contams the Amounes of fucll Debts or Sums of Mnny as carr-y a fimple Intereft till tlle Principal be paid

2, Tlle fecond Co-hlmn anrwerS thenommon CaSe of Debts due by Bond, that by Law are allow'd tSt tO exceed the Pe- nalty whicl iszgenerally dowlble tlie prin6ipal De-bt.

3. Tbe third Colum-n antwersDthe CaSe of a S¢cutity or j-oynt <:)bligor, that i lias duly and confiantly paid tlle lntereft, and at laS the principal Sum of a Debtj f£om wllich lle laas a Cczunter-Bond from tlw principali;E)ebtorto fave llim 11arm-- leSs, sgainS whom lle nlay- make his Charge :from tllis Column.

4. In cafe the Parties flall agrce t1}At ttle Debt Ihall lie for any time certain, or uncertain; and for the much greater eafe, ad^7antage and fatisfadEtion of both of thems no Int-ereR to be call'd for, till the Principal it {elf is paid; but to carry Xnte- reR as it becomes due, tlle tWnder allow'ing for tlle timeand charge he maflc have been at, in receiving and putting OUt his Intere0, the fQurth Columtl Will fit this Cafe, or elSc the fjftll

( ttl )



r ,>:: a as a greattz or lefs d¢ree of Lelity is agrecd upon in favxr of tlie- Bo.rrower

y. The fiftll Golumn is alto proper in tle followingyCafey siz if wit be demanded, vrilae ERate in-reverfi,- after a -certain -number of Years,;any Sum in E!and xvill Pushafe; thefir; 4q?--betng tO -find the Amount Kof t-llat Sum tO tlletume the ReZ rcrfion commences, its llad in thistolusmn.

6. Tile lall Colunln gives tie Amount ef a Sum of Mo ney, according to tlle commollhMetllod of Compound [ntereRs but being computed witll tllat extraordtnary riger as has been faids (except bLeme ;i-all allowance for fiie loFs nf timet by being done<57early3 'tis 11ardly fuitable tO any CaWe.

Other CaGes might be enumeratedS tO wlliCll the foregoing Theorems migllt be equitably applt:>d; beGdes fiuch extraor dinary ones, wshacin itq<may appear to -Atbitrators, or a Court of li:quity,- that either Pariseh deferves Favour, either by way of CompeHation for In3uries fuier'd from the othet by means of any fraudulent or oppreffilare PraiN:tices, or other- wife, for whsch no-other redrefs is provided.

11[. Of IntereR*Accounts.

T1le hlequality of the ufual Metllod of Ratirg IntereSt Accoutlts, as pradtis'd in our Courts of Equity, will belt appear bg n Example, for which I ihall talee the common general Cafe ̂ of an Interell Account eo be fiated on a Mortgage, wiz {uppofe IODO 1* ro i }et out a-t - per Ctnt. on a Mortgage of x 20 /. per aXnXns, payable Half Yearly, and the Mortgagec after fi-ve -Years end, m iliave PoSefElon tilI the Arrear of In- tereft, accruing Interefic and Principal be diScllarged : 2arre, How long tha-t willibe t; Fuppoilng alSo, for the fake of bre- vty in the Account, tl?e Pay-mentstobe equalty, anel punduw ally made as tlley become due lSy the Chancery Meehod, the Rent is firA apply'd to diStharge tt krear -of [ntereR; and then tlle remainder of e57ertt Half-Year's Rent, after deduEing thn Gme Half-¢r-ce lntere. is apply'd towards the I)itcharge

of



( 1t-3 ) oftluePrsncipal, andtl1erebythe Princlpaland rntereRconti nually lellens,till tl1e xvIlole be paid oR. Noxv ly tllis means rlae Mortgagce, afEer t!ze ArLe;K3? LS di(charged, pays CO}P. Inrerell, xvith-the utmoR rigour forio lnucllperanawof tlleRent, as exceeds tlle [ntereft of tlle wllole Principal Mony, and receives tUt Simple [ntereR br llis Debt; wllicll, however hange it may (eem, is eafily prov'd, by applying tl1e proper Tlleorems of Simple and Compound Interell to tllis GaGe, in wllicll tlle Annuity, Principal Money, Rate and Arrear of Tntere(t are givell, and tlle time requir'd ; tlle re(ult being tlle fame Wl-tll

tllat of vlle Chancery Metllod} except a very fmall diSerence onXy wllen any part of tlle time is exprefs d lzy a Fradion: sxz.

pUttillg L for Logaritlltn, 4 p 7 = 3 0s s 1 ot r = 1 ,C3,

t--- time of contradring tlle Arrear- xo HalfLyears, n- any nunlber of : Years [pent in diScllarging tlle wllole or any part, N= number of Years required ; tlle Equation for tlle Arrear will be p r t * p t- N =aa; and for tlle Principal and acZ

crueing Interell prn Jr p-prn g_-x aL. Wllencc

trt f L -=N=16,7X4gYears-tlle timedemanded; 2.c 2Ls

i. e. ' 5 Years = tlle eime of diScllarging tbe Arrear, and

< = x >7X49 years,-tlle time in wl1icil tl1e Prin- zLs

cipal and accrueing IntereS is diScharged; during wllicll itsevi dent, the DvIortgagee pays full Compound gntereIt for 60 1. per annum of tlle Rent. For t1z; Corred;tion of wl icll inequality, in the frll Place to tlle end tllat ncitller Branc11 m-ay ex- ceedfi or be depriv'd of its due Profits; Tllis general Rule is propos'd as necefEary to 1OW always obRerv'd, siz, Tllat A- mounts of tlle Produceen eacll fide be Rated Separately, and fet again(k-each otller in tlle Account, in order to a BallancS And in tlle commotl CaSes of Mortgages, Governmenc, and

S Sto-



( 124 ) Stock secuTitiesw ec. wllere the Debt is paid of by a Rent, Annuitys Penlion, Dividend, O£ otller Payments iSueing inthe fame manner, and with the like trouble, cllarge, eFc as Int terefi Money does; 1 prefume tl iS Rule will alio be eaf1ly ad mitted, wix. Tllat the fame eqr.uitable Advantage b> impartially allow'd on lroth fidES; for whicl tho Method of Simple 11a. tere0^ as corredled under the fioregoing Head-, feems truly aclapred; whereby tlae Original Profits on-;cacll fide are fup%- pos d tO be deem'd, eitb.er as tnteref}, 0r elre as prin£ipal Mo

my; and fince elle Amounts botll of an Annuity and Sum of htony, for the time pa0, as there RatedX on tlle Srfi of thefe Suppoftions (t being tllere --cz) are likewiSe voucla'd by our Laus, and the prad;tice of our Courts, o be good wllen fepat rately us'd ; I think itS very evident, tllat tllt ACCOUI}C OUgl1t to 1w ftated by fetting rhofe Amounts againtt each otlaer thusS

I + t r prt + prn Jr pan, (Nvllence " p-1\E, 2x,6666

Years)- and tllat tlliS MetllOd iS moR proper for genenl Ufie, in tllt Cafes mention'd : Unlefs it dlould be tllougllt Xt, in conGderation of the various Ways found OUt for tlle ready improvanent of Mony, ro allow a furtller Advantage on both fides, lay charging the-Original Profits as Principal Mo- ny, and giViNg a Simple IntereIt tllere tO, whicil Rill falls flzort of tlle Advantage alTowvd tO Rents by vI e Chancery Me- tllod. And tllis is tO l)e done tWO sVaySs wi4. either- by applyZ ang an Amount of Rent tO pay oStlle Arrear firS, and after- >ards anotller Amount of iEtent to diScharge the Principal9 and acerueing [ntereR; or elKe by lerCing vhe Profits witb all Arrears and other CIlarg.es run- on at Simple IntercIl on each fldCx till the end of tlle Tcrm : 3'i4 pUttiNg X = a-[, 7-

t-IOc=a- pr,f -XN-rxQc,gf-2ptt2,-2-er ,--t X p r t _ Arrear of Intere(} ; by tlze firR of rlleSc 'tu7ill bes for tlle Arrea-r, , *t p t z ;3 - -x r + z x X n, and for the- Princi pal asad accruing I£stcr-eR, ;1 pc r i I x X es _ p ; Wleflce

va xcs



t ltS ) i - t

+ 2 v 8 p X r l fX -,f Ag = ff _ I 8>765 3

r

Years. By tlle otller 'twill be. for tlle wlloles y + ttr2S + p-

2^ X r I xx , Wllence sf luGp x 8 XrAgt-g _ N 4 X r

1 8,I 648Years. The Lender will lzave to alledge for tl?e SrIR of tilCfiC tWO Ways, that as tlle Rent is not 11indred by any otller parallel Cbarge from making tlle utmoa produce it can, fo for that reaGon ougllt his Pritlcipal Mony tO have tllC Ad- wantage of the Arrears being firll diScllarged, wllich alio agrees with the Chancery Metllod in this particular.

Lafily, anotller way of Rating thts Account, may be taken from that Notion of Simple lntereR, svhereby tlle Annuity on-

y Is charged as Principal Mony: and then 'twill be, for the Arrear, p r t + p t n-a n, and for the Principal and. accruingq

j _ v 8 p evr Jrf 2 _g

IntereEt, ; X r + 1 x X n p; Whence (-

4^r P_ Jr 2 8 P a r + f f) = N_ I 79307^ Years ; zvhich zas - 4rr appears to be famc with the Chanczry htctl1ed, Ql31y tllat

the Compound Ineereft in that, is turned into Simple ill this X and as it Itill retains part of tlle fiame inequality, tO the 4dS vantage of the Borrower, it feems only 1St tO be obServed in fuch CaSes- wherein the Borrower may be tlloughe to meric hsour^- as when >1zc Debt is paid out of the Profits of "rade, arifingvby extrsordinary RiSque or TnduAry. But ftnce Ellcl] a kllle of diflindion is llardly poffilble to be reducvd tc) getlera.l pradtice, tlle uSe of this Theorerrl feems re[trained to hucb Cafes onlys wlterein the PartKies vlemilves, or a Cours of Jiquity iaX think it reafonable.

For a further illultration of thefe Rules9 tlle follou7leg Spectmen is added, which lhews at flgllt, lloxv tlle RefrXs of the [cveral Methods d jffer, as tlle Rent Arrear, orka*<of lntere0, is great-ct ol lofs, and coniequcatly of l1ow mucll more

$



or 1cX conceril ic is to tIa: Pariic as weli a) to tte due Adnae nz(Rration Q53tXnIC:0 t3 have regard tberetoJ

Tlic t'tit! requirtt ltr. s . tr titt^t;**8t¢^t1 i .4i> t e--}*+; ;r. - .s o t.. ..__ .

<3 dth3rgt a Nto1 r. og - ^Zo ts 9 tNr 3 >nt 5 - o ! g- > 8>g-s+^ % ffi > r +, g-; ,st ttt<¢S,sm,

At tht Rate of Intw per .tzrtt>a of z te; >, | 6tor w1 X lt- tt Cf | .} brr; C No Arear (Jf Intercl+ |#- F s v t ,* ' X,

By the Chanc^rv Wtct <od r &*9 I4 I t t Br rSt At;y $ t 6)4 O 5 I Jt ) 3 ) §§ By the Xme turnint, the 0mptntt 1nto SlffM 5 §

By Simp.Int.cor 7 Princ. arXd j cn t, rl t at nt. 3 t t >^ a p r s ¢> 7n t] *3 ,>r t 3 52O ̂+90

tXieOrignaIPrv >theArrear t6rflidlShart>d t 2

Sts char^,ed as 5 Intere f} T 4} * S z !.l s dSSo z A Xoc.ct 3 ) t t ) 3 t

5 YearsfArrear of Inter^Rtf I | § t

13y the Chancery Metaod _ : 4 +S s y I Cs t ̂ 49 t t ,6 { 3 > n 3 > s8 t i t By the time turnint, the t:omp.Int. into SimX+X<g3tI78re>!z3}JV3s t z +eC? BySlmpfTat.cor.-7Princ*a-ndJ continsat Int. T)})I9I{Igt r&+SX 4>7148 z- f o64 tlleOriglrlal>Pro- StlleArlear) Frli difcharD d 1 5D5897+I8)--l6s:,- zy 4899 o4>8(35tt Sts charged s 3 IntereR I X >23 5: I ̂ X j666 3 T 4 550 4t} t 3 t z

IO Years Arreal of IntereR { -* By the CAsncely Methed IS}ore_Ist 49:8s94°z 38}58)> By the Sme turnint, the Com p. Int + into Sinz l S.oo / n ̂ 1g°7 ^ zO>Oa3 5 4Ot z4O4

B S;mp.Int*cor) Princ and,¢ corutinS vt Int 4OX316{ ^ 5,5998 33?63nX48}08 J?4

theOrlgindPrv- itheArrear fir0dt@ht :It89S ^7S58^ 56Xt95^!5)}81O fits charged as JlllteretE- - i t?4^86^6>6666 v7>)@°C53.3o)

I S Years Artcar of Int

By theCh£wcery Merlled . _ I*6:g) 6 6? * 49; i 7>5 48}58 t)

By the fame turninr theComp, Int. into Stm.*:I*974 r|27Zzo72|O6tt83-S S2. 4O° By Simp.Int cor+7 prtne- enAS contlnq or Tsit.; fit 3 y >;gv}*3>t 3 4 >)p 6q,6y8)86; thcOriginalPlo- t theArrear 1 firR diScharg'd ^ 81 c209* 3 7, 7 rtz$4S, 5 1 59174, y66+ fi+s charged as J Intere. _- "-- -- - ;*- 5,oooo') [)6666 4)}7 50O?d >}) ) ) S!

I XmuLt alfb obServe ir the fike af l;XCll AS are ulllAcqUaintEd wstll fpecious Arithmetics tllat rllougll for brevitys fake tlle foregoing Theoremsand Examples are laid down and wrougllt in AlgebraicTerms: Yet tlle ne Accounts may be Rated afFer the Chancery manner it felf, accordlug tO the {iVeral Principles before deliveSd, andvwitll vlle fame Refuits, witll this only cauuons t11at (infiead of a continual dedudion of the Rent or Annuity ol;lt of rlle Principal an-d ntetell: of tbe Debt, wllicll occafions tlle Error befbre mentiontd) tlle preceding General RuJe of Pratling Weparat: Axasunts be okkrvtdy wllich nlaybe done by continually adding tlie profits togeF ther on each fide} in tlle fame manners as iS rlle Parties were

Q



( t7) to tnake a fi3parac eSl!)argt aga5XEt eacll Ottltt, WiCh iS tlls rather to 1o: tzeLcds as being tlue only Courfe tllat catl be taken, in caSetlleSunls c: Times of payinent {llould dSers lzut elle-- reEpeAi7e ReStllts will notwitbSanding be analogous to the abonre E:xampies.

hll svlaicll is futnzitted to tlze confdiratlotz of more di-Ccer ning Judgnlents,, efpecially the Applicatio.ls c)f tlle Rules to parricular Cafc$, for exemplifyitlg tlle Theerems. But if ant7 of -tbofe Rules or Theorems Ihould be obJeRed agaitz0, meerZ y becaufe they tend tO introd ucc fome AIterations in tt3e

preSet pradice; I {hall for anXzJer onXy add, svith fubtniSion, to wtl)at l5 before faid, tllat in former Agess xvI?en our Laxxs £elating to tllefe matters tlad tTleir rIle, (tlle Profits of England: sriElng claiefly from Husbandry and Tilsage, and little from Trade,3 tlle Ca(ll of tlle Kingdom waras but low, tbeRateso of Tnterell very higll tO the Advantage of Ulurersf and tllofe srays for the ready [mprovement of Monoy accomodated tU all: -Peoples ute, not known; (much like tO NFlliCll we are told xvas tile Srate of the JewiRl ASairs, sallcn titey wrere forbidden tO take Utury of any but Strangers.) But latter Ages llave produc'd vaS Alterations in all theSe ReXpedas, wllicll llavillg happen'd by intenf}ble degrees, may be one teafon why tlei- ther our Leginature, nor Courts of Judicature llave yet taken fucb noeice tllereofs; as time and leifilre, with tlle Tender of proper and pradrical Methods of Compuration} may 11ere after induce tllem to do.

F I N I S

1 ON D O N : Printed for W. I N N Y s at the Prinae's Arms i St. Pasgl's Chgrsh Drd. M DCCX\To



rules for correcting the usual methods of computing amounts and present values, by compound as well...

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