By the fecond and fourth Theorems two useful Questions may be eafily answer'd. 1. As for Inftance: If it be required to find what Annuity, or yearly Rent, &c. may be purchased for any propos'd Sum, to continue any affigned Time, allowing any Rate of Intereft. This Queftion may be anfwer'd by Theorem 2. 2. Again: If it be required to find how long any yearly Rent, Penfion, or Annuity, &c. may be purchafed (or enjoy'd) for any propofed Sum, at any given Rate of Intereft. All Questions of this Kind are easily answer'd per Theorem 4. In thefe Questions it is fuppos'd that the Purchase or yearly Rent is to commence or be immediately enter'd upon. But if it be required to find the Value or Purchase of any Annuity, or yearly Rent, &c. in Reverfion; that is, when it is not to be enter'd upon until after some time, or Number of Years are paft; then firft find what the Sum propos'd to be laid out in the Purchase would amount to, if it were put to Intereft during the time the Annuity, &c. is not to be in prefent Poffeffion; and make that amount the Sum for the Purchase, proceeding with it as in either of the two last Questions, &c. CHA P. II. Of Compound Interest, and Annuities, &c. COMPOUND Intereft is that which arifes from any Principal, and its Intereft put together as the Interest fill becomes due; fo that at every Payment, or at the Time when the Payments became due, there is created a new Principal; and, for that Reafon, it is called Intereft upon Intereft, or compound Intereft. As for Inftance; fuppofe 100l. were lent out for two Years at 6 per Cent. per Annum compound Intereft. Then at the end of the firit Year, it will only amount to 106/. as in fimple Intereft. But for the fecond Year this 1061. becomes Principal, which will amount to 112 l. 75. 2 d. at the fecond Years End, whereas by fimple Intereft it would have amounted to but 112/ And And altho' it be not lawful to let out Money at compound Intereft; yet in purchafing of Annuities or Penfions, &c. and taking Leafes in Reverfion, it is very ufual to allow compound Intereft to the Purchaser for his ready Money; and therefore it is very requifite to understand it. Р Let a Sect. 1. Of Compound Intereft. the Principal put to Intereft. tthe Time of its Continuance. the Amount of the Prin. and Int. R= } as before. the Amount of 1 l. and its Int. for 1 Year at > any given Rate, which may be thus found. Viz. 100.. 106:1..1.06 the Amount of 1 l. at 6 per cent. Or 100.. 107 :: 1 .. 1.09 the Amount of 1 l. at 7 per cent. And fo on for any other affigned Rate of Intereft. Then if Rthe Amount of 11. for one Year at any Rate. the Amount of 1 l. for two Years. the Amount of il. for three Years. the Amount of ii. for four Years. RR RRR R4 the Amount of 11. for five Years. Here t5. For 1..R:: R.. RR :: RR..R::R3.. R4 R4..R3, &c. in As 1, is to the Amount of 1 l. at one Years end That is fo is that Amount, to the Amount of il. at two Years end, &c. 9 Whence it is plain that compound Interest is grounded upon a Series of Terms increafing in geometrical Proportion continued, wherein t (viz. the Number of Years) does always affign the Index of the laft and higheft Terim, viz. the Power of R, which is Rt. Again .. Rp.pRa the Amount of p for the Time that Rt the Amount of il. As 1, is to the Amount of il. for any given That is Time; fo is any propos'd Principal (or Sum), to its amount for the fame 'Time. From the Premises the Reafon of the following Theorems may be very easily understood. Theorem 1. Rt = a. From hence the two following Theorems are cafily deduced. P By these three Theorems all Questions about compound Intereft may be truly refolved by the Pen only, viz. without Tables, tho' not fo readily as by the help of Tables calculated on purpose. But here I will fhew how to folve fuch Questions by the help of a Table of Logarithms: thus Ra, as above; therefore, by the Nature of Logarithms. Theorem 1. Lp+tx LR=La La-Lp Theorem 3. t I.a-Lp Theorem 4. -LR =t. LR Queft. 1. What will 375 l. 10s. amount to in 9 Years at 6 per cent. per annum compound Interest? Here is given p=375.5, t=9, and R=1.06. To find a per Theorem 1. The Logarithm of 375.5 is 2.5746099 9 × Logarithm of 1.06 is .2277 5 3 1 2.8023630=Lp--txLR=La; Therefore a is 634.4 fere 6341. 8s. The Answer required. Queft. 2. What Principal (or Sum) must be put to Interest to raise a Stock of 6341. 8s. in 9 Years at 6 per cent. per annum, &c? Here is given a=634-4, R=1.06, and t9. To find p per Theorem 2. Thus The Logarithm of 634.4 is 2.8023632 9 × Logarithm of 1.06 is .2277531 2.5746101-La-txLR=Lp; Confequently is 375.5+; that is p 3756. 10S. which is the Principal (or Sum) as was required. Queft. 3. In what time will 375 10 s. raise a Stock of (or amount to) 6341. 8s. allowing 6 per cent. per annum compound Interest ? Here is given a = 634.4, ₤=375.5, and R=1.06. To find t by Theorem 4. The The Logarithm of 634.4 is 2.8023632. .0253059) .2277533 (9+= that is t is 9 the Number of Years required. Quest. 4. If 3751. 10s. will amount to (or raise a Stock of) 6341. 8 s. in 9 Years; what must the Rate of Interest be per cent. per annum compound Intereft? Here is given a=634.4, P=375.5, and t=9. Quere R per Theorem 3. The Log. of 634.4 is 2.8023632 The Log. of 375.5 is 2.5746099 9).2277533(.0253059= La-Lp =LR And the natural Number which anfwers the Logarithm .0253059 in the Table is 1.06; Then I.. 1.06-1:: 100..6 the Rate per cent required. Note, If the Logarithm of the given Number; or, if the required Number of the given Logarithm, be not exact enough for your Purpose in your Table of Logarithms, you may find them fufficiently exact by the Appendix to Chap. III. Part XV. or make them as exact as you pleafe by the faid Chapter. Sect. 2. Of Annuities or Penfions in Arrear computed at Compound Intereft. When Annuities, &c. are faid to be in Arrear, fee P. 260. And I fhall here make ufe of the fame Letters to reprefent the fame Things as before in that Page, fave only that the Amount of 1 l. for 1 Year, as in § 1. of this R is here Intereft I the first Years Rent of any Annuity without Then will Ru+u= The Amount of the first Years Rent and its Intereft; more the fecond Years Rent, &c. The Amount of the 1st and 24 Years Rents with their Interest, more the 3d Years Rent, &c. Ru+u a the Amount of any Yearly Rent, or Annuity being_forborn 3 Years. And from hence may And RRu+Ru+u Here RR be deduc'd thefe Proportions. * A a z pis. Viz. u..Ru::Ru..RRu:: Ru..R3 u, and fo on in for any Number of Terms, or Years, denoted by t, wherein the laft Term will always be uR; wherefore, by Part VIII. Chap. II. Step 15. :R-1:xu R-I ; confeq. L:R-1:+Lu-L:R— 'Theor. 1. a 1: La a a Theor. 4.RR+ -10. If this Equation be refolved into Numbers, one of its Roots will fhew the Value of R. Quest. 1. If 201. Yearly Rent, or Annuity, &c. be forborn (viz. remain unpaid)9 Years; what will it amount to at 6 per cent. per annum compound Interest? Here is given u=30, 19, and R1.06; to find a per Theorem 1. LRtxLR9x L1.0698.0253059.2277531: And the *Number anfwering to this Log. is 1.689480=R'; Confeq. R-1.68948, whofe Log. is ...... 7.8385217 LuL201.4771213 Sum 1.3156430 L:R-1:=L.c6=2.7781513 La 2.5374917: And the Number anfwering to this Log, of a is 344.74 =0=344/. 145. 94. The Amount required. Quest. 2. What Yearly Rent, or Annuity, &c. being forborn, or unpaid 9 Years, will raise a Stock of 344/. 145. 94d. 344.741. at 6 pcr Cent, &c ? Here is given a u by Theorem 2. 344.74, t = 9, and R= 1.06; to find See the Appendix to the Logarithms; viz. to Part XV. Chap. III. L:R |