CHAP. XII of Compound Interest, and Annuities, &c. COMPOUND Intereft is that which arifes from any Principal and it's Intereft put together, as the Intereft fo becomes due; so 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; Suppofe 100l. were lent out for two Years, at 6 per Cent. per Annum, Compound Intereft: then at the End of the first Year, it will only amount to 106 7. as in Simple Intereft. But for the fecond Year this 1061. becomes Principal, which will amount to 1127. 75. 2 d. at the fecond Year's End, whereas by Simple Interest it would have amounted to but 1127. 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. Let Sect. 1. Of Compound Interest. Pthe Principal put to Intereft. A the Amount of the Principal and Interest. R= Viz. 100 { the Amount of 1 l. and it's Intereft for 1 Year, at any given Rate, which may be thus found. 106: 1: 1,06 the Amount of 17. at 6 per Cent. Or 100 105 1: 1,05= the Amount of 17, at 5 per Cent. and fo on for any other affigned Rate of Intereft, Then if Rthe Amount of 1. for one Year, at any Rate. R? R3 the Amount of 1 for two Years. R4 the Amount of 11. for four Years. Here t=5 For 1:R:: R:RR:: RR: RRR:: RRR: R4 :: R As one Pound: is to the Amount of one Pound at one That is Year's End :: fo is that Amount: to the Amount of one Pound at two Year's End, &c. Whence Whence it is plain, that Compound Intereft 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 Term: Viz. the Power of R, which is R. Again, As1: R:: P: PRA the Amount of P for the Time, that Rthe Amount of 11. As one Pound; is to the Amount of one Pound for any That is given Time:: fo is any propofed Principal (or Sum) to it's Amount for the fame Time. : From the Premifes (I prefume) the Reafon of the following Theorems, may be very cafily understood. Theorem 1. PRA, as above. 1 From hence the two following Theorems are eafily deduced. Theorem 2. A A =P. Theorem 3.=R. P By these three Theorems, all Queftions 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; as will appear farther on. Question 1. What will 2561. 10s. amount to in feven Years, at 6 per Cent. per Annum, Compound Intereft? Here is given P=256,5; t=7; and R= 1,06 which being involved until it's Index=t (viz. 7.) will become R 1,50363. Then 1,50363 x 256,5=385,6811=A=385 l 13s. 7 d. which is the Anfwer required. Question 2. What Principal or Sum of Money must be put (or let) out to raise a Stock of 3851. 13s. 7 d. in Seven Years, at 6 per Cent. per Annum, Compound Intereft? Here is given A-385,6811;. R=1,06; and t=7; to find P. by Theorem 2. Thus R1,50363) 385,6811=A (256,5=P. That is, P256 10s. which is the Principal or Sum, as was required. Question Question 3. In what Time will 2561, 10s. raife a Stock of (or amount to) 385 l. 13s. 7 d. allowing 6 per Cent. per Annum, Compound Interest? Here is given P=256,5; A= 385,6811; R= 1,06; to A 385,6811 find t by the third Theorem Rp 256,5 = 1,50363, which being continually divided by R= 1,06 until nothing remain, the Number of those Divifions will be 7=t. Thus 1,06) 1,50363 (1,41852. And 1,06) 1,41852 (1,338225. Again 1,06) 1,338225 (1,262477. And fo on until it become 1,06) 1,06 (1. which will be at the feventh Divifion. Therefore it will be t7 the Number of Years required by the Question. Question 4. If 2561. 10s. will amount to (or raife a Stock of) 3851. 138. 7 d. in feven Years Time; what must the Rate of Intereft be, per Cent. per Annum? A Here is given P=256,5; A=385,6811, and t=7, Quære R. By Theorem 3.= R = 1,50363; as before in the laft Question. And if R=R? = 1,50363, then R=√1,50363, which may be thus extracted. Put r+= R, then 2r7+7ro e +21 r' ee=R' = 1,50363=G 27? 37e+ 21.rs ee G-7. 37rs 4re+3ee= =D 7r5 Then: 0,06 :: 100: 6 the Rate per Cent. required. The first three Queftions may be much more eafily performed by the following Table, which is only the Amounts of one Pound for thirty-nine Years. That That is, of R. RR RRR RR Years. The Amounts of 1 L at 6 per 11.06 R and fo on to R39, The Amounts .. of 1-2 at 6 per Cent, &c. Com pound Intereft. 142.2609039557 2724-8223459407 A 1.1236 RR 15 2.3965581931 28 51116866971 3.1.191016 R3 16 2.5403516847 29 5.4183878990. 41.26247696 17 2.6927727857 30 5-7434911729 1.3382255776 18 2.8543391529 31 6.0881006432. 61.4185191122 19 3.0255995021 32 6.4533866818 7 1.5036302590 20 3.2071354722 33 6.8405898828 81.5938480745 213-3995636005 34 7.2510252757 9.6894789590 22 3.6035374166 35 7.6860867923 10 1.7908476965 23 3.8197496616 36 8.1472519998 11 1.8982985583 24 4.0489346413378.6360871198 122.0121964718 25 4.2918707197 38 9.15425 23470 13 2.1329282601264.5493829629 399.7035074878 The Title of this Table fhews it's Construction, and it's Ule will eafily appear by an Example or two. EXAMPLE 1, What will 375 1. 10s. amount to in nine Years, at 6 per Cent. per Annum, &c ? The tabular Number against 9 Years is 1,689479 which being multiplied with the Principal 375,5 will produce 634,3993 &c. viz. 634 1. 8 s. ferè, being the Amount or Answer required. EXAMPLE 2. What Principal (or Sum) must be put to Intereft to raise a Stock of 6341. 8 s. in nine Years Time, at 6 per Cent. per Annum, &c. If the propofed Stock (viz. 634,4) be divided by the tabulat Number that is against the given Number of Years (viz. 9.) the Quotient will be the Principal (or Sum) required. Viz. against 9 is 1,689479. Then 1,689479) 634,4 (375,5 = 375 l. 10:3 the Principal (or Sum) required. EXAMPLE 3. In what Time will 3751. 109. raife a Stock of (or amount to) 634 1. 8 s. at 6 per Cent. &? I Divide Divide the propofed Stock (viz. 634,4) by the given Principal (viz. 375,5) and the Quotient will fhew the tabular Number that ftands over against the Time fought. Thus 375,5) 634,4 (1,689479 &c. this Number being fought in the Table, will be found to ftand against 9 Years, which is the Time required, But if the Quotient cannot be truly found in the Table of Amounts for Years, as above; then take out of that Table the neareft Number that is lefs, and make it a Divifor, by which you muft divide the firft Quotient; and then feek the fecond Quotient in the Table of Amounts for Days (which is inferted a little further on) and it will affign the Number of Days: as in this Example. In what Time will 5631. amount to 8601, at 6 per Cent. per Annum, Compound Interest? Answer. In 7 Years and 99 Days. Thus 563) 860 (1,52753 which fhews the Time to be more (or above) feven Years; for over againft 7 Years is 1,50363 which being made the new Divifor: Viz. 1,50363) 1,52753 (1,01589 &c. this Number is the neareft Amount to 99 Days. Note, If the Stock, Principal, and Time be given; the Rate of Intereft will be beft found by extracting the Root, &c. as before in the fourth Question. The next Thing that I fhall here propofe, is to make this Table (which is only calculated for the Rate of 6 per Cent.) univerfally useful for all the Rates of Compound Intereft, which I may prefume to fay, is a new Improvement of my own, being well fatisfied it never was published before; and not only fo, but I have heard several very good Artifts affirm it was impoffible to be done. The Method of performing it is briefly thus, Let x= the Difference between 1,06 R, the Amount of 11, for one Year (in the Table) and any other propofed Amount of 1 for one Year; which admits of two Cafes. R, : Cafe. If the propofed Rate be greater than the 1,06 then will R+ the true Amount of 1. for one Year at that Rate. Cafe 2. But if the propofed Rate be less than 1,06 R, then it will be Rx the Amount of 1 1. &c. Make St-1=b, t-2 = c, t - 3d, 1-4f, &c. 1} + b = 8, cg=m, id m=n, ƒn= &c. LI Then |