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Here is given A634.4, R1.06 and 9 to find p per Theorem 2. Thus;

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The Logarithm of 634. is 2.8023632 9x Logarithm of 1,06 is =

.2277531

The Difference is 2.5746101 =

=

L, A-txL, RL, p; confequently p is 375.5+; that isp=375 4.10 s. ood. + which is the Principal (or Sum) as was required.

Q. 3. In what time will 3751. 10 s. ood. raise a Stock of (or amount to) 6341. 8s. ood. allowing 6 per Cent. per Annum, Compound Interest?

Here is given 4634.4, p = 375.5 and r1.06 to find by the fourth Theorem.

The Logarithm of 634. 4 is
The Logarithm of 375.5 is

The Difference is

And. 2277533
L, ALP
L, R

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Years required.

2.8023632. 2.5746099

2277533

0.0253059 (the Logarithm of 1,6) is =

Queft. 4. If 375 l. Stock of) 6341. 8 s. Rate of Intereft be per Here is given A

=; that is, is 9 the number of

10s. 00 d. will amount to (or raise a
oo d. in Nine Years; What must the
Cent. pèr Annum ?
634.4, P = 375.5 and t=9.

Quere R per Theorem 3.

The Logarithm of 634.4 is 2.8023632.
The Logarithm of 375. 5 is = 2.5746099.

The Remainder is .2277533;

Which Remainder Divided by 9, gives .0253059 =
L, A — LP — L, R:
Lp =

And the Natural Number that anfwers the Logarithm, 0253059 will be found in the Chiliads to be 1.06

Then 1.06

:: 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

Part XVI may make them as exact as you pleafe by CHAP. 5Part XV.

Now, I think, the Method of folving Questions in Compound Intereft by the foregoing manner (that is, by the help of Logarithms) preferrable to that of folving them by the help of Tables; For, to make a Table for every Rate of Intereft that may be, or that really is Ufed, is too troublefome.

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But my Author's Method is pretty good, which "Mr. Ward. is thus.

He makes a Table for the Amount of 17. at 6 per Cent, per Annum, Compound Intereft, thus ;

The Amount of 1 1. for 1 Year at the aforefaid Rate, being J. 06 (R); the Amount of 14. for two Years, will be 1. 1236 (R); the Amount of 1. for three Years, will be 1.191016 (R3) &c.

And this Table he makes univerfally Useful for all Rates of Compound Intereft. Thus

s;

Let x be the difference between 1.06 R the Amount of 11. for one Year in the Table, and any other propofed Amount of 1 I, for one Year.

CASE 1. Then, if the propofed Rate be Greater than 1.06 = R, R+x will be the true Amount of 11. for one Year at that Rate.

CASE 2. But if the propofed Rate be less than 1.06=R, then it will be R-x=true Amount of 1 I, for one Year at that Rate.

Wherefore (by the 1 Theorem in this §. 1.) px R+x |15=A in Cafe 1. And fx R-x=4 in Cafe 2. A Wherefore, by Sir Ifaac Newton's Theorem,

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This Equation being Solv'd by the converging Series, the Value of x will thereby be found, and then R+ xoc R−x will fhew the Rate of Interest.

And if the Time given or fought be not terminated by whole Years, but by Weeks, Months, Quartets, or Halfyears, &c. For Refolving fuch Queftions, reduce thofe parts of a Year into Days; that done, find an Answer according to the demand of the Queftion (and agreeing to r. as before) for those number of Days; and in order to that, the Amount of 1. for one Day, at 6 per Cent. per Annum (as already found in the latter part of the Logarithms) will be very Ufeful; for by it a Table for the Amount of 1 l. for any number of Days at 6 per Cent, per Annum, may be calculated, thus ;

The Amount of 11. for 1 Day at 6 per Cent. per Annum, being 1.0001596536, &c. The Amount of 1 1. for two Days, will be 1.0003193326, &c.—1.0001596536,&c.; the Amount of il. for three Days, will be 1.0004790372,&c. = 1.00015 96536, &c. 13. ; &c.

Now, in order to make this Table of Amounts for Days, Ufeful for all Rates of Interest, you may proceed as before in that for

Years.

Sea. 2. Of Annuitics or Pensions in Arrear, computed at Compound Intereft.

When Annuities, &c. are faid to be in Arrear; See Pag. 293 And I fhall here make use of the fame Letters to

represent the fame things as before in that Page, save only that R is here the Amount of 1 for one Year, as in §. i. of this Chapter.

Suppose the first Year's Rent of any Annuity without Intereft,

Then will Ru+u=

And R'u+Ru+u=

The Amount of the first Year's Rent

and its Intereft; more the second Year's Rent, &c.

i

The Amount of the first and fecond Years Rents with their Intereft; more the third Year's Rent, &c.

Here

Here R Ru+R+u= A the Amount of any Yearly Rent or Annuity, being forborn three Years; and from hence may be deduced thefe Proportions.

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Viz. u .. Ru:: RuRu :: R3ú ·· R3u, and fo on in

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for any number of Terms or Years denoted by t, wherein

the laft Term will manifeftly be = u R Wherefore by Part 8. Chap. 2.

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t--I

uR Prop. 1.

R-I

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Theorem ri

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If this Equation be continually Divided by Runtil nothing remains, the number of thofe Divifions will bet. See Or rather from the Nature of Log44

Page

rithms.

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If this Equation be refolved into Numbers, one of its Roots will fhew the Value of R.

Quest. 1. If 30. 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, 9, and R1.c6, to find A per Theorem 1.

First, In order to find what R is to in this Question, I proceed thus by the Logarithms.

The Logarithm of R Logarithm of 1.06
Logarithm of I. c6 = 9×.9253059. 2277531.

=9×

And

And the Number in the Chiliads that anfwers this Logarithm is 1.68948 fere Rt

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u Ru=1=3441.

R-i

=1=3441. 14s. 9 d. the Amount required.

1

Queft. 2. What Yearly Rent or Annuity, &c. being For born or Unpaid 9 Years, will raile a Stock of 344/. 145.9d. = 344.7395 at 6 per Cent. per Annum, &c.

Here is given 4 = 344. 7395, 9, and R = 1.06 to find u by Theorem 2.

AR 344 7395×1.06 = 365.42387.

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Quest. 3. In what time will 301. Yearly-Rent raise à Stock of, or amount to 344. 145. 9 d. allowing 6 per Cent. per Annum for the forbearance of the Payments, &c.

Here is given u=30, A = 344.7395, and R= 1.06; to find t per Theorem 3.

First, RA-A+u=365.42387 -344-7395 +30=, .30.68437.

And the Logarithm of 50. 68437 is

1.7048743 The Logarithm of u L, 30= 1.4771212

L, RA — A + u-L," =

227753!

LR= L; 1.06 =

.0253059

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