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therefore, by Theor. 1, log, m (= log. A + log. R2➡ I -log. R-1)=log. 50+ log. 1,041-1,-log.,04 2,596597; and confequently m 395. the value that was to be found.

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Examp. 2. What annuity, forborn feven years, will amount to, or raise a stock of 395 l. at 4 per cent. compound interest.

In this cafe we have given R = 1,04, n =*7, and m395; whence, by Theorem 2, log. A (= log. m

log. RI+ log. R-1)= log. 395- log. 1,04) — 1+log.,041,6989700; and consequently A 50%. which is the annuity required.

Examp. 3. In how long time will 50 7. annuity raise a ftock of 3951. at 4 per cent. per annum, compound intereft?

Here, we have R

therefore, by Theor. 3, n

1,04, A = 50, m = 395; and

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,1192559 = 7, the number of years required.

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Examp. 4. If 120 l. annuity, forborn eight years, amounts to, or raises a stock of 1200 . what is the rate of intereft?

In this cafe we have given n≈ 8, A 120, and m 1200, to find R, therefore, by Theorem 4, we have R3 - 10R+9= 0; from which, by any of the methods in Sect. 13, the required value of R will be found 1,06287; therefore the rate is 6,287, or 6 l. 5 s. 9 d. per cent. per annum.

The folution of the laft cafe, where the rate is required, being a little troublefome, I fhall here put down an approximation (derived from the third general formula, at p. 165) which will be found to answer very near the truth, provided the number of years is not very great.

Let

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Thus, for example, let n = 8, A = 120, and m =

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The preceding examples explain the different cafes of annuities in arrear; in the following ones the rules for the valuation of annuities are illuftrated.

Examp. 1. To find the present value of 100l. annuity, to continue feven years, allowing 4 per cent. per annum compound interest.

Here, we have given R = 1,04, A = 100, and n=7; and therefore, by Theorem 1, log. (= log. A +

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I

R"

1,047

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log.,042,778296; and consequently

v600,2600/. 4 s. which is the value that was to be found.

Examp. 2. What annuity, or yearly income, to continue 20 years, may be purchased for 1000l. at 3 per

cent?

In this cafe, R = 1,035, n = 20, v = 10000; whence, by Theorem 2, we have log. A (= log. v

+ log. R— 1 — log. 1 —

I

I

R

;) = 1,847336; and

confequently A=70,36, or 701. 7 s. 2 d.

Examp. 3. For how long time may one, with 600 L purchase an annuity of 100 l. at 4 per cent.?

In this example we have R 1,04, A = 100, and 600; and therefore, by Theorem 3, n ( =

log. A log. A + v — vR

log. R

years required.

)=7, the number of

Examp. 4. To determine at what rate of interest, an annuity of 50% to continue 10 years, may be purchased, for 400 1.

Here A 50, n = 10, and v = 400; whence, by

Theorem 4,

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have R" — 1,125R1o +,125 = 0; which equation refolved, gives the required value of R = 1,042775; and confequently the rate of intereft, 4,2775 per

annum.

The folution of this laft cafe being fomewhat tedious, the following approximation (which will be found to answer very near the truth when the number of years is not very large) may be of use.

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express the

3000Q― 2n + 1 × 400

6Q.· 5Q— 31 — 4 + 1⁄2 • n + 2 . Iin+13

ate per cent. very nearly.

Thus, for example, let A (as above) be 50

n = 10, and v = 400; then, Q being =

82500-8400

27,5, we have 165 × 103,5 + 246' for the rate, per cent. the fame as before.

10XIIX50

1000-800

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or 4,2775,

of

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PLA

SECTION XVII.

Of Plane Trigonometry.

DEFINITION S.

LANE Trigonometry is the art whereby, having given any three parts of a plane triangle (except the three angles) the reft are determined. In order to which, it is not only requifite that the peripheries of circles, but also that certain right lines, in and about the circle, be fuppofed divided into fome affigned number of equal parts.

2. The periphery of every circle is fuppofed to be. divided into 360 equal parts, called degrees; and each degree into 60 equal parts, called minutes, and each minute into 60 equal parts, called feconds, or fecond minutes, &c. Any part of the periphery is called an arch, and is measured by the number of degrees and minutes, &c. it contains.

3. The difference of any arch from 90 degrees, or a quadrant, is called its complement, and its difference from 180 degrees, or a femicircle, its fupplement,

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ter paffing through the other extremity: thus BF is the

fine of the arch AB, or BD.

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6. The verfed fine of an arch is the part of the diameter intercepted between the arch and its fine: fo AF is the verfed-fine of AB, and DF of DB.

7. The co-fine of an arch is the part of the diameter intercepted between the center and the fine; and is equal to the fine of the complement of that arch. Thus CF is the co-fine of the arch AB, and is equal to BI, the fine of its complement HB.

8. The tangent of an arch, is a right line touching the circle in one extremity of that arch, continued from thence to meet a line drawn from the center through the other extremity; which line is called the fecant of the fame arch: thus AG is the tangent, and CG the secant of the arch AB.

9. The co-tangent and co-fecant of an arch, are the tangent and fecant of the complement of that arch: thus HK and CK are the co-tangent and co-fecant of the arch AB.

10. A trigonometrical canon, is a table exhibiting the length of the fine, tangent, &c. to every degree and minute of the quadrant, with refpect to the radius, which is fuppofed unity, and conceived to be divided into 10000000 or more decimal parts. Upon this table the numerical folution of the feveral cafes in trigonometry depend; it will therefore be proper to begin with, its conftruction.

PROPOSITION I.

The number of degrees and minutes, &c. in an arch being given; to find both its fine and co-fine.

This problem is refolved, by having the ratio of the circumference to the diameter, and by means of the known feries's for the fine and co-fine (hereafter demonftrated). For, the femi-circumference of the circle, whofe radius is unity, being 3,141592653589793 &c. it will therefore be, as the nu nber of degrees or minutes in the whole femi-circle is to the degrees or minutes in the arch propofed, fo is 3,14159265358 to the length of the faid ach; which let be denoted by a; then, by the feries's above quoted, its fine will be ex

.

preffed

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