Then substitute the value of c and d in this

form z=d+(d2 +c3) + 3⁄4/d−√ (d2+c3);

or z=d+√(d2+c3)−

C

3d+v (d2+c3),

and the value of the root z, of the reduced equation 23 +az =b, will be obtained. Lastly, take x-z-p, which will give the value of x, the required root of the original equation x3+px2+qx=r, first proposed.

One root of this equation being thus obtained, then depressing the original equation one degree lower, after the manner described p. 260 and 264, the other two roots of that equation will be obtained by means of the resulting quadratic equation.

Note. When the co-efficient a, or c, is negative, and c3 is greater than d2, this is called the irreducible case, because then the solution cannot be generally obtained by this rule. Ex. To find the roots of the equation 3 — 6x2+10x = 8. First, to take away the 2d term its co-efficient being -6, its 3d part is 2; put therefore x=z+2, then

x3=23+6z2 + 12z+8

-622-242-24

+10z+20

-6x3 = +10x

Theref. /d+(d2+c3)=√//2+√(+− )=1/2+√W=

/2-3=0.42265

then the sum of these two is the value of z=2.

=

Hence x z +2 = 4, one root of x in the eq. x3-6x2+

10x 8.

To find the two other roots, perform the division, &c. as in p. 261, thus:

3

x−4)x3 - 6x2+10x-8(x2 -2x+2=0

Hence x-2x=-2, or x2 −2x+1= −1, and x

√ −1 ; x=1+ √ −1 or = 1-1, the two other sought. Ex. 2. To find the roots of x3-9x2+28x=30.

Ans. x3, or =3+/-1, or 3-1. Ex. 3. To find the roots of x3 7x2+14x-20.

Ans. 5, or 1+/-3, or 1-√3.

OF SIMPLE INTEREST.

As the interest of any sum, for any time, is directly proportional to the principal sum, and to the time; therefore the interest of 1 pound, for 1 year, being multiplied by any given principal sum, and by the time of its forbearance, in years and parts, will give its interest for that time. That is, if there be put

r the rate of interest of 1 pound per annum,

p

any principal sum lent.

t = the time it is lent for, and

@= the amount or sum of principal and interest; ther is prt the interest of the sum p, for the time t, and conseq. p+prt or p× (1+rt)=a, the amount for that time.

From this expression, other theorems can easily be deduced, for finding any of the quantities above mentioned; which theorems collected together, will be as below. 1st, a=p+prt, the amount,

a-P. the time.
"
pr

For Example. Let it be required to find, in what time any principal sum will double itself, at any rate of simple interest.

In this case, we must use the first theorem, a = p + prt, in which the amount a must be made = 2p, or double the principal, that is, p+prt 2p, or prt = p, or rt = 1; and

Here, r being the interest of 17. for 1 year, it follows, that the doubling at simple interest, is equal to the quotient of any sum divided by its interest for 1 year. So, if the rate of interest be 5 per cent. then 100-5-20, is the time of doubling at that rate.

Or the 4th theorem gives at once a-p_2p-p_2- 1

1

the same as before.

r

COMPOUND INTEREST.

BESIDES the quantities concerned in Simple Interest namely,

p the principal sum,

the rate on interest of 11. for 1 year,

a = the whole amount of the principal and interest,
t = the time,

there is another quantity employed in Compound Interest, viz, the ratio of the rate of interest, which is the amount of 11. for 1 time of payment, and which here let be denoted by R. viz.

R=1+r, the amount of 11. for 1 time.

Then the particular amounts for the several times may be thus computed, viz. As 17. is to its amount for any time, so is any proposed principal sum, to its amount for the Same time; that is, as

11.: R: : P
17. R:

:

pᎡ

: PR, the 1st

year's amount,
year's amount,
year's amount,

11. R: PR: PR3, the 3d

and so on,

Therefore, in general, pR'a is the amount for the year, ort time of payment. Whence the following general theo. rems are deduced :

From which, any one of the quantities may be found, when the rest are given.

As to the whole interest, it is found by barely subtracting the principal p from the amount a.

Example. Suppose it be required to find, in how many years any principal sum, will double itself, at any proposed rate of compound interest.

In this case the 4th theorem must be employed, making a=2p and then it is,

t=

log. a-log. plog. 2p-log. p.__log. 2

log. R

log. R

log. R

So, if the rate of interest be 5 per cent. per annum ; then R=1+05=1.05; and hence

t=

log. 2 log. 1.05

that is, any sum doubles itself in 14 years nearly, at the rate of 5 per cent. per annum compound interest.

Hence, and from the like question in Simple Interest, above given, are deduced the times in which any sum doubles itself, at several rates of interest, both simple and compound;

The following Table will very much facilitate calculations of compound interest on any sum, for any number of years, at various rates of interest.

The amount of 11. in

any number of years.

5

6

Yrs. 3 34 | 4

11.0300 1.03501.0400 1.0450 1.0500 1.0600 21.06091.0712.0816 1.0920 1.1025 1.1236 31.09271.1087 1.1249 1.1412 1.1576 1.1910 41.12551.1475 1.1699 1.1925 1.2155 1.2625 51.15931-1877 1.2167 1.2462 1.2763 1.3382 61 19411 22931.2653 1.3023'1.3401 1.4185 71.22991-2723 1 3159 1.3609 1.4071 1.5036 81.2668 1.3168 1.3686 1.4221 1.4775 1.5939 91.3048 -3629 1.4233 1.48611.5516 1.6895 101.3439 1-4106 1.4802 1.5530 1.6289 1.7909 11 1.3842 1-4600 1.5895 1 6229 1.7103 1.8983 12 1.4258 1.5111 1.6010 1.6959 1.7959 2.0122 13 1.4685 1.5640 1.6651 1.7722 1.8856 2.1329 141.5126 1.6187 1.7317 1.8519 1-9799 2.2609 15 1.5580 1.6753 1.8009 1.9353 2.0789 2.3966 16 1.6047 1.7340 1.8730 2.0224 2.1829 2.5404 17 1.6528 1.7947 1.9479 2.1134 2.2920 2.6928 18 1.7024 1.8575 2.02582.2085 2.4066 2.8543 19 1.7535 1 9225 2.1068 2.3079 2.5270 3.0256 20 1.80611.9898 2.19112.41172 6533 3.2071

The use of this Table, which contains all the powers, Rt, to the 20th power, or the amounts of 11. is chiefly to calculate the interest, or the amount of any principal sum, for any time, not more than 20 years.

For example, let it be required to find, to how much 5231. will amount in 15 years, at the rate of 5 per cent. per annum und interest.

compo

In the table, on the line 15, and in the column 5 per cent.

is the amount of 11. viz.

this multiplied by the principal

gives the amount

or

and therefore the interest is

Note 1. When the rate of interest is to be any other time than a year; as suppose to year, &c; the rules are still the same ;

2.0789

523

1087.2647

10871. 5s. 31d. 5641. 5s. 31d.

a

determined to a year, or but then t will express