new divifor; then I seek how often the divifor 48 is coutained in 174, and the anfwer 3 I place both in the quotient and divifor as before, and multiply the divifor alfo thereby, and the product 1449 placed under and fubtracted from the refolvend, there remains 291, to which bringing down the laft period 96, I have 2919 for the laft refolvend, and the double of 243 or 486 for a divifor this last quotient is 6, which I place in the quotient and divifor, and multiply the divifor thereby, and the product is equal to the laft refolvend: thus 2436 is the answer.

Qu. 2. What is the fquare root of 152399025?-Anfwer 12345.

2. 3. What is the fquare root of 22071204?-Answer 4698.

To extract the Cube Root.

Rule 1. Divide the given number into periods of three figures each, by placing a point over every third figure, beginning with the unit figure.

2. Find the greatest cube in the first period, and place its cube root on the right hand of the given number.

3. Subtract the cube from the said period, and to the remainder bring down the next period, which is called the refolvend.

4. Place three times the root under the refolvend, and also three times the fquare of the root, the latter being removed one place to the left; thefe two added together form a divifor: Catego

5.Seek how often this divifor is contained in the refolv end, exclufive of the place of units, and place one lefs than the aufwer in the quotient.

6. Under the divifor place the cube of the last quotient figure, the fquare of it multiplied by three times the rest of the quotient and three times the root multiplied by the fquare of the quotient; each of the two latter numbers being removed one place farther towards the left hand than the

foregoing

foregoing, and the fum of these three numbers is called the Subtrahend.

7. Then fubtract this fubtrahend from the refolvend, and to the remainder bring down the next period for a new refolvend, with which proceed as before, and fo on till the work be finished.

Example 4. What is the cube root of 164566592?

86742)7102592 fecond refolvend

162 three times 54

$748 three times the fquare 54

87642 fecoud divifor

512 cube of 8

10368 fquare of 8 multiplied by 3 times 54 69984 three times 8 multiplied by the fquare of 54 7102592 fecond subtrahend

Answer 548 the cube root requireda

28

Qu. 5. What is the cube root of 389017?—Anf. 73. Qu. 6. What is the cube root of 405 225 — Auf, 73. There are many other methods of extracting the cube root given by different mathematicians; but none of which I have feen are so fimple and eafy to be remembered as the method here laid down.

After what has been faid, it need hardly be mentioned, that the extraction of the fquare root is proved by multiply"ing the root into itself; and the extraction of the cube root, by multiplying the root three times into itself.

Kk 2

CHAP.

CHAP. IV.

OF DECIMAL ARITHMETIC.

SECT. I.

REDUCTION OF DECIMAL FRACTIONS.

A DECIMAL fraction is that fraction whose denominator has an unit in the first place on the left hand, with as many cyphers annexed as neceffary: thus, To Toboo, &c. are decimal fractions. But these fractions are ufually expreffed in writing without a denominator, by writing the numerator, and prefixing as many points or cyphers before it on the left hand, as there are more places of figures in the denominator than in the numerator. Thus the foregoing fractions are written .5, .25, .048, .0572, and expressed five tenths,wenty-five hundredths, forty-eight thoufandths, &c.

Cyphers placed on the right hand of decimal fractions make no alteration in their value: thus, if a cypher be annexed to the foregoing fraction .5, it will be then .50 fifty hundredths or half an integer, as before; if two cyphers .500 it will be five hundred thousandths, or, and the fame of the others.

But qyphers placed on the left hand of a decimal fraction decrease their value, every cypher decreafing it in a tenfold proportion: thus .5, .05, .005, are five tenths, five hun◄ dredths, five thousandths parts respectively.

The

The value of every figure in a decimal fraction increases in a tenfold proportion from the first place on the right hand, as in whole numbers.

The figures in the first place of a decimal fraction on the left hand are called primes, thofe in the fecond place feconds, thofe in the third place thirds, &c.

.

Cafe 1. To reduce a vulgar fraction to a decimal one of the fame value.

Rule. Divide the numerator by the denominator, and the quotient will be the decimal fraction required.

If the numerator be too small to be divided by the deno 'minator, add one or more cyphers thereto, in which cafe as many places must be separated from the quotient for decimals as the difference of the number of places of decimals in the dividend and the divifor, and the remaining figures in the quotients (if any) are integers.

But if the number of places in the quotient be less than the required number of decimal places, as many cyphers muft be prefixed on the left hand thereof as are neceffary. Example 1. What is the decimal fraction of a pound equal to ?

Here