work to be right. Had there been any remainder it must have been added to the square of the root found.

EXAMPLE 2. Let 106929 be a number given, and let the square root thereof be required.

First, point your given number, as before directed, putting a point over the units, hundreds, and tens of thousands; then seek what is the greatest square number in 10 (the first point) which by the little table you will find to be 9, and 3 the root thereof; put 9 under 10, and 3 in the quotient; then subtract 9 out of 10, and there remains 1; to which bring down 69, the next point, and it makes 169 for the resolvend; then double the quotient 3, and it makes 6, which place on the left-hand of the resolvend for a divisor, and seek how often 6 in 16; the answer is twice, put 2 in the quotient, and also on the right-hand of the divisor making it 62. Then multiply 62 by the 2 you put in the quotient, and the product is 124; which subtract from the resolvend, and there remains 45; to which bring down 29, the next point, and it makes 4529 for a new resolvend. Then double the quotient 32, and it makes 64. which place on the left side of the resolv end for the divisor, and seek how often 64 in 452, which you will find 7 times: put 7 in the quotient, and also on the right-hand of the divisor, making it 647, which multiplied by the 7 in the quotient, it makes

4529, which subtracted from the resolvend, there remains nothing So 327 is the square root of the given number.

NOTE. The root will always contain just so many figures, as there are points over the given number to be extracted: And these figures will be whole numbers or decimals respectively, according as the points stand over whole numbers or decimals.-The method of extracting the square root of a decimal is exactly the same as in the foregoing examples, only if the number of decimals be odd, annex a cypher to the right hand to make them even before you begin to point. The root may be continued to any number of figures you please, by annexing two cyphers at a time to each remainder, for a new resolvend.

PRACTICAL EXAMPLES.

3. It is required to extract the square root of

2268741.
4. What is the square root of 7596796 ?

Ans. 1506.23. Rem. 121871.

Ans. 2:56.228. Rem. 3212016.

5. What is the square root of 751427.5745 ?

Ans. 866.84. Rem. 59889.

6. Extract the square root of 656714.37512.

Ans. 810 379. Rem. 251479. 7. What is the square root of 15241578750190521 ?

Ans. 123456789.

8. What is the square root of 75.347 ?

Ans. 8.6802649729. Rem. 24536226559.

9. What is the square root of .4325 ?

Ans. .65764. Rem. 96304,

To extract the Square Root of a Vulgar Fraction.

RULE.

1. Reduce the given fraction to its lowest terms, if

it be not in its lowest terms already; then extract the square root of the numerator for a new numerator, and the square root of the denominator for a new denominator.

2. If the fraction will not extract even, reduce it to a decimal, and then extract the square root.

3. When the number to be extracted is a mixed fraction, reduce the fractional part to a decimal, and annex it to the whole number, then extract the square root.

EXAMPLE 1. Extract the square root of 13

First, is equal to 4 in its lowest terms, the square root of 25 is 5, and the square root of 36 is 6; therefore is the root required.

EXAMPLE 2. Let seven-eighths be a vulgar fraction given, whose square root is required.

Reduce this to a decimal, it makes .875; to which annex cyphers, and extract the square root, as if it was So the root is .9354.

a whole number.

EXAMPLE 3. Let be a vulgar fraction, whose square root is required.

EXAMPLE 4. What is the square root of 15% ? Here reduced to a decimal is .625, which annexed to the 15 makes 15.625, the square root of which is 3.95284. Rem. 559344.

CHAPTER VIII.

EXTRACTION of THE CUBE Roor.

To extract the cube root, is nothing else but to find such a number, as being first multiplied into itself, and then into that product, produceth the given number; which to perform, observe the following direc

tions.

1st, You must point your given number, beginning with the unit's place and make a point, or dot, over every third figure towards the left-hand.

2dly, Seek the greatest cube number in the first point, towards the left-hand, putting the root thereof in the quotient, and the said cube number under the first point, and subtract it therefrom, and to the remainder bring down the next point, and call that the resolvend.

3dly, Triple the quotient, and place it under the resolvend; the unit's place of this under the ten's place, of the resolvend; and call this the triple quotient.

4thly, Square the quotient, and triple the square and place it under the triple quotient; the units of this under the ten's place of the triple quotient, and call this the triple square.

5thly, Add these two together, in the same order as they stand, and the sum shall be the divisor.

6thly, Seek how often the divisor is contained in the resolvend, rejecting the unit's place of the resolvend (as in the square root,) and put the answer in the quotient.