and in the divisor making 81, as also below the same. These multiplied make also 81, set below the dividend, and subtracting, we have 49 remaining, to which the last period 56 being brought down, we have 4956 for the new dividend. Then, for a new divisor, either double the root 4'1, or else, which is easier, to the last divisor add the figure 1 standing below it, and either way gives 82 for the first part of the new divisor. This 82 is 6 times contained in 495, and therefore 6 is the next figure to set in the root and in the divisor, as also below the same; which being then multiplied by it, gives 4956, the same as the dividend; therefore nothing remains, and 4:16 is the root of 17.3056, as required.

Note. When all the periods of the given number are brought down and used, and more periods are required to be found, the operation may be continued by adding as many periods of ciphers as we please, namely, bringing always two ciphers at once to each dividend. And when the root is to be extracted to a great number of places, the work may be much abbreviated: thus, having proceeded in the extraction after the common method, till you have found one more than half the required number of figures in the root, the rest may be found by dividing the last remainder by its

VOL. I.

U

cor

corresponding divisor, annexing a cipher to every dividual, as in division of decimals; or rather, without annexing ciphers, by omitting continually the right hand figure of the divisor, after the manner of contracted division of decimals.

So the operation for the root 2, to 12 or 13 places, may be thus.

1

Here, having found the first seven figures 1'414213 by the common extraction, by adding always periods of ciphers,

the last six figures 562373 are found by the method of contracted division in decimals, without adding ciphers to the remainder, but only pointing off a figure at each time from the last divisor.

The use of the square root will be shown in Mensuration, where it will be more particularly wanted.

PROBLEM XI.

To extract the cube root.

I. Point the given number into periods of three places each, beginning at units; and there will be as many integral places in the root, as there are points over the integers in the given number.

II. Seek the greatest cube in the left hand period; write the root in the quotient, and the cube under the first period; from which subtract it, and to the remainder bring down the next period: call this the resolvend, under which draw a line.

III. Under the resolvend, write the triple square of the root, so that units in the latter stand under the place of hundreds in the former; under the triple square of the root, write the triple root, removed one place to the right; and the sum of these two lines call a divisor, under which draw a line.

IV. Seek how often this divisor may be had in the resolvend, its right hand place excepted, and write the result in the quotient.

V. Under the divisor, write the product of the triple square of the root by the last quotient figure, setting the units place of this line under that of tens in the divisor; under this line, write the product of the triple root by the square of the last quotient figure; let this line be removed one place beyond

v 2

yond the right of the former; and under this line, removed one place forward to the right, set the cube of the last quo, tient figure. The sum of these three lines call the subtra, hend, under which draw a line.

VI. Subtract the subtrahend from the resolvend; to the remainder bring down the next period for a new resolvend; the divisor to this, must be the triple square of all the quo tient added to the triple thereof, as in the third article, &c.

4]

38988 Divisor

15552

Triple square of 36 multiplied by 4

1728 Triple of 36 multiplied by square of 4

64 Cube of 4

1572544 Subtrahend

If the work of this Example be well considered, and compared with the foregoing rule, it will be easy to conceive how any other example of the like nature may be wrought. And here observe, that when the cube root is extracted to more than two places, there is a necessity of doing some work on a spare piece of paper, in order to come at the root's triple square, and the product of the triple root, by the square of the quotient figure, &c.

In this Example, the given number is a cube number, and therefore at the end of the operation there remained nothing; for 364 multiplied by 364, and the product multiplied by 364 again, gives 48228544, the given number.

But if the number given be not a cube number, then to the last remainder always bring down three ciphers, and work anew for a decimal fraction, if needful.

These Examples are all performed in the same manner as

the foregoing one.

The use of the cube root will be shown in Mensuration.

THE