sor, a cipher must be placed in the root, and another period brought down.

From the above illustrations, we see the reasons for the following rule.

RULE

FOR EXTRACTING THE SQUARE ROOT.

1. Point off the given number, into periods of two figures each, beginning at the right.

2. Find the greatest square in the first left hand period, and subtract it from that period. Place the root of this square in the quotient. To the remainder bring down the next period for a dividend.

3. Double the root already found (understanding a cipher at the right) for a divisor. Divide the dividend by it, and place the quotient figure in the root, and also in the divisor.

4. Multiply the divisor, thus increased, by the last figure of the root, and subtract the product from the dividend. To the remainder bring down the next period, for a new dividend. Double the root already found, for a new divisor, and proceed as before.

EXAMPLES.

What is the square root of 998001 ?

998001(999 Root.

81

189)1880

1701

1989)17901

000

A. 26.

Find the sq. root of 784. A. 28. Of 676. Of 625. A. 25. Of 487,204. A. 698. Of 638,401. A. 779. Of 556,516. A. 746. Of 441. A. 21. Of 1024. A. 32. Of 1444. A. 38. Of 2916. A. 54.

What is the rule for extracting the square root?

Of 6241.

A. 79. Of 9801. A. 99. Of 17,956. 134. Of 32,761. A. 181. Of 39,601. A. 199. 488,601. A. 699.

A.

Of

A.

Of 299.

A.

Find the sq. root of 69. A. 8.3066239. Of 83. 9.1104336. Of 97. A. 9.8488578.

17.2916165. Of 222.

A. 14.8996644. Of 282. A.

Of 351.

A.

A.

A. 19.8494332.
A. 26.4386081. Of 979.

16.7928556. Of 394. 18.7349940. Of 699. 31.2889757. Of 989. A. 31.4483704. Of 999. A. 31.6069613. Of 397. A. 19.9248588. Of 687. A. 26.2106848. Of 892. A. 29.8663690.

It was shown in the article on Involution, that a fraction is involved by involving both numerator and denominator, hence to find the root of a fraction, extract the root both of numerator and denominator. If this cannot be done, the fraction may be reduced to a decimal, and its root extracted.

36

249001

What is the square root of 25? A. 5. Of 160801? A. 481. Of 237102? A. 487 Of 430336?

480249

Of $1998? A. 788.

96

693

Of 606841?

942841

Find the sq. root of 2. A. .8660254. Of .645497. Of 173. A. 4.168333. Of

.193649167.

Of 12

.288617394+.

A. .83205.

Of

EXTRACTION OF THE CUBE ROOT.

A Cube is a solid body, having six equal sides, each of which is an exact square. Thus a solid, which is 1 foot long, 1 foot high, and 1 foot wide, is a cubic foot; and a solid whose length, breadth, and thickness are each 1 yard, is called a cubic yard.

The root of a cube is always the length of one of its sides; for as the length, breadth, and thickness of such a body are the same, the length of one side, raised to the third power, will show the contents of the whole.

Extracting the Cube Root of any quantity, therefore, is finding a number, which multiplied into itself, twice, will

produce that quantity ;-or it is finding the length of one side of a given quantity, when that quantity is placed in an exact cube.

To ascertain the number of figures in a cube root, we point off the given number, into periods of three figures each, beginning at the right, and there will be as many figures in the required root as there are periods.

1. What is the length of one side of a cube, containing 32768 solid feet?

32768(3
27

5768

Pointing off as above, we find there will be two figures in the root, a ten and a unit.

Fig. 1.

30

30

We now take the highest period, 32 (thousands), and ascertain what is the largest cube that can be contained in this quantity, the sides of which will be of the order of tens. No cube larger than 27 (thousands) can be contained in 32 (thousands). The sides of this are 3 tens or 30 (because 30X30X30-27,000)

which are placed as the first figure of the root. This cube may be represented by Fig 1.

We now take the 27000 from 32000, and 5000 solid feet remain. These are added to the next period (768), making 5768, which are to be arranged around the cubic figure 1, in such a way as not to destroy its cubic form; consequently the addition must be made to three of its sides.

We must now ascertain, what will be the thickness of the addition made to each side. This will of course depend upon the surface to be covered. Now the length of one side has been shown to be 30 feet, and, as in a cube, the length and breadth of the sides are equal, multiplying

the length of one side into itself will show the surface of one side, and this multiplied by 3, the number of sides, gives the contents of the surface of the three sides. Thus 30X30-900, which multiplied by 3=2700 feet.

Now as we have 5768 solid feet to be distributed upon a surface of 2700 feet, there will be as many feet in the thickness of the addition, as there are twenty-seven hundreds in 5768. 2700 is contained in 5768 twice; therefore 2 feet is the thickness of the addition made to each of the three sides.

By multiplying this thickness, by the extent of surface (2700X2) we find that there are 5400 solid feet contained in these additions.

But if we examine Fig. 2, we shall find that these additions do not com. plete the cube, for the three corners a a a need to be filled by blocks of the same length as the sides (30 feet) and of the same breadth and thickness as the previous additions (viz. 2 feet).

Now to find the solid contents of these blocks, or the number of feet required to fill these corners, we multiply the length, breadth, and thickness of one block together, and then multiply this product by 3, the num ber of blocks. Thus, the breadth and thickness of each block has been shown to be 2 feet; 2X2=4, and this multiplied by 30 (the 30 length)=120, which is the solid contents of one block. But in three, there will be three times as many solid feet, or 360, which is the number required to fill the deficiencies.

30

meet.

Fig. 3.
30

In other words, we square the last quotient figure (2) 2 multiply the product by the first figure of the quotient (3 tens) and then multiply the last product by 3, the 30 number of deficiencies.

But by examining Fig. 3, it appears that the figure is not yet complete, but that a small cube is still wanting, where the blocks last added

The sides of this small cube, it will be seen, are each equal to the width of these blocks, that is, 2 feet. If each side is 2 feet long, the whole cube must contain 8

solid feet (because 2×2×2 =8), and it will be seen by Fig. 4, that this just fills the vacant corner, and completes the cube.

32 We have thus found, that the additions to be made around the large cube (Fig. 1) are as fol.

lows.

5400 solid feet upon three sides, (Fig. 2).
66 to fill the corners a a a.

360 66

8 66

"to fill the deficiency in Fig. 3.

Now if these be added together, their sum will be 5768 solid feet, which subtracted from the dividend leave no remainder and the work is done. 32 feet is therefore the length of one side of the given cube.

The proof may be seen by involving the side now found to the third power, thus; 32x32x32-32768; or it may be proved by adding together the contents of the several parts, thus,

27000 feet-contents of Fig. 1.

5400" addition to the three sides.

=