OPERATION.

Bolidity of cube AD,

3

80

Area of the Sfigs. AC, BD, &CE, 802=x8-19200, trial divisor. 192969 1st dividend 66 "AB, BC, &CD, 80x3x9= 2160

Area of figure XY,

92

81

Area of the 7 pieces added to the cube, 21441, true divisor,

EXPLANATION.-We first find

the greatest cube contained in the left-hand period. We know that this number must be more than 80, since 803-512000, which is less than 704969; also, that it must be less than 900, since 90" =729000, which is greater than 704969. Hence, the first, or lefthand figure of the root, is 8; whose cube is 512, which is the greatest cube contained in 704, the first or left-hand period.

Hence, each side of the cube AD, represented by Fig. 1, is 80 linear feet; therefore, its cubical contents is 803-512000 cubic feet; and 704963-512000-192969 cubic feet, which is still to be added to the ube AD.

We first add the three square slab pieces AB, BC, and CD, whose length and breadth are each respectively 80 feet, (the side of the cube AD.) The area of the face of the first piece, AB, is 802= 6400 square feet. Since these are 3 pieces, 3x6400-19200 square feet is the are of the three pieces added. As many times as the 1st dividend contains the trial divisor, (the area of these three pieces mentioned,) so many feet in thickness they must be, which we find to be 9 feet.

Figure 2, represents the cube AD, with the three pieces AB BC, and CD, added.

A

192969

0

D

B

B

D

Fig. 2.

2. What is the cube root of 69934528?

(We will perform the above example, leaving off the useless siphers, &c.)

3. What is the cube root of 185193?

4. What is the cube root of 1860867?

Ans. 57

Ans. 123.

5. What is the cube root of 257259456? Ans. 636. 6. What is the cube root of 41673648563 ?

By reversing the process of involution under ART. 214, we obtain for extracting the cube root the following

GENERAL RULE.

Commencing at units, separate the number into periods of three figures each.

Then find the largest digit, the cube of which shall not exceed the left-hand period. Place this digit, which is called the first figure of the root, on the right, in the form of a quotient; also, on the left, for the first term of a first column, and its square for the first term of a second column, and from the left-hand period of the given number, subtract its cube. Then to the remainder, annex the next period, for the FIRST DIVIDEND. Now double the term in the first column, for its second term, and add its product into the root already found, to the first term of the second column, for the first TRIAL DIVISOR. Annex two ciphers to the trial divisor, and write the number of times it is contained in the dividend, for the next figure of the root; also, annex it to the sum of the last term in the first column, and the first figure of the root;this will be the next term of the first column. Add the product of this term into the digit of the root last found, advancing it two places to the right, to the last term of the second column, for its next term, this will be the TRUE DIVISOR. From the DIVIDEND, subtract the product of the

1

true divisor into the last digit of the root found; and to the remainder annex the next period. for the second dividend.

Proceed in a similar way until all the periods have been used.

REMARK. By carefully examining the foregoing involution, the pupil wil be able to deduce other rules for the extraction of the cube root, some of which may, perhaps, appear more plain than the one I have just given, as this is more readily deduced from Algebraic involutions. I have given this rule as it will be less laborious to extract the cube root of large numbers by it than by many other rules usually given; also, because it keeps distinct the three geometrical magnitudes-lines, surfaces, and solids.

The first rule, however, is the most simple, and will be found of much im. portance in reducing surd quantities to their simplest form, (as will here after be explained,) or in determining the roots of rational quantities.

EXPLANATION.-We first find the greatest cube contained in the left-hand period. We know that this number must be more than 80, since 80512000, which is less than 704969; also, that it must be less than 903, sino 903729000, which is greater than 704969. Hence, the first, or left-har figure of the root, is 8; whose cube is 512, which is the greatest cube c tained in 704, the first or left-hand period.

We first add the three square slab pieces AB, BC, and CD, whose length and breadth are each respectively 80 feet, (the side of the cube AD.) The area of the face of the first piece, AB, is 8026400 square feet. The length of the other two pieces, BC, and CD, is 80+80=160 feet, and their width 80 feet. Hence, their superficial contents is 160 X 80 12800 square feet, which added to 6400 square feet, the superficial contents of the piece, AB, gives 19200 square feet, the superficial contents of the three pieces, AB, BC, and CD. As these three square slabs make up by far the greatest amount of the whole increase, if we divide 192969, (the number of cubic feet remaining to be added,) by 192000, the number of square feet in the three pieces, AB, BC, and CD, (which may be called the trial divisor,) it will give their thick mess; which we find to be 9 feet.

Figure 2, represents the cube AD, with the three pieces AB, BC, and CD, added.

We now add the three corner-pieces, EG, HF, and HX, whose lengths are respectively 80 feet, (the side of the cube AD) and whose width and thickness are each 9 feet respectively; also, the cornerpiece AW, whose length, width, and thickness, are each 9 feet. Therefore, the length of the three pieces, EG, HF, and HX, is 240 fest; which being increased by the length