20 feet. We now take the highest peri. od 7 (hundreds), and ascertain Fig. 1. how many feet there will be in the largest square that can be B made of this quantity, the sides of which must be of the order of 20 tens. No square larger than 4 20 (hundreds) can be obtained in 7 (hundreds), the sides of which 400 will be each 20 feet (because 20x 20=400). These 20 feet (or 2 20 feet. tens) being sides of the square, are placed in the quotient as the first figure of the root. This square may be represented by Fig. 1. We now take out the 400 from 700, and 300 square feet remain. These are added to the next period (84 feet), making 384, which are to be arranged around the square B, in such a way as not to destroy its square form ; consequently the additions must be made on two sides. To ascertain the breadth of these additions, the 384 must be divided by the length of the two sides (20+20), and as the root already found is one side, we double this root for a divisor, making 4 tens or 40, for as 40 feet is the length of these sides, there will be as many feet in breadth as there are forties in 384. The quotient arising from the division is 8, which is the breadth of the addition to be made, and which is placed in the quotient, after the 784(28 Root. 4 tens; But it will be seen by Fig. 2, that to complete the square, the corner E must be filled by a small square, the sides of which are each equal to the width of C and D, that is, 8 feet. Adding this to the 4 tens, or 40, we find that the whole length of the addition to be made around the square B, is 48 feet, instead of 40. This multiplied by its breadth, 8 feet (the quotient figure), gives the contents of the whole addition, viz. 384 feet. As there is no remainder, the work is done, and 28 feet is the side of the given square. The proof may be seen by involution, thus ; 28X28= 784 ; or it may be proved, by adding together the several parts of the figure, thus ; B contains 400 feet. 160 160 E 64 Proof 784 If, in any case, there is a remainder, after the last period is brought down, it may be reduced to a decimal fraction, by annexing two ciphers for a new period, and the same process continued. Whenever any dividend is too small to contain the divi. 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 of 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 divi. dend. Double the root already found, for a new divisor, and proceed as before. EXAMPLES. What is the square root of 998001 ? 998001(999 Root. 189) 1880 1701 1989)17901 000 Find the sq. root of 784. A. 28. Of 676. A. 26. 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. A. 99. Of 17,956. A. 134. Of 32,761. A. 181. Of 39,601. A. 199. Of 488,601. A. 699. Find the sq. root of 69. A. 8.3066239. Of 83. A. 9.1104336. Of 97. A. 9.8488578. Of 299. A. 17.2916165. Of 222. A, 14.8996644. Of 282. A. 16.7928556. Of 394. A. 19.8494332. Of 351. A. 18.7349940. Of 699. A. 26.4386081. Of 979. A. 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. What is the square root of h? A. Of 160801? A. 4 of 17208? A. 483 of 43533$? A. . Of 414495 ? A.706. Of 09441 ? A. 777 Find the sq. root of 4. A. .8660254. Of . .645497. Of 173 A. 4.168333. .193649167. A. .83205. A. .288617394+. 2 4 9001 3 80 A. 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 l 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? 3276863 5768 Fig. 1. Pointing off as above, we find there will be two figures in the root, a ten and a unit. We now take the highest 30 period, 32 (thousands), and ascertain what is the largest cube that can be contained 30 in this quantity, the sides of which will be of the order of 30 tens. No cube larger than 27 (thousands) can be contained in 32 (thousands). The sides of this are 3 tens or 30 (be. 30 cause 30 X 30X30=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), ma. king 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 |