(Fig. III). This number is obtained in the operation by doubling the 7 and annexing one cipher, the result being written at the left of the dividend. Dividing 517, the area, by 140, the approximate length, we obtain 3, the probable width of the addition, and second figure of the root. Since 3 is also the side of the little square, we can now find the entire length of the addition, or the complete divisor, which is 70 + 70 + 3 = 143 (Fig. III). This number is found in the operation by adding 3 to the trial divisor, and writing the result underneath. Multiplying the complete divisor, 143, by the trial quotient figure, 3, and subtracting the product from the dividend, we obtain another remainder of 88 square feet. 70 70 Trial Divisor = 140. Complete Divisor = 143. Fig. III. 3 With this remainder, for the same reason as before, we must proceed to make a new enlargement; and we bring down two decimal ciphers, because the next figure of the root being tenths, its square will be hundredths. The trial divisor to obtain the width of this new enlargement, or the next figure in the root, will be, for the same reason as before, twice 73, the root already found, with one cipher annexed. But since the 7 has already been doubled in the operation, we have only to double the last figure of the complete divisor, 143, and annex a cipher, to obtain the new trial divisor, 146.0. Dividing, we obtain .6 for the trial figure of the root; then proceeding as before, we obtain 146.6 for a complete divisor, 87.96 for a product; and there is still a remainder of .04. Hence, the side of the given square plot is 73.6 feet, nearly. The above is the geometrical explanation of square root. The following is another clear explanation of the same process: We find the greatest square in 5400 as before, which is 4900, and place its root at the right. Since the square of a number divided into any two parts is equal to the square of the first part, plus twice the product of the first by the second, plus the square of the second part (423), having found the square of the first part, which is 4900, the remainder 517 must be equal to twice the product of the first part 70 by the second part (?) plus the square of the second part. Since we do not know the second part, we take as a trial divisor twice the first, which is 140, and we find the second part to be about 3. Twice the product of the first by the second would be 3 times 140, and the square of the second, 3 x 3; or, since 3 x 140 + 3 × 3 = 3 × 143, we take 143 as a complete divisor and multiply it by the quotient figure 3. In the same manner we proceed with the remainder 88. and we find the root to be 73.6+ RULE.-I. Point off the given number into periods of two figures each, counting from unit's place toward the left and right. II. Find the greatest square number in the left-hand period, and write its root for the first figure in the root; subtract the square number from the left-hand period, and to the remainder bring down the next period for a dividend. III. At the left of the dividend write twice the first figure of the root, and annex one cipher, for a trial divisor; divide the dividend by the trial divisor, and write the quotient for a trial figure in the root. IV. Add the trial figure of the root to the trial divisor for a complete divisor; multiply the complete divisor by the trial figure in the root, and subtract the product from the dividend, and to the remainder bring down the next period for a new dividend. V. To the last complete divisor add the last figure of the root, and to the sum annex one cipher, for a new trial divisor, with which proceed as before. 1. If at any time the product is greater than the dividend, diminish the trial figure of the root, and correct the erroneous work. 2. If a cipher occurs in the root, annex another cipher to the trial divisor, and another period to the dividend, and proceed as before. 2. What is the square root of 406457.2516? 3. The decimal points in the work may be omitted, care being taken to point off in the root according to the number of decimal periods used. 4. The pupil will acquire greater facility, and secure greater accuracy, by keeping units of like order under each other, and each divisor opposite the corresponding dividend, by the use of the lines, as shown in the operation. 3. What is the square root of 576? 4. What is the square root of 6561? 5. What is the square root of 444889? 6. What is the square root of 29855296? Ans. 24. Ans. 81. Ans. 667. Ans. 5464. 7. What is the square root of 3486784401? Ans. 59049. 8. What is the square root of 54819198225 ? Ans. 234135. 5. The cipher in the trial divisor may be omitted, and its place, after division, may be occupied by the trial root figure, thus forming only complete divisors. 9. What is the square root of 2 ? Extract the square roots of the following numbers. 10. √3. Ans. 1.7320508+. 13. √7. 11. √5. Ans. 2.2360679+. 14. √8. 12. √6. Ans. 2.4494897+. 15. √10. What is the square root of: Ans. 2.6457513+. 6. The square root of a common fraction may be obtained by extracting the square roots of the numerator and denominator separately, provided the terms are perfect squares; otherwise, the fraction may first be reduced to a decimal. 7. Mixed numbers may be reduced to the decimal form before extracting the root; or, if the denominator of the fraction is a perfect square, to an improper fraction. APPLICATIONS OF SQUARE ROOT. 435. An Angle is the opening between two lines that meet each other. Thus, the two lines AB and AC meeting, form an angle at A. 436. A Right-Angled Triangle is a triangle having one right angle, as at C. The Base is the side on which it stands, as AC. The Perpendicular is the side forming a right angle with the base, as BC. The Hypotenuse is the side opposite the right angle, as AB. B C 437. Similar Figures are figures which have the same form and differ only in size. 438. The following principles, which are demonstrated in geometry, afford applications of square root. PRINCIPLES. I. The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides; therefore, II. The hypotenuse is equal to the square root of the sum of the squares of the other two sides. III. The areas of two circles are to each other as the squares of their radii, diameters, or circumferences. IV. The base or perpendicular of a right-angled triangle is equal to the square root of the difference of the hypotenuse and that of the other side. V. The ratio of the area of two similar figures, is equal to the square of the ratio of any two like dimensions of them; therefore, VI. The ratio of any two like dimensions of two similar figures is equal to the square root of the ratio of their areas. EXAMPLES. 1. The two sides of a right-angled triangle are 3 and 4 feet. What is the length of the hypotenuse? 2. A circular skating rink has a diameter of 75 feet. What would be the diameter of a similar rink with the area? Ans. 43.3 feet. 3. If an army of 55225 men is drawn up in the form of a square, how many men will there be on a side? Ans. 235. 4. The diagonals of two similar rectangles are as 6 to 13. How many times does the larger contain the smaller? Ans. 4 times. 5. The sides of two square blocks are 5 feet and 10 feet respectively. How do they compare in area? Ans. 5 foot block is of other. 6. A man has 200 yards of carpeting 14 yards wide. What is the length of one side of the square room which this carpet will cover? Ans. 45 feet. 7. How many rods of fence will be required to inclose 10 acres of land in the form of a square? Ans. 160 rods. 8. The end of a May-pole, broken 39 feet from the top, struck the ground 15 feet from the foot. What was the height of the pole? Ans. 75 feet. 9. A ladder 40 feet long is so placed in a street, that without being moved at the foot, it will reach a window |