yards on a side, which 625 square yards of carpeting will exactly cover. The proof may be seen by adding together the several parts of the figure, thus : The square A contains 400 yards. The figure B 100 Or we may prove it by involution, thus :25 × 25 = 625, as before. From this example and illustration we derive the following general RULE FOR THE EXTRACTION OF THE SQUARE ROOT. I. Point off the given number into periods of two figures each, by putting a dot over the units, another over the hundreds, and so on. These dots show the number of figures of which the root will consist. II. Find the greatest square number in the left band period, and write its root as a quotient in division. Subtract the square number from the left hand period, and to the remainder bring down the next period for a dividend. III. Double the root already found for a divisor; seek how many times the divisor is contained in the dividend, excepting the right hand figure, and place the result in the root, and also at the right hand of the divisor; multiply the divisor, thus augmented, 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. IV. Double the root already found for a new divisor, and continue the operation as before, until all the periods are brought down. Note 1. If we double the right hand figure of the last divisor, we shall have the double of the root. Note 2. As the value of figures, whether integers or decimals, is determined by their distance from the place of units, so we must always begin at unit's place to point off the given number, and, if it be a mixed number, we must point it off both ways from units, and if there be a deficiency in any period of decimals, it may be supplied by a cipher. It is plain, the root must always consist of so many integers 211 and decimals as there are periods belonging to each in the given number. EXAMPLES FOR PRACTICE. 2. What is the square root of 10342656 ? OPERATION. 10342656 (3216, Ans. 9 62) 134 124 641) 1026 6426) 38556 3. What is the square root of 43264 ? OPERATION. 43264 (208, Ans. 4 408) 3264 4. What is the square root of 998001 ? 7. What is the square root of '001296 ? Ans. 999 Ans. 15'3. Ans. 31'05671. Ans. '54. 108. In this last example, as there was a remainder, after bringing down all the figures, we continued the operation to decimals, by annexing two ciphers for a new period, and thus we may continue the operation to any assigned degree of exactness; but the pupil will readily perceive, that he can never, in this manner, obtain the precise root; for the last figure in each dividend will always be a cipher, and the last figure in each divisor is the same as the last quotient figure; but no one of the nine digits, multiplied into itself, produces a number ending with a cipher; therefore, whatever be the quotient figure, there will still be a remainder. 11. What is the square root of 3? Ans. 173+. Ans. 3'16+. Ans. 13'57+. Note. We have seen, (T 105, ex. 9,) that fractions are squared by squaring both the numerator and the denominator. Hence it follows, that the square root of a fraction is found by extracting the root of the numerator and of the denominator. The root of 4 is 2, and the root of 9 is 3. When the numerator and denominator are not exact squares, the fraction may be reduced to a decimal, and the approximate root found, as directed above. a 1. What is involution? 2. What is understood by a power? 3. the first, the second, the third, the fourth power? 4. What is the index, or exponent? 5. How do you involve a number to any required power? 6. What is evolution? 7. What is a root? 8. Can the precise root of all numbers be found? 9. What is a surd number? 10. rational? 11. What is it to extract the square root of any number? 12. Why is the given sum pointed into periods of two figures each? 13. Why do we double the root for a divisor? 14. Why do we, in dividing, reject the right hand figure of the dividend? 15. Why do we place the quotient figure to the right hand of the divisor? 16. How may we prove the work? 17. Why do we point off mixed numbers both ways from units? 18. When there is a remainder, how may we continue the operation? 19. Why can we never obtain the precise root of surd numbers? 20. How do we extract the square root of vulgar fractions? EXERCISES. 1. A general has 4096 men; how many must he place in rank and file to form them into a square? Ans. 64. 2. If a square field contains 2025 square rods, how many rods does it measure on each side? Ans. 45 rods. 3. How many trees in each row of a square orchard containing 5625 trees? Ans. 75. 4. There is a circle, whose area, or superficial contents, is 5184 feet; what will be the length of the side of a square of equal aren? 518472 feet, Ans. 5. A has two fields, one containing 40 acres, and the other containing 50 acres, for which B offers him a square field containing the same number of acres as both of these; how many rods must each side of this field measure? Ans. 120 rods. 6. If a certain square field measure 20 rods on each side, how much will the side of a square field measure, containing 4 times as much? /20 X 20 X 4 = 40 rods, Ans. 7. If the side of a square be 5 feet, what will be the side of one 4 times as large? 9 times as large? times as large? large? 25 times as large? 16 36 times as Answers, 10 ft.; 15 ft.; 20 ft.; 25 ft.; and 30 ft. 8. It is required to lay out 288 rods of land in the form of a parallelogram, which shall be twice as many rods in length as it is in width. Note. If the field be divided in the middle, it will form two equal squares. Ans. 24 rods long, and 12 rods wide. 9. I would set out, at equal distances, 784 apple trees, so that my orchard may be 4 times as long as it is broad; how many rows of trees must I have, and how many trees in each row? Ans. 14 rows, and 56 trees in each row. 10. There is an oblong piece of land, containing 192 square rods, of which the width is as much as the length; required its dimensions. Ans. 16 by 12. 11. There is a circle, whose diameter is 4 inches; what is the diameter of a circle 9 times as large? Note. The areas or contents of circles are in proportion to the squares of their diameters, or of their circumferences. Therefore, to find the diameter required, square the given diameter, multiply the square by the given ratio, and the square root of the product will be the diameter required. No4X4 X 9 = 12 inches, Ans. 12. There are two circular ponds in a gentleman's pleasure ground; the diameter of the less is 100 feet, and the greater is 3 times as large; what is its diameter? Ans. 173'2+ feet. 13. If the diameter of a circle be 12 inches, what is the diameter of one as large ? Ans. 6 inches. ¶ 109. 14. A carpenter has a large wooden squarc; one part of it is 4 feet long, and the other part 3 feet long; what is the length of a pole, which will just reach from one end to the other? Note. A figure of 3 sides is called a triangle, and, if one of the corners be a square corner, or right angle, like the angle at B in the annexed figure, it is called a right-angled triangle, of which the square of the longest side, A C, (called the hypotenuse,) is equal to the sum of the squares of the other two sides, A B and B C. = 42 16, and 329; then, 9+16 5 feet, Ans. 15. If, from the corner of a square room, 6 feet be measured off one way, and 8 feet the other way, along the sides of the room, what will be the length of a pole reaching from point to point? Ans. 10 feet. 16. A wall is 32 feet high, and a ditch before it is 24 feet wide; what is the length of a ladder that will reach from the top of the wall to the opposite side of the ditch? Ans. 40 feet. 17. If the ladder be 40 feet, and the wall 32 feet, what is the width of the ditch? Ans. 24 feet. 18. The ladder and ditch given, required the wall. Ans. 32 feet |