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, (¶ 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. 15. What is the square root of? 16. What is the square root of 10%? 17. What is the square root of? 18. What is the square root of 201? Ans. Ans. . Ans. . Ans.. 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. 19. What is the square root of 675? Ans. '866+. Ans. '912+. SUPPLEMENT TO THE SQUARE ROOT. QUESTIONS. 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 ? do we extract the square root of vulgar fractions? 20. How 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 área, or superficial contents, is 5194 feet; what will be the length of the side of a square of equal area? 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? . 16 times as large? 25 times as large? 36 times as large? 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. 4X4X9= 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. T 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. 4216, and 32 = 9; 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? 18. The ladder and ditch given, required the wall. Ans. 24 feet. Ans. 32 feet 19. The distance between the lower ends of two equal rafters is 32 feet, and the height of the ridge, above the beam on which they stand, is 12 feet; required the length of each rafter. Ans. 20 feet. 20. There is a building 30 feet in length and 22 feet in width, and the eaves project beyond the wall 1 foot on every side; the roof terminates in a point at the centre of the building, and is there supported by a post, the top of which is 10 feet above the beams on which the rafters rest; what is the distance from the foot of the post to the corners of the eaves? and what is the length of a rafter reaching to the middle of one side? a rafter reaching to the middle of one end? and a rafter reaching to the corners of the eaves? Answers, in order, 20 ft.; 15'62 ft.; 18'86 ft.; and 22'36+ ft. 21. There is a field 800 rods long and 600 rods wide; what is the distance between two opposite corners ? Ans. 1000 rods. 22. There is a square field containing 90 acres ; how many rods in length is each side of the field? and how many rods apart are the opposite corners ? Answers, 120 rods; and 1697+rods. 23. There is a square field containing 10 acres; what distance is the centre from each corner? Ans. 28'28 + rods. EXTRACTION OF THE CUBE T110. A solid body, having six equal sides, and each of the sides an exact square, is a CUBE, and the measure in length of one of its sides is the root of that cube; for the length, breadth and thickness of such a body are all alike; consequently, the length of one side, raised to the 3d power, gives the solid contents. (See ¶ 36.) Hence it follows, that extracting the cube root of any number of feet is finding the length of one side of a cubic body, of which the whole contents will be equal to the given number of feet. 1. What are the solid contents of a cubic block, of which each side measures 2 feet? Ans. 23 = 2 X 2 X 28 feet. 2. How many solid feet in a cubic block, measuring 5 feet on each side? Ans. 53 125 feet. the length of each side, which is expressed by the former quotient figure, 2, (tens.) 3 times 2 (tens) are 6 (tens) 60; or, what is the same in effect, and more convenient in practice, we may multiply the quotient figure, 2, (tens,) by 30, thus, 2 X 30= 60, as before; then, 60 X 16960, contents of the three deficiencies n, n, n. Looking at Fig. III., we perceive there is still a deficiency in the corner where the last blocks meet. This deficiency is a cube, each side of which is equal to the last quotient figure, 4. The cube of 4, therefore, (4 X 4 X 4=64,) will be the solid contents of this corner, which in Fig. IV. is seen filled. Now, the sum of these sev eral additions, viz. 4800 + 960645824, will make the subtrahend, which, subtracted from the dividend, leaves no remainder, and the work is done. Fig. IV. shows the pile which 13824 solid blocks of one foot each would make, when laid together, and the root, 24, shows the length of one side of the pile. The correctness of the work may be ascertained by cubing the side now found, 243, thus, 24 X 24 X 24 = 13824, the proved by adding together he contents of all the several parts, thus, Feet. 8000 contents of Fig. I. 4800 960 addition to the sides a, b, and c, Fig. 1. addition to fill the deficiencies n, n, n, Fig. II. 64 addition to fill the corner e, e, e, Fig. IV. 13824= contents of the whole pile, Fig. IV., 24 feet on ach side |