IV. Multiply the divisor by the figure in the root last found, and subtract the product from the dividend. To the remainder, bring down the next period, for a new dividend. Double the root now found, for a new divisor, and proceed in the operation as before, until all the periods are brought down. OBS.-Doubling the right-hand figure of the last divisor, observing to add 1 to the place of tens, when the double of the unit figure is over ten, is the same as doubling the root, or quotient. EXAMPLES. Art. 228.-2. What is the square root of 119716? figures are brought OBS. 1.-When there is a remainder, after all the down, ciphers may be annexed, and the operation continued to any assigned degree of exactness. Ans. 259.46+. Ans. 159.02+. Ans. 145.006+. 6. What is the square root of 67321? Ans. 82.7207+. OBS. 2.-When there are whole numbers and decimals in the given sum, point off both ways from the units' place; if the decimals be an odd number, annex ciphers, and make them even. 10. What is the square root of 10.4976? 11. What is the square root of 336.234? 12. What is the square root of .108241 ? 13. What is the length of a square field square rods? Ans. 3.24. Ans. 18.333+. Ans. .329. containing 7744 Ans. 88 rods. 14. What is the square root of? Ans. . OBS. 3.-The square root of a fraction may be found by extracting the root of the numerator and denominator. OBS. 4.-When the numerator and denominator are surd numbers, reduce the fraction to a decimal, and extract the root as above directed. 19. What is the square root of ? 20. What is the square root of ?? Ans. .866+. Ans. .9355+. 21. How many rows on one side of a square cornfield, containing 15376 hills? Ans. 124. 22. An army of 242064 men are drawn up in a solid body, in the form of a square. What is the number of men in rank and file? Ans. 492. 23. A man has 841 peach-trees, which he wishes to plant in the form of a square. How many must be planted in each row? Ans. 29. 24. There is a circular pond, containing 110889 square rods. What will be the length of a square field containing the same number of rods? Ans. 333 rods. 25. A number of men gave £22 1s. for a charitable purpose, each giving as many shillings as there were men. What was the number of men? 1 Ans. 21. 26. What is the length of one side of a square acre of land? 27. The diameter of a circle is 6 inches. eter of a circle 4 times as large? Ans. 12.64+. What is the diam Ans. 12. OBS. 5.-Circles are to one another as the squares of their diameter; therefore, to find the required diameter, square the given diameter, multiply the square by the given ratio, and the square root of the product will be the diameter required. 28. The diameter of a circle is 24 feet. eter of a circle one-fourth as large? What is the diam Ans. 12 feet. 29. In the right-angled triangle ABC, the side AC is 9 feet, and the side BC 12 feet. What is the length of the side AB? 30. What is the distance between the opposite corners of a room, 20 feet in length and 15 in width? Ans. 25 feet. 31. If the distance between the opposite corners of a room be 25 feet, and the width of the room be 15 feet, what is the length? Ans. 20 feet. 32. If a room be 20 feet in length, and 25 feet between the opposite corners, what is the width ? Ans. 15 feet. 33. Two men owning a pasture 32 rods in width, and 50 rods between the opposite corners, agreed to divide said pasture into two equal parts by a wall running through it lengthwise. Suppose they pay 50 cents a rod for building the wall, what does it cost them? Ans. $19.209. 34. Suppose a ladder 50 feet long, to be so placed as to reach a window 30 feet from the ground on one side of the street, and without moving it at the foot, will reach a window 20 feet high, on the other side; what is the width of the street? Ans. 85.825+ feet. 35. Two men travel from the same place-one due east, the other due north. One travels 40 miles the first day, the other 30. What is the nearest distance between them at night? Ans. 50 miles. 36. A. and B. set out together, and travel in the same direction on parallel courses, which are 20 miles apart. A. travels 45 miles, and B. 25. What is the distance between them at night? Ans. 28+ miles. 37. Suppose a pine-tree to stand 25 feet from the end of a house 40 feet in length, the foot of the tree being on a level with the foundation of the chimney, which stands in the centre of the house, and a line reaching from the foot of the tree to the top of the chimney, be 75 feet, what is the height of the chimney? and if the height of the tree beofofof 14 of the height of the chimney, what will be the length of a line reaching from the top of the chimney to the top of the tree? 60 feet, height of the chimney. 75 feet, length of the line. Ans. { Art. 229.-To find a mean proportional between two numbers. RULE. Multiply the given numbers together, and the square root of their product is the mean proportion sought. 1. What is the mean proportional between 3 and 12? Operation. 3×12=36, and √/36=6 Ans. It is evident, that the ratio of 3 to 6 is the same as the ratio of 6 to 12; for, and 2. What is the mean proportional between 12 and 48? Ans. 24. 3. What is the mean proportional between 9 and 81 ? Ans. 27. 4. What is the mean proportional between 25 and 625 ? Ans. 125. EXTRACTION OF THE CUBE ROOT. Formation of the Cube, and Extraction of the Cube Root. Art. 230.-The third power, or cube of any number, is the product of that number multiplied into its square; and the cube root is a number which, multiplied into its square, will produce the given number. Roots and powers are correlative terms; that is, if 3 is the cube root of 27, then 27 is the third power, or cube, of 3. There are but nine perfect cubes among numbers expressed by one, two, or three figures; each of the other numbers has for its cube root a whole number, plus a fraction. Thus 64 is the cube of 4, and 27 is the cube of 3; therefore, the cube root of each number between 27 and 64 must be 3 plus a fraction. What is the cube of 24? tens. units. 24=2+4 2+ 4 8+16 4+ 8 4+16+16 2+ 4 16+64 +64 8+32+32 8+48+96+64=13824 It will be perceived, from the above process, that the cube of a number composed of tens and units, is made up of four parts, viz: 1. The cube of the tens, (8 thousands.) 2. Three times the product of the square of the tens into the units, (48 hundreds.) 3. Three times the product of the tens into the square of the units, (96 tens.) 4. The cube of the units, (64 units.) To extract the cube root is to find a number which, multiplied into its square, will produce the given number. What is the cube root of 13824? Operation. 8 22x3=12)58 24 As this number is greater than 1000, which is the cube of 10, but less than 1,000,000, its root will consist of two figures, tens and units; but the cube of tens cannot be less than thousands; therefore, the three figures, 824, on the right, cannot form a part of it. Hence we separate these from 13 by a point, and look for the cube of tens in 13, the left-hand period. The root of the greatest cube contained in 13 is 2, which is the tens in the required root; for the cube of 20, which is 8000, is less, and the cube of 30, which is 27000, is greater than the given number; |