2. Begin at the left hand, and find the quotient root in that riod, and place it on the right of the given sum in the quo nt, and its square under said period, which subtract from the number above. 3. Then bring down the next period of 2 figures, and place it on the right of the remainder, as in division, and this forms a new dividend. 4. Now double this figure or root in the quotient, and place it on the left of the new dividend for a divisor. 5. Then consider how often the divisor is contained in the dividend, omitting the last figure, and place the result on the right of the root in the quotient, and then place this figure on the right of the number produced by doubling for a divisor, and multiply as in division, until the periods are all brought down. For Decimals.-When decimals occur in the given number, it must be pointed both ways from the decimal point, and the root must consist of as many figures, of whole numbers and decimals respectively, as there are periods of integers or decimals in the given number. When a decimal alone is given, annex one cipher, if necessary, so that the number of decimal places shall be equal; and the number of decimal places in the root will be equal to the number of periods in the given decimal For Vulgar Fractions.-1. Reduce mixed numbers to im proper fractions, and compound fractions to simple ones, and then reduce the fraction to its lowest terms. 2. Extract the square root of the numerator and denominator separately, if they have exact roots; but if they have not, reduce the fraction to a decimal, and then extract the root, as above, &c. Proof: square the root and add in the remainder. EXAMPLES. Illustration.-A square number can not have more places of figures than double the places of the root, and at least but one less. A square is a figure of four equal sides, each pair meeting perpendicularly, or a figure whose length and breadth are equal. As the area, or number of equal feet, inches, &c., in a square, is equal to the products of the two sides, which are equal, the second power is called the square. Let the follow ing figure represent a board one foot square, and one inch in thickness, which being sawn or cut into square or solid inches, will make 144 inches, or 144 blocks one inch square; and the square root of 144 is 12; because 12×12=144, which in this case will be inches one side of the square. the 9 under the first period (15), which subtract and you have 6; then bring down the next period (46), which place to the right of the remainder (6); now double the quotient figure, 3 × 2=6, and place it in the divisor, one place to the left; now consider how many times 6 in 64, which for trial call 9; write 9 in the quotient, and to the right of 6 in the divisor, which makes the divisor 69; then multiply 69 by 9, and place the result under the dividend; subtract as before, and bring down the next period (82); now double the two quotient figures, 39+2=78, which place in the divisor; consider how many times 78 is contained in 258, suppose 3 times; place 3 in the quotient and in the divisor, as before; then multiply the divisor by the last ' quotient figure, and write the result as above directed, and the remainder is 233; which by annexing ciphers may be continued. Ans. 8647. 2. What is the square root of 74770609 ? 234.5. 57.19+ 121.968175. 9817+ 4225 Ans. 2.091+ 7. What is the square root of 4.372594? 8. What is the square root of .00103041? 9. What is the square root of 2794? 10. What is the square root of 275 11. What is the square root of ? 2. What is the square root of 61? 13. What is the difference between 81 and 812? 341 350 14. What is the square root of 224? 15. What is the square root of 11 Ans. .89802+ Ans. Ans. 2. Ans. 6552. Ans. .9574+ Ans. 1.5. Ans. 99.9. Ans. 1.414213. 17. What is the square root of 21? 144 ? 15129 Ans..09756+ Ans. 31.05671. Ans. 2.63. Ans. 327. Ans. 4. Ans. .7745. Ans. 4. To find a mean proportional between two numbers. RULE. Multiply the given numbers together, and extract the square root of the product, which will be the mean proportional sought. 30. What is the mean proportional between 24 and 96 ? Ans. √96×24=48. To find the side of a square equal in area to any given superficies whatever. RULE. Find the area, and the square root is the side of the square sought. 31. If the area of a circle be 184.125, what is the side of a square equal in area thereto ? Ans. 184.125-13.569+ 32. A general has an army of 5625 men; how many must he place in rank and file to form them into a square? Ans. /5625=75. 33. Suppose that Napoleon, at the battle of Marengo, commanded an army of 256036 men; how many did he place in rank and file to form them into a solid square? Ans. 506. Having the area of a circle to find the diameter. RULE. Multiply the square root of the area by 1.12837, and the product will be the diameter. Or multiply the area by 1.2732 and take the square root of the product. 34. Required the diameter of a circle whose area is 82 feet 84 inches. Ans. 10 feet 3.13 inches. 35. Admit a leaden pipe of an inch in diameter will fill a cistern in 3 hours; I demand the diameter of another pipe which will fill the same cistern in 1 hour. Thus 3.75 and .75 × .75.5625; then 3 hours : 5625 :: 1 hour: 1.6875 and 1.6875 1.3 inches pearly, Ans. (inversely.) 36. If a circular pipe of 1.5 inches diameter fill a cistern in 5 hours, in what time would it be filled by one 3.5 inches diameter ? Ans. 55 minutes, 4.8 seconds. 37. If 784 trees be planted in a square orchard, how many must be in a row? Ans. 28. The square of the longest side or hypotenuse of a right-angled triangle, is equal to the sum of the squares of the other two sides; consequently, the difference of the square of the longest, and either of the others, is the square of the remaining side. 38. The wall of a fort is 17 feet high, which is surrounded by a ditch 20 feet in breadth; required the length of a ladder to reach from the outside of the ditch to the top of the wall. Ans. 17×17=289: 20×20=400+289=√/689=26.2+ 39. Two ships leave the same port, one sails due east 40 miles' and the other due north 50 miles; required the distance from each other. 40. There is a church 48 feet in height from the ground te the eaves or roof, and the street is 64 feet in width; what length of cord would it require to reach from the opposite side of the street to the eaves of the house? Ans. 80 feet. 41. In a circle, whose area or superficial content is 4096 feet, I demand what will be the length of one side of the square containing the same number of feet. Ans. 64 feet. 42. A certain square garden measures 4 rods on each side what will be the length of one side of a garden containing 4 times as many square rods? Ans. 8 rods. ; 43. If one side a square piece of land measure 5 rods, what will be the side of one measuring 4 times as large? 16 times as large? 36 times as large? Ans. 10, 20, 30. 44. In a load of 120 cherry-boards of one inch in thickness, 27 inches in width, and 11.5 feet in length, how many square feet, and what is the value of the boards at D22 per M.? Ans. 3105 feet; value D68.31. 45. In a tract of land of 640 acres, required the length of one side; and suppose the tract to contain 16 times as much, what would be the length of one side? and suppose each rod valued at D75, what would be the amount? Ans. 1st, 320 poles, length of one side; 2d, 1280 po., or 16 times as much; 3d, D122880000 value. 46. One side of a square field is 400 rods in length; required the number of acres in the field. Note. For examples in square and cub. measure, see appendix. What is evolution? What is the cube root? REVIEW. What is the square root of a number? root of a number, what must first be done? After you have counted off the given number into periods, how will you pro |