3 If 484 trees be planted in a square orchard, how many must be in a row each way ? answer 22 Note I. The fquare of the longest fide of a right angled triangle is equal to the fum of the fquares of the other two fides; and confequently the difference of the fquare of the longeft, and either of the other, is the fquare of the remaining fide. 2. The fquare root of a vulgar fraction is found by reducing it to its lowest terms, and extracting the root of the numerator, for a numerator, and of the denominator, for a denominator. If it be a furd, reduce it to its equivalent decimal, &c. 3. A mixt number may be reduced to an improper fraction, or a decimal, and the root thereof extracted as before. 14 The wall of a fortress is 17 feet high, which is sur rounded by a moat 20 feet in breadth; query the length of a ladder to reach from the outside of the moat to the top of the wall? answer 26.2 feet. 15 A line of 36 yards long will exactly reach from the top of a fort to the opposite bank of a river, known to be 24 yards broad; the height of the wall is required? answer 26.83+ yards. 16 Suppose a ladder 60 feet long be so planted as to reach a window 37 feet from the ground on one side of the street, and without moving it at the foot, will reach a window 23 feet high on the other side; what breadth was the street of? 17 What is the square-root of THE answer 103,64 feet. THE CUBE ROOT. answer 71528 63 4 8,7649+ HE Cube of a number is the product of that number multiplied into its square. Extraction of the cube root is the finding of such a number, as, being multiplied into its square, will produce the number proposed. RULE. First, Distinguish the proposed number into periods of three figures each, beginning at the units place, or decimal N 2 point; point and when the decimal does not consist of a complete period or periods, annex a cipher or ciphers to make it so ; and the places of the root will be as many as the periods of the given cube in whole numbers and decimals refpectively. Secondly, Find the great froot of the left hand period, which place to the right of the given number, and subtract the cube thereof from said period; and to the remainder bring down the next period for a dividual. Thirdly, Take the triple square of the ascertained root for a defective divisor. Fourthly, Reverse mentally the units and tens of the dividual, and try how often the defective divisor is contained in the rest; place the result of this trial to the root, and its square to the right of said divisor, supplying the place of tens with a cipher, if the square be less than 10 Fifthly, Complete the divisor, by adding thereto the product of the last figure of the root by the rest, and by 30. Sixthly, Multiply, subtract, and bring down the next period for a dividual, for which find a divisor as before; and so proceed with every period. Note. Defective divisors, after the first, may be more concisely found by addition, thus: To the last complete divisor, add the number which completed it, with twice the square - of the last figure in the root; the sum will be the next defective divisor. EXAMPLES. 1 What is the cube root of 441194,917-2 444191,917(76,3 ans. 343 Defec. div. & sqr. of 6=14736)101194 +1260 complete divisor 15996) 95976 f Defec. div. & sqr. of 31732809)5218947 +6810complete divisor 1739649)5218947 5 answer 325 439 638 2805 8765 2 What is the cube-root of 34328125 ? 3 What is the cube-root of 84504519? 4 What is the cube-root of 259594072 ? What is the cube-root of 22069810125? 6 What is the cube-root of 673373097125? What is the cube-root of 12,977875? 8 What is the cube-root of ,001906624? 9 What is the cube-root of 15926,972504? 10 What is the cube-root of 171,46775406? II What is the difference between half a solid foot, and a solid half foot? 2,35 ,124 25,16+ 5.555+ answer 3 half feet. 12 In a cubical foot, how many cubes of 6 inches, and how many of three, are contained therein ? answer 8 of 6in. and 64 of 3in. 13 The content of an oblong cellar is 1953,125 cubic feet; required the side of a cubical cellar that shall contain just as much? answer 12,5 feet. 14 A stone of a cubic form contains 474552 solid inches; what is the superficial content of one of its sides? answer 6084 inches: 15 A merchant laid out 6911 4s. in cloths, but forgot the number of pieces purchased, also how many yards were in each piece, and what they cost him per yard; but remembers, that they cost him as many shillings per yard as there were yards in each piece, and that there was just as many pieces; query the number purchased? answer 24 Note I. The cube root of a vulgar fraction is found by reducing it to its lowest terms and extracting the root of the numerator for a numerator, and of the denominator for a denominator. If it be a furd, extract the root of its equivalent decimal. 2. A mixt number may be reduced to an improper fraction, or a decimal, and the root thereof extracted. 16 What is the cube-root of 32 ? ans. 763 949+ 2,3908+ 1,966+ 2,092+ GENERAL F GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. IRST, if the index of the power be even, extract the square-root of the given number; whereby it will be depressed to a power half as high; or, if the index will divide by 3 without remainder, take the cube root for a power as high; thus proceed till the required root be obtained, or an odd power result, the index of which will not divide evenly by 3. II. The root of fuch an odd power may be extracted thus: First, Beginning at units, point the given number into periods of as many figures each as are expressed by its index. Secondly, Find such a figure or figures, by the table of powers or by trial, as will be nearest the first of the root, whether greater or less. Thirdly. Involve the part of the root so found to the power, and take the difference between this power and as many periods of the given number as there are figures obtained of the root, and multiply this difference by the faid figures for a dividend, Fourth'y, Multiply the sum of the same periods and pow. er by the integral half of the index (i. e. for a 5th power, by 2, a 7th. by 3, &c and to the product add the said power for a divifor. Fifthly, Apply the quotient, as a correction to the part of the root before found, by addition or subtraction, accordingly as that part is less or more than just. Sixthly Repeat the operation, if greater accuracy, or more figures in the root be desired; using the root so corrected instead of the fire or figures first found, &c. 2 What is the cube root of ? 3 What is the fourth root of 97,41 ? answer 7937005 A 3,1415999 5,254037 answer 32017 5 What is the seventh root of 34487717467307513182 492153794673 ? 6 What is the eighth root of 11210162813204762362464 97942460481 ? 7 What is the ninth root of 9763796029890739602796 30298890 ? 8 What is the 365th root of 1.05? answer 13527 answer 2148,7201 1.0001336 ARITHMETICAL PROGRESSION. RITHMETICAL Progression is a rank, or series of numbers, which increase or decrease by a common difference, in which five particulars are to be observed, viz. First, The first term; Secondly, The common excess, or difference; Fourthly, The number of terms; Fifthly, The sum of all the terms. Note. In any series of numbers in arithmetical progression the sum of the two extremes will le equal to the sum of any tavo terms equally distant therefrom: as, 2, 4, 6, 8, 10, 12; where 2+12=14; so 4+10=14; and 6+8=14; or 3, 6, 9, 12, 15; where 3+15=18; also 6+12=18; and 9+9=18. CASE 1. The first term, common difference, and number of terms given, to find the last term, and sum of all the terms; RULE. First, Multiply the number of terms, less 1, by the common difference, and to that product add the first term, the sum is the last term... Secondly, Multiply the sum of the two extremes by the number of terms, aud half the product will be the sum of the series. EXAMPLES. |