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 respectively. Secondly, Find the great root of the left hand period, which place to the right of the given number, and subtrac 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 di vidual, 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 ro. Fifthly, Complete the divisor, by adding thereto the pro duct 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 concise. ly 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 I What is the cube root of 444194,947 ? 444194,947(76,3 any 343 Defec. div. & sqr. of 6-14736) 101194 Defec. div. & sqr, of 31732809)5218947 2 What is the cube-root of 34328125! What is the cube-root of 22069810125? answer 325 439 638 2805 8765 2,35 ,124 25,16+ to What is the cube-root of 171,46776406? 5.555+ II What is the difference between half a solid foot, and a solid half foot? 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 bin, and 64 of 3in. 13 The content of an oblong cellar is 1953,125 cubie 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 691/ 45. 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 1. 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. Ifit 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. 1,966+ 2,092 GENERAL &c . GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. a square-root of the given number; whereby it will be depressed to a power halt as high ; or if the index will divide by 3 without remainder, take the cube root for a power, j 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 such an odd power may be extracted thus First, Beginning at units, point the given number into per riods 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 ob tained of the roots, and multiply this difference by the said: figures for a dividend. Fourthly, Multiply the sum of the same periods and pow. er by the integral half of the index (ie. for a sth power, by 2, a 7th by 3, &c.) and to the product add the said. power for a divisor.. Fifthly, Apply the quotient, as a correction to the part of the root before foand, 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 figure or figures first found, &c. EX A M P L E S. 1 What is the 5th root of 1,2461819? 1,24618 2 1,00000 ,045 ,24618 1,0 Divide 5:49/236),24611806.045 2197 265 279 1,045 Roots 1,00000 2 What is the cube root of ? answer »7937005 3. What is the fourth root of 97,41 ? 3,1415999 4 What is the sixth root of 21035,8 ? 5,254037 5 What is the seventh root of 34487717467307513182 492153794673? answer 32017 6 What is the eighth root of 11210162813204762362464 97942460481 ? answer 13527 7. What is the ninth root of 9763796029890739602796 30298890 answer 2148,7201 8 What is the 365th root of 1.05? 1.0001336 ARITHMETICAL PROGRESSION. A a 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 ; Fifthly, The sum of all the terms. sum of the two extremes will be equal to the sum of any fuo terms equally distant therefrom: os, 2, 4, 6, 8, 10, 12: where 2+12=14; 1 4+10=14; and 6+8=14; or 3. 6, 9, 12, 15; where 3+15=18 atre 6+12=18; and 9+9=18. CASE 1. The first term, common difference, and number of terras given, to find the last term, and sum of all the terms : RULE. First, Multiply the number of terras, legs 1, by the conte mon difference, and to that produ&t add the first term, the sum is the last term. Secondly, Multiply the sum of the two extremes by the number of terms, and half the product will be the sum of the series. EXAMPLE EXAMPLE S. Bought 19 yards of shalloon, at 1d. for the first yard, 3d. for the second, 5d. for the third, &c. increasing 2d, every yard; what did they amount to? 19-1=18 1+37=38 The last term 37 2)722 12)361 sum of the terms.. 2,0)3,021 £ 1 10 1 answer. 2 Sixteen persons bestowed charity to a poor man; the first gave 5d. the second 9d. and so on in arithmetical progression; what did the last person give; and what sum did the indigent person receive? answer the last gave 55 5d. sum received 21 6s 8d. 3 A merchant sold too yards of cloth; for the first yard he received Is. for the second 25. for the third 35. &c. what sum did he receive? answer 2528 105. 4 Admit roo stones were laid two yards distant from each other in a right line, and a basket placed two yards from the first stone; what distance must a person travel, to gather them singly into the basket? answer 11 M. 3fur. 180yd. 5 Sold 54 yards of cloth; the price of the first yard was 25. of the second s. &c. what was the price of the last yards and sum for all? the last yard 81. 1. answer whole sum 2201, 1s. 6 H covenanted with K to serve him 14 years, and to have 5 the first year, and his wages to encrease annually 21. during the term, what had he the last year, what on an a verage yearly, and what for the whole time? answer 31. the last year 18%. annually. 252/. whole time. CASE |