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 greatest root 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, Reserve 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 cisely 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. be more con. EXAMPLES, 1 What is the cube root of 444194,947 ? 444194,947)76,3 ans. . 543 {71260 Defec. div. & sqr. of 6 14736)101194 +1260 = complete divisor 159976) 95975 { 5 Defec. div. & sqr. of S= 1782809) 5218947 +6840 = complete divisor 1739649 5218947 151 answer 325 499 638 2 What is the cube-root of 34328125 ? 2805 6. What is the cube-root of 673373097125 ? 8765 7. What is the cube-root of 12,977875 ? 2,35 8 What is the cube-root of ,001906624? ,124 9 What is the cube-root of 15926,972504? 25,16+ 10 What is the cube-root of 171,46677406 ? . -5,555+ 11 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 thereein ? answer 8 of 6in, and 64 of sin, 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 ? answem 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 1. The cube root of a vulgar fraction is found by reducing it to its low est terms an: extracting the root of the numerator for a numerator, and of the denominator for a denominator. If it be a surd, extract tbe root of its equivalent decimal. 2. A mixt imber may be reduced to an improper fraction, or a decimal, and the root thereof extracted. 16 What is the cube-roof of 354 ? answer 17 What is the cube-root of 648? 18. What is the cube-root of ? ,763 19 What is the cube-root of ? ,949+ 20 What is the cube-root of 13ş? 2,3908+ 21. What is the cube-root of 4241 ? 33 22 What is the cube-root of 5197 ? 23 What is the cube-root of 40528? 24 What is the cube-root of 7} 1,966+ 25 What is the cube-root of 91 ? 2,092 + GENERAL 123 GENERAL RULE FOR EXTRAOTING The ROOTS OF ALL POWERS. PIRST, 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 such 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 inany periods of the given number as there are figures obtained of the root, and multiply this difference by the said figures for a dividend. Fourthly, Multiply the suin of the same periods and power 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 divisor. 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 yreater accuracy, or more figures in the root be desired; using the root so corrected instead of the figure or figures first found, &c. EXAMPLES 1,24618 ,045 1,0 Divide 5),49|236) '2461180(9045 ,2197 2 What is the cube-root of i? answer ,7937005 3 What is the fourth root of 97,41 ? 3,1415999 4 What is the sixth root of 21035,81 5,254037 5. What is the seventh root of 34487717467507513182 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. 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. the sun of the two extremes will be equal to the sum of any two 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, and half the product will be the sum of the series. EXAMPLES. EXAMPLES 1 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? 19-13 18 1 +37=38 2 19 number of terms. f. 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 gsve 5s 5d. sum received 21 6s 8d. 3 A merchant'sold 100 yards of cloth ; for the first yard he received 1s. for the second 2s. for the third 3s. &c. what sum did he receive ? answer 2521 10s. 4 Admit 100 stones were laid two yards distant from each other in a rightline, and a basket placed two yards from the first stone; what distance musta person travel, to gather them singly into the basket ? answer 11 M. 3fur. 180yds. 5 Sold 54 yards of cloth ; the price of the first yard was 2s. of the second 5s. &c. what was the price of the last yard, and sum of all ? S the last yd. 81 1s. answer whole sum 2201 is. 6 H covenanted with K to serve him 14 years, and to have 5l. the first year, and his wages to increase annually 21. during the term, what had hè the last year, what on an average yearly, and what for the whole tiine ? 31l. the last year. answer 181. annually. 2521. whole time. CASE { |