Find how many times the divisor is contained in the dividend, and place the result in the root (quotient); then multiply the divisor by this quotient figure, placing the product under the dividend. Multiply the former quotient figure, or figures, by 30, and this product by the square of the last quotient figure, and place the product under the last; under these two products place the cube of the last quotient figure, and call their amount the subtrahend. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, and so on, until the whole is finished. Note 1. When the subtrahend happens to be larger than the dividend, the quotient figure must be made one less, and we must find a new subtrahend. The reason why the quotient figure will be sometimes too large, is, because this quotient figure inerely shows the width of the three first additions to the original cube; consequently, when the subsequent additions are made, the width (quotient figure) may make the solid contents of all the additions more than the cubic feet in the dividend, which remain after the solid contents of the original cube are deducted. 2. When we have a remainder, after all the periods are brought down, we may continue the operation by annexing periods of ciphers, as in the square root. 3. When it happens that the diviser is not contained in the dividend, a cipher must be written in the quotient (root), and a new dividend formed by bringing down the next period in the given sum. A. 26. A. 83. 9. What is the cube root of 17576? 10. What is the cube root of 571787? 11. What is the cube root of 970299? A. 99. 12. What is the cube root of 2000376? A. 126 13. What is the cube root of 3796416? A. 156. 14. What is the cube root of 94818816? A. 456. 15. What is the cube root of 175616000? A. 560. 16. What is the cube root of 748613312? A. 908. 17. What is the cube root of 731189187729? A. 9009. 18. What is the cube root of ? .. If the root be a surd, reduce it to a decimal before its root is extracted, a the square root. 21. What is the cube root of? A. ‚13+. 22. What is the cube root of? A.,34+· 23. What is the length of one side of a cubical block, which contains 1728 solid or cubic inches? A. 12. 24. What will be the length of one side of a cubical block, whose contents dhall be equal to another block 32 feet long, 16 feet wide, and 8 feet thick? 3 32 X 16 X 816 feet, Ans. 25. There is a cellar dug, which is 16 feet long, 12 feet wide, and 12 feet deep; and another, 63 feet long, 8 feet wide, and 7 feet deep; how many solid or cubic feet of earth were thrown out? and what will be the length of one side of a cubical mound which may be formed from said earth? A. 5832. 18. 26. How many solid inches in a cubical block which measures 1 inch on each side? How many in one measuring 2 inches on each side? 3 inches on each side? 4 inches on each side? 6 inches on each side? 10 inches on each side? 20 inches on each side? A. 1. 8. 27. 64. 216. 1000. 8000. 27. What is the length of one side of a cubical block, which contains I solid or cubic inch? 8 solid inches? 27 solid inches? 64 solid inches? 125 solid inches? 216 solid inches? 1000 solid inches? 8000 solid inches? A. 1. 2. 3. 4. 5. 6. 10. 20. By the two preceding examples, we see that the sides of the cube are as the cube roots of their solid contents, and their solid contents as the cubes of their sides. It is likewise true, that the solid contents of all similar figures are in proportion to each other as the cubes of their several sides or diameters. Note. The relative length of the sides of cubes, when compared with their solid contents, will be best illustrated by reference to the cubical blocks accompanying this work. 28. If a ball, 3 inches in diameter, weigh 4 pounds, what will a ball of the same metal weigh, whose diameter is 6 inches? 33 63 4: 32: Ratio, 23 x 432 lbs., Ans. 29. If a globe of silver, 3 inches in diameter, be worth $160, what is the value of one 6 inches in diameter ? 33: 63: $160: $1280, Ans. 30. There are two little globes; one of them is 1 inch in diameter, and the other 2 inches; how many of the smaller globes will make one of the larger? Ꭿ, 8. 31. If the diameter of the planet Jupiter is 12 times as much as the diameter of the earth, how many globes of the earth would it take to make one as large as Jupiter? A. 1728. 32. It the sun is 1000000 times as large as the earth, and the earth is 8000 miles in diameter, what is the diameter of the sun? A. 800000 miles. Note. The roots of most powers may be found by the square and cube roots only; thus the square root of the square root is the biquadrate, or fourth root; and the sixth root is the cube of this square root. ARITHMETICAL PROGRESSION. ↑ LXXXVIII. Any rank or series of numbers more than 2, increasing by a constant addition, or decreasing by a constant subtraction of some given number, is called an Arithmetical Series, or Progression. The number which is added or subtracted continually is called the common difference. When the series is formed by a continual addition of the common difference, it is called an ascending series; thus, 2, 4, 6, 8, 10, &c., is an ascending arithmetical series; but 10, 8, 6, 4, 2, &c., is called a descending arithmetical series, because it is formed by a continual subtraction of the com mon difference, 2. The numbers which form the series are called the terms of the series or progression. The first and last terms are called the extremes, and the other terms the means. In Arithmetical Progression there are reckoned 5 terms, any three of which being given, the remaining two may be found, viz. 1. The first term. 2. The last term. 3. The number of terms. The First Term, the Last Term, and the Number of Terms, being given, to find the Common Difference ;— 1. A man had 6 sons, whose several ages differed alike; the youngest vs 3 years old, and the oldest 28; what was the common difference of their ages. The difference between the youngest son and the eldest evidently shows the increase of the 3 years by all the subsequent additions, till we come to 28 years; and, as the number of these additions are, of course, 1 less tha.. the number of sons (5), it follows, that, if we divide the whole difference (28 −3 =), 25, by the number of additions (5), we shall have the difference between each one separately, that is, the common difference. Thus, 28-3=25; then, 25÷55 years, the common difference. Hence, To find the Common Difference; Divide the difference of the extremes by the number of terms, less 1, and the quotient will be the common auference 2. If the extremes be 3 and 23, and the number of terms 11, wha, is the com mou difference? A. 2. 3. A man is to travel from Boston to a certain place in 6 days, and to go only 5 miles the first day, increasing the distance travelled each day by an equal excess, so that the last day's journey may be 45 miles; what is the daily inerease, that is, the common differeuce? A. 8 miles. 4. If the amount of $1 for 20 years, at simple interest, be $2,20, what is the rate per cent.? In this example, we see the amount of the first year is $1,06, and the last year $2,20; consequently, the extremes are 106 and 220, and the number of terms 20. A. $,066 per cent. 5. A man bought 60 yards of cloth, giving 5 cents for the first yard, 7 for the second, 9 for the third, and so on to the last; what did the last cost? Since, in this example, we have the common difference given, it will be easy to find the price of the last yard; for, as there are as many additions as there are vards, less 1, that is, 59 additions of 2 cents, to be made to the first yard, it follows, that the last yard will cost 2 x 59 118 cents more than the first, and the whole cost of the last, reckoning the cost of the first yard, will be 118 +5= $1,23. A. $1,23. Hence, When the Common Difference, the First Term, and the Number of Terms, are given, to find the Last Term ;— Multiply the common difference by the number of terms, less 1, and add the first term to the product. 6. If the first term be 3, the common difference 2, and the number of terms 11, what is the last term? A. 23. 7. A man, in travelling from Boston to a certain place in 6 days, travelled the first day 5 miles, the second 8 miles, travelling each successive day 3 miles farther than the former; what was the distance travelled the last day? A. 20. 8. What will $1, at 6 per cent., amount to, in 20 years, at simple interest? The common difference is the 6 per cent.; for the amount of $1, for 1 year, is $1,06, and $1,06+$,06 = $1,12, the second year, and so on. A. $2,20. 9. A man bought 10 yards of cloth, in arithmetical progression; for the first yard he gave 6 cents, and for the last yard he gave 24 cents; what was the amount of the whole? In this example, it is plain that half the cost of the first and last yards will be the average price of the whole number of yards; thus, 6 cts. +24 cts. =30÷ 215 cts., average price; then, 10 yds. X 15 150 cts. $1,50, whole cost. A. $1,50. ence, When the Extremes, and the Number of Terms, are given, to find the Sum of all the Terms; Multiply half the sum of the extremes by the number of terms, and the product will be the answer. 10. If the extremes be 3 and 273, and the number of terms 40, what is the sum of all the terms? A. 5520. 11. How many times does a regular clock strike in 12 hours? A. 78. 12. A butcher bought 100 oxen, and gave for the first ox $1, for the second $2, for the third $3, and so on to the last; how much did they come to at that ate? A. $5050. 13. What is the sum of the first 1000 numbers, beginning with their natural order, 1, 2, 3, &c.? A. 500500. 14. It'a board, 18 feet long, be 2 feet wide at one end, and come to a point at the other, what are the square contents of the board? A. 18 feet. 15. If a piece of land, 60 rods in length, be 20 rods wide at one end, and at the other terminate in an angle or point, what number of square rods does it contain? A. 600. 16. A person, travelling into the country, went 3 miles the first day, and increased every day's travel 5 miles, till at last he went 58 miles in one day; how many days did he travel? We found, in example 1, the difference of the extremes, divided by the number of terms, less 1, gave the common difference; consequently, if, in this example, we divide (58-3=) 55, the difference of the extremes, by the common difference, 5, the quotient, 11, will be the number of terms, less 1; then, 1+ 12, the number of terms. A. 12. Hence, When the Extremes and Common Difference are given, to find the Number of Terms ;— Divide the difference of the extremes by the common difference, and the quotient, increased by 1, will be the answer. 17. If the extremes be 3 and 45, and the common difference 6, what is the number of terms? A. 8. 18. A man, being asked how many children he had, replied, that the youngest was 4 years old, and the eldest 32, the increase of the family having been 1 in every 4 years; how many had he? A. 8. GEOMETRICAL PROGRESSION. ¶ LXXXIX. Any rank or series of numbers, increasing by a constant multiplier, or decreasing by a constant divisor, is called Geometrical Progression. Thus, 3, 9, 27, 81, &c., is an increasing geometrical series; And 81, 27, 9, 3, &c., is a decreasing geometrical series. There are five terms in Geometrical Progression; and, like Arithmetical Progression, any three of them being given, the other two may be found, viz. 1. The first term. 2. The last term. 3. The number of terms. 4. The sum of all the terms. 5. The ratio; that is, the multiplier or divisor, by which we form the series. 1. A man purchased a flock of sheep, consisting of 9; and, by agreem was to pay what the last sheep came to, at the rate of $4 for the a $12 for the second, $36 for the third, and so on, trebling the price" the lasf what did the flock cost him? |