APPLICATION. 36. What is the difference between the cube root of 729 and the square root of the same number? Ans. 37. What is the difference between the cube root of 1331 and the square root of 121? Ans. NOTE. The cube root of any number which is a complete cube, may be found by a very simple mental process; when the root does not exceed two figures, thus in the cube of 1, 4, 6, and 9, the right-hand figure in the cube will be the same as the root. But 3 in the cube and 7 in the root reciprocate, and the same is true of 2 and 8. Hence if the units figure of the cube be 1, 4, 6, or 9, the units figure of the root will be the same; but if the units figure of the cube be 3, that of the root will be 7, and if the units figure of the cube be 7, that of the root will be 3, and 2 and 8 have the same property. Here the root of 38. What is the cube root of 103.823? the greatest cube in the left-hand period is 4, and the units figure of the cube is 3, therefore the units figure of the root must be 7, hence the root is 47. 39. What is the cube root of 389,017, of 132,651, of 658,503, of 970,299, of 2197, of 15,625, of 9261, of 531,441, of 175,616, of 148,877, of 238,328, of 405,224, of 24,389, of 97,336, of 185,193, of 17,576, of 474,552, of 91,125, of 195,112, of 912,673 ? 40. The contents of a cubical cellar is 21,952, what are the length, breadth, and depth? Ans. 28 feet. 41. A certain stone of a cubical form contains 474,552 solid inches, what is the superficial contents of one of its sides? Ans. 6084. 42. Required the side of a cubical vessel that shall contain 80 gallons, each. 231 cubic inches. Ans. 26.43+in. All solid bodies have the same ratio to each other, as the cubes of their diameters, or similar sides. 43. If a globe of gold one inch in diameter be worth $920, what is the value of a globe 3 inches in diameter ? Ans. $5145. 44. There are two marble statues of the same form but differing in size, one is 5 feet high and weighs 750 pounds, the other is 7 feet high; what will be its weight? Ans. 2058 lbs. 46. If a ball 3 inches in diameter weighs 4 pounds, what will the weight of another ball of the same metal 9 inches in diameter be ? Ans. 108 lbs. lb. 9 Ans. $108 lbs. 47. If a ball 4 inches is diameter weighs 9 pounds, what is the diameter of a ball weighing 72 pounds? Ans. 8 inches. lb. 48-64 #p 512-8 Ans. 48. There are two stacks of hay of precisely the same shape, one is 12 feet high, and contains 7 tons, the other is 184 feet high; how many tons does it contain? 16 Ans. 25,5 tons. 49. If a man can dig a cellar, which will measure 8 feet every way, in 2 days, how long time should he require to dig a similar cellar 12 feet every way? Ans. 8,7 days. 16 50. What is the difference between half a solid foot and a solid half foot? Ans. 648 cubic inches. To find two mean proportionals between two given numbers, divide the greater by the less, and find the cube root of the quotient, then multiply this root by the least of the given numbers for the least mean, and the least mean by the same root for the greater mean. 51. What are two mean proportionals between 4 and 108 ? Ans. 12 and 36. 52. What are two mean proportionals between 7 and 448 ? Ans. 28 and 112. 53. What are two mean proportionals between 8 and 216? Ans. 24 and 72. 54. What are two mean proportionals between 2 and 128? Ans. 8 and 32. 55. What are two mean proportionals between 17 and 2125? Ans. 85 and 425. 56. What are two mean proportionals between 56 and 12,096 ? Ans. 336 and 2016. 57. What are two mean proportionals between 76 and 166,972? Ans. 988 and 12,844. 58. The solid contents of a globe 21 inches in diameter, are 4849.0596; what is the diameter of a globe whose solid contents are 11494.0672 solid inches? Ans. 28 inches. 59. In a cubical foot, how many cubes of 3 inches, of 4 inches, of 6 inches, and 2 inches? Ans. 64, 3 inch cubes; 27, 4 inch cubes; 8, 6 inch cubes; and 216, 2 inch cubes. 60. What must be the length of a side of a cubical box to contain just one bushel? Ans. 12.907+inches. 61. What are the inside dimensions of a cubical bin, that will hold 190 bushels of grain? Ans. 6 feet, 2.204+inches. NOTE-The roots of the 4th, 6th, 8th, 9th, and 12th powers may be extracted as follows: For the 4th root take the square root of the square root. For the 9th root take the cube root of the cube root, and for the 12th root take the cube root of the 4th root. In the 5th and 9th powers the units figure will be the same as the units figure of the root. ARITHMETICAL PROGRESSION. A series of numbers which uniformly increase or decrease by the addition or subtraction of the same number, is called an arithmetical progression, or progression by difference; and the number by which the terms increase is called the common difference. The numbers composing 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. When the numbers increase they form an ascending series, but when they decrease, a descending series, or progression. Thus, 1, 2, 3, 4, 5, 6, 7, &c., form an ascending series, and 7, 6, 5, 4, 3, &c., form a descending series. In any series of numbers in arithmetical progression, the sum of the two extremes is equal to the sum of any two terms equally distant from the extremes, or to twice the middle term, when the number of terms is odd. Thus in the series 1, 2, 3, 4, 5, 6, 7, 8, 9, &c., the sum of the extremes 1 and 9, is equal to the sum of 2 and 8, or 3 and 7, or 4 and 6, or twice 5, the middle term. The most important application of the principle of progression, is the finding of the sum of the series. NOTE. A curious property of odd numbers, and one that is not generally known, (nor has it ever that we are aware of, been applied to any practical purpose,) will be of considerable service, by way of enabling us to find the sum of any series in arithmetical progression. It is this, the sum of any series of odd numbers, commencing with 1, gives the square of the number of terms in that series. Thus, 1 and 3 are 4-the square of 2-(the number of terms,)—1, 3, and 5 are 9-the square of 3-(the number of terms,)—1, 3, 5, and 7 are 16-the square of 4(the number of terms,)—1, 3, 5, 7, and 9 are 25, the square of 5(the number of terms, &c.) Hence if we wish to find the sum, say of 13 terms of such a series, we have only to recollect that the square of 13 is 169, and we have the sum required. What is the sum of 18 terms of the series, 1, 3, 5, &c., of 14 terms, of 17 terms, of 24 terms, of 35 terms, of 47 terms, of 29 terms, of 39 terms, of 34 terms, of 15 terms, of 45 terms, of 57 terms, of 75 terms? Take now a series of numbers, commencing with 2, instead of 1, (increasing by the same common difference,) and compare the result. Thus, 2+4+6+8+10+12=42=sum of the series. 21+3+5+7+9+11=36=sum of the series. 6 differ. of the sums. equal to the number of terms in either series. Again, take a series, commencing with 3, instead of 1, increasing by the same common difference, and compare the results. Thus, S3+5+7+9+11+13+15=63=sum of series. 21+3+5+7+9+11+13=49-sum of series. 14 difference is equal to twice the number of terms in either series. Now observe that when the series commences with 2, instead of 1, the sum of any two terms in this series, will exceed either of those terms by one more than the excess of the sum of the corresponding terms in the first series above either of the terms added; and when the first term is 3, instead of 1, the sum of any two terms in this series will exceed either of those terms by 2 more than the excess of the sum of the corresponding terms in the first series above either of the terms added; and if the first term were 4, this difference of increase would be 3, and if the first term were 5, it would be 4, &c. Hence the sum of any number of terms of a series whose common difference is 2, will exceed the sum of an equal number of terms of the natural series of odd numbers, by as many times the number of terms in the series as the first term of the given series is greater than 1; or since the sum of the natural series of odd numbers is equal to the square of the number of terms, the sum of any other series whose common difference is 2, will exceed the square of the number of terms, by as many times the number of terms as the first term is greater than 1. From the above illustrations we deduce the following rule for finding the sum of any number of terms of an ascending series in arithmetical progression. RULE. To the square of the number of terms add the number of terms for the sum of the series, when the first term is 2, and the common difference the same; but when the common difference exceeds 2, increase the sum thus found in the ratio of that excess, and to or from the result add or subtract, as many times the number of terms, as the first term |