in each of its three dimensions; hence, 23=8, is the solidity of that corner; and this exactly disposes of the remaining timber. The cube completed is seen at fig. 5. The contents of the different parts of the completed block are, fig. 2, 8000+; fig. 3, 2400+; fig. 4, 240+; fig 5, 8=10648 feet. Note 1st.-In Square Root we were directed to point off the given number into periods of two figures each; and in Cube Root, the direction is, to allow three figures to each period. The following is the reason the square of any number always consists of twice as many figures as the number itself, or one less than twice as many. The cube of any number always consists of three times as many figures as the number itself, or one or two less than three times as many. That is, in Square Root, the left hand period may consist of one or two figures; and in Cube Root, the same period may consist of one, two, or three figures. Illustration.—132-169, one less than twice the number of figures squared, and to extract its root it would be thus pointed, 169. 462-2116, twice the number of figures squared. 133— 2197, two less than three times the figures cubed, and the left hand period consists of one figure only. 25°-15625, one less than three times the figures cubed, and the left hand period consists of two figures. 993-970299, three times the number of figures cubed, and the left hand period consists of three figures. In the following solution, the scholar will carefully compare each step of the operation with the rule. The first thing to be done, is to form the periods, (Art. 1st, Rule.) The first figure of the root is then to be determined, its cube subtracted, and to the remainder, the next period of three figures to be brought down. (Art. 2d, Rule.) He must then proceed to obtain a divisor, as directed by Art. 3d, and, lastly, to determine the solidity of the several additions made, and to make the result a subtrahend. (Art. 4th.) Ex. 2. What is the cube root of 12812904? It is obvious from the first division by 12, that the divisor is not contained in the dividend, in all instances, as many times as it would be in simple division. In dividing it must be remembered to omit the first two figures on the left hand of the dividend. 3. What is the cube root of 250047? Ans. 63. Proof, 633 =250047. 4. What is the cube root of 970299? Ans. 99. Proof as before. Ans. 1.25. Ans. 2805. Ans. 568. 5. What is the cube root of 1.953125? Ans. 4.38. 10. What is the cube root of 205379? Ans. 59. 11. What is the cube root of 432081216? Ans. 756. 11. There is a cubic rock containing 8000 solid feet. What is the distance from corner to corner? Ans. 20. 13. What is the difference between half of a solid foot, and a solid half foot? Ans. 3 solid half feet. 14. What is the superficial area of one of the faces of a cubic block containing 4096 solid feet? Ans. 256 square feet. 15. What is the side of a cubical mound, equal to one, 144 feet long, 108 feet broad, and 24 feet deep? Ans. 72 feet. Multiply together the several dimensions of the given mound, and extract the cube root of their product. Note 2d. All solid bodies are to each as the cubes of their similar sides or diameters. 16. If a ball weighing 8 lb. be 6 inches in diameter, what will be the diameter of another ball of the same material, weighing 64 lb.? 3 8:64: 63: 1728 1728 Ans. 12 inches. = 17. If a ball 6 inches in diameter, weigh 8 lb. what is the weight of another ball of the same kind, measuring 12 inches in diameter ? Ans. 64 lb. 63: 123::8:64. 18. What would be the value of a globe of silver, one foot in diameter, if a globe of the same, one inch in diameter, be worth $6? Ans. $10368. 19. If a globe of silver, one inch in diameter, be worth $6, what is the diameter of another globe of the same metal, worth $10368? Ans. 12 inches. 20. How many globes one foot in diameter, would be required to make one globe 27 feet in diameter? Ans. 19683. 21. Suppose the diameter of the sun to be 110 times as large as that of the earth; how many bodies like the earth would be required to make one as large as the sun? Ans. 1331000. 22. If a man dig a square cellar, that will measure 5 feet each way, in one day, how long would it take him to dig one measuring 15 feet each way? Ans. 27 days. QUESTIONS.-What is a cube? How is the area of each face of a cubic body found? How is the content of a cubic body found? What is the extraction of the cube root? How is the number whose root is to be extracted, to be pointed? Of which period is the root first found? What is then done with this root? How many figures are to be brought down to what remains? How is a divisor found? By what do you multiply the divisor? How many cyphers do you place on the right of the product? Where do you place the product? What farther is done with the quotient or root figure? What does the sum of all these products form? What is the fifth step of the rule? How is a second divisor obtained? What is Note 1st? Can we know of how many figures the root will consist? Of what is the root figure the linear measure? How much of the whole timber is disposed of by subtracting the cube of the quotient figure from the left hand period, in the operation taken for explanation? How many feet remain to be added? To how many sides of a cube must equal additions be made to preserve its cubic form? Why is three times the square of the root taken for a divisor? What is determined by dividing? How many solid feet are disposed of by the first addition made to the three faces of the cube? Explain how the solid content of the addition to each face is obtained. Why, in multiplying the divisor by the root figure, are two cyphers placed on the right of the product? How is the deficiency of the divisor made up in dividing? What operation as directed by the rule fills up the corners left vacant ? Why is the square of the quotient figure multiplied by the previous root figures and a cypher placed on the right hand of the product? How much of the remaining timber is required to fill the three vacant corners? What is the measure of the corner left vacant, after the preceding additions were made? How is it filled? Why in Square Root do we divide the given number into periods of two figures each? Why is the given number divided into periods of three figures each, in Cube Root? Of how many figures does the square of any number consist? Of how many does the cube? How are the roots proved? Ans. By raising the root to a power of the same name as the root itself. ARITHMETICAL PROGRESSION. Any series of numbers more than two, increasing or decreasing by a constant and uniform difference, is called Arithmetical Progression, or Arithmetical Series. The series formed by a continual addition of any number, (called the common difference,) is called the ascending series. Thus, 3, 5, 7, 9, 11, 13, &c. is an ascending series, of which the common difference is 2. The reverse of this forms the descending series; that is, a series decreasing by a continual subtraction of the common difference. Thus, 13, 11, 9, 7, 5, 3, &c. is a descending series. The numbers constituting the series are called terms. The first and last term of the series are called extremes; all the intervening terms are called the means, and the number constantly added or subtracted, is called the common difference. When the first term and common difference are given, the series may be indefinitely extended. In every arithmetical progression, five particulars are to be noticed; of which, if any three be given, the remaining two may be found. The five particulars are, the first term, the last term, the common difference, the number of terms, and the sum of all the terms. CASE 1st.-THE FIRST TERM, THE NUMBER OF TERMS, AND THE LAST TERM GIVEN, TO FIND THE COMMON DIFFERENCE. Ex. 1. The first term of an arithmetical series is 2; the number of terms, 13; and the last term, 38. What is the common difference? The series commences with 2, as the first term; therefore, 38-2-36, is the amount of all the additions made to this number. But the whole number of terms being 13, and one of these being given, 12 terms must be formed by adding the common difference. Therefore, 36÷12=3, the common difference required. The whole series, therefore, is, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38; thirteen in number. We have then the following rule for solving sums like the preceding: RULE.-Divide the difference of the extremes by the number of terms, less one. The quotient will be the common difference. 2. A man in feeble health, commenced a journey and traveled 9 days. On the first day he traveled only 3 miles, but afterwards continued to gain each day an equal number of miles on the journey of the preceding day, till the last day, on which he traveled 43 miles. The daily increase is required? Ans. 5 miles per day; that is, 43-3÷8=5. 3. I owe a debt which by agreement I am to pay at 17 different periods. The first payment is to be $20, and the last, $100. Required the common difference of the several payments. Ans. $5. What 4. A man had 10 sons whose ages differed alike; the youngest of whom was 2 years old, and the oldest 29. was the difference of their ages? Ans. 3 years. CASE 2d.-THE FIRST TERM, THE COMMON DIFFERENCE, AND THE LAST TERM GIVEN, TO FIND THE NUMBER OF TERMS. Ex. 1. If the extremes be 3 and 51, and the common difference 6, what is the number of terms? |