its root 2, which we place in the quotient or root. As 2 is, in the place of tens, because there is to be another figure in the root, its value is 20, and it represents the side of a cube (Fig. 1), the contents of which are 8000 cubic feet; thus 20 X 20 X 20 = 8000. Fig. 1. 20 20 We now subtract the cube of 2 (tens) = 8 (thousands) from the first period, 17 (thousands), and have 9 (thousand) feet remaining, which, being increased by the next period, makes 9576 cubic feet. This must be added to three sides of the cube, Fig. 1, in order that it may remain a cube. To do this, we must find the superficial contents of the three sides of the cube, to which the additions are to be made. Now, since one side is 2 (tens) or 20 feet square, its superficial contents will be 20 X 20 400 square feet, and this multiplied by 3 will be the superficial contents of three sides; thus, 20 X 20 X 3 = 1200, or, which is the same thing, we multiply the square of the quotient figure, or root, by 300; thus, 22 X 300 1200 square feet. Making this number a divisor, 20 Fig. 2. 20 = we divide the dividend 9576 by it, and obtain 6 for the quotient, which we place in the root. This 6 represents the thickness of each of the three additions to be made to the cube, and their superficial contents being multiplied by it we 20 have 1200 X 6 = = 7200 cubic feet for the contents of the three additions, A, B and C, as seen in Fig. 2. Having made these additions to the cube, we find that there are three other deficiencies, n n, o o, and rr, the length of which is equal to one side of the additions, 2 (tens) or 20 feet; and their breadth and thickness, 6 feet, equal to the thickness of the additions. Therefore, to find the solid contents of the additions, necessary to supply these deficiencies, we multiply the product of their length, breadth and thickness by the number of additions; thus, 6 X 6 X 20 X 3=2160, or, which is the same thing, we multiply the square of the last quotient figure by the former figure of the root, and that product by 30; thus, 62 X 2 QUESTIONS. - What is done with this greatest cube number, and what part of Fig. 1 does it represent? What is done with the root? What is its value, and what part of the figure does it represent? How are the cubical contents of the figure found? What constitutes the remainder after subtracting the cube number from the left hand period? To how many sides of the cube must this remainder be added? Why? How do you find the divisor? What parts of the figure does it represent? How do you obtain the last figure of the root? What part of Fig. 2 does it represent? Why do you multiply the divisor by the last quotient figure? What parts of the figure does the product represent? What three other deficiencies in the figure? X 30 2160 cubic feet for the contents of the additions ss, u u, and vv, as seen in Fig. 3. Fig. 3. These additions being made to the cube, we still observe another deficiency of the cubical space x x x, the length, breadth, and thickness of which are each equal to the thickness of the other additions, which is 6 feet. Therefore, we find the contents of the addition necessary to supply this deficiency by multiplying its length, breadth, and thickness together, or cubing the last figure of the root; thus, 6 X 6 X 6 216 cubic feet for the contents of the addition z z z, as seen in Fig. 4. 26 26 RULE. Fig. 4 = The cube is now complete, and if we add together the several additions that have been made to it, thus, 7200+ 2160 +2169576, we obtain the number of cubic feet remaining after subtracting the first cube, which, being subtracted from 26 the dividend in the operation, leaves no remainder. Hence, the cubical pile formed is 26 feet on each side; since 26 X 26 X 26 = 17576, the given number of blocks, and the sum of the several parts of Fig. 4. Thus, 8000 + 7200 2160 216 =17576. Hence the following 1. Separate the given number into periods of three figures each, by placing a point over the unit figure, and every third figure beyond the place of units. 2. Find by the table the greatest cube in the left hand period, and put its root in the quotient. 3. Subtract the cube, thus found, from this period, and to the remainder bring down the next period; call this the dividend. 4. Multiply the square of the quotient by 300 for a divisor, by which divide the dividend and place the quotient, usually diminished by one or two units, for the next figure of the root. 5. Multiply the divisor by this last quotient figure, and write the product under the dividend; then multiply the square of the last quotient figure by the former quotient figure or figures, and this product by 30, and place the product under the last; under all, set the cube of the last quotient figure, and call their sum the subtrahend. QUESTIONS.-How do you find their contents? What parts of Fig. 3 does the product represent? What other deficiency do you observe? To what are its length, breadth and thickness equal? How do you find its contents? What part of Fig. 4 does it represent? What is the rule for extracting the cube root? 6. 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, till the whole is completed. NOTE. The observations made in Notes 1, 2, and 3, under square root, are equally applicable to the cube root, except in pointing off decimals each period must contain three figures, and two ciphers must be placed at the right of the divisor when it is not contained in the dividend. 43147522d subtrahend. 2. What is the cube root of 74088 ? 11. What is the cube root of 8.144865728 ? 12. What is the cube root of .075686967? 13. What is the cube root of 25? QUESTION. How many ciphers must be placed at the right of the divisor when it is not contained in the dividend? ART. 282. When it is required to extract the cube root of a vulgar fraction, or a mixed number, it is prepared in the same manner as directed in square root. (Art. 269.) ART. 283. THE cube root may be applied in finding the dimensions and contents of cubes and other solids. 1. A carpenter wishes to make a cubical cistern that shall contain 2744 cubic feet of water; what must be the length of one of its sides? 2. A farmer has a cubical box that will hold 400 bushels of grain; what is the height of the box? 3. There is a cellar, the length of which is 18 feet, the width 15 feet, and the depth 10 feet; what would be the depth of another cellar of the same size, having the length, width, and depth equal? ART. 284. A SPHERE is a solid bounded by one continued convex surface, every part of which is equally distant from a point within, called the centre. A B The diameter of a sphere is a straight line passing through the centre, and terminated by the surface; as A B. ART. 285. A CONE is a solid having a circle for its base, and its top terminated in a point, called the vertex. QUESTIONS. Art. 282. How is a vulgar fraction or a mixed number prepared for extracting the square root? Art. 283. To what may the cube root be applied? - Art. 284. What is a sphere? What is the diameter of a sphere? Art. 285. What is a cone? C A The altitude of a cone is its perpendicular height, or a line drawn from the vertex perpendicular to the plane of the base; as B C. ART. 286. Spheres are to each other as the cubes of their diameters, or of their circumferences. Similar cones are to each other as the cubes of their altitudes, or the diameters of their bases. All similar solids are to each other as the cubes of their homologons or corresponding sides, or of their diameters. ART. 287. To find the contents of any solid which is similar to a given solid. RULE.-State the question as in Proportion, and cube the given sides, diameters, altitudes, or circumferences, and the fourth term of the proportion is the required answer. ART. 288. To find the side, diameter, circumference, or altitude of any solid, which is similar to a given solid. RULE. · State the question as in Proportion, and cube the given sides, diameters, circumferences, or altitudes, and the cube root of the fourth term of the proportion is the required answer. · EXAMPLES FOR PRACTICE. 1. If a cone 2 feet in height contains 456 cubic feet, what are the contents of a similar cone, the altitude of which is 3 feet? Ans. 1539 cubic feet. OPERATION. 23456:1539. 2. If a cubic piece of metal, the side of which is 2 feet, is worth $6.25; what is another cubical piece of the same kind worth, one side of which is 12 feet? 3. If a ball, 4 inches in diameter, weighs 50lb., what is the weight of a ball 6 inches in diameter ? QUESTIONS. What is the altitude of a cone? Art. 286. What proportion do spheres have to each other? What proportion do cones have to each other? What proportion do all similar solids have to each other? - Art. 287. What is the rule for finding the contents of a solid similar to a given solid? - Art. 288. What is the rule for finding the side, diameter, &c., of a solid similar to a given solid? |