From the preceding analysis we derive the following RULE. Separate the given number into periods of three figures each, beginning at the right hand. Find the greatest number whose cube is contained in the first left-hand period, for the first figure of the root. Subtract the cube of this figure from the left-hand period, and to the remainder annex the next period for a dividend. For a trial divisor, annex one cipher to the figure of the root just found, square the number thus formed, and multiply it by 3, find how often it is contained in the new dividend, and the quotient will be the second figure of the root. To the trial divisor add 3 times the product of the first figure of the root with a cipher annexed, by the quotient figure just found, and to the result add the square of the figure last found; the sum will be the complete divisor. Multiply the complete divisor by the second figure of the root, and subtract the product from the dividend. If there are any more periods to be brought down, annex the next period to the last remainder, and for a trial divisor annex a cipher to the figures of the root, square the number thus formed, multiply it by 3, and proceed as before. Note 1.-If the trial divisor is not contained in the dividend, write a cipher in the root, annex two ciphers to the trial divisor, and bring down the next period for a new dividend. Note 2.—If there should be a remainder after all the periods have been brought down, annex periods of ciphers, and find the root to any required number of decimal places. Note 3.—If the given number contains a decimal, point off the periods to the right in the decimal, and, when any period is not complete, supply the deficiency with ciphers. 2. What is the cube root of 2 to three decimal places ? 2.000000000(1.259 13 Complete Divisor 364 728 Trial Divisor 1202 x3 = 43200) 272000 120 X3x5= 1800 25 Complete Divisor 45025 225125 Trial Divisor 12502x3 = 4687500) 46875000 1250 x3x9= 33750 = 81 Complete Divisor 4721331 42491979 Remainder 4383021 PROOF.-Cube the root 1.259 and add the remainder; the result should be 2.. 3. What is the cube root of 997002999? 4. Extract the cube root of each of the following numbers: 91125, 67917312, and 78402752. 5. Find the value of 0.8, 3.70, 3.125, and x 7. 6. Find the cube root of 4. ANALYSIS.Changing the fraction to its lowest terms, 1, we find the V1=1, and the 3/8=2; therefore the vž=. PROOF.—1 x x 1 = 5. 7. Extract the cube root of each of the following: 2100, 55, .0065, 21.17, 18192.3145. 8. What is the cube root of 88998.7654? 9. Find the cube root of .0021 to 4 decimal places. 10. What is the cube root of 14 50? 11. If a cubical box holds a bushel of wheat, what is the depth of the box? 12. A can in the form of a cube holds just one gallon; what is the length of one of its sides? 13. What is the depth of a cubical vessel that will hold 10 barrels of water? 14. A water tank will contain 1500 gallons; what must be the length of the side in feet, if the form is that of a cube? 15. What is the length of a cubical pile of wood which contains 15 cords? 16. In excavating a cellar 24 ft. long, 15 ft. wide, and 6 ft. deep, 80 cubic yards of earth were removed; how deep would the cellar have been, if the three dimensions had been equal, the same quantity of earth being removed? 17. Find the value of each of the following: 2:4 :: 864 : what? 5:10 :: V125 : the cube root of what? Note.-The like sides of similar solids are to each other as the cube roots of their solidities. Thus, if the solidities of two cubes are 8 and 64, their sides are a 8 and a 64, or, if the ratio of the solidities is 8 : 64, the ratio of the sides is 08:064. 18. If a cubical box contains 27 bushels, how many times larger must it be to contain 243 bushels? 19. If a coal-bin has a capacity of 64 tons, how many times larger must it be to contain 512 tons ? 20. A box which measures 4 ft. long, 3 ft. wide, and 21 ft. in depth, contains 27 cubic feet; what are the sides of a similar box which contains 216 cubic feet? 21. There are two numbers which are to each other, as the : /343, and the less number is .05; what is the greater number? MENSURATION. Mensuration is the art of measuring lines, surfaces, and solids. In measuring lines, surfaces, etc., surveyors make use of the following tables. LINEAR MEASURE. 4 rods make 1 chain, (ch.) 80 chains make 1 mile, (mi.) Gunter's chain is 4 rods, or 66 feet, in length. Measurements are recorded in chains and hundredths. SQUARE MEASURE, 625 square links (sq. 1.) make 1 pole, (P.) 16 poles make 1 square chain, (sq. ch.) 10 square chains make 1 acre, (A.) 640 acres make 1 square mile, (sq. mi.) 36 square miles make 1 township, (Tp.) For the purpose of facilitating computations of the weight of materials used in the arts and trades, in farming, engineering, etc., the following tables have been prepared. A Table showing the number of pounds in a bushel. A Table showing the weight in pounds of a cubic foot of each substance named therein. SUBSTANCE. WEIGHT. SUBSTANCE WEIGHT. 150 ei Alcohol, pure . .. 491 Lead . . . . 711 6 common. il 52 Lignum-vitæ .. 83 Ale . 197 . . . . . Limestone . . . " broken . 54 Manganese . . 500 Bituminous coal, solid 80 Maple . . . . 1. 47 « « broken 50 Marble . . . . 168 Brick, pressed ... Mercury . . . | 849 a common hard . | 125 Milk . . . . . 641 66 soft 100 Nitre .... | 119 Butter . . . . . 59 Poplar . . . . 24 Cedar . . . . . 35 Quartz . . . . 165 Cherry . . . . . 44 Salt . . . . . 133 Clay · · · 125 131 · Saltpetre .... Copper . . . . . 547 Sand, dry ... 94 Cork . . . . . | 156 moist .. 112 Earth, solid. . . " wet . . . 130 " loose.. Silver . . . . . 655 Glass, green ... Slate, average weight 175 Gold . . . . . 1204 489 Steel, hard. . . 166 Tin . . . . . 456 Gravel . . . . . 120 Turpentine,spirits of 54 Ice . · · · 58 Vinegar . . . . 651 Tron, cast . . . . 450 Walnut . . . 42 " wrought. . . 480 Water . . . Lard . . . . . ! 59 Wine, Burgundy 62 of the irming ushele 165 |