By examining Fig. III. fig. 3, it will be a seed, that we jo. O shall have 3 o sides, each 14 inches long, and SH a 10 inches wide, C besides a small face 4 inches square, upon which additions are to be made to complete the cube. ' Let cu repreP sent a layer extending to the bottom of the pile, and rising 4 inches higher than the part a b in fig, 23, if," u be 10 inches, that layer will be 14 inches high and 10 inches wide. Let's c and a d be each 4 inches; also, let it be 10 inches from a to the bottom of the pile, then s a will be 14 inches long and 10 inches high, and ow b d 14 inches long and 10 inches wide. Now if the additions be made 4 inches thick, the corner m o c s will be 4 inches Square, which is not included in either of the other parts to which the additions are to be made. The number of square inches contained in the three faces cu, sa, 0 whd, which have all been shown to be equal, added to the number contained in m oc s, will constitute the perfect divisor, since by adding to them, the form of the pile will be a cube. The width of each of the large faces is 10 inches, and the length 14. b be 10 inches, a s 14, and all the angles right ones; also, let b fand fr each equal a t, the whole figure will represent the whole surface upon which the additions are to be made. inches, that is, 3 times the whole root, except the right hand figure, by s a, 14 inches, the whole root, the product will be the number of square inches contained in a s a r, and the square of m o, which is 4 inches, the last quotient figure, the number contained in the small square m o c s ; therefore, three times the whole root, except the right hand figure, multiplied by the whole root, and added to the square of the last quotient figure, will always form a perfect divisor, as this will always be the case, let the length of a r, a s, and m o, be any other corresponding numbers. By this method, in the present example, we find the perfect divisor to be 436, the number of square inches contained in m o c s and s a ra: ; and if we multiply the number of square inches contained in these surfaces by the thickness of the additions, the product will express the number of blocks required to complete the additions. Operation completed. 2 7 4 4 ( 1 4 436 can be taken 4 times 1 from 1744, the remainin - number of blocks, there4 4 fore the last figure of the 7 4 4 root will be 4, and the - length of one side of the cube formed by the whole number of blocks, 14 inches, the cube root of 2744. When the number of periods are more than two, the other additions are made in the same manner. JNote 4.—It will be difficult for young students to understand this demonstration without the aid of blocks. Every instructer should be furnished with them, 20. If 59'876°471 cubical blocks, each one foot on a side, be placed in the form of a cube, what will be the length of one side 2 Ans. 391 feet. 21. How long is the side of a cubical box, which will contain 110'592 cubical feet of grain, if 3 inches be allowed for the thickness of the sides of the box * Ans. 48 ft. 3 in. 22. If the earth excavated in digging a trench one mile in length, 40 feet wide, and 6 feet deep, be thrown into a cubical form, how high will the cube be, allowing the earth to occupy 4 of the space which it did before it was removed ? Ans. 103,5+ feet. 23. If a cubical box contain 71 cubical feet, what will be the length of a cubical box which shall eon Remainder to question 24th is 1°252'672’906,066264. 25. What is the height of a cubical box, which will contain nine times as much as one whose height is ten feet ! Ans. 20,8+feet. 26. If the length of one side of a cubical block be 1 inch, what will be the length of one which will contain half as much Ans. ,79-Hin. 27. What is the cube root of one thousand million ? - Ans. One thousand. 28. What is the cube root of 1 ? Ans. 1. 29. If the diameter of a globe be 15 inches, what will be the diameter of a globe twice as large 2 Explanation.—Multiply the cube of the given diameter by 2, and extract the cube root of the product. Ans. 18,89-Finches. 30. If the diameter of a globe be 2, what is the diameter of another half as large 2 Ans, 1,58+. A Series of Numbers is several numbers following each other in succession. Progression, is a series of numbers regularly increasing or decreasing. Arithmetical Progression is a series of numbers increasing or decreasing regularly by the constant adding or subtracting of some particular number. 2, 4, 6, 8, 10, 12, is a series of numbers in arithmetical progression, made by the constant addition of the number 2; or, 15, 12, 9, 6, 3, is a series decreasing by the constant subtraction of 3. Common Difference, the number added or subtracted to make out the series. Extremes, the first and last terms of a series. JMeans, the terms between the extremes. ARITHMRTICAL PROGRESSION. PROBLEM I. 1. If a man work 10 years, and clear 100 dollars the first year, I 11 the second, and so on, clearing 11 dollars each year more than he did the preceding one,—how many dollars did he clear the 10th year. Illustration.—By adding 11 together 9 times, the sum is the number of dollars he would gain more than the 100, the tenth year, and by adding the 100, the sum is his whole gain the tenth year; but if we multiply 11 by 9, the product will be the same as the sum found by adding. When a series of numbers in arithmetical progression, is made out at full length, the common difference is added to, or subtracted from all the terms except the last one; consequently, the last term in an increasing series, or the first term in a decreasing, contains the common difference as many times as the series has terms wanting one. But as the product of a number, multiplied by any number of units, is cqual to the multiplicand added together as many times as the multiplier contains units, we obtain the same number whether we add the common difference as many times as the series has terms, wanting one, or multiply it by the same number. |