feet :: 6 feet : 12 feet. Again, A B will be to D E as A C to DF. That is, 6 feet : 12 feet :: 10 feet : 20 feet. Circles are to each other as the squares of their diameters. If the diameter of a circle be multiplied by 3.14159, the product is the circumference. If the square of the diameter of a circle be multiplied by .785398, the product is the area. If the square root of half the square of the diameter of a circle be extracted, it is the side of an inscribed square. If the area of a circle be divided by .785398, the quotient is the square of the diameter. EXAMPLES. 25. A certain general has an army of 141376 men. How many must he place in rank and file to form them into a square ? Ans. 376. 26. If the area of a circle be 1760 yards, how many feet must the side of a square measure to contain that quantity ? Ans. 125.857+ feet. 27. If the diameter of a round stick of timber be 24 inches, how large a square stick may be hewn from it ? Ans. 16.97+ inches. 28. I wish to set out an orchard of 2400 mulberry-trees, so that the length shall be to the breadth as 3 to 2, and the distance of each tree, one from the other, 7 yards. How many trees must there be in the length, and how many in the breadth ; and on how many square yards of ground will they stand ? Ans. 60 in length; 40 in breadth ; 112749 square yards. 29. If a lead pipe of an inch in diameter will fill a cistern in 3 hours, what should be its diameter to fill it in 2 hours ? Ans. .918+ inches. 30. If a pipe 14 inches in diameter will fill a cistern in 50 minutes, how long would it require a pipe that is 2 inches in diameter to fill the same cistern? Ans. 28m. 7 sec. 31. If a pipe 6 inches in diameter will draw off a certain quantity of water in 4 hours, in what time would it take 3 pipes of four inches in diameter to draw off twice the quantity ? Ans. 6 hours. 32. If a line 144 feet long will reach from the top of a fort to the opposite side of a river that is 64 feet wide, what is the height of the fort ? Ans. 128.99+. 33. A certain room is 20 feet long, 16 feet wide, and 12 feet high ; how long must a line be to extend from one of the lower corners to an opposite upper corner ? Ans. 28.28 feet. 34. Two ships sail from the same port; one goes due north 128 miles, the other due east 72 miles ; how far are the ships from each other? Ans. 146.86+. 35. There are two columns in the ruins of Persepolis left standing upright ; one is 70 feet above the plane, and the other 50; in a straight line between these stands a small statue, 5 feet in height, the head of which is 100 feet from the summit of the higher, and 80 feet from the top of the lower column. Required the distance between the tops of the two columns. Ans. 143.537+ feet. 36. The height of a tree, growing in the centre of a circular island, 100 feet in diameter, is 160 feet; and a line extending from the top of it to the farther shore is 400 feet. What is the breadth of the stream, provided the land on each side of the water be level ? Ans. 316.6+ feet. 37. A ladder 70 feet long is so planted as to reach a window 40 feet from the ground, on one side of the street, and without moving it at the foot it will reach a window 30 feet high on the other side ; what is the breadth of the street ? Ans. 120.69+ feet. 38. If an iron wire to inch in diameter will sustain a weight of 450 pounds, what weight might be sustained by a wire an inch in diameter ? Ans. 45,000lbs. 39. A tree 80 feet in height stands on a horizontal plane ; at what height from the ground must it be cut off, so that the top of it may fall on a point 40 feet from the bottom of the tree, the end where it was cut off resting on the stump? Ans. 30 feet. 40. Four men, A, B, C, D, bought a grindstone, the diameter of which was 4 feet; they agreed that A should grind off his share first, and that each man should have it alternately until he had worn off his share ; how much will each man grind off? Ans. A 3.22+,,B 3.81+, C 4.97+, D 12 inches. 41. What is the length of a rope that must be tied to a horse's neck, that he may feed over an acre ? Ans. 7.136+ rods. 42. Required the greatest possible number of hills of corn that can be planted on a square acre, the hills to occupy only a mathematical point, and no two hills to be within three and a half feet of each other. Ans. 4165. 43. James Page has a circular garden, 10 rods in diameter ; how many trees can be set upon it, so that no two shall be within ten feet of each other, and no tree within two and a half feet of the fence inclosing the garden ? Ans. 241. 44. I have a board whose surface contains 495 square feet; the board is 14 inches thick, and I wish to make a cubical box of it. Required the length of one of its equal sides. Ans. 36 inches.* 45. A carpenter has a plank 1 foot wide, 2231 feet long, and 24 inches thick; and he wishes to make a box whose width shall be twice its height, and whose length shall be twice its width. Required the contents of the box. Ans. 5719 cubic inches.* * The operation may be found on page 203 of the last edition of the Key. Section LXII. EXTRACTION OF THE CUBE ROOT. A CUBE is a square prism, being bounded by six equal sides, which are perpendicular to each other. A number is said to be cubed when it is multiplied into its square. To extract the cube root is to find a number, which, multiplied into its square, will produce the given number. RULE. — 1. Separate the given number into periods of three figures each, by putting 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, calling it the triple square; multiply also the quotient by 30, calling it the triple quotient ; the sum of these call the divisor. 5. Find how many times the divisor is contained in the dividend, and place the result in the quotient. NOTE. — One or two units, and sometimes three, must be allowed. 6. Multiply the triple square by the last quotient figure, and write the product under the dividend ; multiply the square of the last quotient fig. ure by the triple quotient, and place this product under the last ; under all, set the cube of the last quotient figure, and call their sum the subtrahend. 7. 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 same rule must be observed for continuing the operation, and pointing for decimals, as in the square root. ILLUSTRATION. We suppose we have 46,656 cubic blocks of granite, which measure one foot on each side. With these we wish to erect a cubical monument. It is required to ascertain how many blocks, or feet, will be the length of one side of the monument. It is evident that the number of blocks will be equal to the cube root of 46,656. As the given number consists of five figures, its cube root will contain two places ; for the cube of any number can never contain more than three times that number, and at least but two less. We therefore separate the given number into periods of three figures each, putting a point over the unit figure, and every third figure beyond the place of units ; thus, 46,656. We find by the table of powers, or by trial, the greatest power in the left-hand period, 46 (thousand), is 27 (thousand), the root of which is 3. This root we write in the quotient; and, as it will occupy the place of tens, its real value is 30. If this be considered the side of a cube, it will contain 27,000 cubic feet, 30 x 30 x 30 = 27,000 feet. Fig. 1. Let this cube be represented by fig. ure 1, each of whose sides measures 30 feet; therefore its contents will be 30 x 30 x 30 = 27,000 feet, as above. We subtract the contents of this cube from 46,656, and there remain 19656 cubic feet. OPERATION. 27,000 2700) 19,656 16,200 216 Or we might have subtracted the cube of 3,= 27, from the first period, and to the remainder have brought down the next period, and the result would have been the same. (See operation. The cubic blocks that remain must be applied to the three sides of figure 1. For, unless a cube be equally increased on three sides, it ceases to be a cube. To effect this, we must find the superficial contents of three sides of the cube, and with these we must divide the remaining number of cubic feet or blocks, and the quotient will show the thickness of the additions. As the length of a side is 30 feet, the superficial contents will be 30 X 30=900 square feet, and this multiplied by 3, the number of sides, will be 900 X 3=2700 feet. With this as a divisor, we inquire how many times it is contained in 19,656, and find it to be 6 times (one or two units, and sometimes three, must be allowed on account of the other deficiencies in enlarging the cube). This 6 is the thickness of the additions to be made to the three sides of the cube, and by multiplying their superficial contents by it, we have the solid contents of the additions to be made 2700 X 6= 16200; that is, we multiply the triple square by the last quotient figure, and this may be represented by the three superficies ABCD, EFGH, and I K L M. (See figure 2.) Having applied these additions to our cube, we find there are three other deficiencies, abcd, e f g h, ij kl, the length of which is equal to that of the additions, 30 feet, and the height and breadth of each are equal to the thickness of the ad. ditions, 6 feet. To find the contents of these, we multiply the product of their length, breadth, and thickness by their num. ber; thus, 6 x 6 x 30 x 3=3240; or, which is the same thing, we multiply the triple quotient by the square of the last quotient figure ; thus, 90 x 6 x 6 = 3240. See rule. Having made these additions to Fig. 2. the cube, we still find one other deficiency, N. (See figure 2) The length, breadth, and thickness of which are equal to the thickness of the former additions, viz. 6 feet. The contents of this are found by multiplying its length, breadth, and thickness together ; that is, cubing the last quotient figure ; thus, 6 x 6 x 6 = 216. By making this last addition, we find that our cubical A monument is finished, and that the first figure together with the several additions is equal to the cubical blocks, 46,656. - Gil BI |