EXAMPLES. EXAM. I. Required the diameter of a 641b leaden ball? Ans 6 684 inches. EXAM. 2. What is the diameter of an 8lb leaden bail? Ans. 3.343 inches. PROBLEM V. To find the Weight of an Iron Shell TAKE of the difference of the cubes of the external and internal diameter, for the weight of the shell. That is, from the cube of the external diameter, take the cube of the internal diameter, multiply the remainder by 9, and divide the product by 64. EXAMPLES. EXAM. 1. The outside diameter of an iron shell being 12.8, and the inside diameter 9.1 inches; required its weight? Ans. 188.941lb. EXAM. 2 What is the weight of an iron shell, whose external and internal diameters are 9 8 and 7 inches ? Ans. 84 lb. PROBLEM VI. To find how much Powder will fill a Shell. DIVIDE the cube of the internal diameter, in inches, by 57-3, for the lbs of powder. EXAMPLES. EXAM. 1. How much powder will fill the shell whose internal diameter is 9 1 inches? Ans. 13lb nearly. EXAM 2. How much powder will fill a shell whose internal diameter is 7 inches? Ans. 6lb. PROBLEM VII. To find how much Powder will fill a Rectangular Box. FIND the content of the box in inches, by multiplying the length, breadth, and depth all together. Then divide by 30 for the pounds of powder. EXAMPLES. EXAM. 1. Required the quantity of powder that will fill a box, the length being 15 inches, the breadth 12, and the depth 10 inches? Ans 60lb. EXAM 2: EXAM. 2 How much powder will fill a cubical box whose side is 12 inches ? PROBLEM VIII. Ans. 57 lb. To find how much Powder will fill a Cylinder. MULTIPLY the square of the diameter by the length, then divide by 38 2 for the pounds of powder. EXAMPLES. EXAM. 1. How much powder will the cylinder hold, whose diameter is 10 inches, and length 20 inches? Ans. 523ib nearly. EXAM 2. How much powder can be contained in the cylinder whose diameter is 4 inches, and length 12 inches? Ans. 5lb. PROBLEM IX. To find the Size of a Shell to contain a Given Weight of Powder. MULTIPLY the pounds of powder by 57.3, and the cube root of the product will be the diameter in inches. EXAMPLES. EXAM. 1. What is the diameter of a shell that will hold of powder ? PROBLEM X. To find the Size of a Cubical Box to contain a given Weight of MULTIPLY the weight in pounds by 30, and the cube root of the product will be the side of the box in inches. EXAMPLES. EXAM. 1. Required the side of a cubical box, to hold 50lb of gunpowder? Ans. 11.44 inches. EXAM. 2. Required the side of a cubical box, to hold 400lb of gunpowder ? PROBLEM XI. Ans. 22 89 inches. To find what Length of a Cylinder will be filled by a given MULTIPLY the weight in pounds by 38.2, and divide the product by the square of the diameter in inches for the length. EXAMPLES, EXAMPLES. EXAM. 1. What length of a 36-pounder gun, of 6 inches diameter, will be filled with 12lb of gunpowder? Ans. 10-314 inches. EXAM. 2 What length of a cylinder, of 8 inches diameter, may be filled with 20lb of powder ? 104 Ans. 11 inches. OF THE PILING OF BALLS AND SHELLS. IRON Balls and Shells are commonly piled by horizontal courses, either in a pyramidical or in a wedge-like form; the base being either an equilateral triangle, or a square, or a rectangle. In the triangle and square, the pile finishes in a single ball; but in the rectangle, it finishes in a single row of balls, like an edge. In triangular and square piles, the, number of horizontal rows. or courses, is always equal to the number of balls in one side of the bottom row. And in rectangular piles, the number of rows is equal to the number of balis in the breadth of the bottom row. Also, the number in the top row, or edge, is one more than the difference between the length and breadth of the bottom row. PROBLEM I. To find the Number of Balls in a Triangular Pile. MULTIPLY Continually together the number of balls in one side of the bottom row, and that number increased by 1, also the same number increased by 2; then of the last product will be the answer. That is, n.nl.n+ 2 is the number or sum, where n is the number in the bottom row. EXAMPLES. EXAM. 1. Required the number of balls in a triangular pile, each side of the base containing 30 balls? Ans. 4960. EXAM. 2. How many balls are in the triangular pile, each side of the base containing 20 ? Ans. 1540. PROBLEM PROBLEM II, To find the Number of Balls in a Square Pile. MULTIPLY Continually together the number in one side of the bottom course, that number increased by 1, and double the same number increased by 1; then of the last product will be the answer. 1 EXAMPLES. EXAM. 1. How many balls are in a square pile of 30 rows? EXAM. 2. How many balls are in a square pile of 20 rows? PROBLEM III. To find the Number of Balls in a Rectangular Pile. From 3 times the number in the length of the base row subtract one less than the breadth of the same, multiply the remainder by the same breadth, and the product by one more than the same, and divide by 6 for the answer. That is, b.b+1.31 − b + 1 is the number; where is the length, and 6 the breadth of the lowest course. Note. In all the piles the breadth of the bottom is equal to the number of courses. And in the oblong or rectangular pile, the top row is one more than the difference between the length and breadth of the bottom. EXAMPLES. EXAM. 1. Required the number of balls in a rectangular pile, the length and breadth of the base row being 46 and 15? Ans. 4960. EXAM. 2. How many shot are in a rectangular complete pile, the length of the bottom course being 59, and its breadth 20? Ans. 11060. PROBLEM IV. To find the Number of Balls in an Incomplete Pilc. FROM the number in the whole pile, considered as com plete, subtract the number in the upper pile which is want ing at the top, both computed by the rule for their proper form; and the remainder will be the number in the frustum, or incomplete pile. EXAMPLES. EXAM. 1. To find the number of shot in the incomplete triangular pile, one side of the bottom course being 40, and the top course 20? Ans. 10150. EXAM. 2. How many shot are in the incomplete triangular pile, the side of the base being 24, and of the top 8 ? Ans. 2516. EXAM. 3. How many balls are in the incomplete square pile, the side of the base being 24, and of the top 8 ? Ans. 4760. EXAM. 4. How many shot are in the incomplete rectangular pile, of 12 courses, the length and breadth of the base being 40 and 20 ? Ans. 6146. OF DISTANCES BY THE VELOCITY OF SOUND. By various experiments it has been found, that sound Aies: through the air, uniformly at the rate of about 1142 feet in 1 second of time, or a mile in 4 or 14 seconds. And therefore, by proportion, any distance may be found corresponding to any given time; namely, multiplying the given time, in seconds, by 1142, for the corresponding distance in feet; or taking of the given time for the distance in miles. Or dividing any given distance by these numbers, to find the corresponding time. Note. The time for the passage of sound in the interval be tween seeing the flash of a gun, or lightning, and hearing the report, may be observed by a watch, or a small pendulum. Or, it may be observed by the beats of the pulse in the wrist, counting, on an averge, about 70 to a minute for persons in moderate health, or 5 pulsations to a mile; and more or less according to circumstances. EXAMPLES. EXAM. 1. After observing a flash of lightning, it was 12 seconds before the thunder was heard; required the distance of the cloud from whence it came ? Ans. 24 miles EXAM. 2. How long, after firing the Tower guns, may the |