EXAMPLES.

EXAM. 1. How many balls are in a square pile of 30 rows? Ans. 9455.

EXAM. 2. How many balls are in a square pile of 20 rows? Ans. 2870.

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

6

is the number; where lis

the length, and b 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 Pile.

FROM the number in the whole pile, considered as complete, subtract the number in the upper pile which is wanting 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.

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 flies, through the air, uniformly at the rate of about 1142 feet in 1 second of time, or a mile in 4 or 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 between 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 average, about 70 to a minute for persons in moderate health, or 54 pulsations to a mile; and more or less aecording to circumstances.

EXAMPLES.

EXAM. 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 report be heard at Shooter's-Hill, supposing the distance to be 8 miles in a straight line? Ans. 374 seconds.

EXAM. 3. After observing the firing of a large cannon at a distance, it was 7 seconds before the report was heard; what was its distance? Ans. 14 mile.

EXAM. 4. Perceiving a man at a distance hewing down a tree with an axe, I remarked that 6 of my pulsations passed between seeing him strike and hearing the report of the

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T2

blov

blow; what was the distance between us, allowing 70 pulses to a minute? Ans. 1 mile and 198 yards.

EXAM. 5. How far off was the cloud from which thunder issued, whose report was 5 pulsations after the flash of lightning; counting 75 to a minute? Ans. 1523 yards.

EXAM. 6. If I see the flash of a cannon, fired by a ship in distress at sea, and hear the report 33 seconds after, how far is she off? Ans. 74 miles.

PRACTICAL EXERCISES IN MECHANICS, STATICS, HYDROSTATICS, SOUND, MOTION, GRAVITY, PROJECTILES, AND OTHER BRANCHES OF NATURAL PHILOSOPHY.

QUESTION 1. REQUIRED the weight of a cast iron ball of 3 inches diameter, supposing the weight of a cubic inch of the metal to be 0-258lb avoirdupois? Ans. 3.647391b.

QUEST. 2. To determine the weight of a hollow spherical iron shell, 5 inches in diameter, the thickness of the metal being one inch ? Ans. 13-23871b.

QUEST. 3. Being one day ordered to observe how far a battery of cannon was from me, I counted, by my watch, 17 seconds between the time of seeing the flash and hearing the report; what then was the distance ? Ans. 3 miles.

QUEST. 4. It is proposed to determine the proportional quantities of matter in the earth and moon; the density of the former being to that of the latter, as 10 to 7, and their diameters as 7930 to 2160. Ans. as 71 to 1 nearly.

QUEST. 5. What difference is there, in point of weight, between a block of marble, containing 1 cubic foot and a half, and another of brass of the same dimensions ?

Ans. 496lb 14oz.

QUEST. 6. In the walls of Balbeck in Turkey, the ancient Heliopolis, there are three stones laid end to end, now in sight, that measure in length 61 yards; one of which in particular is 21 yards or 63 feet long, 12 feet thick, and 12 feet broad: now if this block be marble, what power would balance it, so as to prepare it for moving?

Ans, 683 tons, the burden of an East-India ship. QUEST. 7. The battering-ram of Vespasian weighed, suppose 10,000 pounds; and was moved, let us admit, with such

such a velocity, by strength of hand, as to pass through 20 feet in one second of time; and this was found sufficient to demolish the walls of Jerusalem. The question is, with what velocity a 32lb ball must move, to do the same execution? Ans. 6250 feet.

QUEST. 8. There are two bodies, of which the one contains 25 times the matter of the other, or is 25 times heavier; but the less moves with 1000 times the velocity of the greater: in what proportion then are the momenta, or forces, with which they moved?

Ans. the less moves with a force 40 times greater. QUEST. 9. A body, weighing 20lb, is impelled by such a force, as to send it through 100 feet in a second; with what velocity then would a body of 8lb weight move, if it were 'impelled by the same force? Ans. 250 feet per second.

QUEST. 10. There are two bodies, the one of which weighs 100lb, the other 60; but the less body is impelled by a force 8 times greater than the other; the proportion of the velocities, with which these bodies move, is required? Ans. the velocity of the greater to that of the less, as 3 to 40.

QUEST. 11. There are two bodies, the greater contains 8 times the quantity of matter in the less, and is moved with a force 48 times greater: the ratio of the velocities of these two bodies is required?

Ans. the greater is to the less, as 6 to 1.

QUEST. 12. There are two bodies, one of which moves 40 times swifter than the other; but the swifter body has moved only one minute, whereas the other has been in motion 2 hours: the ratio of the spaces described by these two bodies is required?

Ans. the swifter is to the slower, as 1 to 3.

QUEST. 13. Supposing one body to move 30 times swifter than another, as also the swifter to move 12 minutes, the other only 1: what difference will there be between the spaces described by them, supposing the last has moved 5 feet? Ans. 1795 feet.

QUEST. 14. There are two bodies, the one of which has passed over 50 miles, the other only 5; and the first had moved with 5 times the celerity of the second; what is the ratio of the times they have been in describing those spaces? Ans. as 2 to 1.

QUEST. 15. If a lever, 40 effective inches long, will, by a certain power thrown successively on it, in 13 hours, raise a weight 104 feet; in what time will two other levers,

each 18 effective inches long, raise an equal weight 15 Ans. 10 hours 8 minutes.

feet?

QUEST. 16. What weight will a man be able to raise, who presses with the force of a hundred and a half, on the end of an equipoised handspike, 100 inches long, meeting with a convenient prop exactly 71⁄2 inches from the lower end of the machine? Ans. 2072lb.

QUEST. 17. A weight of 14lb, laid on the shoulder of a man, is no greater burden to him than its absolute weight, or 24 ounces: what difference will he feel, between the said weight applied near his elbow, at 12 inches from the shoulder, and in the palm of his hand, 28 inches from the same ; and how much more must his muscles then draw, to support it at right angles, that is, having his arm stretched right out? Ans. 241b avoirdupois.

QUEST. 18. What weight hung on at 70 inches from the centre of motion of a steel-yard, will balance a small gun of 91⁄2 cwt, freely suspended at 2 inches distance from the said centre on the contrary side? Ans. 30lb.

QUEST. 19. It is proposed to divide the beam of a steelyard, or to find the points of division where the weights of 1, 2, 3, 4, &c, lb, on the one side, will just balance a constant weight of 95lb at the distance of 2 inches on the other side of the fulcrum; the weight of the beam being 10lb, and its whole length 36 inches?

Ans. 30, 15, 10, 7, 6, 5, 4, 34, 35, 3,2,2, &c. QUEST. 20. Two men carrying a burden of 200lb weight between them, hung on a pole, the ends of which rest on their shoulders; how much of this load is borne by each man, the weight hanging 6 inches from the middle, and the whole length of the pole being 4 feet ?

:

Ans. 125lb and 75lb.

QUEST. 21. If, in a pair of scales, a body weigh 90lb in one scale, and only 40lb in the other; required its true weight, and the proportion of the lengths of the two arms of the balance beam, on each side of the point of suspension? Ans. the weight 60lb, and the proportion 3 to 2.

QUEST. 22. To find the weight of a beam of timber, or other body, by means of a man's own weight, or any other weight. For instance, a piece of tapering timber, 24 feet long, being laid over a prop, or the edge of another beam, is found to balance itself when the prop is 13 feet from the less end; but removing the prop a foot nearer to the said end, it takes a man's weight of 210lb, standing on the less end,