ing two passengers to weigh 3 cwt. and the body of the chaise cwt. more, what will the horse, in both these cases, bear, more than his harness? S 1163 lb. in the first. 777 lb. in the second.

Ans.

4. Of the Wheel and Arle.

337. RULE.-As the diameter of the axle is to the diameter of the wheel, so is the power applied to the wheel to the weight suspended on the axle.

1. If the diameter of the axle be 6 inches, and that of the wheel be 48 inches, what weight applied to the wheel will balance 1268 lb. on the axle ? 48:6: 1268: 158 lb. Ans. .

2. If the diameter of the wheel be 50 inches, and that of the axle 5 inches, what weight on the axle will 2 lb. on the wheel balance? 5:50:2:20 lb. Ans. 3. If the diameter of the wheel be 60 inches, and that of the axle 6 inches, what weight at the axle will balance 1 lb. on the wheel? Ans. 10 lb.

5. Of the Screw.

338. The power is to the weight which is to be raised, as the distance between two threads of the screw, is to the circumference of a circle described by the power applied at the end of the lever. To find the circumference of the circle; multiply twice the length of the lever by 3.1416; then say, as the circumference is to the distance between the threads of the screw, so is the weight to be raised to the power which will raise it.

1. The threads of a screw are 1 inch asunder, the lever by which it is turned, 30 inches long, and the weight to be raised, 1 ton-2240 lb.; what power must be applied to turn the screw? 30×2=60, and 60×3.1416—188.496 inches, the circ.

Then 188.496: 1 :: 2240: 11.88 lb. Ans. 2. If the lever be 30 inches (the circumference of which is 188.496), the threads 1 inch asunder, and the power 11.88 lb., what weight will it raise?

1: 188.496: 11.88: 2240 lb. nearly, Ans. 3. Let the weight be 2240 lb., the power 11.88 lb., and the lever 30 inches; what is the distance between the threads?

4. If the power be 11.88 lb., the threads 1 inch sunde- what is the

Ans. 1 inch, nearly. weight 2240 lb., and the length of the lever? Ans. 30 inches, nearly.

339. 1. What number taken from the square of 48 will leave 16 times 54 ? Ans. 1440. 2. What number added to the 31st part of 3813, will make the sum 200 ?

Ans. 77.

3. What will 14 cwt. of beef cost, at 5 cents per pound?

Ans. $78.40. 4. How much in length that is 83 inches wide, will make a square foot ? Ans. 171 inches.

5. What number is that to which if of be added, the sum will be 1?

Ans.

6. A father dividing his fortune among his sons, gave A 4 as often as B 3, and C 5 as often as B 6; what was the whole legacy, supposing A's share $5000? Ans. $11875.

7. A tradesman increased his estate annually by £100 more than part of it, and at the end of 4 years found that his estate amounted to £10342 3s. 9d.; what had he at first? Ans. £4000. 8. A person being asked the time of day, said the time past noon is equal to of the time till midnight; what was the ime? Ans. 20 minutes past 5.

9. The hour and minute hand of a clock are together at 12 o'clock; when are they next together? Ans. 1h 5m.

10. A young hare starts 40 yards before a greyhound, and is not perceived by him till she has been up 40 seconds; she scuds away at the rate of 10 miles an hour, and the dog on view makes after it at the rate of 18. In what time and distance will the dog overtake the hare?

Ans. 60s. time, 530 yds. distance. 11. What part of 3d. is part of 2d. ? Ans. 12. A hare is 50 leaps before a greyhound, and takes 4 leaps to the greyhound's 3; but 2 of the greyhound's leaps are as much as 3 of the hare's; how many leaps must the hound take to catch the hare? If 3:1:1: the hare's gain.

2

:1:1: the hound's gain. Then, and ::: 50: 300-300, Ans.

H

13. A post is in the sand, in the water, and 10 feet above the water; what is its length?

Ans. 24 feet.

14. A man being asked how many sheep he had, said, if he had as many more, half as many more, and 7 should have 20; how many had he?

15. In an orchardthe trees bear apples, plums, and 50 of them cherries; how many trees in all?

sheep, he

Ans. 5. pears, are there

Ans. 600.

16. A can do a piece of work alone in 10 days, B can do it in 13; in what time will both together do it? Ans. 515 days. 17. What is the difference between the interest of £350 at 4 per cent. for 8 years, and the discount of the same sum at the same rate, and for the same time? Ans. £27 38.

18. Sound moves at the rate of 1142 feet in a second; if the time between the lightning and thunder be 20 seconds, what is the distance of the explosion? Ans. 4.32 miles.

19. If the earth's diameter be 7911 miles, and that of the moon be 2180, how many moons will be required to make one earth? Ans. 47.788+. 20. If a cubic foot of iron were drawn into a bar of an inch square, what would be its length, supposing no waste of metal? 12×12×12 27648in.-2304ft. Ans.

.25X.25

21. A lent B a solid stack of hay, measuring 20 feet every way; some time after B returned a quantity measuring every way 10 feet; what proportion of the hay is yet due? Ans. .

22. A general disposing his army into a square, finds he has 284 soldiers over and above, but increasing each side by one soldier, he wants 25 to fill up the square; how many soldiers had he? Ans. 24000.

340. 23. Supposing a pole 75 feet high to stand on a horizontal plane, at what height must it be cut off, so as that the top of it may fall on a point 55 feet from the bottom, and the end where it was cut off, rest on the stump or upright part?

75X75-55X55=71 ft. Ans. 75X2

RULE. From the square of the length of the pole (i. e. the sum of the hypothenuse and perpendicular) take the square of the base; then divide the remainder by twice the length of the pole, and the quotient will be the height at which it must be cut off.

24. Suppose a ship sail from lat. 43° N. between N. and E. till her departure from the meridian be 45 leagues, and the sum of her distance and difference of latitude be 135 leagues; what is the distance sailed, and the latitude come to?

135X135-45X45 lea. m.

135X2

135-60-751. dis. s'd. 60-180-3° of lat. 43°+3° 46° come to.

Ала.

341. 25. Four men bought a grindstone 60 inches in diameter; how much of its diameter must each grind off to have an equal share of the stone, if one grind his share first, and then another, till the stone is ground away, making no allowance for the eye?

sub

RULE.-Divide the square of the diameter by the number of men, tract the quotient from the square, and extract the square root of the remainder, which is the length of the diameter, after the first share is taken off; and by repeating the latter part of the process, all the several shares may be found.'"

60X60-4-900, the subtrahend.

3600—900—51.96 and 60-51.96—8.04, 1st share. /2700-900—42.42+ and 51.96-42.42-9.54, 2d share. 1800-900-30. and 42.42-30—12.42, 3d share.

and 30, 4th's share.

26. Suppose one of those meteors called fireballs to move parallel to the earth's surface, and 50 miles above it, at the rate of 20 miles per second; in what time will it move round the earth?

The earth's diameter being 7964 miles, the diameter of the orbit will be 7964+50×2=8064, and 8064×3.1416-25333.8624, its circumference. Then 25333.8624-20-1266.69312s.-21 6" 41" 35" 13 55 Ans.

27. When first the marriage knot was tied betwixt my wife and me,
My age with hers did so agree as nineteen does with eight and three;
But after ten and half ten years we man and wife had been,
Her age came up as near to mine as two times three to nine.

Ans. 57 and 33.

What were our ages at marriage? 28. A body weighing 30 lb. is impelled by such a force as to send it 20 rods in a second; with what velocity would a body weighing 12 lb. move, if it were impelled by the same force? Ans. 50 rods. 29. In a thunder storm I observed by my clock that it was 6 seconds between the lightning and thunder; at what distance was the explosion? Ans. 6852 ft. 113 miles.

30. There is a square pyramid, each side of whose base is 30 inches, and whose perpendicular height is 120 inches, to be divided into three equal parts by sections parallel to its base; what will be the perpendicular height of each part?

30X30X40 36000, the solidity in inches. Now of this is 24000, and is 12000. Therefore, as 36000: 120X120X120 24000 1152000

::

12000: 576000

}

Also,

Then, 31152000-104.8. $576000-83.2. Then 120-104.8—15.2, length of the thickest part, and 104.8-83.2—21.6, length of the middle part; consequently, 83.2 is the length of the top part.

31. I have a square stick of timber 18 inches by 14, but one with a third part of the timber in it, provided it be 8 inches deep, will serve; how wide will it be? Ans. 10 inches.

32. There are 4 spheres, each 4 inches in diameter, lying so as to touch each other, in the form of a square, and on the middle of this square is put a fifth ball of the same diameter; what is the distance between the two horizontal planes passing through the centres of the balls?

42+42÷22.828+ inches, Ans.

33. There are 2 balls, each 4 inches in diameter, which touch each other, and another ball of the same diameter is so placed between them that their centres are in the same vertical plane; what is the distance between the horizontal planes which pass through their centres? 42-22-3.46+in. Ans.

34. A military officer drew up his soldiers in rank and file, having the number in rank and file equal; on being reinforced with three times his first number of men, he placed them all in the same form, and then the number in rank and file was just double what it was at first; he was again reinforced with three times his number of men, and after placing the whole in the same form as at first, his number in rank and file was 40 men each; how many men had he at first? Ans. 100. 35. If a weight of 1440 lb. be placed 1 foot from the prop, at what distance from the prop must a power of 160 lb. be applied to balance it? Ans. 9 feet.

36. Three men wishing to carry a stick of timber, which is of uniform size and density, and 30 feet long; if one man takes hold at one end of the stick, how far from the other end must the other two take hold together, that each may bear an equal portion? Ans. 7 feet.

The centre of gravity being in the middle of the stick, we may regard its weight as all accumulated in that point, and the stick itself as a lever supporting it; and then the parts borne will be inversely as the distances from the middle, and the reverse, i. e. the man at the end being 15 feet from tiddle, the 2 must be of 15, or 7.5 feet from the middle, and 15-7.5=the distance from the end.

Wheright the 2 men to take hold in order to carry of the stick?

The one being 15 feet from the middle, the two, in order to carry 3 times as much, must be 1-3d of 15-5 feet from the middle, and 15—5— 10 ft., the distance from the end.

37. Suppose a pole 100 feet high, to be 24 inches in diameter at the ground, and 4 in. do. at the top, and a vine 1 inch in diameter at the ground to run up this pole, winding round every 3 feet, and gradually diminishing so as to come to a point at the top of the pole, what is the length of the vine?

Ans. 162 feet, 11.94 inches.