2. The mean dimensions of square and round timber are found in a similar

manner.

3. The method for finding the solidity of round timber makes an allowance of about for waste in hewing. (Arts. 640, 258. Obs. 3.)

18. Find the area of a board 12 ft. long, and the ends 14 in. and 12 in. wide. 19. Find the solidity of a joist 16 ft. long, the ends being 8 in. and 4 in. sq. 20. Find the solidity of a log 50 ft. long, the circumferences of the ends being 6 ft. and 4 ft.

GAUGING OF CASKS.

645. The process of finding the contents or capacities of casks and other vessels is called GAUGING.

646. The contents of casks are found by multiplying the square of the mean diameter into the length; then this product multiplied by .0034 will give the wine gallons, and multiplied by .0028 will give the beer gallons.

OBS. The mean diameter of a cask is found by adding to the head diameter .7 of the difference between the head and bung diameters when the staves are very much curved; or by adding .5 when very little curved; and by adding .55 when they are of a medium curve.

21. How many wine gallons in a cask but little curved, whose length is 45 in., its bung diameter 40 in., and its head diameter 36 in. ?

22. How many beer gallons in a cask much curved, whose length is 64 in., its bung diameter 52 in., and head diameter 46 in. ?

TONNAGE OF VESSELS.

647. Government Rule.-I. If the vessel be double-decked, take the length from the fore part of the main stern to the after part of the stern-post, above the upper deck; then the breadth at the broadest part above the main wales, half of which breadth shall be accounted the depth of such vessel; from the length deduct three-fifths of the breadth, multiply the remainder by the breadth and the product by the depth; divide the last product by 95, and the quotient shall be deemed the true tonnage of the vessel.

II. If the vessel be single-decked, take the length and breadth as above directed, deduct from the length three-fifths of the breadth, and take the depth from the under side of the deck plank to the ceiling in the hold, then multiply and divide as before, and the quotient shall be deemed the tonnage.

Carpenter's Rule.-The continued product of the length of the keel, the breadth at the main beam, and the depth of the hold in feet, divided by 95 will give the tonnage of a single-decked vessel. For a double-decker, instead of the depth of the hold, take half the breadth at the beam.

23. What is the government tonnage of a double-decker, whose ength is 150 ft., the breadth 35 ft., and the depth 25 ft. ?

24. What is the carpenter's tonnage of the same vessel?

MECHANICAL POWERS.

648. The Mechanical powers are six, viz: the lever, the wheel and axle, the pulley, the inclined plane, the screw, and the wedge.

649. When the power and weight act perpendicularly to the arms of a straight lever, the power is to the weight, as the distance from the fulcrum to the weight is to the distance from the fulcrum to the power.

1. If the power is 100 lbs., the long arm 10 ft., and the short arm 2 ft., what weight can be raised?

2. The arms of a lever are 15 ft. and 4 ft., and the weight raised 500 lbs.: what is the power?

650. When a weight is sustained by a lever resting on two props,

The long arm the short arm :: the weight supported by the long arm : the weight supported by the short arm.. Hence,

The whole length: short arm :: whole weight : weight on s. a. (Leg. III. 16.)

3. A and B carry 256 lbs. suspended upon a pole 5 ft. from A and 3 ft. from B: how many pounds does each carry?

4. A and B carry 90 lbs. upon a lever 12 ft. long: where must it be placed, that B may carry of it?

651. The wheel and axle operate on the same principle as the lever; the semi-diameter of the wheel answers to the long arm, and the semi-diameter of the axle to the short arm.

5. If the diameter of a wheel is 6 ft., and that of the axle 1 ft., what weight will 100 lbs. raise?

6. A wheel is 8 ft. diameter, an axle 1 ft.: what weight will 200 lbs. raise ?

652. In the application of movable pulleys,

The POWER: the WEIGHT: 1: twice the NUMBER of pulleys.

7. What weight can a power of 200 lbs. raise with 4 movable pulleys? 8. What power with 8 pulleys will raise a pillar of granite weighing 10 tons ? 653. The perpendicular height of an inclined plane is to its length, as the power to the weight.

9. What power will draw a train of cars weighing 100000 pounds up an inclined plane which rises 60 ft. to a mile?

.654. The screw acts upon the principle of the inclined plane. Hence, The distance between the threads is to the circumference of a circle described by the power, as the power is to the weight.

10. What weight can be raised by a power of 1000 lbs. applied to a screw whose threads are 1 inch apart, at the end of a lever 12 ft. long?

655. The power applied to the head of a wedge is to the weight, as half the thickness of the head is to the length of its side. In the use of the wedge, not less than half the power is lost by fraction against the si les.

720705

MISCELLANEOUS EXAMPLES.

1. The sum of two numbers is 980, and their difference 62: what are the numbers?

2. The product of two numbers is 4410, and one is 63: what is the other? 3. What number multiplied by 285, will produce 145?

4. What number multiplied by 61, will be equal to 74 multiplied by 5 ? 5. If an army of 24000 men have 120000 lbs. of bread, how long will it last them, allowing each man 13 lbs. per day ? ***

6. What is the interest of $5256 for 60 days, at 7 per cent.?

7. What is the amount of $16230 for 4 months, at 6 per cent.?

8. What is the bank discount on $1200 for 90 days, at 6 per cent.?

9. For what sum must a note be made, payable in 4 months, the proceeds of which shall be $1800, discounted at a bank at 7 per cent. ?

10. A capitalist sent a broker $25000 to invest in cotton, after deducting his commission of 21 per cent.: what amount of cotton ought he to receive?

11. A merchant bought 500 yards of cloth for $1800: how must he retail it by the yard to gain 25 per cent. ?

12. A man bought 640 bbls. of beef for $5000, and sold it at a loss of 12 per cent. how much did he get a barrel?

13. If a man buys 1000 geographies, at 37 cents apiece, and retails them at 50 cents, what per cent. will he make?

14. A grocer bought 180 boxes of lemons for $360, and sold them at 10 per cent. less than cost: what did he lose?

15. How many dollars, each weighing 412 grs., can be made from 16 lbs. 5 oz. of silver?

16. How many eagles, weighing 258 grs. apiece, will 21 lbs. 10 oz. make? 17. How long a thread can be spun from 1 ton of flax, allowing 5 oz. will make 100 rods of thread?

18. How many revolutions will the hind wheel of a carriage 5 ft. 6 in. in circumference, make in 2 miles 4 furlongs?

19. How many revolutions will the fore wheel of a carriage 4 ft. 7 in. in circumference, make in the same distance?

20. Bought 1500 doz. buttons for $187.50: what was that per gross?

21. A man paid $132 for 40 bbls. of cider: what is that a quart?

22. A man paid $150 for 10 rods of land, what was that per acre ?

23. A man having $2500, laid out of it in flour, at $5 per barrel: how many barrels did he buy?

24. The commander of an exploring expedition found that of his provisions were exhausted in 28 months: how much longer would they last?

25. What cost 15 lbs. of cheese, at $85 per hundred ?

26. How many yards of carpeting yd. wide will it take to cover a floor 18 ft. long and 15 ft. wide?

27. If yard of calico costs., what will of an ell English cost?

28. How long will 468256 lbs. of beef last an army of 8245 soldiers, allowing each man 1 lb. per day?

29. How long would the same quantity of beef last the army, if reinforced ɔy 2500 men, allowing each man 14 lb. per day?

30. Bought of a pipe of wine for $126: what was that per gallon?

31. If a man can walk 17 miles in 5 hours, 12 minutes, 31 seconds, how far can he walk in 3 hours, 40 minutes, 33 seconds?

32. If a man traveling 14 hours per day, performs half his journey in 9 days, how long will it take him to go the other half traveling 10 hours a day? 33. If you lend a man $700 for 90 days, how long ought he to lend you $1200 to requite the favor?

34. A milkman's measure was deficient half a gill to a gallon: how much did he cheat his customers in selling 8720 gallons?

35. If 85 yds. of calico cost $10.20, what will 1500 yds. cost?

36. If 1650 lbs. of sugar cost $206.25, what will 87 lbs. cost?

37. If 1424 gals. of cil cost $1062, what will 210 gals. cost?

38. If wind moves 24 miles per hour, how long is it in moving from the pole to the equator, a distance of 6214 miles?

39. If of a barrel of flour costs of a dollar, what will 1⁄2 of a bbl. cost?

16

40. If 3 of a ton of chalk cost £4, what will of a ton cost?

41. If of a bushel of wheat cost, how much will of a bushel cost?

42. If 5 of a ship cost $16000, what will of her cost?

43. If 164 bbls. of mackerel cost $653, what will 484 bbls. cost?

44. If 284 gals. of oil cost $31.25, how much will $250 buy ?

45. At $3 for 40 doz. eggs, what will 460 doz. cost?

46. The President's salary is $25000 per annum: how much can he spend per day, and lay up $10000 of it?

47. If one person lies in bed 9 hours per day, and another 6 hours, how much time will the one gain over the other in 20 years?

48. A cistern has three faucets; the 1st will empty it in 10 min., the 2d in 20 min., the 3d in 30 min.: in what time will they all empty it?

49. A man and his wife drink a barrel of beer in 30 days, and the man alone can drink it in 40 days: how long will it last the wife?

50. A teacher being asked how many scholars he had, replied † study Arithmetic, study Latin, study Algebra, study Geometry, and 24 study French: how many scholars had he?

51. A man having spent and of of his money, had £483 left: how much had he at first?

52. A man bequeathed of his property to his wife, to his son, to his daughter, and the remainder, which was $1500, to the Bible Society: what did his whole property amount to?

53. What is that number Z of which exceeds of it by 45?

54. Pewter is composed of 112 parts of tin, 15 of lead, and 6 of brass: how much will it take of each ingredient to make 555 pounds of pewter?

55. Two travelers start at the same time from Boston and Washington to meet each other; one goes 5 miles an hour, the other 7 miles; the whole distance is 436 miles: how far will each travel?

56. A grocer divided a barrel of flour into 2 parts, so that the smaller con tained as much as the other: how many pounds were there in each?

57. A, B, and C, build a ship together; A advanced $1000, B $12000, and C$13000; they gain $5000: what is the gain of each?

58. A, B, and C, entered into partnership; A furnished $600, B and C together $1800; they gained $960, of which B took $280: how much did A and C gain; and B and C put in respectively?

59. The liabilities of a bankrupt are $63240, and his assets $12648: what per cent. can he pay?

60. A bankrupt compromises with his creditors for 37 per cent.: how much will he pay on a claim of $3656?

61. How much will he pay on a debt of $12680.375 ?

62. A owns 3 and B of a ship; A's share is worth $10000 more than B's: what is the value of the ship?

63. A man gave his oldest son of his property less $50; to the second, he gave; and to the youngest he gave the remainder, which was less $10: what was the amount of his property?

64. A man and boy together can frame a house in 9 days; the man can frame it alone in 12 days: how long will it take the boy to frame it ?

65. A cistern has a receiving and a discharging pipe; when both are running it takes 18 hours to fill it; if the latter is closed it requires 15 hours to fill it: if the former is closed, how long will it take the latter to empty it?

66. Four men, A, B, C, and D, spent £255, and agreed that A should pay ; B; C; and D : how much must each pay?

67. A, B, and C, formed a joint stock of £820, and gained £640, in the division of which A received £5 as often as B did £7, and C £8: how much did each put in and receive?

68. A, B, and C, gained a certain sum, of which A and B received $640, B and C $880, and A and C $800: what was the gain of each?

69. What number is that and of which being multiplied together, will produce the number itself?

70. A club spent £2, 12s. 1d.; on settling, each paid as many pence as there were individuals in the party: how many were there in the party?

71. The sum of two numbers is 120, and the difference of their squares is 4800: what are the numbers?

72. The difference of two numbers is 53, and the difference of the squares is 10759: what are the numbers?

73. The diagonal of a square is 80 ft.: what is its side?

74. The diagonal of a square field is 120 rods: what is its areɛ ?

75. Find the side of the greatest square beam which can be hewn from log 5 ft. in diameter ?

76. The mainmast of a ship is 95 ft. long, the diameter of the base is 3 ft., that of the top 24 ft.: what is its solidity?

77. A man wished to tie his horse by a rope so that he could feed on just an acre of ground: now long must the rope be?