MEASUREMENT OF SOLIDS.

632. In the measurement of solids it is customary to assume a cube as the measuring unit, whose sides are squares of the same name. (Art. 258. Obs. 2.)

633. To find the solidity of bodies whose sides are perpendicular to each other.

Multiply the length, breadth, and thickness together. (Art. 286.) OBS. When the contents of a solid body and two of its sides are given, the other side is found by dividing the contents by the product of the two given sides. (Art. 159.)

1. What are the contents of a stick of timber 4 ft. square, and 851 ft. long? 2. What is the capacity of a cubical vessel, 14 ft. 8 in. deep?

634. To find the solidity of a prism.

Multiply the area of the base by the height. (Leg. VII. 12.) OBS. This rule is applicable to all prisms, triangular, quadrangular, pentagɔnal, &c., also to all parallelopipedons, whether rectangular or oblique.

3. Find the solidity of a prism 46 ft. high, whose base is 7 ft. square?

635. To find the lateral surface of a right prism.

Multiply the length by the perimeter of its base. (Leg. VII. 5.) OBS. If we add the areas of both ends to the lateral surface, the sum will be the whole surface of the prism.

4. What is the surface of a triangular prism, whose sides are each 3 ft., and its length 12 ft. ?

636. To find the solidity of a pyramid and cone.

Multiply the area of the base by of the height. (Leg. VII. 18.) 5. What is the solidity of a pyramid 100 ft. high, whose base is 40 ft. square? 6. What is the solidity of a cone 150 ft. high, whose base is 15 ft. in diameter ?

637. To find the lateral or convex surface of a regular pyramid, or cone. (Leg. VII. 16, VIII. 3.)

Multiply the perimeter of the base by the slant-height.

What is the lateral surface of a regular pyramid, whose slant-height is, 15 ft., and base is 30 ft. square?

8. What is the convex surface of a right cr ne, whose slant-height is 94 ft. and the perimeter of its base 37 ft.'?

638. To find the solidity of a frustum of a pyramid and cone. To the sum of the arcas of the two ends, add the square root of the product of these areas; then multiply this sum by of the perpendicular height. (Leg. VII. 19. Sch., VIII. 6.)

9. If the two ends of the frustum of a pyramid are 3 ft and 2 ft. square, and the height is 12 ft., what is its solidity?

639. The convex surface of a frustum of a pyramid and cone is found by multiplying half the sum of the circumferences of the two ends by the slant-height. (Leg. VII. 17, VIII. 5.)

10. If the circumferences of the two ends of the frustum of a cone are 18 ft. and 14 ft., and its slant-height 11 ft., what is its convex surface?

640. To find the solidity of a cylinder.

Multiply the area of the base by the height. (Leg. VIII. 2.)

11. Find the solidity of a cylinder 10 ft. in diameter, and 35 ft. high. 12. Find the solidity of a cylinder 100 ft. in circumference, and 150 ft. high641. To find the convex surface of a cylinder.

Multiply the circumference of the base by the height. (Leg. VIII. 1.) 13. Find the convex surface of a cylinder 5 yds. in diameter, and 5 yds. long. 642. To find the convex surface of a sphere or globe. Multiply the circumference by the diameter. (Leg. VIII. 9.) 14. What is the surface of a globe 18 inches in diameter ?

15. If the diameter of the moon is 2162 miles, what is its surface?

643. To find the solidity of a sphere or globe.

Multiply the surface by of the diameter. (Leg. VIII. 11.)

16. Find the solidity of a globe 15 inches in diameter.

17. The diameter of the moon is 2162 miles: what is its solidity?

MEASUREMENT OF LUMBER.

644. The area of a board is found by multiplying the length into the mean readth. (Arts. 622, 623.)

The solid contents of hewn or square timber are found by multiplying the ength into the mean breadth and depth.

The solid contents of round timber are found by multiplying the length by ‡ the mean girt or circumference.

Obs. 1. The mean breadth of a tapering board is found by measuring it in the middle, or by taking the sum of the breadths of the two ends.

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 dc uble-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 ower?

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 friction against the si les.

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 284, will produce 145?

4. What number multiplied by 61, will be equal to 74 multiplied by 51 ?

5. If an army of 24000 men have 520000 lbs. of bread, how long will it last them, allowing each man 11⁄2 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 hẹ 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: hów many barrels did he buy?

24. The commander of an exploring expedition found that 4 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 3.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 h. per day?