RULE.

Multiply together the diameters of the two extremities, and to the product add one-third of the square of the difference of the diameters. Multiply this sum by .785398, and the product will be the mean area between the two extremities. The mean area multiplied by the length, will give the solid contents.

23. What are the solid contents of a stick of timber, whose length is 50 feet, the diameter of the larger end 36 inches, and the diameter of the smaller end 30 inches?

24. What are the solid contents of a ship's mast, whose length is 35 feet, the diameter at the base 24 inches, and the smaller diameter 18 inches?

PROBLEM X.

To find the solid contents of a sphere.

RULE.

Multiply the cube of the diameter by .5236.

25. The diameter of a globe is 43 feet. solid contents?

What are the

26. What are the solid contents of the earth, and what does it weigh, supposing the mean weight to be twice that of water?

PROBLEM XI.

To gauge, or find the dimensions of a cask.

RULE.

Find the diameter at the bung, the diameter at the head, and the length of the cask, all in inches. Subtract the head diameter from the bung diameter, and note the dif. ference.

If the staves of the cask be much curved, multiply the difference by .7; if little curved, by .6; if of a medium curve, by .65; and if nearly or quite straight, by .55, and

add the product to the head diameter. The sum will be a mean diameter, by which the cask is reduced to a cylinder.

Multiply the square of the mean diameter by the length, and divide the product by 359 for the contents in beer gallons, or by 294 for the contents in wine gallons.

27. How many wine gallons will fill a cask whose bung diameter is 40 inches, the head diameter 30 inches, and the length 50 inches?

28. How many beer gallons will a cask contain, which measures 31 inches at the head, 33 inches at the bung, and 47 inches in length?

29. What are the contents in wine measure of a tub, whose inner diameter at the bottom is 29 inches, at the top 36 inches, and the height 30 inches? (The tub is a frustrum of a cone, and the solid contents are found by Problem IX.)

30. How many gallons will fill a churn, that is 18 inches in diameter at the bottom, 12 inches at the top, and 3 feet in height?

PROBLEM XII.

To find the carpenters' tonnage of a vessel.

RULE.

Multiply the breadth at the main beam, half the breadth, and the length, together. Divide the product by 95 and the quotient is the tonnage.

This is probably the best general rule for forming estimates; but no rule can be given that will produce a perfectly accurate result. The rule employed by government, in the collection of revenue, gives about ៖ of the tonnage thus obtained.

31. What is the tonnage of a vessel, whose length is 70 feet, and breadth 25 feet?

32. The length of a vessel is 163 feet, and the breadth 31 feet. Required the tonnage.

33. Find the tonnage of a vessel that is 113 ft. long, and 24 ft. wide.

34. What is the tonnage of a ship that is 116 feet long, and 31 feet wide?

NATURAL PHILOSOPHY.

PROBLEM I.

To find the specific gravity of a body.

RULE.

If the body is heavier than water, weigh it both in water and out of water, and the difference will be the weight lost in the water. Then, the weight lost in the water: the whole weight the specific gravity of water* : the specific gravity of the body.

But if the body is lighter than water, attach to it another body heavier than water, so that the two may sink together. Weigh the two together, and the heavier by itself, both in water and in the air, and find the loss of each in the water. Subtract the less loss from the greater, and say, the last remainder the weight of the body in air :: the specific gravity of water: the specific gravity of the body.

1. A piece of dwt. in the air.

gold weighed 364 dwt. in water, and 38 What was the specific gravity?

2. What is the specific gravity of a body that weighs 13 pounds in the air, and 93 pounds in water?

3. What is the weight of a block of oak, that contains 132 cubic feet, the specific gravity being .925 ?

PROBLEM II.

To find the distance at which bodies may be seen at sea, or on level ground, the height being known.

*The specific gravity of water is 1. A cubic foot of water weighs about 1000 oz., Av. Therefore the specific gravity of any body in thousandths, will represent the weight of a cubic foot in ounces.

RULE.

To the earth's diameter, (41815224 feet,) add the height of the eye, and multiply the sum by the height of the eye. The square root of the product is the distance at which an object ON THE SURFACE of the earth or water can be seen.

Work in the same way with the height of the object, and the sum of the two results is the distance at which the object may be seen.

How far may a mountain, that is 1 miles high, be seen from the mast-head of a ship, 50 feet above the surface of the water?

=

(41815224 + 50) × 50 45724 ft. or 83 ms. ✔(41815224 + 7920) × 7920 = 575534 ft. or 109 Ans. 117 ms.

ms.

4. How far can Bunker Hill Monument, which is 282 feet above the level of the sea, be seen from the deck of a vessel, the spectator's eye being 15 feet above the water?

5. How far may a mountain 23 miles high, be seen from the mast-head of a vessel, 40 feet above the water?

PROBLEM III.

To determine the distance of a gun, or a thunder cloud, from seeing the flash, and hearing the report.

RULE.

Multiply the number of seconds that elapse between the flash and the report by 1142, for the distance in feet.

6. Four and a half seconds after seeing the flash of a cannon, the report was heard. What was the distance?

7. What is the distance of an electrical cloud, if the thunder is heard in 2 seconds after the flash is seen?

PROBLEM IV.

To find the pressure of water against the banks of a stream or the dam of a pond.

RULE.

Multiply the area of the bank by one half the depth of the water, for the cubical contents of a column of water equivalent to the pressure.

8. The gate of a floom is 18 feet deep and 16 feet wide. What pressure does it sustain ?

9. What amount of pressure is sustained by a bank whose area is 5694 feet, the average depth of water being 10.5 feet?

QUESTIONS FOR REVIEW.

EVERY PRINCIPLE INVOLVED IN THE FOLLOWING QUESTIONS SHOULD BE FULLY AND CLEARLY EXPLAINED BY THE PUPIL.

What is ARITHMETIC? What is a number? In how many ways are written numbers expressed? How many figures are employed for the purpose ? What are they called? Why are they so called? What is a unit? What is an abstract number?—an applicate number? What is an integer?—a fraction? Of how many operations does Arithmetic consist? What are they called? What is the object of each?

What is NUMERATION? How many modes of numeration are now in use? Describe the Roman method? What is the leading principle of the Arabic method? How many characters are employed to represent numbers? What does zero represent? What is its use? What is the decimal point? How many places are embraced in a period? Repeat the numeration table for integers-for decimals.* How are decimals read? What is the effect of zeroes at the right of decimals? How may this be shown? Write and explain on the board, a number containing seven periods of integers, and twenty places of decimals. How many figures were embraced in a period, in the ancient English system of Numeration?

* The table may be continued to any extent. The denominations above duodecillions, to the twenty-second period, are: Tredecillions, Quatuordecillions, Quindecillions, Sexdecillions, Septendecillions, Octodecillions, Novemdecillions, Vigintillions.