?

22. If 4 cwt. be carried 36 miles for $51, how many pounds can be sent 20 miles for the same money 23. A person owning of a coal mine, sells of his share for 570 dollars. What is the whole mine worth?

24. If the discount on $106, for a year, be $6, what is the discount on $477, for the same time?

XI.

MEASUREMENT

OF SURFACES, SOLIDS AND CAPACITIES.

It has already been taught, that surfaces are measured in squares, and, that solid bodies are measured in cubes. A SQUARE is a figure, that has four equal sides, and four equal angles. Its angles are called right angles: angles more pointed are called acute angles; and those less pointed, obtuse angles. To find the area of a square, in smaller squares-Multiply one side into itself.

1. How many square feet are there in a table that measures 4 feet on every side? How many square inches? A PARALLELOGRAM is a four-sided figure, having opposite sides equal, and having four right angles. To find the area of a parallelogram-Multiply the length into the breadth.

2. How many square rods in a garden measuring 4 rods in length, and 3 in breadth? How many square

A TRIANGLE is a figure, that has three sides and three angles. A triangle, which has one right angle, is called a

RIGHT-ANGLED TRIANGLE.

To find the

area of a right-angled triangle-Multiply the base by half the perpendicular.

KYPOTENUSE.

feet?

PERPENDICULAR

3. How many square rods are there in a right-angled triangular field, measuring 98 rods on the base, and 75 ods on the perpendicular? How many acres?

A CIRCLE is a plane surface, bound ed by one curve line, called the circum ference. The diameter being known, to find the circumference-Multiply the diameter by 3.14159. Then, to find the area-Multiply half the circumference by half the diameter.

Thiameter

4. How many square inches are there in the head of a barrel, the diameter of which measures 17 inches? A CUBE is a regular solid body, having six equal, square sides. To find its contents in smaller cubes-Multiply the breadth of a side twice into itself. The product of the length, breadth, and thickness is the contents of any thing, whose opposite sides are equal.

5. How many cubic inches are there in a box measuring 34 inches in length, 26 in width, and 18 in depth? A CYLINDER is a round body, with equal, circular ends. To find its cubical contents-Find the area of one end, and multiply this by the length.

6. How many cubic inches are there in a drum measuring 16 inches across the head, and 18 inches in length?

PLASTERING AND PAVING are charged by the square

yard.

Their surface is first found in square feet, and then reduced to square yards.

7. How many square yards of plastering in the ceiling and four sides of a room, that is 15 feet long, 12 feet wide, and 10 feet high; deducting two doors, 7 by 4 feet each, and four windows, 5 by 36 feet each?

8. How many bricks are required to pave a cellar, that is 48 feet long and 30 feet wide; allowing each brick to be 8 inches long, and 3.8 inches wide? Here find the area of the cellar in square inches, and divide it by the square incnes in the area of a brick.

SHINGLES AND CLAPBOARDS are of various dimenTherefore, to know how many are requisite to cover

sions

$

*

210

a building, we find the number of square inches in the roof or side to be covered, and divide this number by the number of square inches, that one shingle or clapboard will cover.

9 If shingles 4 inches in width be laid so that 6 inches of their length is exposed to the weather, how many are required to cover a roof 45 by 32 feet?

10. How many clapboards, each covering 46 by 4 inches, are sufficient for the side of a house 45 by 22 feet?

BOARDS are sold by the thousand square feet, and each board is measured thus-Multiply the length in feet by the width in inches, and divide the product by 12; the quotient will be square feet.

11. How many square feet are there in 17 boards, each board being 21 feet long, and 18.5 inches wide?

12. How many square feet of boards will floor a room 14 by 18 feet, allowing of the stuff for waste?

PLANK AND JOIST are measured by finding how many square feet of boards, one inch in thickness, they are equal to. Therefore-Multiply the length in feet by the width in inches, and this product by the depth in inches; then divide the last product by 12, for the square feet.

13. How many square feet in a plank that is 9 feet in length, 14 inches in width, and 2.4 inches in depth? 14. How many square feet in a joist that is 13 feet long, 4 inches wide, and 8.2 inches deep?

To measure hewn

TIMBER is sold by the cubic ton. timber-Multiply the length in feet by the width in inches, and this product by the depth in inches; divide by 144, for the cubic feet, and then by 50 for the tons.

To measure round timber-Take the circumference in inches, by girding the log, one-third of the way from the but to the top; then multiply the length in feet, by the square of of the circumference; divide by 144 for the cubic feet, and then by 40 for the tons.

15. How much hewn timber in a stick measuring 25 feet in length, 19 inches in width, and 20 inches in depth? 16. How much round timber in a log, 30 feet long, and 55 inches in circumference?

OF EARTH.

CELLARS, WELLS, and other pits, are measured by the cube of six-feet side; and this cube is called a SQUARE To measure a cellar-Add together the depth of the four corners, divide the sum by 4, multiply the quotient by the length, and this product by the width, all in feet, for the cubic feet; then divide by 216 for the squares. To measure a well-Proceed as with a cylinder to find the cubic feet, and divide by 216 for the squares.

17. How many squares in a cellar, the length being 30 ft., width 22 ft., depth at corners, 12 ft., 9 ft., 7 ft., and 4 ft.? 18. At $1.08 a square, what is to be paid for digging a well, 60 feet deep, and 8 feet in diameter?

STONE WALLS are measured by the perch, of 243 cubic feet. To measure a straight wall-Multiply, in feet, the length by the height, and this product by the thickness, for the cubic feet; then divide by 24.75 for the perches.

To measure a circular wall-Take the diameter, to the centre of the thickness of the wall, and compute the circumference, in feet. Then multiply the circumference, height, and thickness together, all in feet, for the cubic feet, and divide by 24.75, for the perches.

19. How much wall, of 2 feet thickness, and 8 feet height, in a cellar measuring 36 feet on every side within the clear? 20. How much wall in a well 40 feet deep; the wall being 2 feet thick, and the diameter being 4.5 feet?

BINS, BOXES, &c., holding commodities sold by the gallon or bushel, are measured thus-Find the contents in cubic inches, as already taught; then divide by 231 for wine gallons, or, by 2150.4 for bushels.

21. How many gallons in a vat, measuring 60 inches in length, 36 inches in breadth, and 72 inches in depth?

22. How many bushels of grain in a bin, 84 inches in ength, 32 inches in breadth, and 48 inches in depth?

CYLINDRIC VESSELS, such as tubs and cisterns for holding water, are measured thus-Multiply, in inches, the diameter of one end into itself, and this product into the height then divide by 294 for the wine gallons.

If the ends of the vessel be unequal-Multiply the greater diameter by the less, and to the product add of the square of their difference; multiply this sum by the height, and divide by 294, for the gallons.

23. How many gallons will a tub hold, the diameter of which is 18 inches, and the height 22 inches?

24. How many gallons of water will a cistern hold, measuring 72 inches across the bottom, 60 inches across the top, and 84 inches in height?

THE CAPACITY OF CASKS is found as followsTake the interior dimensions as nearly as possible. SubIract the diameter of the head from the diameter at the bung. Multiply the difference by .7, if the staves be MUCH curved; or by .6, if LITTLE curved; or by .65, if they be of MEDIUM curve. Add the product to the head diameter, and the sum will be the mean diameter. Square the mean diameter; multiply the square by the length of the cask, and divide this product by 294, for wine gallons.

25. Find the number of gallons in a cask of medium curve, 47 inches in length, 31 inches diameter at the bung, and 26 inches diameter at the head.

26. What is the capacity of a cask, much curved, measuring 32.5 inches in length, 19 inches at the bung, and 15.4 inches at the head?

XII.

DUODECIMALS.

DUODECIMALS are compound numbers, the value of whose denominations diminishes in a uniform ratio of 12. They are applied to square and cubic measure.

The denominations of duodecimals are the foot, (f.), the prime or inch, ('), the second, ("), the third, ("), the fourth, (""), the fifth, (), and so on. Accordingly, the expression, 3 1'7" 9 6 denotes 3 feet 1 prime 7 seconds 9 thirds 6 fourths.

The accents, used to distinguish the denominations be ow feet, are called indices.