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17. What must be the hight of a cubical bin that will hold 1000 bu. of wheat?

18. The width and hight of a crib of unshelled corn are equal, and each is one-third of its length. If the contents of the crib are 7465 bushels, what is its length ?

19. If the hight of an oat bin is twice its width, and its length is three and onehalf times its hight, what must be its dimensions, if the bin holds 1750 bushels?

20. A cubical cistern contains 630 barrels. How deep is it?

21. A square cistern, the capacity of which is 420 barrels, has a depth equal to only one-half its width. Find its dimensions.


475. Duodecimals are denominate fractions of either linear, square, or cubic measure. They are found by successive divisions of the unit by 12, and are added, subtracted, multiplied, and divided in the same manner as compound numbers, though they may be treated as fractions, 12 being the uniform denominator. The scale is uniformly 12.

476. The Unit of measure in Duodecimals is the foot. Its first division by 12 gives primes ( "); primes divided by 12 give seconds ("), seconds divided by 12 give thirds ("), and so on.

REMARK.-Duodecimals are but little used.


477. A Triangle is a plane figure bounded by three straight lines.

478 To find the area of a triangle, the base and hight being given. RULE. Multiply the base by one-half the hight.

To find the area of a triangle, when the three sides are given. RULE.Find one-half of the sum of the three sides; from this subtract each side separately; multiply together the four results thus obtained, and extract the square root of the product.

To find the area of any plane figure, the opposite sides of which arequal and parallel. RULE.Multiply the base by the perpendicular night.

To find the area of a plane figure, whose opposite sides are parallel but of unequal length. RULE.-Obtain the average length, and multiply by the per.. pendicular hight. CIRCUMFERENCE

479. A Circle is a plane figure bounded by a curved line, every part of which is equally distant from a point within called the center.



480. The Circumference of a circle is the curved line bounding it.


481. The Diameter of a circle is a straight line passing through the center and terminating in the circumference.

482. The Radius of a circle is a straight line passing from the center to any point of the circumference.

483. To find the circumference of a circle, the diameter being given. RULE.-Multiply the diameter by 3.1416.

To find the diameter of a circle, the circumference being given. RULE.Divide the circumference by 3.1416.

To find the area of a circle, the circumference and diameter being given. RULE.Multiply the circumference by the diameter, and divide the product by 4.

To find the side of a square equal in area to a given circle. RULE.Multiply the circumference by .2821.

To find the area of a square that can be inscribed within a given circle. RULE.Multiply the square of the radius by 2, and extract the square root of the result.


484, A Cylinder is a circular body of uniform diameter, the ends of which are parallel circles.

REMARK.—The convex surface of a cylinder is cqual to the surface of a rectangular body, the length and hight of which are equal to the circumference and hight of the cylinder. See the figure, A, B, C, D, back of the cylinder in the acompanying diagram,



485. To find the surface or area of a cylinder. RULE.Multiply the circumference by the hight.

To find the contents of a cylinder. RULE.—Multiply the area of the base by the hight.

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488. To find the surface of a regular pyramid or cone. RULE.- Multiply the perimeter or circumference of the base, by one-half the slant hight.

To find the contents of a pyramid or cone. RULE. - Multiply the area of the base by one-third the perpendicular night.


489. A Sphere is a solid bounded by a curved surface, all points of which are equally distant from a point within called the center.

490. The Diameter of a sphere is a line drawn through its center, terminating each way at the surface.


491. To find the surface of a sphere. RULE.Multiply the square of its diameter by 3.1416.

To find the volume of a sphere. RULE.Multiply the cube of the diameter by .5236.

To find how large a cube may be cut from any given spherc, or may be inscribed within it. RULE.Divide the square of the diameter of the sphere by 3, and extract the square root of the quotient; the root thus found will be the length of one side of the cube.

To gauge or measure the capacity of a cask. RULE.- Multiply the square of the mean diameter in inches by the length in inches, and this product by .0034; the result will be the capacity in gallons.

REMARK.— In case the cask is only partly full, stand it on end, find the mean diameter of the part filled, multiply its square by the hight, and that product by .0034.


REMARK.-In giving one example under each of the several preceding rules in measurements, the object is as much for reference as for practice in solving.

492 1. How many square feet in the gable end of a house 24 ft. wide and 6 ft 6 in. high ?

2. Find the number of square yards in a triangular sail, the sides of which are 36 ft., 45 ft., and 48 ft. respectively.

3. How many acres in a rectangular field 108 rods long and 48 rods wide ?

4. A farm stretches across an entire section, being 200 rods wide on the west. line and 160 rods wide on on the east line. How many acres in the farm ?

5 How many feet of fence will inclose a circular pond 82.5 ft. in diameter ? 6. What is the diameter of a circle, the circumference of which is 90 rods ?

7. The diameter of a circular park is 50 rods. How many acres does the park cover ?

8. What is the side of a square having an area equal to that of a circle 100 ft. in diameter ?

9. What is the largest square timber that can be hewn from a log 42 inches in diameter ?

10. What will be the cost of a sheet-iron smoke-stack 40 ft. high and 2 ft. in diameter, at 154 per square foot ? 11. Find the capacity in gallons of a tank 14 ft. deep and 18 ft. in diameter?

A pyramid has a triangular base 3 ft. on each side, and a slant hight of of 10 ft. Find the number of square feet in its surface.




13. A tent is in the form of a cone; if its slant hight is 16 ft. and its base circumference 30 ft., how many square yards of duck were used in making it ?

14. How many square inches of leather will cover a foot ball 8 in. in diameter? 15. How many cubic feet in the contents of a globe 4 ft. in diameter ?

16. The diameter of the earth is 7901 miles, and that of the planet Jupiter 85390 miles. How many spheres like the earth are equal to Jupiter ?

17. What will be the length of the largest cube that can be cut from a sphere 7901 miles in diameter ?

18. A cask 28 in. at each end, and 34 in. at the bilge, is 3 ft. long. How many gallons of water will it hold ?

19. If a cask 24 inches at the chime, 30 inches at the bung and 3 feet long, is ; full, how many more gallons may be put into it ?


493. Paper in the stationery trade is sold by the following

Table. 24 sheets

=1 quire. 20 quires

1 ream. 2 reams

1 bundle.
5 bundles

= 1 bale.
A bale contains 200 quires, or 4800 sheets.

REMARKS.-1. In copying, a folio is usually 100 words.

2. In type-setting, an em is the square of the body of a type, used as a unit by which to measure the amount of printed matter on a page.

Sheet folded into.

494. Books are sometimes classified by their size, or the number of pages in a sheet. Name.

2 leaves,

Quarto, 4to.
4 leaves,

Octavo, 8vo.
8 leaves,

Duodecimo, 12mo. 12 leaves,

16 leaves,

18 leaves,

24 leaves,

32 leaves,


Table for Counting.
12 units = 1 dozen.

12 dozen = 1 gross.
20 units
1 score.

gross 1 great gross.

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Table for Land and Lot Measures. 104 feet square

1 acre. 4 of an acre. 10 rods X 16 rods 1471 fect square

1 acre. * of an acre. 8 rods x 20 rods 20816 feet square

1 acre. 1 acre.

40 yards X 121 yards


495. The Metric System is a decimal system of denominate numbers. It is in use in nearly all the European States, in South America, Mexico, and Egypt. It is also used somewhat in Asia, and is authorized by law in the United States; but its use here is so limited as to justify only a reference to it, and the presentation of its unit equivalents in our weights and measures, as a reference for interested parties.

496. The Unit of Length and basis of the system is the Meter= 39.37+ inches, being one ten-millionth of the distance from the equator to the pole. The unit of area is the Ar (A.); the unit of solidity is the Ster (S.); the unit of weight is the Gram (G.); the unit of capacity is the Liter (L.). Higher denominations are called Dek'a (10), Hek'to (100), Kil'o (1000), and Myr'ia (10000). Lower orders are called Dec'i (tenths), Cen'ti (hundredths), Mil'li (thousandths).


dm =

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Metric Linear Table.
10 mil'li-me'ters (mm) =1 cen'ti-me'ter...
10 cenʼti-me'ters = 1 dec'i-meter
10 dec'i-me'ters = 1 Meter

10 me'ters
=l dek'a-me'ter

Dm 10 M.
10 dek'a-me'ters = 1 hek’to-me'ter.

Hm 100 M. 10 hek’to-me'ters = 1 kil'o-me'ter

Km 1000 M. 10 kil'o-me'ters = 1 myr'ia-me'ter

Mm = 10000 M. REMARKS.-1. All tables are formed in a similar manner. 2. In naming units, abbreviations are commonly used.

3. The system being on a decimal scale, the full mastery of the names of the higher and lower denominations, with unit equivalents, will be sufficient for practical use.

497. An Act of Congress requires all reductions from the Metric to the common system, or the reverse, to be made according to the following

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SQUARE MEASURE. 1 sq. inch = 6.452 sq. centimeters. 1 sq. centimeter = .155 of


inch. 1 sq. foot =.0929 of a sq. meter.

1 sy. decimeter = .1076 of a sq. foot. 1 sq. yard =.8361 of a sq. meter.

1 sq. meter

= 1.196 sq. yards. 1 sq. rod = 25.293 of a sq. meter. 1 ar = 3.954 sq. rods. 1 acre = 40.47 ars.

1 hektar 2.471 acres. 1 sq. mile = 259 hektars.

1 sq. kilometer = .3861 of a sq. mile.

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