Townships are subdivided into Sections, and sections into Half-Sections, Quarter-Sections, Half-Quarter-Sections, Quarter-Quarter-Sections, and Lots.

Diagram No. 1 shows the sub-divisions of a Township into Sec tions, and how they are numbered, commencing at the N. E. corner. Diagram No. 2 shows the sub-divisions of a Section, on an enlarged scale, and how they are named.

A Lot is a sub-division of a section, usually of irregular form, on account of bordering upon a navigable river or lake-containing as near as may be the area of a Quarter-Quarter-Section, and described as lot No. 1, 2, 3, etc., of a particular section.

City and village plats are usually sub-divided into Blocks, and these into Lots.

1. If a township of land is equally divided among 288 families, how many acres does each receive? What part of a section ?

2. What number of rails will enclose a quarter-section of land with a fence 6 rails high, and 3 lengths for every 2 rods; and what will be the cost of the rails, at $40 per thousand?

3. A man bought the S. of a section of land at $21 an acre, and afterward sold the E. of what he bought at $4.37 an acre. What was his gain?

4. If I buy the N.E. and the E. of N. W. † of a section of land, how many acres do I purchase? What part of a whole section? How are the parts located in respect to each other?

5. A speculator bought of the Ill. Central R. R. Co., the S. of Section 4, township 10 north, range 6 east, at $2 an acre. He afterward sold the E. of S. E. at $2.75 an acre; the N.W. † of S.E. † at $3 N. of S. W. at $3.84 an acre. he left? What was his gain on the purchase price of the whole? Draw diagram.

an acre; and the How many acres has

6. A man having purchased a section of land from the U. S. Government at $1.25 an acre, sold the S. of S. W. at $2.50 an acre; the N.E. of N. W. at $1.75 an acre; the W. of S. E. at $2 an acre; and the W. 11 of S. W. of N.E. at $3 an acre. How many acres has he remaining, and what is his gain, provided the remainder is sold at $24 an acre? Draw diagram and explain.

RECTANGULAR SOLIDS.

469. A Rectangular Solid is a body bounded by six rectangular plane faces.

The opposite sides are equal and

parallel.

It has three dimensions-length, breadth, and thickness.

When all its faces are equal, it is called a Cube.

470. The Volume or Solid Contents of a body is the space included within the surfaces which bound it, and is expressed by the number of times it contains a given unit of measure.

471. The Unit of Measure for solids is a cube, the edge of which is a unit of some known length.

Thus, the unit of cubic inches is a cube the edge of which is 1 inch, or 1 cubic inch; of cubic feet, 1 cubic foot, etc.

contains 3 such sections, or 3 times 9 cu. ft., which are 27 cu. ft. Hence the volume of 1 cu. yd. is 27 cu. ft.

So the volume of a solid, formed of two adjacent sections, is ex pressed by 3 cu. ft. x 3 x 2 = 18 cu. ft.

472. To find the volume of a rectangular solid

RULE. Find the product of the numbers denoting the three dimensions, expressed in the same denomination ; this result is the volume.

473. To find a required dimension of a rectangular solid:

RULE.-Divide the volume by the product of the numbers denoting the other two dimensions; the quotient will be the required dimension.

474. 1. What are the contents of a rectangular solid 6 ft. long and 4 ft. square?

2. What is the volume of a solid 9 ft. long, 4 ft. wide, and 3 ft. thick ?

3. A vat 12 ft. square contains 1224 cu. ft. What is its depth?

4. What is the volume of a bin, the inside dimensions of which are 8 ft. 6 in. by 6 ft. by 4 ft. 4 in.?

5. How many cubic yards of earth must be removed in digging a cellar 36 ft. long, 24 ft. wide, and 6 ft. deep? Find the volume of rectangular solids having the following dimensions :

6. Of a cube the edge of which is 1 yd. 1 ft. 9 in.

7. Of a solid 6 yd. 2 ft. 7 in. by 3 ft. 4 in. by 2 ft. 11 in. 8. Of a solid 5 ft. square and the height 6.4 ft. Find the required dimension of rectangular solids, the volumes and two dimensions being as follows: 9. Volume, 6 cu. ft.; length, 8 ft.; width, 8 ft. 10. Volume, 20 cu. ft.; length, 36 ft.; width, 10 in.

11. Volume, 13 cu. yd. 14 cu. ft. 900 cu. in.; width, 7 ft. 3 in.; height, 5 ft. 6 in.

12. How many cubic feet of air in a room that is 24 ft. 9 in. long, 18 ft. 4 in. wide, and 10 ft. 8 in. high?

13. How many cords in a pile of wood 30 ft. long, 8 ft. wide, and 6 ft. 6 in. high?

14. A pile of wood containing 67 cords, is 90 ft. long and 12 ft. wide. How high is it?

15. What will be the cost of a pile of wood 12 ft. 6 in. long, 8 ft. wide, and 4 ft. 6 in. high, at $3.75 a cord?

16. What will it cost to dig a cellar 45 ft. long, 28 ft. wide, and 8 ft. 6 in. deep, at $.42 a cubic yard?

17. What must be the length of a load of wood that is 3 ft. high and 5 ft. 4 in. wide, to contain a cord?

18. How many cans, 8 in. by 6 in. by 3 in., can be packed in a box 32 in. by 24 in. by 15 in. in the clear?

19. At $3 a cord, what is the value of the wood that can be piled under a shed 50 ft. long, 25 ft. wide, and 12 ft. high?

20. In building a house, 200 joists 10 in. by 3 in. were used, which together amounted to 1000 cu. ft. What was the length of each?