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3. The hypothenuse is 110 ft. and the base is 19.5 ft. Find the perpendicular to two decimal places.

4. The hypothenuse is 86 ft. and the base is equal to the perpendicular. Find both of the wanting terms to two decimal places.

5. The hypothenuse is 127 ft. and the base is equal to of the perpendicular. Find wanting terms to three decimal places.

REMARKS.—1. Observe, in example 4, that the square root of ļ the square of the hypothenuse is equal to the base; and in example 5, that the square root of f of the square of the hypothenuse is equal to the base.

2. Carry all roots to two decimal places.

6. What is the length of one side of a square field, the area of which is one acre?

7. How many feet of fence will enclose a square field containing five acres?

8 I wish to lay out ten acres in the form of a square. What must be its, frontage in feet and inches?

9. What is the distance from the top of a perpendicular flag-staff 105 ft. high to a point 4 rods from the base and on a level with it?

10. What is the width of a street in which a ladder 60 ft. long can be so placed that it will reach the eaves of a building 40 ft. high on one side of the street, and of another building 50 ft. high on the opposite side of the street?

11. What length of line will reach from the lower corner to the opposite upper corner of a room 64 ft. long, 27 ft. wide, and 21 ft. high?

12. If a farm be one mile square, how far is it diagonally across from corner to corner? Find the answer in rods, feet, and inches.

13. How many rods of fence will enclose a square field containing 20 acres?

14. A farm of 180 acres is in the form of a rectangle, the length of which is twice its width. How many rods of fence will enclose it?

15. What will be the base line of a farm of 136 A. 40 sq. rd. if it is in the form of a right-angled triangle, with the base equal to the perpendicular?


447. The Unit of measure used by land surveyors is Gunter's Chain, 4 rods, or 66 feet, in length, and consisting of 100 links.

RFMARK. — Rods are seldom used in Surveyor's Measure, it being customary to give distances in chains and links or hundreths.


7.92 inches..

1 link 1. 25 links

1 rod rd.
4 rods, or 66 feet ---

80 chains, or 320 rods 1 mile --- mi.
Scale, descending, 80, 4, 25, 7.92; ascending, 7.92, 25, 4, 80,

1 chain -


448. 1. Reduce 3 mi. 27 ch. 19 1. 4 in. to inches.
2. Reduce 14841 l. to higher denominations.
3. Reduce 16 of a chain to lower denominations.
4. Reduce .953 of a mile to links.

5. A lot having a frontage of 4 rods contains of an acre. What is its depth in chains, links, and inches?

6. A field 37 ch.42 1. long, and 30 ch. 21 l. wide, will require how many feet of fence to enclose it?

7. How many rods of fence wire will enclose a farm 21 ch. 50 1. long and 18 ch. 60 l. wide, if the fence be made 6 wires high?

8. A garden is 3074 feet long and 2504 feet wide. What is the girt, in chains, links, and inches, of a wall surrounding it ?

An errand boy goes from his starting point east 33 ch. 50 1. 3 in., thence north 14 ch..90 1. 2 in., and returns. How many full steps of 2 feet 4 inches did he take, and what was the remaining distance in inches ?


449. The Unit of land measure

the acre.


sq. rd.


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sq. mi.

625 square links (sq. l.) = 1 square rod.
rods ---

1 square
chain.. sq.

10 square chains, or

=1 acre


640 acres.


mile --- sq. REMARK.-In surveying United States lands, a selected North and South line is surveyed as a Principal Meridian, and an East and West line, intersecting this, is surveyed as a Base Line. From these, other lines are run at right angles, six miles apart, which divide the territory into Townships six miles square. N.

The surface of the earth being convex, these merid6 miles.

ians converge slightly. The townships and sections are, therefore, not perfectly rectangular; thus is created the necessity for occasional offsets called Correction Lines.

Each township (Tp.) is divided into 36 equal squares of 1 square mile each, as shown in the first diagram. These squares are called sections (Sec.), and are divided into halves and quarters; each quarter-section, 160 acres, is in turn divided into halves, or lots of 80 acres, and quarter or half lots of 40 acres each, as shown in the second diagram.

The row of townships running north and south is called a Range; the townships in each range are numbered north and south from the base line, and the ranges numbered east and west from the principal












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1 Mile,

N. 42 Section.

320 A cres.

1 Mile.


meridian. The numbering of the sections in every township is as in the township diagram given, and the corners of all quarter-sections are permanently marked by monuments of stone or wood, and a description of each monumeột and its location (surroundings) made in the field notes of the surveyor.

The advantages of the United States survey over all others are. 1st, its official character and uni. formity; and 2d, its simplicity. Any one having a sectional map of the United States may place a pencil point upon any described land, thus knowing absolutely its exact location.

For example, Sec. 26, Tp. 24, N. of Range 8, E. of the 5th Principal Meridian, describes a section in the 24th tier of townships north of the base line, and 8th range east of the fifth principal meridian,

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1. Make a diagram of a township, and locate S. of Sec. 21, and mark its acreage.

2. Make a diagram of a township, and locate S. E. of Sec. 16, and mark its acreage.

3. Make a diagram of a township, and locate N. W. I of S. W. I of Sec. 12, and mark its acreage.

4. Make a diagram of a township, and locate Secs. 35, 26, and E. f of 27, and mark their acreage.

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1 FT.

451. Cubic Measure is used in measuring solids or volume.

1 FT.


452. A Solid is that which has length, breadth, and thickness; as the walls of buildings, bins of grain, timber wood, stone, etc.

3 FT.

3 FT.

453. A Cube is a regular solid bounded by six equal square sides, or faces ; hence its length, breadth, and thickness are equal.

454. The Measuring Unit of solids w is a cube, the edge of which is a linear unit.

Thus a cubic foot is a cube, each edge of which is 1 foot; a cubic yard is a cube, each edge of which is 1 yard. See the accompanying diagrams.


455. To Find the Volume of a Solid.
Rule.—Multiply together its length, breadth, and thickness.

1728 cubic inches (cu. in.) = 1 cubic foot..
27 cubic feet.

1 cubic yard -----cu. yd.
128 cubic feet.

= l cord of wood --cd. ft.

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458. To Find the Cubical Contents of Square Timber.

Rule.—Multiply together the feet measurements of length, width, and depth.

459. To Carry Timbers, one person supporting an end and two others with bar.

DIRECTIONS.—Let the two with the bar lift at a point | the length from the end.

REMARK. -1. Formerly a perch of masonry was 241 cu. ft.; but the perch of 164 cu. ft., which is 164 ft. long, 1 ft. high, and 1 ft. wide, is now in gencral use.

2. A cubic yard of carth is called a load.

3. Mechanics estimate their work on walls by the girt, and no allowance is made for windows or doors. In estimating the amount of material required, such allowances are made.

Formulas for Rectangular Solids.
Length x Breadth x Hight = Volume.
Volume = (Length x Ercadth) Ilight.
Volume = Length x Ilight) Breadth.

Volume = (Breadth x llight) Length.
REMARK.—The three given dimensions must be expressed in units of the same denomination,

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460. To Find the Number of Ericks for a Wall
Rule.-Multiply the cubic feet by 23%, and add odo.

REMARK.— For guide in purchasing material the above will be found correct for bricks 8 in. X 4 in. X 2 in., after allowing for mortar.

461. To Find the Number of Perches in a Wall.
Rule.-Divide the contents of the wall, in feet, by 16%.


462. 1. Reduce 468093 cu. in. to higher denominations.
2. Reduce 132 cu. yd. 11 cu. ft. 981 cu. in. to cubic inches.

3. What is the volume of a solid 8 ft. 3 in. long, 5 ft. 10 in. high, and 4 ft. 6 in, wide ? 4. How


cubic feet of air in a room 26 ft. 8 in. long, 22 ft. 6 in. wide, and 12 ft. high ?

5. How many cubic yards of earth must be removed in digging a cellar 60 ft. long, 304 ft. wide, and 71 ft. deep ?

6. How many perches of masonry, of 164 feet each, in a wall 85 ft. long, 32 ft. high, and 11 ft. thick ?

7. Reduce of a cubic inch to the fraction of a cubic yard.
8. What decimal part of a cubic yard is 7 cu. ft. 108 cu. in. ?
9. What fraction of a cubic foot is 220 cu. in. ?
10. Reduce .525 of a cubic yard to lower denominations.

11. What will be the cost, at 21% per cubic yard, of excavating for a reser. voir 180 ft. long, 105 ft. 3 in. wide, and 15 ft. 9 in. deep ?

12. What will be the cost of building the walls of a block 140 ft. long, 66 ft. wide, and 57 ft. high, at $1.40 per perch of 161 cu. ft., if the wall is 16 in. thick, and no allowance be made for openings ?

13. How many common bricks will be required for the above wall, allowance being made for 28 windows each 31 ft. wide and 8 ft. high, 48 windows each 3 ft. 9 in. wide and 8 ft. high, and 4 doors each 8 ft. wide and 11 ft. high ?

14. A room 28 ft. long, 18 ft. wide, and 12 ft. high, will store how many cords of wood ?

15. How many cords of wood in a pile 108 ft. long, 7 ft. 9 in. high, and 6 ft. wide ?

16. From a pile of wood 71 ft. 6 in. long, 9 ft. 4 in. wide, and 6 ft. 8 in. high, 21} cords were sold. What was the length of the pile remaining?

17. At $4.75 per cord, what will it cost to fill with wood a shed 34 ft. long, 18 ft. wide, and 10 ft. high ? 18.

What is the weight of a block of granite 11 ft. 3 in. long, 3 ft. 6 in. thick, and 8 ft. 4 in. wide, if it weighs 166 lb. per cubic foot ?

19. What is the weight of a white oak timber 15 in. square and 40 ft. long, if the weight per cubic foot be 72.5 lb. ?

20. How many cubes 1 in. on each edge can be cut from a cubic yard of wood, if no allowance be made for waste by sawing?

21. Find the contents of a cube, each edge of which is 2 yd. 77 in.

22. How many perches of masonry in a wall 77 ft. high and 2 ft. thick, enclosing a yard 12 rods long and 9 rods wide? How many bricks will be required, and if bricks cost $6.50 per M and laying them cost $1.60 per M, what will be the cost of the wall ?

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