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Examples.

1. What is the superficial content of a board, that is 1 ft. 7 in. wide at one end, and 7 in. at the other; and 23 ft. 11 in. long?

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1 7+7=2

2÷1

21

1 mean width.

1 ft. 1' x 23 ft. 11'25 ft. 10′ 11′′ Ans.

2. What is the area of a piece of land that is 30 rds. long; 20 rds. wide at one end, and 18 rds. at the other? Ans. 570 rds. 3. What is the area of a hall that is 32 ft. long; 22 ft. wide at one end, and 20 at the other? Ans. 672 ft. 4. A man has a farm lying 300 rds. on the road, and the width of it at one end is 80 rds. and at the other 60: I demand the content of the farm.

Ans. 21000 rds. or 131 acres, 40 rds.

CASE III.

To measure the surface of a rightangled triangle.

NOTE.-A right angled triangle is formed by a right line falling perpendicularly on another line, as the line A B falling upon the line C B, makes a right angle at B.

A

B

RULE-Multiply the base C B, by half the perpendicular A B, and the product will be the area; or mut

N

tiply the base and perpendicular together, and half the product will be the area: also, the perpendicular multiplied by half the base, the product is the area.

NOTE.-All triangles, not having one right angle, are in general terms called oblique angled triangles, &c. Figure second represents an oblique angled triangle; and the same rules that apply in measuring a right angled triangle will apply in measuring an oblique angled triangle.

Figure 2.

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EXPLANATION-It is evident that the area of any triangle may be found by multiplying the base by half the perpendicular; or by multiplying the base and perpendicular together, and taking half the product; or by multiplying the perpendicular by half the base. By multiplying half of A B (fig. 1) by C B, the parallelogram DFCB is measured; EFA is in the triangle, and is not measured; C D E is not in the triangle and is measured in the parallelogram; C D E being equal to EFA there can be no loss sustained: in figure 2, multiplying half the perpendicular D C, by the base A B, reduces the triangle A D B to the parallelogram HPB A; T2 D is equal to T H A, and D 2 O is equal to OPB. Therefore it is evident that the parallelogram HP BA is equal in area to the triangle A D B.

Examples.

1. What is the content, of a piece of land 13 rods on the base, and 8.8 rods on the perpendicular, in form of figure first?

Half perp.

44×13.5-5914 rds. Ans.

2. What is the area of a piece of ground in form figure 2d, base 27.4 rds.; perpendicular 8.2 rds.? Ans. 112.34 rods.

3. What is the area of the gable end of a house, beam 40 feet, and height 12 feet 6 inches? Ans. 250 sq. ft.

4. What is the superficial content of a piece of board, in form of figure first: the length of which is 16 ft. 9 in. and the wide end 1 ft. 10 in. and the narrow end coming to a point?

ft.

1

"

in. in. ft. in. ft. in.
10+11x16 915 4 3 Ans.

5. What is the content of the second figure, A B 20 ft. and C D 6 ft. 4 in,? Ans. 63 ft. 4 in.

NOTE. The method of measuring every board with the pen would be too tedious for common practice; I therefore shall show the method of making a board rule, and measuring boards with the same.

CASE IV.

To make a board rule, and measure boards with it,

RULE. Prepare a hard piece of wood two feet long and about 3 inches wide, and about half an inch thick; divide the sides of the rule into lines about half an inch apart; number these lines beginning with 10 and so on till all are numbered; (it is more convenient to make the rule two feet and one inch long, fixing a steel point one inch from the end of the rule, and numbering the divisions of the rule on the odd inch, and when measuring the length of a board with a rule the steel point will scratch every two feet) divide a foot on cach line on the rule into as many equal parts as is expressed by the figures on the end of the rule; thus 1 foot on the line numbered 10 at the end must be divided into 10 equal. parts, and the line numbered 11, into eleven equal parts. &c. number each of these parts beginning with 1 and so on till all are numbered. See the following figure

ONE SIDE OF THE RULE.

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9 10 11 12 13 14 15 16 17 18 19 20 21 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

B

NOTE. At 1, 2, are the steel points; from 2 to A, and from 1 to B is 2 feet, and the feet are separated by the line 0, 0.

CASE V.

11.12 13 14 15 16

2

To measure boards with
this rule.

RULE. Find first the length of the board, and then look on the end of the rule for the number corresponding to the length of the board, apply that line of the rule to the middle of the board, and the width of the board will extend on that line to a number expressing its

area.

Examples.

1. What is the superficial content of a board, that is 11 feet long, and 12 in. wide? Ans. 11 ft.

NOTE.-The board being 11 ft. long, I look for the line numbered 11 at the end, and apply that line to the middle of the board, and the width extends to that division numbered 11, which is its content. Ans. 11 ft.

CASE VI.

To measure boards with Gunter's sliding rule.

RULE. Bring the width of the board in inches, on the slider, against 12 on the line above; then

look along on the line above for the number expressing the length of the board; and against that number, (on the slider) stands the number expressing the area of the board.

Example.

1. What is the content of a board that is 18 feet long and 10 inches wide? Ans. 15 feet.

NOTE.-I first brought the width of the board in inches, against 12, on the line above; I then looked on the line above for the length of the board, vis, 18; and against the length (on the slider) stood 15 the number, answering to the area of the board.

CASE VII,

To measure boards with Gunter's scale and dividers!

RULE. On the line of numbers extend from 1 to the width, and that extent will reach from the length to the superficial content or area; any right angled parallelogram may be measured by the same rule.

Example.

1. What is the superficial content of a board that is 20 feet long and 14 feet wide?

Ans. 25 feet.

NOTE. On the line of numbers I extend from 1, to 2 tenths, or of the distance from 1 to 2; and that extent reaches from 20, (the length of the board) to the superficial content 25.

CASE VIII.

To measure joist, plank, &c.

DEFINITION.-Joists are of different dimensions, sometimes 3 by 3, or 3 by 4, &c. plank are also of different thickness; and both plank and joists are reduced to board, or superficial measure, and are. measured thus,

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