Thus, a map of the United States which is drawn so that 200 miles measured anywhere across the country is represented on the map by a distance of 1 inch, is said to be drawn to the scale of 200 miles to an inch (scale: 200 mi. = 1 in.).

EXERCISES.

1. Consult a map of the United States. Find the scale to which it is drawn. Find from this map the number of miles in a straight line from Boston to San Francisco.

2. A map of Illinois drawn to the scale of 200 mi. to an inch is 1 in. long. How many miles long is the state?

3. On a map drawn to the scale of 240 mi. to an inch, the distance from Chicago to Denver is 3 in. How many miles is it from Chicago to Denver?

4. In the house of which the floor plan is shown in § 140, the width of the living room is 15 ft. By measuring the width of the living room in the plan, find the scale to which the plan is drawn.

5. From the scale found in Ex. 4, determine the number of feet in the width of the porch. Find the length of the porch.

6. How many feet wide is the dining room of this house?

7. Draw a rectangle representing a rectangular field that is 1200 ft. long and 480 ft. wide to a scale of 240 ft. to an inch. What are the dimensions of the drawing?

8. The distance from A to the inaccessible

point B may be obtained as follows: Measure a base line AC. Measure ∠ACB and ∠BAС. Then construct a map of the measurements to scale, and determine from the map the distance from A to B.

C

B

A

If AC = 960 ft., ∠ ACB = 40°, and ∠BAC = 90°, draw a map of the measurements to the scale of 160 ft. to an inch, and compute AB from the map.

9. In order to find the height of a church spire CD, the base line AB is measured 75 ft. long toward the foot of the spire D. It is found that DAC = 50° and ∠DBC=80°. Make a drawing of these measurements to the scale of 25 ft. to an inch, and compute the height CD of the spire.

C

10. A and B are two forts in the A lines of the enemy, and it is desired to know their distance apart and their distances A from our lines. From point C in our lines, we measure ∠ACD and ∠BCD. Then we go to a second point D and measure CDA and ∠ CDB. ZACD = 120°, ∠BCD = 50°, ∠CDA = 45°, ∠ CDB = 100°, and CD = 2000 ft. Draw a plan to the scale of 500 ft. to the inch, and find the distances AC, BD, and AВ.

141. The plane table. A plane table consists of a drawing board mounted upon a tripod. A sheet of paper is pinned on the board, and a straightedge is laid on it for sighting and drawing lines. The instrument is used for finding the distances between inaccessible points and for making maps of small areas.

In using the plane table for finding the distance between two points Mand N, the instrument is set up at any convenient point A. A sheet of paper is fastened on the board, and a pin stuck through the paper into the board at a point a directly over A. The straightedge is placed against the pin, and a base line ab drawn on the paper toward a sec

pin is then removed to a point 6 directly over B, in line ab. Lines on and bm are then drawn toward N and M, respectively. The distance AB is measured. Lines an and bn meet at n, and am and bm meet at m. Then mn is drawn and measured. Also ab is measured. From these measurements the distance MN is computed by proportion.

In using the plane table for making a map or for other kinds of measurements, the procedure is very similar to that described above.

EXERCISES

1. Prove that the line MN in the figure of § 141 may be found from the proportion AB

MN mn

ab

SUGGESTION. - For proving this proportion use the proportion

3. The following drawing shows the method of making a map cdefg of the river bank CDEFG by use of the plane table. Study the drawing, and explain how the points c,

NOTE. - It will be found a very interesting and valuable experience, if the school will provide, or the students make, a plane table and use it out of doors in doing such work as is suggested in the exercises above.

CHAPTER VII

INEQUALITIES: METHODS OF ATTACK

142. Inequalities. - An inequality is a statement that two numbers or magnitudes are unequal.

The symbols of inequality are >, meaning "is greater than," and <, meaning "is less than."

Thus "a is greater than 6" is written a > b.
And "a is less than 6" is written a < b.

The inequalities a > b and x > y are of the same order.
The inequalities a > b and x < y are of the reverse order.

143. Theorem. - If two sides of a triangle are unequal, the angles opposite these sides are unequal, the angle opposite the greater side being the greater.

2. Then let CD be a part of AC equal to BC. Draw DB.