GRAPHICAL REPRESENTATIONS

217. Graphical methods of representing relations between different measurements are so extensively used in many lines of work that it seems best to give a brief treatment of the subject here. Such graphical representations as are given in the following exercises show relations pictorially in a much clearer manner than can be shown by a mere statement of figures.

1 in.

Ex. 1. Explain graphically the relation between an inch and a centimeter. The two lines drawn accurately to scale represent graphically the relation between the inch and the centimeter.

1 cm.

Ex. 2. Draw a line 1.5 in. long and find the number of centimeters in it.

Ex. 3. Explain graphically the relation between the pound and the kilogram, given 1kg 2.2 lb.

=

1 lb.

1Kg

= 2.2 lb.

Ex. 4. Explain graphically the relation between a pint and a liter, given 11 = 1.76 pt.

Ex. 5. From a diagram find (a) the number of centimeters in 4 in., (b) the number of liters in a gallon, (c) the number of pounds in 5Kg.

Ex. 6. The values of manufactures produced in the United States, Germany, France and Great Britain in 1860 were $1907000000, $1995000000, $2092000000,

$2808000000 respectively, and in and in 1894 they were $9498000000, $3357000000, $2900000000, $4263000000 respectively.

These facts may be represented graphically as follows:

These measurements, drawn accurately to a scale, show at a glance the comparative growth in manufactures produced in the different countries mentioned from 1860 to 1894.

Ex. 7. The areas of England and Michigan are 50839 and 58915 square miles respectively. The populations are approximately 31000000 and 2421000. Represent graphically the comparative sizes and the comparative density in population of the two.

The square roots of the numbers representing the areas correct to units' place are 225 and 243 respectively. The ratio between these

two numbers reduces to 5 to 5.4. If some convenient unit of measure be taken, and squares be constructed with sides equal to 5

to 5.4 of these units, these squares will represent graphically the comparative areas. The comparative density in population will be represented by the number of dots that appear in each square, it being assumed that a dot represents 100000 in population. There will then be 310 dots in the square representing England and 24 in the square representing Michigan.

Ex. 8. On a certain day between 6 A.M. and 7 P.M. the thermometer registers as follows: 6 A.M., 20°; 7 a.m., 22.5°; 8 A.M., 27°; 9 A.M., 35°; 10 A.M., 42.5°; 11 A.M., 48°; 12 M., 52°; 1 P.M., 55°; 2 P.M., 60°; 3 P.M., 62°; 4 P.M., 60°; 5 P.M., 50°; 6 P.M., 42°; 7 P.M., 35o. Illustrate graphically this variation in temperature.

Draw two straight lines perpendicular to each other. Measure off on the horizontal line OX equal spaces, each representing 1 hr., and on the perpendicular line OY equal spaces, each one representing 10°. The temperature at 6 A.M. is shown at 0; at 7 A.M. at A, a distance

of 1 unit along OX and of a unit above OX parallel to OY; at 8 A.M. at B, a distance of 2 units along OX and .7 of a unit above OX parallel to OY. In the same way points may be located showing the temperature at each hour. A continuous curve drawn through these points is the temperature curve for the day from 6 A.M. to 7 P.M. This curve

shows at a glance the variation in temperature between the hours given.

Ex. 9. Two trains leave a certain place traveling in the same direction, one at the rate of 20 mi. an hour, and the other at the rate of 40 mi. an hour. If the second train leaves 3 hr. after the first, when and where will it pass the first?

Let each space along OX represent 20 mi., and each space along OY represent 1 hr. At the end of the first hour the first train is at A ; at the end of the second

6 hr. Y

hour at B; and at the
end of the sixth hour
at P. At the end of 5 hr.
the fourth hour the sec-
ond train, which starts r.
from O', spaces above
O, since it starts 3 hr.
later, is at A'; at the
end of the fifth hour at
B'; and at the end of
the sixth hour at P.
The point P, where the

line OP and O'P cross,
is the place where the

3 hr. '

2 hr.

1 hr.

second train overtakes the first. If from P perpendiculars PX and PY are dropped upon OX and OY, then the distances OX and OY will represent the space traveled and the time that has elapsed since the starting of the first train till the second one overtakes it. OX contains 6 distance spaces, and represents 120 mi., while OY contains 6 time spaces, and represents 6 hr.

EXERCISE 47

For convenience in constructing the graphical representations required in the following exercises, the student should provide himself with paper ruled in small squares.

1. Illustrate graphically the comparative areas and the comparative density in population in the following cases:

2. On Jan. 1, 1904, the thermometer registered the temperature at 1 A.M., and at each succeeding hour till midnight, at Ypsilanti, Michigan and Havana, Cuba, respectively as follows:

3. The mean temperature for January (average for the 31 da. of the month) for the same hours and places as in Ex. 2 was as follows:

1 2 3 4 5 6 7 8 9 10 11 N. 14.2° 14.3° 14.2° 14.2° 14.1° 14.3° 14.2° 14.5° 15.3° 17.1° 18.1° 19.7° 67.2° 67.0° 66.7° 66.4° 66.0° 65.6° 65.4° 66.6° 68.9° 71.0° 73.1° 74.0°

1

3

4

5

6

2 9 10 11 Mt. 20.4° 20.7 20.6° 19.9° 18.8° 18.1° 17.5° 16.5° 16.0° 15.4° 14.5° 16.6° 74.7° 74.9° 75.0° 74.7° 74.2° 72.9° 71.6° 70.6° 69.8° 69.0° 68.4° 67.9°

8

Illustrate graphically.