these are drawn to an enlarged scale.

When a drawing is made to

scale, the scale should always be indicated on it. This may be done in various ways. Thus a mere statement may be made, e.g.,

1 inch to 10 feet;

or, the scale may be indicated by a fraction which gives the ratio of any line in the drawing to the actual line represented. The scale can also be shown graphically by means of a specially marked line. Both the latter methods are illustrated, for instance, on the map of the Kingdom of Saxony in The Times Atlas:

The scale should be expressed fractionally, that is, by express ing the ratio of a line in the drawing to the actual line represented. Thus in the first example above the scale is 1:120; in the second the scale is 1:870000.

When a drawing is made to scale, the distance between two objects can be measured directly, by merely measuring the distance between their corresponding points on the drawing. For instance, if 1 inch represents 120 feet, then 2.5 inches represents 300 feet. Another example: On the map of Saxony referred to above, the distance between Leipzig and Dresden is, approximately, 4 inches, and 41 inches 870000 gives about 62 miles as the distance in an airline between these cities. This method of finding distance can be used in solving many of the problems in trigonometry.* To find the length of the representative line in the drawing when the scale and the actual length are given, is an exercise in simple proportion; so, also, to find the actual length of a line when the scale and the length of the representative line are known.

* This is one of the methods which will be employed in this book in problems involving distance. Proficiency in drawing will be very helpful to the student.

10.]

EXAMPLES.

EXAMPLES.

17

1. When an inch represents 10 ft., how long must the lines be that will represent 3 in., 6 in., 1 ft., 2 ft., 5 ft., 15 ft., 7.5 ft., 30 ft., 40 ft., 55 ft.? What is the scale ?

2. When an inch represents 5 yd., how long must the lines be that will represent 2 yd., 4 yd., 7 yd., 11 yd., 3 yd. 2 ft., 4 yd. 1 ft. 8 in.? What is the scale ?

3. When an inch represents 150 ft., what distances are represented by in., in., in., 1 in., 2 in., 4.8 in., 5.3 in. ? What is the scale ?

4. When an inch represents 10 mi., what distances are represented by 3 in., 7 in., † in., 1⁄2 in., 31 in. ? What lengths on the drawings will represent 7 mi., 18 mi., 25 mi. ? What is the scale ?

5. What are the scales when 1 in. represents 100 ft., in. represents a mile, in. represents 20 ft.,in. represents 15 yd., 1 in. represents 1 mi., 10 mi., 100 mi. ?

6. Draw to a scale 1:240 (20 ft. to the inch) the circles in Exs. 2, 3, 4, 5, Art. 9.

7. On a map in Baedeker's Guide to Paris the distance between the nearest corners of the Eiffel Tower and Notre Dame Cathedral is 71⁄2 in. What is the distance between those points, the map being drawn to a scale 1:20000 ?

8. Make the comparison of angles asked for in Exs. 8, 9, 10, Art. 8; Exs. 11, 12, 13, Art. 9.

SUGGESTED EXERCISES. Make drawings to scale of the floor plan of a dwelling house, of some other building, of some grounds. Find the distances between various points, such as diagonally opposite corners, by making measurements in the drawing and applying the scale. Compare the results obtained in this way with the results obtained by other methods. Other methods that may be used are: (1) making an off-hand estimate of the distance; (2) actually measuring the distance by "pacing off" or by using a rule or tape line; (3) making a computation. Let the student, from his own experience, form a judgment as to which of the four methods referred to is the easiest, and which the more exact. Find the air-line distances between places by measuring the distances between them on maps. Several maps may by used so as to have a variety of scales.

NOTE. The word scale also has another meaning in drawing and measurement. Engineers and draughtsmen use various kinds of rules called scales. The faces of these rules contain different numbers of divisions to an inch, one 10 divisions, one 20, one 30, and so on; and generally, one inch on each face is subdivided so that a small fraction of an inch may be set off or read. Some paper scales are on the protractor inserted in this book.

in

11. Degree measure. The protractor. It has been seen geometry that: (1) When one line is perpendicular to another line, each of the angles made at their intersection is a right angle; (2) All right angles are equal to one another. In some geometrical propositions angles are compared, and one angle is shown to be greater or less than another. But geometry, with the exception of a few cases, does not show by exactly how much the one angle is greater or less than the other. In order to show this, measurement is necessary; and in order to measure, a unit angle of measurement must be chosen. The unit of angular magnitude which is generally used in practical work is the angle that is oneninetieth part of a right angle. This unit angle is called a degree. All degrees are equal to one another, since all right angles are equal to one another. Each degree is subdivided into 60 equal parts called minutes, and each minute is subdivided into 60 equal parts called seconds. Hence comes the following table of angular

measure:

60 seconds 1 minute,

60 minutes = 1 degree,

90 degrees = 1 right angle.

The magnitude of an angle containing 37 degrees and 42 minutes and 35 seconds, say, is written thus: 37° 42′ 35′′, read 37 degrees, 42 minutes, 35 seconds. This system of measurement is sometimes called the rectangular system, sometimes the sexagesimal system. In this chapter only acute angles, that is, angles which contain between 0° and 90°, are considered. Chapter V. considers angles of all magnitudes.

NOTE 1. An angle 1o is subtended by 1 in. at a distance 4 ft. 9.3 in., and by 1 ft. at a distance 57.3 ft. An angle 1' is subtended by 1 in. at a distance 286.5 ft., and by 1 ft. at a distance 3437.6 ft., about two-thirds of a mile. An angle 1" is subtended by 1 in. at a distance of nearly 33 mi., by 1 ft. at a distance a little greater than 39 mi., by a horizontal line 200 ft. long on the other side of the world, nearly 8000 mi. away. These facts can be verified later. See Ex. 3, Art. 83.

NOTE 2. Another system of angular measurement was advocated by Briggs and other mathematicians (see Art. 1), and was introduced in France at the time of the Revolution. In this system, which is a decimal one and called the centesimal system, a right angle is divided into 100 equal parts called grades, each grade into 100 equal parts called minutes, and each minute into 100 equal parts called seconds. It has not been generally adopted, on account of the immense amount of labour that would be necessary in order to change the mathematical tables computed for the other system.

11.]

THE PROTRACTOR.

19

NOTE 3. The sexagesimal system (from sexagesimus, sixtieth) was invented by the Babylonians, who constructed their tables of weights and measures on a scale of 60. Their tables of time (1 day = 24 hr., 1 hr. = 60 min., 1 min. = 60 sec.) and circular measure have come down to the present day. It has been suggested that their adoption of the scale of 60 is due to the fact that they reckoned the year at 360 days. "This led to the division of the circumference of a circle into 360 degrees, each degree representing the daily part of the supposed yearly revolution of the sun around the earth Probably they knew that the radius could be applied to the circumference as a chord six times, and that each arc thus cut off contained 60 degrees. Thus the division into 60 parts may have suggested itself. .. Babylonian science has made its impress upon modern civilization. Whenever a surveyor copies the readings from the graduated circle on his theodolite, whenever the modern man notes the time of day, he is, unconsciously perhaps, but unmistakably, doing homage to the ancient astronomers on the banks of the Euphrates." Cajori, History of Elementary Mathematics, pp. 10, 11.

NOTE 4.

See Art. 71.

Another system of angular measure is described in Chapter IX.

The protractor. The protractor is an instrument used for measuring given angles and laying off required angles on paper. Protractors are of various kinds, of which the semicircular and the full-circled are the most common. The degrees are marked all round the edge. A paper protractor is inserted in this book for use in solving problems.* In order to draw a line that shall make a given angle with a given line at a given point, proceed as follows: Place the centre of the protractor at the given point and bring its diameter into coincidence with the given line, keeping the semicircle on the side on which the required line is to be drawn; prick off the required number of degrees with a sharp pencil or fine needle. The line joining the point thus fixed and the given point, is the line required. In order to measure a given angle with the protractor, place the centre at the vertex of the angle, and place the diameter in coincidence with one of the boundary lines of the angle; the number of degrees in the arc intercepted between the boundary lines of the angle is the measure of the angle.

* A horn protractor costs about 25 cents, and a small metal one about 50 cents. One who is neat and handy can make a paper protractor.

NOTE. Before proceeding further, the student should be able to draw with ease a right-angled triangle, having been given: (a) The hypotenuse and a side; (b) the two sides about the right angle; (c) the hypotenuse and one of the acute angles; (d) one of the sides about the right angle and the opposite angle; (e) one of the sides about the right angle and the adjacent angle. It is here taken for granted that these problems have been considered in a course in plane geometry or in a course of geometrical drawing.

EXAMPLES.

N. B. The student is advised to do Exs. 1-6 carefully, and to preserve the results, for they will soon be required for purposes of illustration.

1. Draw to scale the triangles considered in Exs. 8, 9, 10, Art. 8, and Exs. 11, 12, 13, Art. 9, and measure the angles.

2. Make drawings, on two different scales, of a right-angled triangle whose base is 20 ft. and adjacent acute angle is 55°. In each drawing measure the remaining parts and thence deduce the unknown parts of the original triangle. In each drawing calculate the ratios specified in Ex. 8, Art. 8.

3. Same as Ex. 2, for a right-angled triangle whose hypotenuse is 30 ft. and angle at base is 25°.

4. Same as Ex. 2, for a right-angled triangle whose base and perpendicular are 30 ft. and 45 ft. respectively.

5. Same as Ex. 2, for a right-angled triangle whose hypotenuse is 60 ft. and base is 45 ft.

6. Same as Ex. 2, for a right-angled triangle in which the base is 50 ft. and the angle opposite to the base is 40°.

7. What angles of a whole number of degrees can easily be constructed geometrically without the aid of the protractor? Make the constructions.

12. Trigonometric ratios defined for acute angles. The ratios referred to at the beginning of Art. 8 will now be explained so far as acute angles are concerned. (Before proceeding, the student