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CHAPTER I

INTRODUCTORY

1. Lines.-Euclid defines a line as "that which has length without breadth." A line, therefore, in the strict mathematical sense has no material existence, and the finest "line" that can be drawn on paper is only a rough approximation to a mathematical line. A straight line is defined by Euclid as "that which lies evenly between its extreme points." An important property of straight lines is that two straight lines cannot coincide at two points without coinciding altogether. This property is made use of in the applications of a "straight-edge." A line is also defined as the locus or path traced by a moving point. If the direction of motion of the point is constant the locus is a straight line.

In geometry when the term "line" is used it generally stands for "straight line."

2. The Circle-Definitions.-A circle is a plane figure contained by one line, which is called the circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference are equal to one another: and this point is called the centre of the circle.

The meanings of the terms radius, diameter, chord, arc, sector, and segment of a circle are given in Fig. 1.

The term "circle" is very frequently used when the "circumference " of the circle is meant. Euclid, for example, states that

[graphic]

FIG. 1.

"one circle cannot cut another at more than two points."

"A tangent to a circle is a straight line which meets the circumference, but, being produced, does not cut it" (see also Art. 13, p. 12). 3. Angles." A plane angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line."

"When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle." If it is required to test whether the angle BAC of a set-square, which is nominally a right angle, is really a right angle,

B

ARC

apply the set-square to a straight-edge DE as shown in Fig. 2, and draw the line AB. Next turn the set-square over into the position AB'C' and draw the line AB'. If AB' coincides with AB, then the angle BAC is equal to the angle BAC, and therefore by the definition each is a right angle.

An angle which is less than a right angle is called an acute angle, and an angle which is greater than a right angle and less than two right angles is called an obtuse angle.

C

A

E

FIG. 2.

If two angles together make up a right angle they are said to be complementary, and one is called the complement of the other.

If two angles together make up two right angles they are said to be supplementary, and one is called the supplement of the other.

In measuring angles in practical geometry the unit angle is generally the 1-90th part of a right angle and is called a degree. The 1-60th part of a degree is called a minute, and the 1-60th part of a minute is called a second. 47° 35′ 23" is to be read 47 degrees, 35 minutes, 23 seconds. The single and double accents which denote minutes and seconds respectively are also used to denote feet and inches respectively, but this double use of these symbols seldom causes any confusion.

AOB (Fig. 3) is an

Angles are also measured in circular measure. angle, and AB is an arc of a circle whose centre is O. The circular measure of the angle AOB is the ratio

arc = 0. For two right angles the length of the

radius

arc

is πT,

where π=

3.1416, and the circular measure of two right angles is therefore equal to .

When

the arc is equal to the radius the angle is the unit

angle in circular measure and is called a radian.

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If n is the number of degrees in an angle whose circular measure

=

n Ө 180 π

since each of these is the fraction which

is radians, then
the angle is of two right angles.

1800

Hence n =

= 57.29580, and 0 =

π

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80

90 100 110 120 130

70 6050

90

910

6/0,'5/0

120, 130 140

FIG. 4.

B

30 20 10

150 160 170

30 40 50 60 70 80

A scale of chords is a convenient instrument for measuring angles. The way in which the scale is constructed is shown in Fig. 5. ADF is a right angle, and the arc ACF is described from the centre D. The arc ACF is shown divided into 18 equal parts, each part therefore subtends at D an angle of 5°. On the scale of chords AB, lengths are marked off from A equal to the chords of the arcs of the different angles. For example, AE is the chord of the arc which subtends an angle ADE of 40° at the centre D, and this length is marked off on AB by describing the arc EH from the centre A. It will be found that the length of the chord AC for 60° is equal to the radius of the arc ACF Hence in using the scale of chords the radius of the arc is made equal to the distance between the points marked 0 and 60.

[graphic]

Ao 10 20 30 40 50 60 70 80 90
SCALE OF CHORDS

FIG. 5.

B

The scale shown in Fig. 5 only shows the chords for angles which are multiples of 5° up to 90°, but by making the scale large enough, single degrees, and even fractions of a degree, may be shown. It may also be extended beyond 90°. But when an angle is greater than 90° its supplement, which will be less than 90°, should first be determined. It should be noticed that the divisions on the scale of chords are unequal.

By using the ordinary 45° and 60° set-squares separately and together angles which are multiples of 15° may be constructed with great accuracy. Angles may also be measured and constructed by making linear measurements taken from tables of the trigonometrical ratios of angles.

4. Trigonometrical Ratios of Angles.-The angle A in the diagrams, Fig. 6, is contained by the lines OP and OX. The line OP

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is supposed to start from the position OX and sweep out the angle A by moving round O as centre in the direction opposite to that of the hands of a watch. An angle so described is called a positive angle. If the motion takes place in the opposite direction the angle described is called a negative angle.

PN is perpendicular to OX. PN is positive (+) when it is above

OX, and negative (-) when it is below OX. ON is positive when it is to the right of O, and negative when it is to the left of O. OP is always positive.

The most important trigonometrical ratios of an angle are, the sine, the cosine, the tangent, and the cotangent. The sine of the angle A is

the ratio of PN to OP. Thus, sine A =

PN
OP'

If PN is half of OP,

then sine A = . If PN = 2, and OP = 3, then, sine A =

ON

The other important ratios are, cosine A = tangent A =

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In addition to
OP
PN'

cosecant A

=

OP'

PN

ON'

the above the following ratios are also used: OP secant A = ON'

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coversed sine A = 1 − sine A, and

The names of these ratios are abbreviated as follows: sin, cos, tan, cot, cosec, sec, covers, and vers.

The following results are obvious from the definitions:

1

tan A =

1 cot A'

1

1

cosec A =

sec A =

sin A'

cos A'

cot A = tan A' The complement of A is 90° - A, and the supplement of A is 180° A.

The following relations between the trigonometrical ratios of an angle and those of its complement and

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It may be noticed that the sine and cosine of an angle can never be greater than 1, but the tangent and cotangent can have any magnitudes.

The problem of constructing an angle which has a given trigonometrical ratio evidently resolves itself into the simple one of constructing the right-angled triangle OPN (Fig. 6) having given the ratio of one side to another. But in general there are two angles less than 360° which have a given trigonometrical ratio.

5. Accuracy in Drawing.-A good serviceable line by an ordinary draughtsman is about 0-005 inch wide, but an expert draughtsman can work with a line of half that width. A good exercise for the student is to try how many separate parallel lines he can draw between two parallel lines which are 0.2 inch apart, as shown in Fig. 7, which is twice full size for the sake of clearness. When the lines are drawn as close as possible together, but distinctly separate, it will be found that the distance between them is about equal to the width of the lines. Hence, if the lines and spaces are about 0.005 inch wide, there will be about 20 lines in a width of 0.2 inch.

The probable error in measuring the distance between two points

on a straight line may be as small as the thickness of the division lines on the measuring scale, or say 0·005 inch.

The error e in the position of the point of intersection O of two straight lines OA

and OB (Fig. 8),

when the error in the lateral position of one of them

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B

0-2

A

FIG. 8.

is the angle between the lines. If x = 0·005 inch, e = 0.286 inch when = 1, e = 0·057 inch when

0 = 10°.

=

5°, and e = 0·029 inch when

Using a radius of 10 inches, an angle may be constructed by marking off the chord from a table of chords with a minimum probable error of from 1-10th to 1-15th of a degree.

Great accuracy in drawing can only be obtained by exercising great care, and by having the best instruments kept in the best possible condition.

Exercises Ia

To test accuracy in drawing and measuring

1. On a straight line about 7 inches long mark a point O near one end. Mark off the following lengths on this line: OA = 1.20 inches, AB = 0.85 inch, BC 1:35 inches, CD 1.55 inches, and DE = 0.95 inch, all measured in the same direction along the line. Measure the length OE, which should be 5.90 inches.

=

2. Take twenty strips of paper about 24 inches long, and number them 1 to 20. On an edge of No. 1 mark off a length of 2 inches. On an edge of No. 2 mark off the length from No. 1. On an edge from No. 3 mark off the length from No. 2, and so on to No. 20. Then compare

the length on No. 20 with the length on No. 1.

3. Draw a straight line AB (Fig. 9) and mark two points A and B on it 2 inches apart. Draw BC, making the angle ABC 60°. Make BC 4 inches long. Draw CD, making the angle BCD 30°. = Make CD = CA. Draw DE parallel to AB. Make DE 4 inches long. Draw EF, making the angle DEF = 30°. Make EF = CA. Let DE cut BC at O. A circle with centre O and radius OB should pass through the points D, F, C, E, and A. Also the chords of the arcs BD, DF, FC, CE, and EA should each be 2 inches long.

FIG. 9.

4. On a straight line AB (Fig. 10) 2 inches long construct an equilateral triangle ABC, using the 60° set-square. Produce the sides of this triangle as shown. Through the angular points of the triangle ABC draw parallels to the opposite sides forming the triangle DEF. Using the 30° and 90° angles of the set-square, draw lines through D, E, and F, to form the hexagon DHEKFL as

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