INTRODUCTORY COURSE OF DRAWING AND

MEASUREMENT.

The instruments required in this course are compasses, a flat ruler, set squares, and a protractor; dividers and parallel rulers will also be found useful. The ruler should have a scale of inches and tenths of an inch on one edge, and a scale of millimetres on the other edge.

Numerical answers are given in many cases to three significant figures, but with these instruments the third figure can seldom be estimated correctly. In many of the problems it may be advisable for the teacher to help the class by rough sketches on the board.

1. The student must first learn the meanings of the words circle, centre, radius, diameter, triangle, base, side, vertex. See Definitions 15, 16, 17, and 21 on pages 4 and 6. Look at the different shapes of the triangles on page 6.

We shall frequently use the symbol for triangle, and the symbol" for inches; also the abbreviations cm. and mm. for centimetres and millimetres.

Problem I.—To construct a triangle, given its three sides.

Construct a triangle having its

base 2 inches long, its left side 1 inch long, and its right side 1.6 inches long.

Draw the base AB of length 2 inches [Fig. 1]. Describe a circle with centre A and radius 1 inch. Describe a circle with centre B and radius 1.6 inches. Take the point C above AB where the circles cut one another, and join it to A and B by straight lines. required triangle.

16"

A

2"

B

Fig. 1.

ABC will be the

For the side AC is a radius of the left hand circle and is therefore 1 inch long; and the side BC is a radius of the right hand circle and is therefore 1.6 inches long.

66

Fig. 2.

It is not necessary to draw the complete circles, but only the parts of the circles (or arcs") in the neighbourhood of the point C. See Fig. 2.

Note that by turning a triangle round we may regard any side as the base.

Examples I.

Describe triangles whose sides have the following lengths :

1. Base, 2"; left side, 2"; right side 1".

2. Base, 2"; left side, 1.5"; right side, 2.5′′.

3. Base, 3"; left side, 2"; right side, 1.5".

4. Base, 30 mm.; left side, 70 mm.; right side, 50 mm. 5. Each of the three sides to measure 27 mm.

6. Base, 2.6"; left side, 1"; right side, 2.4".

7. Base, 30 mm.; left side, 39 mm.; right side, 60 mm.

8. It will be found impossible to draw a ▲ if one of its sides is greater than the other two together. Try to draw a Δ whose sides measure 3, 1.5 and 1 inches; and another whose sides measure 20, 30, 60 mm.

2. On angles.-Any triangle has three corners. These corners are called angles. If the student examines the triangles which he has just drawn and the triangles on page 6 of this book, he will see that some of the corners are very sharp, so that if the triangle were made of metal these corners could be used for piercing. These sharp corners are small angles, and the more blunt corners are large angles. Thus, in the triangle given in Definition 28 on page 6, the angle at the right hand end of the base is small, and the angle at the left hand end is large.

Any two lines drawn from one point form an angle. Thus in Fig. 3 four angles are drawn of which the angle A is the smallest, the angle D the largest, and the angle C is larger than the angle B.

When the two legs of a pair of compasses are opened they form an angle; and the wider we open them, the larger is the angle which they form. We could set the two legs of our compasses into each of the positions represented by the angles in Fig. 3. Read Definitions 8 and 9 on page 2.

The two lines which form an angle are called the arms of the angle, and the point where the two arms meet is called the vertex or sometimes the angular point. Thus in Fig. 3 the lines AH and AK are the two arms of the angle A, and the point A is the vertex of the angle or the angular point,

Up to the present we have been naming an angle by a letter placed at the angular point. This method is satisfactory so long as there is only one angle at that point; but in other cases it may lead to confusion. The correct way to name the first angle in Fig. 3 is "the angle HAK;" notice the order of the letters; in the middle we place the angular point A, while the letters H and K are points one on each arm of the angle. It does not matter whether we place H or K first or last so long as we keep A in the middle; so that we may call the angle "KAH" if we choose. Similarly the fourth angle in Fig. 3 may be called either QDR or RDQ. In Fig. 2, the angle at the left hand end of the base should be called the angle BAC (or CAB), and the angle at the top of the triangle should be called the angle ACB (or BCA).

3. On equality of angles.-It is very important to notice that the size of an angle does not depend on the lengths of its arms but merely on the shape at the corner. The size of an angle depends only on the difference in the directions of its arms. In Fig. 3 if we lengthen or shorten one or both of the arms of the angle HAK we should not alter the size of the angle, so long as we do not alter the shape at the corner A.

Two angles are equal if we can place one angle on the other in such a way that the corners fit. Thus the angle at Q in the triangle PQR [Fig. 4] is equal to the angle at Y in the triangle XYZ; for we can place the angle at Y on to the angle at Q in

such a way that the corners at Y and Q exactly fit (as shown in Fig. 5). But the angle at R is smaller than the angle at Z (Fig. 4) for if we place the corner Z on to the corner R we obtain Fig. 6, from which it is evident that the arms RQ and RP are less wide apart than the arms ZY and ZX.

Examples II.

(For answers see page xxiv.) Turn to page 14, and look at the triangle DEF ;— 1. Name the angle at the vertex of the triangle. 2. Name the angle on the left side of the triangle.

Turn to the triangles in Fig. 4:—

3. What are the arms of the angle whose vertex is P?

4. What are the arms of the angle whose vertex is Y?

5. Name the angle whose arms are QP and QR.

6. Name the angle whose arms are ZY and YX.

7. Which do you consider the smallest angle (i.e. the "sharpest corner") in the triangle PQR?

8. Which do you consider the largest angle in the triangle PQR?

9. Which do you consider the largest angle in the triangle XYZ?

10. Is the angle at P greater or less than the angle at X?

4. Now consider Fig. 7. We have eleven different lines drawn from P, and they form ten small equal angles, viz. the angles APB, BPC, CPD, DPE, etc. The arms of these angles of different lengths, but the angles themselves have been drawn equal.

are

Any two of the lines which meet at P form an angle, for example the lines PC and PE form an angle CPE. Also it is obvious that this angle CPE

K

E

H

F

M

Fig. 7.

B

is made up of two of the small equal angles, viz. CPD and DPE. Thus this angle CPE is twice as large as the angle CPD or twice as large as the angle APB.

In the same way the angle EPH is made up of three of the small equal angles,—viz. EPF, FPG, and GPH. Hence the angle EPH is three times as large as the angle APB.

We shall frequently use the symbol

Using Fig. 7:—

1. How many times is

Examples III.

for angle.

APB contained in each of the following angles ?—

(a) DPF, (b) FPL, (c) APG, (d) MPA.

2. Name the line which divides KPE into two equal parts.

3. Name the line which divides KPC into two equal parts.

4. PM is one arm of an angle which is five times as great as APB; which is the other arm?

5. How many times is 4 BPD contained in LPF?

5. On Right Angles. In Fig. 8 the line CD is said to "stand upon" the line AB. In such a figure there are two angles formed at C, viz. the angles BCD and ACD. In Fig. 8 ZACD is obviously larger than ZBCD; but in Fig. 9 where