measure the arc G H, and set it up from L to M. Join CM. Then the angle D C M is equal to the angles A and B.

EXAMPLES.

EX. 13.-Draw any irregular four-sided figure, no side less than 11′′, and draw a second figure having equal angles and two of its sides 3" and 21".

EX. 14.—Draw any irregular triangle and show, by adding the three angles together, that the three angles of a triangle together equal two right angles.

(Add two of the angles to the third angle, and the first and last lines should form one straight line.)

EX. 15.-Draw an isosceles triangle, base 3′′, vertical angle 45°. (The sum of the base angles will be 180° - 45° = 135°, .'. draw a line and a perpendicular to it, giving two right angles, bisect one of these right angles, thus giving an angle of 90° + 45° = 135°, then bisect this angle for one of the base angles.)

EX. 16.-Draw any line, A B, and mark a point, C, outside it. Through C draw a line parallel to A B.

(Join C by a line to any point D in A B, and from C draw a line making the same angle with CD as CD makes with A B.-Euclid i., 23.)

Construction of Angles and Protractors.-Lines may be readily drawn at different angles to a given line, by drawing a semicircle upon the line, and knowing that a semicircle contains 180°, dividing the semicircumference to obtain the desired angles. This method of setting off angles is much facilitated by remembering that the radius of a circle steps round the circumference exactly six times, and that if any two of these points next one another are joined to the centre by lines, the angle between the lines is 60°, for the whole angle at the centre is four right angles, or 360°, and the construction gives exactly one-sixth, or 60°. E Hence a line may be quickly and accurately drawn at 60° to any given line as follows:

Let A B be the given line, and let a line be required at 60° to A B, starting from the end A. With A as centre and any radius, draw an arc cutting A B in C, and from Ċ set up the same radius to the point D, and join AD. Then the angle BAD is 60° (Fig. 8).

Fig. 8.

B

This construction suggests an easy method of trisecting a right angle; for if in Fig. 8 the lines A B, AE are at right angles, then the angle E A D is one third of a right angle, and by setting off the same radius as before from the point E to F,

the angle BA F is made one-third of a right angle. Hence the lines A D and A F trisect the right angle.

=

The following angles are thus easily obtained :-30° by bisecting 60°; 75° 60° + half of 30°; 120° by setting off two 60°, and 135° = 90° + 45° (half a right angle). Also 108° = % of 180°, and is therefore found by drawing any semicircle, dividing it into five equal parts and joining the centre to the second division point from one end; the two angles thus formed will be 72° and 108°. To obtain 135° which = ‡ of 180°, divide the semicircle into four equal parts, and join to the first division from one end. These angles are important as being those of certain useful regular polygons, the construction of which will be described further on.

Protractors.-An extension of this method is employed to construct protractors, which enable angles of any degree of measurement to be set off. A semicircle of 6" diameter is drawn and its semicircumference is accurately divided into 180 equal

*

C

Fig. 9.

parts called degrees, any further subdivision into minutes and seconds not being possible on so small a scale. The most common protractor is of oblong form, 6" long and 1" wide, and is shown in its finished form in Fig. 9 (divided to show 2° only), which also clearly shows the method of construction. Notice that the divisions are marked both ways to allow of using from either end, and that the lines showing each 10° are longer than the lines showing the smaller divisions. In constructing a protractor, the semicircumference should be divided by continual bisection as in Fig. 2, as repeatedly as possible, and then the dividers used to obtain the small divisions.

* This protractor is made of boxwood and its divisions are not usually very reliable. A more accurate instrument is the horn protractor, which is generally of semicircular form.

The protractor is used by placing the edge AB to coincide with the line from which the angle is to be drawn, and the middle point C against the point in the line from which the angle is to start.

EXAMPLES.

EX. 17.-Draw lines meeting at the following angles :(a) 60°, (b) 75°, (c) 90°, (d) 105°, (e) 108°, (ƒ) 120°, (g) 135°, (h) 140°.

EX. 18.-Construct a triangle base 21", one base angle 60°, verticle angle 45°.

EX. 19.-Construct a protractor 6′′ x 21′′, to show divisions of 5°.

MISCELLANEOUS EXAMPLES.

(1) Draw a line, A B, 3" long, and find three points beyond B through which A B would pass if produced.

(2) Draw a parallelogram, base 3", diagonals 44" and 63".

(The base, and half of each diagonal form a triangle.)

(3) Draw a line, A B, 31′′ long, and produce it to a point C, so that BC shall be of A B.

(Divide A B into four by bisection, and add one piece on.)

(4) Draw two lines meeting at a point A at 135°, and bisect the angle, using only the parallel edges of a rule and pencil.

(Place one edge of rule coinciding with one arm of angle, and draw line along other edge, do the same with other arm of angle, the two lines drawn will meet in a point, which when joined to vertex bisects the angle.)

(5) Draw a circle of any diameter between 4′′ and 6′′, and find its centre (as though unknown) using only the parallel edge of a rule, a measuring rule and a pencil.

SECTION

III.

DIVISION OF LINES AND CONSTRUCTION
OF SCALES.

Division of Lines.-In the division of lines and angles by the method of bisection as explained in the preceding section, it was seen that the construction only applied to obtaining division into certain numbers of parts, and did not admit of general application. There are, however, other methods by which lines can be accurately divided into any desired number of equal

parts, with which it is very necessary for the student to become familiar.

The most common method of division in practical mechanical drawing is known as "division by trial, or with dividers." Thus to divide a given line into any number of equal parts, the dividers are set to what the draughtsman considers to be approximately the right distance, and then, starting from one end this distance is stepped off along the line the required number of times. If the last step just reaches to the line end, the division is accurate, but if not, the dividers must be opened or closed until the equal division is obtained. Circles, arcs, and angles can be divided in the same way since although the dividers then really measure the length of chords, yet the arcs are proportional to them (Euclid iii., 28). For accuracy in division the use of dividers in a practised hand is more reliable than geometrical methods. It is, however, necessary to use some form of spring dividers, and to avoid making holes through the paper at each step.

The following is the geometrical construction for the division of lines into any required number of parts. It is based upon the properties of similar triangles, and is particularly useful, as permitting of division into fractional parts, or into parts proportional to a given ratio, or to the divisions of a given line.

PROBLEM VI. (Fig. 10a, b).—To divide a line into any number of equal parts.

B

B

Fig. 10a.

Fig. 106.

Fig. 10a.-Let A B be the line to be divided into five equal parts. From one end, A, draw a line, A C, of any length, and at any angle to A B. Mark off upon this line five equal parts, as at 1, 2, 3, 4, 5. Join 5 to the end B, and through the points 1, 2, 3, 4, draw lines parallel to B5, meeting AB as shown. Then A B is divided into five equal parts.

(The five triangles thus formed, each having A for a vertex, are similar; therefore, since A 5 is divided into five equal parts, A B is similarly divided (Euclid vi., 4). The equal parts set off down the line A5 may be of any convenient length, but a little practice will show that the greatest accuracy is obtained when the angle BAC is small, as drawn, and the length is approximately equal to the fraction required of the given line.)

In order to divide a given line A B into three and a-half equal parts, it is only necessary to set off down the line corresponding to A C. 3 x 2 = 7 equal parts, and then draw parallels to B7, through every other one to the given line.

=

By the same method a line can be divided into parts having a desired proportion to each other, or similarly to another divided line which may be either longer or shorter. Suppose we require to divide a given line A B (Fig. 106) in the proportion of 3 : 2 : 1. Set off down the line A C, 3 + 2 + 1 6 equal parts of convenient length, and draw parallels as before from the points 3 and 5 to the line 6 B, then A B is divided into three parts in the required proportion. When the given line A B is to be divided proportionately to the divisions of another given line, it should be drawn from one end of the divided line, at an angle to it as before. Then by joining the ends of the two lines, and drawing parallels through the points in the divided line, the line will be divided similarly to the given divided line. Numerous useful problems in proportion can be worked in this way; the method is in fact a part of graphic arithmetic.

EXAMPLES.

EX. 1.-Divide a line 5′′ long into four equal parts in three different ways.

EX. 2.-Draw a circle 4" diameter, and draw any diameter. Divide half the circle by continued bisection, and the other half with dividers, each into eight equal parts. Mark the points 1, 2, 3, to 16, and join the points 2 and 10; 7 and 15; 12 and 4. If accurate, these lines should pass through the centre of the circle.

EX. 3.-Draw two lines at any angle to each other and meeting, and divide the angle into three equal parts.

EX. 4.-Divide a line 7′′ long into two parts, in the proportion of 11: 2.

EX. 5.—Divide a line 6′′ long into three parts, in the tion of 2 3 4.

propor

EX. 6.—Find by construction the eighth part of 2.5′′. EX. 7.-Draw two lines at any angle, meeting at point A, and find a point, P, 21′′ from A, its distance from one line of the angle being twice its distance from the other line.

EX. 8.-Draw any angle and bisect it by using only a parallel rule and pencil.

EX. 9. Draw a line, A B, 3" long. Find by construction three points through which AB would pass if produced in a straight line.

EX. 10.-Draw any irregular five-sided figure, no side less