Your ability to divide lines and areas into equal parts is necessary in arriving at many of the common geometric forms required in sketching. The simplest method of bisecting lines is by visual comparison, as shown in figure 5-60. The entire line is first observed and weighed optically to determine its fulcrum or point of balance. Each half is compared visually before the bisecting point is placed. This procedure can be repeated any number of times to divide a line into any number of equal divisions, merely by dividing and redividing its line segments.

Centers of rectangular areas are easily determined by drawing their diagonals. If necessary the halves can be divided with diagonals for smaller divisions, as shown in figure 5-61.

The 90-degree angle is predominant in the majority of your sketches. Thus it is important that you learn to sketch right angles accurately, even if it entails checking them with the triangle occasionally. Frequently, the perpendicular edges of your paper can serve as a visual guide for comparison. It is also helpful to turn your sketch upside down; non-perpendicular tendencies of horizontal and vertical lines will become evident. Shaping right angles correctly will give your sketch stability, without which effectiveness is lost.

A 45-degree angle is made by dividing a right angle by visual comparison; and a 30-degree or 60-degree angle, by dividing the right angle into three equal parts. The 30-degree or 45-degree angle may be divided into equal parts in the same manner. (See fig. 5-62.) Always start with the right angle for the most accurate estimation of angle shape.

Perfectly round circles are the most difficult to draw freehand. Figure 5-63 shows methods of

Figure 5-63.-Methods of sketching circles.

drawing circles and curves using straight lines as construction lines. First, draw two straight lines crossing each other at right angles, as in figure 5-63, view A. The point at which they cross will serve as the center of the circle. The four lines radiating from this center will serve as the radii of the circle. You can use a piece of marked scrap paper to measure an equal distance on each radius from the center. Sketch a square, with the center of each side passing through the mark defining a radius. (See fig. 5-63, view B.) Now sketch in your circle, using the angles of the square as a guide for each arc. When larger circles are required, you can add 45-degree angles to the square to form an octagon. This will provide four additional points of tangency for the inscribed circle.

In figure 5-63, view C and view D, four lines, instead of two, are sketched crossing each other. The radii are measured as in constructing the other circle, but a square is not drawn. For this method, you will find it helpful to rotate the paper and sketch the circle in one direction.

For drawing large circles, you can make a substitute for a compass with a pencil, a piece of string, and a thumbtack. Tie one end of the string to your pencil near the tip. Measure the radius of the circle you are drawing on the string, and insert your tack at this point. Now swing your pencil in a circle, taking care to keep it vertical to the paper.

Another technique for drawing circles is shown in figure 5-64. In view A of figure 5-64, observe how the pencil is held beneath the four fingers with the thumb. This grip tends to produce a soft or easy motion for sketching large circles or curves and also makes it possible to sketch small circles, as shown in figure 5-64, views B and C. You notice in figure 5-64, view B, that the second finger rests at the center of the circle

and forms the pivot about which the pencil lead can swing. The distance from the fingertip to the pencil lead determines the radius of the circle. To draw smaller circles, you need to assume a somewhat different grip on the pencil, as shown in view C of figure 5-64, but the principle is the same.