Part 15

If there be no room below the line, the intersection may be taken above the line; that is to say, between the line and the given point.

Second Method, Fig. 104. From any two points B C at some distance apart, in the given line, and with the radii B A, C A, respectively, describe arcs cutting at A D. Draw the perpendicular A D.

EX. 5.--To draw a parallel line through a given point, Fig. 105. With a radius equal to the given point C from the given line A B, describe the arc D from B, taken considerably distant from C. Draw the parallel through C to touch the arc D.

Second Method, Fig. 106. From A, the given point, describe the arc F D, cutting the given line at F; from F, with the same radius, describe the arc E A, and set off F D, equal to E A. Draw the parallel through the points A D.

When a series of parallels are required perpendicular to a base line A B, they may be drawn as in fig. 107 through points in the base line set off at the required distances apart. This method is convenient also where a succession of parallels are required to a given line C D, for the perpendicular may be drawn to it, and any number of parallels may be drawn on the perpendicular.

EX. 6.--To divide a line into a number of equal parts, Fig. 108.

To divide the line A B into, say, five parts. From A and B draw parallels A C, B D on opposite sides; set off any convenient distance four times (one less than the given number), from A on A C, and on B on B D; join the first on A C to the fourth on B D, and so on. The lines so drawn divide A B as required.

Second Method, Fig. 109. Draw the line at A C, at an angle from A, set off, say, five equal parts; draw B 5, and draw parallels to it from the other points of division in A C. These parallels divide A B as required.

EX. 7.--Upon a straight line to draw an angle equal to a given angle, Fig. 110. Let A be the given angle and F G the line. With any radius from the points A and F, describe arcs D E, I H, cutting the sides of the angle A and the line F G.

Set off the arc I H, equal to D E and draw F H. The angle F is equal to A as required.

EX. 8.--To bisect an angle, Fig. 111. Let A C B be the angle; on the center C cut the sides at A B. On A and B as centers describe arcs cutting at D dividing the angle into two equal parts.

EX. 9.--To find the center of a circle or of an arc of a circle. Fig. 112. Draw the chord A B, bisect it by the perpendicular C D, bounded both ways by the circle; and bisect C D for the center G.

EX. 10.--Through two given points to describe an arc of a circle with a given radius, Fig. 113. On the points A and B as centers, with the given radius, describe arcs cutting at C; and from C, with the same radius, describe an arc A B as required.

Second, for a circle or an arc, Fig. 114. Select three points A, B, C in the circumference, well apart; with the same radius describe arcs from these three points cutting each other, and draw two lines D E, F G, through their intersections according to Fig. 107. The point where they cut is the center of the circle or arc.

EX. 11.--To describe a circle passing through three given points, Fig. 114. Let A, B, C be the given points and proceed as in last problem to find the center O, from which the circle may be described.

This problem is variously useful; in finding the diameter of a large fly-wheel, or any other object of large diameter when only a part of the circumference is accessible; in striking out arches when the span and rise are given, etc.

EX. 12.--To draw a tangent to a circle from a given point in the circumference, Fig. 115. From A set off equal segments A B, A D, join B D and draw A E, parallel to it, for the tangent.

EX. 13.--To draw tangents to a circle from points without it, Fig. 116. From A with the radius A C describe an arc B C D, and from C with a radius equal to the diameter of the circle, cut the arc at B D, join B C, C D, cutting the circle at E F, and draw A E, A F, the tangents.

EX. 14.--Between two inclined lines to draw a series of circles touching these lines and touching each other, Fig. 117. Bisect the inclination of the given lines A B, C D by the line N O. From a point P in this line draw the perpendicular P B to the line A B, and on P describe the circle B D, touching the lines and cutting the center lines at E. From E draw E F perpendicular to the center line, cutting A B at F, and from F describe an arc E G, cutting A B at G. Draw G H parallel to B P, giving H, the center of the next circle, to be described with the radius H E, and so on for the next circle, I N.

EX. 15.--To construct a triangle on a given base, the sides being given.

First. An equilateral triangle, Fig. 118. On the ends of a given base A B, with A B as a radius describe arcs cutting at C, and draw A C, C B.

Second. Triangle of unequal sides, Fig. 119. On either end of the base A D, with the side B as a radius describe an arc; and with the side C as a radius, on the other end of the base as a center, describe arcs cutting the arc at E; join A E, D E.

This construction may be used for finding the position of a point C or E at given distances from the ends of a base, not necessarily to form a triangle.

EX. 16.--To construct a square rectangle on a given straight line.

First. A square, Fig. 120. On the ends B A as centers, with the line A B as radius, describe arcs cutting at C; on C describe arcs cutting the others at D E; and on D and E cut these at F G. Draw A F, B G and join the intersections H I.

Second. A rectangle, Fig. 121. On the base E F draw the perpendiculars E H, F G, equal to the height of the rectangle, and join G H.

EX. 17.--To construct a parallelogram of which the sides and one of the angles are given, Fig. 122. Draw the side D E equal to the given length A, and set off the other side D F equal to the other length B, forming the given angle C. From E with D F as radius, describe an arc, and from F, with the radius D E cut the arc at G. Draw F G, E G. Or, the remaining sides may be drawn as parallels to D E, D F.

EX. 18.--To describe a circle about a triangle, Fig. 123. Bisect two sides A B, A C of the triangle at E F, and from these points draw perpendiculars cutting at K. On the center K, with the radius K A draw the circle A B C.