In presenting this chapter to the student no attempt has been made to give a complete list of geometrical problems, but all those have been selected which can be of any assistance to the pattern draftsman, and especial attention has been given in their solution to those methods most adaptable to his wants. They are arranged as far as possible in logical order and are classified under various sub-heads in such a manner that the reader will have no difficulty in finding what he wishes by simply looking through the pages, the diagrams given with each being sufficient to indicate the nature of the problem and, as it were, form a sort of index.

1. To Draw a Straight Line Parallel to a Given Line and at a Given Distance from it, Using the Compasses and Straight-Edge. - In Fig. 131, let C D be the given line parallel to which it is desired to draw another straight line. Take any two points, as A and B, in the given line as centers, and, with a radius equal to the given distance, describe the arcs x x and y y. Draw a line touching these arcs, as shown at E F. Then E F will be parallel to C D.

2. To Draw a Line Parallel to Another by the Use of Triangles or Set-Squares. - In Fig. 132, let A B be the line parallel to which it is desired to draw another. Place one side of a triangle or set-square, F', against it, as indicated by the dotted lines. While holding F1 firmly in this position, bring a second triangle, or any straightedge, E, against one of its other sides, as shown. Then, holding the second triangle firmly in place, slide the first away from the given line, keeping the edges of the two triangles in contact, as shown in the figure. Against the same edge of the first triangle that was placed against the given line draw a second line, as shown by CD. Then C D will be parallel to A B. In drawing parallel lines by this method it, is found advantageous to place the longest edges of the triangles against each other, and to so place the two instruments that the movement of one triangle against the other shall be in a direction oblique to the lines to be drawn, as greater accuracy is attainable in this way.

3. To Erect a Perpendicular at a Given Point in a Straight Line by Means of the Compasses and StraightEdge. - In Fig. 133, let A B represent the given straight line, at the point. C in which it is required to erect a perpendicular. From C as a center with any convenient radius strike small arcs cutting A B, as shown by D and B. With D and B as centers, and with any radius longer than the distance from each of these points to C, strike arcs, as shown by xx and y y. From the point at which these ares intersect, E, draw a line to the point C, as shown. Then E C will be perpendicular to A B.

4. To Erect a Perpendicular at or near the End of a Given Straight Line by Means of the Compasses and Straight-Edge. - First Method. - In Fig. 131, let A B be the given straight line, to which, at the point P, situated near the end, it is required to erect a perpendicular. Take any point (C) outside of the line A B. With C as center, and with a radius equal to the distance from C to P, strike the are. as shown, cutting the given line A B in the point P, continuing it till it also cuts in another point, as at E. From E. through the center, C, draw the line E F, cutting the are, as shown at F. Then from the point F, thus determined, draw a line to P, as shown. The line F P is perpendicular to A B.

5. To Erect a Perpendicular at or near the End of a Given Straight Line by Means of the Compasses and Straight-Edge - Second Method. - In Fig. 135, let B A be the given straight line, to which, at the point P, it is required to erect a perpendicular. From the point P, with a radius equal to three parts, by any scale, describe an are, as indicated by x x. From the same point, with a radius equal to four parts, cut the line B A in the point C. From the point C, with a radius equal to five parts, intersect the arc first drawn by the are y y. From the point of intersection D draw the line D P. Then D P will be perpendicular to B A.

6. To Draw a Line Perpendicular to Another Line by the Use of Triangles or Set Squares - In Fig. 136. let C D be the given line, perpendicular to which it is required to draw another line. Place one aide of a triangle, B. against the given line, as shown. Bring another triangle, A. or any straight edge, against the long side or hypothenuse of the triangle B, as shown. Then move the triangle B along the straight edge or triangle A, as indicated by the dotted lines, until the opposite side of B crosses the line C, D at the required point. When against it, draw the line E F, as shown. Then E F is perpendicular to C D. It is evident that this rule is adapted to drawing perpendiculars at any point in the given line, whether central or located near the end. Its use will he found especially convenient for erecting perpendiculars to lines which run oblique to the sides of the drawing hoard.

Fig. 131 - To Draw a Straight Line Parallel to a Given Straight Line, and at a Given Distance from it, Using the Com-pases and a Straight-Edge.

Fig. 132. - To Draw a Line Parallel to Another by the Use of Triangles or Set Squares.

Fig. 133 - To Erect a Perpendicular at a Given Point in a Straight Line, Using the. Compasses and Straight-Edge.