Plates 4 and 5
Only those geometric solutions which the draftsman will be most likely to need in his work are given here.1
To divide a line A -B into two equal parts, Fig. 28, Plate 4, set the compasses at a radius larger than one-half of the line and with first A and then B as a center, draw intersecting arcs lightly at C and D. Draw a light straight line from C to D. This will bisect A-B.
To divide a line E-F into any number of equal parts, say seven, Fig. 29, draw a light line slanting in any convenient direction, such as E-G. Now lay the scale along E-G and mark off seven equal parts using any convenient length as a unit, say seven-eighths, seven-quarters, seven-halves or seven inches. From the last mark, " 7, " draw a light line to F; then from each mark on line E-G, draw lines parallel to 7-F and cutting line E-F, These last lines will divide line E-F into seven equal parts.
To divide the space between two lines into any number of equal parts, say five, Fig. 30, lay the scale with the zero end on one line and swing the other end around until any multiple of five coincides with the other line. In the illustration 3/8 inch has been chosen as the unit. Mark off the five units along the scale and through each mark draw a line parallel to the two given lines. These will divide the space equally into five parts. This method is very valuable in laying out stair steps, etc.
To draw a perpendicular to a given line at a given point, Fig. 31, lay either triangle with the hypothe-nuse along the given line as shown by the dotted lines and place the T-square blade against one side of it as indicated. Now hold the T-square firmly and turn the triangle around, keeping its square corner against the T-square blade, then slide it along the blade until the hypothenuse passes through the given point, when the required perpendicular may be drawn against the hypothenuse of the triangle.
To draw lines parallel to any given line L-M, Fig. 32, place the triangle against the line L-M as shown and place the T-square blade against the triangle. Holding the T-square in this position, slide the triangle along to positions as shown by dotted lines and any number of lines may thus be drawn parallel to the original line L-M.
1 If other problems are met, consult "Kidder," The Architects' and Builders' Pocket Book.
PLATE 4. GEOMETRIC METHODS.
To bisect any angle N, O, P, Fig. 33, Plate 4, set the compass at any convenient radius "a" and with O as a center draw arcs at N and P lightly. With N and then P as centers and the same radius "b" from either center, draw the arcs intersecting at Q. Through O and Q draw the bisector.
To divide the circumference of a circle into six equal parts, Fig. 34, Plate 4, set the dividers equal to the radius and step off the parts directly, or use the 30-degree triangle as shown in the illustration.
To draw tangent circle arcs, Fig. 35, Plate 4. The point of tangency of two circle arcs is always on a line joining the centers of the two arcs, see lines A-B and A-C. First locate the centers, then connect them by straight lines as indicated, then swing the circle arcs stopping each exactly at its tangent point. Notice that the distance between centers is equal to the sum or the difference of the radii of the circle arcs as the case may be.
To find the center for a given circle or circle arc, Fig. 36, Plate 4, draw any two chords and then draw their perpendicular bisectors by the method of Fig. 28. These bisectors will intersect at the required center if the work is carefully done.
To draw the arc of a circle when given the chord A-B and the rise C-D and when the center of the circle is not on the board, Fig. 37, Plate 5, first draw E-F through D parallel to A-B; then draw A-B. and B-K perpendicular to A-B. Draw A-D and D-B then draw A-E perpendicular to A-D and B-F perpendicular to D-B. Now divide A-H, B-K, E-D, D-F, A-C and C-B into the same number of equal parts (in this case we have chosen six). Draw lines connecting the points as shown and draw the circle arc through their intersections.
To draw a true ellipse having the length of the major and minor axes given, Fig. 38, Plate 5, mark off on a strip of paper a length A-B equal to one-half of the minor axis and A-C equal to one-half of the major axis. Now move this "trammel," as it is called, into successive positions, always keeping point B on the major axis and point C on the minor axis. When in each of these positions, mark location of point A by a dot. After locating enough of these points, draw the ellipse through them with a French curve.
To draw a true ellipse by concentric circles, Fig. 39, Plate 5, draw first the major axis D-E then the minor axis F-G intersecting at center H. Then with H as a center draw a circle of radius H-D and another of radius H-F. Divide these two circles into the same number of parts by drawing lightly the radial lines from H. From the intersections of each radial line with the circles, draw the short lines parallel to D-E and F-G as shown. Where these last lines intersect will be points on the ellipse. Notice that each radial line will fix two points on the ellipse. Locate as many points as accuracy demands.
To draw an approximate ellipse by the three center method, Fig. 40, Plate 5, lay off from the end of the major axis the distance L-K equal to the radius chosen for the end of the ellipse. Locate X the same distance from M on the minor axis as K is from L and draw K-N. Draw the perpendicular bisector of K-N until it intersects the minor axis at O. Draw a line through O and K to P. With A' as a center and radius K-L, draw the circle arc Q-L-P forming the end of the ellipse. Then with 0 as a center and radius 0-P, draw the circle arc P-M-S forming the top of the ellipse. Complete the figure similarly. It will be found that by this method the ellipse will often be much distorted.