To erect a perpendicular to the line b c at the point a. With a as a center, and any radius, as a 6, strike arcs cutting the line at b and c. From b and c as centers, and any radius greater than 6 a, strike arcs intersecting at d. Draw d a, which will be perpendicular to b c at a.

To draw a line parallel to a b. At any points a and b, with a radius equal to the required distance between thelines,draw arcs at c and d. The line c d, tangent to the arcs, will be the required parallel. To bisect the angle bac. With a as a center, strike an arc cutting the sides of the angle in b and c. With 6 and c as centers, and any radius, strike arcs intersecting, as at d. Draw d a, the bisector.

To erect a perpendicular at the end of aline. Take a center anywhere above the line, as at 6. Strike an arc passing through a and cutting the given line at c. Draw a line through c and 6, cutting the arc at d. Draw the line ad, which will be the required perpendicular. To divide a line into any number of equal parts. Let it be required to divide the line a b into 5 equal parts. Draw any line a d, and point off 5 equal divisions, as shown. From 5 draw a line to b and draw 4-4', 3-3', etc. parallel to 5b.

To divide a space between two parallel lines or surfaces (for example, the spacing of risers in a stairway). Draw a b and ce the given distance apart. Then move a scale along them, until as many spaces are included along a d as there are number of divisions required. Mark the points 1,2,3, etc., and draw lines through them parallel to a 6 and ce.

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The magnitude of an angle depends not upon the length of its sides, but upon the number of degrees contained in the arc of a circle drawn with the vertex as a center. The circle is divided into 360 equal parts, called degrees. To divide a quadrant as shown in the figure, first divide it into 3 parts by the arcs at e and d, chords c d and a e being equal to the radius. Then subdivide with dividers.

An inscribed angle has its vertex (as c or d) in the circumference of a circle. Any angle inscribed in a semicircle is a right angle, as a c 6, or a d b.

To draw a circle through three points not in a straight line, as a, b, and c. Bisect ab and also bc. The two bisectors will intersect in a point d, which will be the center of the required circle.

To find the center of a circular arc, as a b c, take a point, as 6 on the curve, and draw ba and b c. Bisect these lines by perpendiculars; the intersection d will be the required center.

To find a straight line nearly equal to a semi-circumference, as a b c. On the diameter construct the equilateral triangle a c h. Through a and c draw h e and hf. Then ef is the length of the semi-circumference. Draw any line, as h k l; then If is almost exactly the length of arc k c, and b1 that of arc b k.

To construct a hexagon from a given side. Describe a circle with a radius ab equal to the given side. Draw a diameter as cb. From c and 6 as centers, and a radius equal to the given side, draw arcs cutting the circle at k, d,f, and e. Connect c, k,f, b, e, and d.

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To inscribe an octagon in a square. Draw the diagonals ac and bd. With a as a center, and ae as a radius, strike an arc cutting the sides of the square at f and h. Repeat the operation at b, c, and d, and draw lines connecting the eight points thus found to form the figure required.

To draw any regular polygon in a circle. Divide 360° by the number of sides; the quotient will he the angle aob. Lay off this angle at the center with a protractor, and draw its chord, a side of the required polygon. Step this side around on the circumference, and connect the points found. To draw a segment of a circle, having given the chord a b and height c d. Draw ef, through d, parallel to a b; also, a d and d b. Draw a e and bf perpendicular to a d and d b; also a h and b k perpendicular to 06. Divide ed, df, ac, cb, ah, and b k into the same number of equal parts. Draw lines connecting the points as shown, and trace the curve through the intersections.

To draw a segment of a circle by means of a fixed triangle. Let a 6 be the required chord, and d c the rise. Drive nails at a and b. Make a triangle, as shown, from thin strips, so that the vertex comes at c, and stiffen it with the cross-brace. Now, by moving the triangle, always keeping the sides touching the nails at a and b, the arc may be traced by a pencil held at c. To draw an ellipse, having given the axes. Draw concentric circles whose diameters are equal to the axes ab and cd. From o draw any radius, as o e. From g, where o e cuts the inner circle, draw gf parallel to the major axis a b. From e, draw ef parallel to the minor axis dc. The intersection f gives a point on the ellipse. Other points are similarly found.

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To draw an ellipse with a string, having given the axes ab and cd. With c as a center, and a radius equal to ob, strike arcs cutting the major axis at e and f, the foci of the ellipse. stick pins at e and f, and attach a string as shown, the length of the string being equal to the length of the major axis. Keep the string stretched with a pencil point, and sweep around the ellipse. To draw a parabola. Having given the coordinates ab and bc, to draw a parabola, complete the rectangle abcd. Divide 6c and cd into the same number of equal parts. From 1', 2', 3', etc., draw lines through a. Through 1, 2, 3, etc., draw lines parallel to ab. The intersections of 1' a and 1-1", of 2' a and 2-2", etc. are points on the required parabola.

To draw a hyperbola. Having given the coordinates a b and 6 c, to draw the curve. Complete the rectangle abcd, and divide bc and c d into the same number of equal parts. Select any point e on the line b a prolonged; as e is taken farther from a, the hyperbola will approach a parabola in form. Connect I, 2, 3 to e, and 1', 2', 3' to a. The intersections of le and l'a, of 2e and 2'a, etc. are points on the required hyperbola. To draw a spiral. From a point, as o, draw radiating lines, oa,ob,oc,etc., making equal angles with each other. At any point, as d, on oa, start the spiral by drawing d e perpendicular to o a; from e where d e intersects o b, draw ef perpendicular to ob; etc. By making the angle between the lines smaller, the spiral may be made to make more turns, and the broken line will approach more nearly to a curve.

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To draw a spiral, second method. Draw a small circle, as bfa. Divide the circumference into any number of equal parts, eight or more. Draw tangents at the points of division, as b c, d e, etc. "With 6 as a center, and 6 a as a radius, strike the arc a c. With d as a center, and dc as a radius, strike the arc ce, etc. This method is a very-close approximation, though not mathematically correct.

The Roman Ionic volute. For the eye, draw the small circle, taking 1/4 the distance between ab and cd, in (6), as its diameter. In (a) is shown the eye enlarged. Mark points 1 and 2, at the middle of f c and fd. Divide 2-12 into three parts by points 6 and 10. At a distance below cd equal to 2 1/2 times the space between 2-1 and 6-5, draw 3-4. Draw 1-12, 10-9, 2-3, and 5-4. Draw 45° lines from 3 and 4; draw 6-7, 10-11, 9-8, 11-12, and 7-8. Use the points thus found as centers, 1 being the center for arc ae; 2 for arc eg; 3 for arc gf; etc.

To draw a helix. A helix is the curve assumed by a straight line, as df, drawn on a plane, when the plane is wrapped around a cylindrical surface. To draw a helix, having given the plan of the cylinder a b c and the rise c'c", divide the arc abc into any number of equal parts, as at 1,2, b, etc. Divide c' c" into the same number of equal parts. Draw verticals through the divisions of the arc a b c, and horizontals through the divisions of c' c", intersecting in points 1', 2', b", etc. Other points may be similarly found and the curve drawn.

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To develop the surface of a cylinder cut by a plane oblique to the axis. Let abc be the plan of the cylinder, and a' c" the inclination of the cutting plane. Divide the arc abc into any number of equal parts, as at 1,2, b, etc. Draw d e equal to the semicircumference, and mark the same divisions as on the arc, as d 2", l"-2", etc. Draw horizontal lines through points 1', 2', etc., intersecting verticals drawn through 1", 2", etc., as shown. Trace a curve through the points d, l'"t 2"', etc., and the figure dfe will be the development of the half cylinder abc. ______

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