line from the centre, and make gc-ths of radius as in Fig. 31.

The Special Rules will follow.

46. PENTAGON, or figure with 5 equal sides. (Fig. 32.) With any radius (ab) draw circle adc, draw diameter ac, and from centre make bfab.

With radius fd and centre ƒ draw arc dg, cutting the diameter at g, and with centre d and radius dg draw arc gh, and draw line dh, which will be one of the 5 sides required.

47. HEXAGON (6 sides). Fig. 33.

With any radius (ab) draw the circle bfdc. From any point (f) in the circumference, and with radius ab, step the circumference, which will be divided into 6 equal parts.

By drawing with the same radius arcs as in the figure, the circumference will be divided into 12 equal parts.

48. HEPTAGON (7 sides). Fig. 34.

With any radius (ab) draw the circle as in the figure. With a as centre and same radius draw the arc cbd, cutting radius ab at f.

With centre c and radius cf draw arc fg.

Then line cg will give the chord, giving the seventh part required.

49. OCTAGON (8 sides).

There are several modes of drawing this figure which it may be of use to understand.

(1) In a circle. (Fig. 35.)

With any radius (ba) draw the circle adef, and the perpendicular and horizontal diameters df and ac.

Bisect the 4 right angles by diameters gh and ij. Then lines between the points a, g, d, i, &c., will give the 8 equal sides.

Let abcd be the square, with cross diagonals ac and bd.

With af as radius, and successively with a, b, c, and d as centre, draw arcs cutting the sides of the square as marked by lines 1 to 8.

Then those lines will give the 8 equal sides.

(3) From a given base (ab). Fig. 37.

From a and b raise the indefinite perpendiculars ax and by; continue ba to d, making ad = ab; and bisect the angle nda, and make acab (second side).

From c draw of parallel to ax and = ab (third side), and from f

as centre and radius ab draw arc cutting perpen

dicular ax at g, and draw fg (fourth side).

Parallel to ab draw ghab (fifth side).

Parallel to ac and

ab draw hi (sixth side).

ab, draw ij (seventh side).

Parallel to of and

= ab].

Join j and b (eighth side) [parallel to fg and =

50. NONAGON (9 sides). Fig. 38.

With any radius (ba) draw the circle as in the figure, and with centre a and same radius draw the arc cbd.

Draw chord cd, and continue same indefinitely to x, and draw perpendicular radius ab, bisecting chord at ƒ, and from ƒ make fg = radius :—and with ƒ and g successively as centre and previous radius, draw arcs gh and hf, cutting at point h.

From h draw line hb.

From intersection of the chord with the line hb at k draw line kd.

Then line kd will give one of the equal ninth parts.

51. DECAGON (10 sides). Fig. 39.

(1) With any radius (ba) draw the circle adcf, and draw the perpendicular and horizontal diameters df and ac.

With half ab as radius draw circle ahbn, and from d through f (centre of that circle) draw line dg (or df), cutting the smaller circle at h.

With d as centre and dh as radius draw arc hi and draw line di.

Then di will be one of the equal 10 sides.

(2) By bisecting the angle dba in Fig. 32, the line di will also give a side of a decagon.

52. UNDECAGON (11 sides).

The general rule in Sect. 45 should be applied in this case.

53. DODECAGON (12 sides). Fig. 33.

With the circle and cross diameters as in Fig. 33, and same radius (ab), with successively f, b, c, and d as centre, draw the four arcs, cutting the circumference, which will by those arcs and the cross diameters divided into 12 equal parts and chords, gl, bh, hi, ic, &c., give the 12 equal sides.

54. DRAWING FIGURES with sides of any required length. As the same mode is adopted with any number of sides, and the showing the operation with two sides will show how all the other sides are to be dealt with, the Fig. (40) will

simply give the two sides bc and cd of a hexicon, or figure of six sides, which it is required to expand to a given length (fg), or contract to a given length (kb).

(1) To expand to length (fg).

From the centre (a), through the ends of all the sides (b, c, d, &c.), draw the indefinite lines ax, ay, az.

Parallel to be, and at any point (f) in which the proposed line fg may be drawn without cutting line ay, draw fg equal to the required side.

From g, parallel to af, draw line gh, cutting line ay at h; and from h, parallel to fg (or be), draw hi, cutting line

ax at i.

Then hi will be one of the sides.

By drawing hj parallel to cd, line hj will give another side.

In like manner from j, &c., the third and remaining sides may be drawn.

To contract to length bk.

From k, parallel to ba, draw kl, and from 1, parallel to bc, draw la, which will give two of the contracted sides; and in like manner the other sides may be drawn.

CONSTRUCTION OF CIRCLES in relation to other Figures in Sectns. 55 to 57, post.

abc.

55. A CIRCLE within a Triangle. Fig. 41, with triangle

Bisect any two angles (say at a and c) (Sect. 16) by lines cutting at d, and, with df or dg as radius, draw the circle.

56. A CIRCLE without a Triangle. Fig. 42, with triangle abc.

Bisect any two sides of the triangle (say at d and g) (Sect. 15), and draw the perpendiculars df and gf, cutting at f.

Then ƒ will be the centre of the circle, and fa, fb, or fc the radius.

57. A CIRCLE within or including a Square. Fig. 27. If within the square, take square abdc and make half the perpendicular height radius.

If including the square, take square ghik and make half the diagonal radius.

58. TO DRAW A TANGENT to a Circle. Fig. 39.

(1) From a point within Circumference.

Let c be the point. Draw from it the diameter ca, and from c the perpendicular ck (by Sect. 12), and which will be the tangent required.

с

(2) From a point without a Circle. Fig. 43, point c. Draw line de, joining the centre of the circle to the given point, and, with cd as diameter, draw the semicircle cfd, cutting the circle at f.

Then cfg will be the tangent required, for join fd, then cfd, being an angle within a semicircle, will be a right angle (E, Sect. 26), and cf will be perpendicular to radius, as in case 1.

59. A TANGENT to two Circles. Fig. 44. Circles A and B. Through the centres a and b draw the indefinite line ax. From a at any angle draw radius ac, and parallel to it from b draw radius bd.

From c through d draw cdf, cutting line ax at f. With af as diameter draw arc ag. Draw radius ag and parallel to it radius bh.