772

THEORY AND PRACTICE OF ENGINEERING.

To investigate the general property of polygons, we mustdivide them into two classes, convex and concave, the firstbeing those in which all the interior angles are less than tworight angles, and the second those which have one or morere-entering angles. If we term those the interior angles ofthe polygon which belong to the interior of the figure,whether less or greater than two right angles, and thoseexterior angles which are obtained by subtracting each in-terior angle from four right angles, we shall have the twoclasses.

The Decagon. Describe a circle, and divide the radiusinto two equal parts, as at M : from this point, with theradius M II, mark the point N; the distance G N will be theside of the decagon.

By a reference to the pentagon we have also the meansto set out this figure, the operation being nearly the same.

The Undecagon. Draw a circle, and cut it in the centre, P,by two lines, Q R and S T, drawn at right angles from thepoint R; with the radius RP, markon the circumference thepoint V, and draw the right line V Q; it will cut the radiusP S in X; the length P X will be one side of the undecagon.

The area of the circle w'as computed by means of in-scribed and circumscribed polygons ; and of all plane figureshaving equal perimeters, the circle contains the greatestarea; and consequently, of all plane figures containing equalareas, it has the least perimeter. The circle, therefore, is amaximum of area and a minimum of perimeter.

The Dodecagon is formed by carrying the length of onehalf the radius round the circle.

Another method may be described for the setting out o«these figures.

On the line A B set out two equal parts, and at their di-vision, or the point D, elevate the perpendicular line CD;then from the point A, with the radius A B, describe the arcB E, which will cut the perpendicular C D in the point F.

Then describe from the point F, with the radius FA orFB, a circle: the length AB carried six times round formsthe hexagon. By dividingB F into six equal partsin the points L, M, N, O, P,and F, and from the pointF taking the radius F P,and describing the arc PQ,which cuts the perpendicu-lar C D in It: from this pointR as a centre, with the ra-dius R A, describe a circle,and carry the length A Bseven times round it, and itwill form a heptagon.

To form an octagon takethe second division F O andwork before, and so on withall the other figures. Thusthe whole of the regularpolygons may be drawn byincreasing the diameters ofthe circles by one divisioneach time, which will giveseveral points on the per-pendicular C D, whence cir-cles may be described onwhich may be carried theline A B as many times asmay be necessary to con-struct the required poly-gon.

Fig. 833.

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Fig. 834

Fig. 835.

Fig. 836.