Chap. VIII.

GEOMETRY.

771

Fig. 828.

Fig. 829.

Method of drawing Multilateral Figures. —The Pen-tagon A is drawn by describing a circle from the point B ofthe given size : then from the centre B draw the two diametersC D and E F at right angles, and from the point G, whichis half the radius of E B, describe the arc CH; the lengthCH will be the side of the pentagon, which may be tracedround the circle.

To inscribe a regular pentagon within a circle, divide theradius BF medially in the point H, so that BH may bethe greater segment: draw the radius B C, at right angles toB F, and join C II: then because the square of C H is greaterthan the square B II by the square of the radius C B, andthat B H is the side of the inscribed decagon, C H is theside of the inscribed pentagon. Therefore a chord equal toC H will subtend a fifth part of the circumference, and ifthe circumference be divided into five parts with chords, eachequal to CII, a regular pentagon will be inscribed. To in-scribe a regular decagon, divide the radius medially, anddivide the circumference into ten parts with chords eachequal to the greater segment of the radius so divided.

The Hexagon has its sides equal to its radius, and is madeup by six equilateral triangles; the diameter of the circlewhich contains it, when cut by a perpendicular passingthrough the centre, forms right angles with the two sides ittouches.

The Heptagon M. On a circle of any given diameter fix thepoint N: with the radius NO describe the arc POQ, anddraw the chord line P Q; the half of this chord will be theside of the figure required.

It must be admitted that we have no exact rule for settingout this figure, and we can only inscribe it within a circleapproximatively. This is sometimes done by continuing theseries 4, 8, 16, &c., which represents the number of partsinto which the circumference may be divided by continuedbisections, until a number be found which is greater or lessby 1 than a multiple of 7. 64 is such a number, being

greater by 1 than 9x7. Now, if the circumference bedivided into 64 parts, and an arc be taken equal to 9 ofthose parts, which is less than a seventh part of the circum-ference by a seventh part, the error may be made up by alittle calculation, and the side obtained near enough formost practical purposes.

The Octagon S. Describe a circle, and draw the twodiameters VX and YZ at right angles through its centre;then divide one quarter of its circumference into two equalparts, and so on with the rest.

The octagon may also be set out by two squares, soplaced that their diagonals are at right angles with eachother, and also by bisecting the arcs which are subtendedby the sides of a square.

The Nonagon B. Describe a circle, and carry two-thirds of its radius nine times round it.

The same method may be adopted for this figure as de-scribed for the heptagon: seven times the arc, which is as-sumed as the seventh of the circumference, falls short butlittle of the whole circumference; and 9 times the arc byabout the same, therefore both are near enough for allpractical purposes.

Any polygon may be decomposed into triangles, bydrawing straight lines from one of its angular points to eachof its opposite angles, and the area of the polygon is the sumof the areas of all the component triangles : their area maybe found without this process, which, when the number ofsides is considerable, leads to some labour; the theorem was established by LTIuilier, whofound that the double of the surface of any rectilinear figure is equal to the sum of therectangles of its sides, taken two and two, excepting one , multiplied by the sine of the sumof the supplements of the interior angles contained between each pair of sides.

3 d 2

Fig 830.

Fig. 832.