Chap. VIII.

GEOMETRY.

741

Infixed Figures are bounded both by lines and partsof circles, as Q 11S.

Regular Figures are all formed of equilateral and equi-angular polygons; circles can be described within and aboutsuch figures; such can also be explained by geometricalmethods in particular instances. General expressions forthe radii of the circles explained within and about them,and for their areas and angles can be given: thus if wedenote the number of the sides of the polygon T by theexpression n, and if u° represent the n tu part of 1800, weshall have a being the side lt = £ a cosec. u°, r?=.\a cot. u°,area= \n a* cot u°.

That figure which has its sides all of equal length, andits angles equal, as those of the hexagon T, &c., is regular.

Figure , in geometry, is often used in two differentsenses: in one it implies a space bounded on all sides,whether by lines or planes; in another it signifies therepresentation only of the object of a theorem or problem,and enables us to render its demonstration or solutionmore easily understood.

An Irregular Figure is that whose sides are unequal,and its angles various, as V, the sides X Y, Y Z, Z 5,being all unequal.

Triangles are figures contained within three sides,and form three angles, as A B C.

The following are some of the properties of plane tri-angles : the greater side is opposite the greater angle, andthe difference of any two is less than the third side.

Compasses have been formed with three legs for theconstruction of maps, by which three points can be takenoff at one time; these have two legs that open in theusual manner, and the third made to turn round an ex-tension of the central pin of the other two, besides havinga motion of its own on the central joint.

A Rectilineal Triangle is formed by three right lines, asAB C.

A Spherical Triangle , T, is that which has its threesides curved.

Equilateral figures inscribable in circles are necessarilyequiangular, but the converse does not always hold true :when the number of sides is odd, the equiangular figureinscribed in a circle is always equilateral; but when thenumber of sides is even, they may either be all equal, orone half equal to each other, and the other half equal toeach other, though not to the former, the two sets beingplaced alternately : this was well understood by themasons of the middle ages, as we see expressedin the tracery of the windows, and the mosaic patternsthey have left us on the pavements and walls of theirseveral buildings. Pisa is rich in such illustrations, andit seems to have been a favourite study to construct equi-lateral and other angles within the circle when the cathe-dral in that city was built.

An Equilateral Triangle has its three sides equal,as DEF.

An Isosceles Triangle has two of its sides equal, andof the same length, the third being either greater orless: V and Y are mixed triangles. Among the pro-perties of the isosceles triangle is one in particular, viz.the angles at the base are always equal, and as the demon-stration given by Euclid is the first, and somewhat in-tricate and difficult for learners, it has been termed thepons asinorum .

A Scalene Triangle has its three sides unequal, as G H I.

A cone or cylinder is said to be scalene if its axis isinclined towards its base; but the term oblique would bemore appropriate.

3b 3

Fig. 654.

Fig. 656.

\

Fig. 657.

Fig. 658

\

An

Fig. 659.

Fig. 660.

1 4

Fig. 661.