744
THEORY AND PRACTICE OF ENGINEERING.
Book II.
Heptagon is bounded by seven sides, E , and as many equalangles; it is called irregular, F, when these are not equal.Ileptagonal numbers are those where the difference ofthe terms of the corresponding arithmetical progression is5. Thus arithmetical being called 1,6, 11, 16, the hep-tagonals written under them would be 1, 7, 18, 34, See., thelatter being formed by the continual addition of the termsof the first: among the properties of these numbers is onevery remarkable, viz. that if any heptagonal number ismultiplied by 40, and 9 added to the product, the sum isa square number.
Octagon is bounded by eight sides and angles, as A, andwhen these vary it is termed an irregular one, as B. Ithas been found that any regular figure, which has thenumber of its sides denoted by 2 n +■ 1 and prime, maybe inscribed in a circle without any other aid than that ofplane geometry, that is, by the intersections of the straightline and circle only ; and it is clear that by dividing thesubtended arcs into two, four, or more equal parts, a re-gular figure of twice four times, &c., the number of sidesof any may be inscribed. An octagon for instance may bedrawn by bisecting the arcs which are subtended' by thesides of a square.
Nonagon is bounded by nine sides, as C, and has nine equalangles; when irregular, as the figure D, these all vary.This figure, by some geometricians called the Enneagon ,has not yet had any rule laid down for its construction,and can only be inscribed approximatively.
Decagon has ten equal sides and angles, E ; when theyvary, as in the figure F, it is not regular.
This figure is the double of the pentagon ; and Euclid has shown in his fourth book of the Elements, that theside of a regular decagon is equal to the greater segmentof the radius of the circumscribing circle, divided by amedial section, or so that the rectangle contained by thewhole radius and one of the parts is equal to the squareof the other part.
Undecagon has eleven equal sides and angles, and thismay be the form of such a figure as II, which also haseleven sides, arranged neither within a circle nor after anyparticular form.
This figure, also termed the endecagon , has no regularrule laid down for its construction ; it can only be set outor inscribed within the circle by approximation.
Duodecagon has twelve sides and angles equal, and isregular when so drawn as H, and irregular as shown in theside figure.
A regular Polygon has all its sides equal, and likewise allits angles equal; and the centre is the same with thecommon centre of the inscribed and circumscribed circles,and the perpendicular, which is drawn from the centre toany one of the sides, is called the apothem: if any twoadjoining angles of a regular polygon be bisected, the inter-section of the bisecting lines will be the common centreof two circles, the one circumscribing, the other inscribedin the polygon. The area of a regular polygon is equalto half the rectangle under the perimeter and apothem.
An Equilateral Figure has all its sides equal, as in thesquare, pentagon, and hexagon, A, B, C.
Those figures of four, five, and six sides when lines arcdrawn from their several angles to the centres, or whenthey are inscribed within circles, may have their separateand relative values easily calculated.
Equiangular Figures are those which have their relativeangles equal; the figures RST and VXY are so, the angleIt being equal to the angle V, the angle S to that of Y,$id that of T to X.
Fig. 676.
Fig. 679.
Fig. 680.
R
Fig. 683.