1000 



REGULAR FIGURES, &c. 



REGULAR FIGURES, &c. 



1010 



twelve sides. The term quindecagon is in use to express the polygon 

 of fifteen sides. 



Let the polygon be described, having sides : let its side be a, its 

 area v, and let r and K be the radii of the inscribed and circumscribed 

 circles. The formulae which connect these quantities are then as 

 follows : Let stand for the nth part of 180", then 



a = 2 R sin v 2r tan v, 

 no? cot v 



which are enough to determine the remaining three of v, a, R, r, when 

 one of them is given. To facilitate the determination and construction 

 of any regular polygon not having more than 12 sides, we take the 

 following table from James Dodson's 'Calculator' (1747), which is 

 correct to every figure so far as we have thought it necessary to 

 examine it The author generally corrected errata with his own pen 

 in every copy, and the one before us has his corrections : 



When the Length of Side = 1. 



When Radiiu of Circumxribed Circle = 1. 



When Radiiu of Inxribed Circle = 1. 



WlunArta = 1. 



By means of these tables the construction of any figure is imme- 

 diately reduced to a short calculation, the drawing of a circle, and 

 setting off equal chords on that circle, the compasses and a scale of 

 equal parts being all the instrumental aid necessary. It is required to 

 construct, for example, a regular heptagon, or figure of seveu sides, 

 with an area of 225 times the square on one of the larger divisions 

 of the scale. The side and radii must therefore be increased in the 

 fourth table in the proportion of ^225 to \/l, or of 15 to 1. And 



5245813x15 = 7-8687 side. 

 6045183 x 15 = 9-0678 rad. circum. 

 5446520 x 15 = S'1698 rad. inscr. 



If the two circles be carefully drawn from the same centre, and chords 

 equal to the side taken oft', the compasses will be found to be earned 

 exactly seveu times upon the larger circle, and the chords, being drawn, 

 | will be found to touch the inner circle, and any little error of construc- 

 tion will be better shown by failure of touching the inner circle 

 correctly than by any other means. 



The above presumes that it is desired to proceed as accurately as 

 possible ; but for rough work, and when the circumscribed circle is 

 known, the proportional compasses, or even a common pair of com- 

 passes and trial, will succeed perfectly well. The proportional com- 

 passes have a scale for the adjustment of the pivot in such manner 

 that when the opening at one end is the radius of a circle, that at the 

 other end shall be the side of the inscribe 1 polygon of a given number 

 of sides. 



The regular polygons hitherto treated have been those of Euclid, 

 without any re-entering angles. The star-shaped polygons (which, 

 though equilateral and equiangular, do not come within Euclid's 

 definition) are described by drawing a regular polygon of the same 

 number of sides, and drawing successive diagonals so as to cut off 

 a number of sides which is prime to the number of sides of the 

 polygon. 



3 



OK AVD CL DIV. VOL. VI. 



Thus, if 12, 23, 34, &c., be the sides of a regular uonagou, or nine- 

 sided polygon, it follows that there are two regular star-shaped nona- 

 gons, one made by diagonals which cut off 2 or 7 sides, and one made 

 by diagonals cutting off -1 or 5 sides. Diagonals cutting off three sides 

 would give three equilateral triangles, but no nonagon at all. These 

 nonagons are 1357924681, and 1594837261. Star-shaped dodecagons 

 are also only one in number, since 5 and 7 are (except 1 and 11, which 

 would only give the dodecagon of Euclid) the only numbers less than 

 12 which are prime to 12. But a regular polygon of 13 sides has 5 

 star-shaped polygons, made by diagonals cutting off 2 and 11, or 3 and 

 10, or 4 and 9, or 5 and 8, or 6 and 7 sides. 



We now come to the subject of regular polyhedrons, presuming the 

 reader to know the contents of the article POLYGON AND POLYHEDRON. 

 A great many properties of these solids have been investigated, but as 

 they are of little use, it will be unnecessary to do more than give tables 

 for constructing them of given dimensions. Let a solid be contained by 

 / faces, each of which is a regular polygon of n sides. Let c be the 

 number of corners or solid angles, e the number of edges, and TO the 

 number of angles which meet at a corner. Then since there are c 

 corners with m angles at each, the number of edges, counting each edge 

 as often as it meets a comer, is m c ; or, as each edge meets a corner 

 twice, 4 me =e, the number of distinct edges. Again, since there are 

 / faces, of n sides each, and every edge is the union of two faces, we 

 havei/=e. But/ 1- c = e + 2, or 



which must be a whole number. And neither m nor n can be less 

 than 3, nor greater than 5, for there are no figures of fewer sides than 

 3, and [POLYGON AND POLYHEDRON] spaces cannot be inclosed entirely 

 by figures of more than five sides. The rest follows from the pro- 

 perties of conjugate solids in the same article. 



Let = 3, or e=6re-r(6 m). This is a whole number (1) when 

 m = 2 : this must be rejected : (2) when m = 3, giving n = 3, m = 3, e = 6, 



ST 



