§ 44. The regular hexagon is more easily constructed, as shown in Prop. 15. The result is that the side of the regular hexagon inscribed in a circle is equal to the radius of the circle.
For this polygon the other three problems mentioned are not solved.
§ 45. The book closes with Prop. 16. To inscribe a regular quindecagon in a given circle. If we inscribe a regular pentagon and a regular hexagon in the circle, having one vertex in common, then the arc from the common vertex to the next vertex of the pentagon is 1⁄5th of the circumference, and to the next vertex of the hexagon is 1⁄6th of the circumference. The difference between these arcs is, therefore, 1⁄5 − 1⁄6 = 1⁄30th of the circumference. The latter may, therefore, be divided into thirty, and hence also in fifteen equal parts, and the regular quindecagon be described.
§ 46. We conclude with a few theorems about regular polygons which are not given by Euclid.
The straight lines perpendicular to and bisecting the sides of any regular polygon meet in a point. The straight lines bisecting the angles in the regular polygon meet in the same point. This point is the centre of the circles circumscribed about and inscribed in the regular polygon.
We can bisect any given arc (Prop. 30, III.). Hence we can divide a circumference into 2n equal parts as soon as it has been divided into n equal parts, or as soon as a regular polygon of n sides has been constructed. Hence—
If a regular polygon of n sides has been constructed, then a regular polygon of 2n sides, of 4n, of 8n sides, &c., may also be constructed. Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides. It follows that we can construct regular polygons of
| 3, | 6, | 12, | 24 | sides |
| 4, | 8, | 16, | 32 | ” |
| 5, | 10, | 20, | 40 | ” |
| 15, | 30, | 60, | 120 | ” |
The construction of any new regular polygon not included in one of these series will give rise to a new series. Till the beginning of the 19th century nothing was added to the knowledge of regular polygons as given by Euclid. Then Gauss, in his celebrated Arithmetic, proved that every regular polygon of 2n + 1 sides may be constructed if this number 2n + 1 be prime, and that no others except those with 2m (2n + 1) sides can be constructed by elementary methods. This shows that regular polygons of 7, 9, 13 sides cannot thus be constructed, but that a regular polygon of 17 sides is possible; for 17 = 24 + 1. The next polygon is one of 257 sides. The construction becomes already rather complicated for 17 sides.
Book V.