REGULAR FIGURES, POLYGONS, SOLIDS, POLYHEDRONS. We have here to add to what is said in POLYGON AND PoLrusratos all that concerns the regular figures or solids, not as to their general properties, but as to the proportions of their parts and the mode of describing them. We shall take first the plane figures, and then the solids.
A regular polygon, meaning one of which all the sides are equal and all the angles are equal, may have any number of sides from three upwards. The Greek terms trigon, tetragon, pentagon, hexagon, hep tagon, octagon, nonagon, decagon, undecagon, dodecagon, are in use (except the first two) to express polygons of three, four, &c., up to twelve sides. The term quindecagon is in use to express the polygon of fifteen sides.
Let the polygon be described, having a sides : let its side be a, its area v, end let r and R be the radii of the inscribed and circumscribed circles. The formulae which connect these quantities are then as follows :—Let v stand for the nth part of 180°, then which are enough to determine the remaining three of v, a, a, 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'a ' 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 :— 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 seven 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 V225 to J1, or of 15 to I. And
If the two circles be carefully drawn from the same centre, and chords equal to the side taken off, the compasses will be found to be carried exactly seven 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 inscribed 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.
Thus, if 12, 23, 34, &c., be the sides of a regular nonagon, or nine aided 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 4 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 II, or 3 and 10, or 4 and 9, or 5 and 8, or 6 and 7 sides.