The polygons adopted were of 20 or more sides approximating to a circular form.
These have faces which are all regular polygons, but not all of the same kind, while all their solid angles are equal.
" Regular polyhedra " are such as have their faces all equal regular polygons, and all their solid angles equal; the term is usually restricted to the five forms in which the centre is singly enclosed, viz.
Partial Polygons of Resistance.In a structure in which there are pieces supported at more than two joints, let a polygon be con-.
- These solids have all their faces equal regular polygons, and the angles at the vertices all equal.
They bear a relation to the Platonic solids similar to the relation of " star polygons " to ordinary regular polygons, inasmuch as the centre is multiply enclosed in the former and singly in the latter.
- These solids are characterized by having all their angles equal and all their faces regular polygons, which are not all of the same species.
Hero's expressions for the areas of regular polygons of from 5 to 12 sides in terms of the squares of the sides show interesting approximations to the values of trigonometrical ratios.
The latter, as we know, calculated the perimeters of successive polygons, passing from one polygon to another of double the number of sides; in a similar manner Gregory calculated the areas.
As regards the funicular diagram, let LM be the line on which the pairs of corresponding sides of the two polygons meet, and through it draw any two planes w, w.
And the corresponding polygons in the other figure by the same letters; a line joining two points A, B in one figure will then correspond to the side common to the two polygons A, B in the other.
The stresses produced by extraneous forces in a simple frame can be found by considering the equilibrium of the various joints in a proper succession; and if the graphical method be employed the various polygons of force can be combined into a single force-diagram.
By constructing several partial polygons, and computing the relations between the loads and resistances which are determined by the application of that theorem to each of them, with the aid, if necessary, of Moseleys principle of the least resistance, the whole of the relations amongst the loads and resistances may be found.
In drawing these polygons the magnitude of the vector of the type Wr is the product Wr, and the direction of the vector is from the shaft outwards towards the weight W, parallel to the radius r.
Relations between Polygons of Loads and of Resistances.In a structure in which each piece is supported at two joints only, the well-known laws of statics show that the directions of the gross load on each piece and of the two resistances by which it is supported must lie in one plane, must either be parallel or meet in one point, and must bear to each other, if not parallel, the proportions of the sides of a triangle respectively parallel to their directions, and, if parallel, such proportions that each of the three forces shall be proportional to the distance between the other two,all the three distances being measured along one direction.
Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere.