368 Dr. Rankine on Isoperimetric Regular Polygons. 



mations to the ratio which the diameter and circumference of a 

 circle bear to each other. 



2. Problem. — Given any regular polygon : to construct a re- 

 gular polygon of equal circumference and of double the number of 

 sides. 



3. Construction. — Let be the centre of the given polygon, 

 A the radius of the inscribed circle, 

 and OB that of the circumscribed 

 circle (which we may call respectively 

 the internal radius and the external 

 radius), so that AB is a half-side of 

 the given polygon. Bisect B in C ; 

 draw C D parallel to A and equal to 

 C B ; from D let fall D E perpendi- 

 cular to A produced, and D F per- 

 pendicular to OB; join D. Then 

 will E == F be the internal radius, 

 and D the external radius of the re- 

 quired polygon ; and DE = DF will 

 be two of its half-sides. 



4. Demonstration. — Let G be the 

 point where C D cuts A B. Then be- 

 cause OB is bisected in C, and the triangles AB and C GB 

 are similar, A B is bisected in G ; and DE = GA=|AB. Also 

 because CD = CB, and F and G are right angles, the triangles 

 C GB and FCD are similar and equal; and DF = BG = i AB. 

 So that D E and D F are two half-sides of a polygon having 

 twice as many sides as the given polygon and of equal circum- 

 ference. 



5. Corollary, — Because the triangles CGB and CFD are 

 equal, the rectangle AEDG is the excess of the area of the 

 quadrilateral E D F above that of the triangle A B ; and if 

 that rectangle be multiplied by the number of sides in the new 

 polygon, the product will be the excess of the area of the new 

 polygon above that of the original polygon. Hence it follows 

 that if there be a series of regular polygons of equal circumfe- 

 rence, each having double the number of sides of the preceding- 

 polygon, each polygon of that series is of a greater area than the 

 preceding polygon ; and the circle of equal circumference, being 

 the limit towards which the polygons approximate, must have a 

 greater area than any of them. 



6. Formulcefor calculation. — Let c be the common circumfe- 

 rence of the polygons ; let a denote the internal radius A, and 

 b the external radius B of the original polygon ; and let a 1 de- 

 note the internal radius E, and b' the external radius D of 

 the new polygon. Let n and 2n be the numbers of sides in the 



