Dr. Rankine on Isoperimetric Regular Polygons. 367 

 original polygon and the new polygon respectively. Then 



*=a/(« 2+ £)> 0) 



(2) 



a-rb 



*=V(*+i5?)- 



</<Jb ; 



(3) 



and by repeating the calculations (2) and (3) again and again, 

 the internal and external radii may be calculated of a series con- 

 sisting of an indefinite number of polygons of equal circumfe- 

 rence, each having double the number of sides of the precediug 

 polygon. It is easy to see that the internal radii form an in- 

 creasing series, and the external radii a diminishing series, both 

 converging towards the radius of the circle of equal circum- 

 ference as a limit. 



7. The linear function of the internal and external radii which 

 approximates nearest to the limit of both series is 



a + 2b 



8. Example. — Taking the semicircumference of the polygons 

 as the unit of length (that is, making c = 2) and calculating to 

 six places of decimals, we obtain the following results for a series 

 of isoperimetric regular polygons of 6, 12, 24, and 48 sides : — 



6 

 12 



24 



48 



0-288675 

 0-311004 

 0-316490 

 0-317855 



b. . 



0-333333 

 0-321975 

 0-319221 

 0-318537 



«+26 

 3 

 0-318447 

 0-318318 

 0-318310 + 

 0-318310- 



The reciprocal of 0-318310 

 figures is 3*14159 + . 



Glasgow Universit}', 

 October 10, 1867. 



to six places of significant 



