of the Heptagon. 125 



concentric with but exterior to both ; p' a segment of the side p ; 

 and q, s, t, u, v five other derived lines. The result is, that in 

 the right-angled triangle of which the inner diameter 2r is the 

 hypotenuse, and u, v supplementary chords, the former chord (u) 

 is very nearly equal to a side of a regular heptagon, inscribed in 

 the interior circle ; while the latter chord (v) makes with the 

 diameter (2r) an angle <£, which is very nearly equal to the ver- 

 tical angle of an isosceles triangle, whereof each angle at the base 

 is triple of the angle at the vertex. In symbols, if we write 

 u = 2r sin <j>, v — 2r cos cf>, 



then cj> is found to be very nearly = -r. It will be seen that the 



equations can all be easily constructed by right lines and circles 

 alone, having in fact been formed as the expression of such a con- 

 struction ; and that the numerical ratios of the lines, including 

 the numerical values of the sine and cosine of cf>, can all be arith- 

 metically computed*, with a few extractions of square roots. 



(r + r') 2 = 5r 2 - =1-2360680 



' r 



f = 4(r' 2 -r 2 ) £ =1-4530851 



1-0514622 

 p r + r' " r 



q* =p*-p'- 2 2 = 1*0029374 



s' 2 +ps=(p-q + ry - =0-8954292 



r"*=V-M 2 , -=1-3423090 



t* = ^ r ^ 2 -( ? -"- r )2 -=1-6234901 



a 2 =2r(2r-/) - =0-8677672 



v 1 =2rt - =1-8019379 



r 



u =2/-sin^> sin</> =0-4338836 



v = 2rcos<£ cos<£ =0-9009689 



* The computations have all been carried to several decimal places 

 beyond what are here set down. Results of analogous calculations have 

 been given by Rober, and are found in page 16 of the first-cited publica- 

 tion of his son, with the assumption p-z=*/3, and with one place fewer of 

 decimals. 



p f __ r+ ^r 1 p[_ 



w< 



