126 Sir W. R. Hamilton on libber's Construction 



3. On the other hand, the true septisection of the circle may 

 be made to depend on the solution of the cubic equation, 



^+4fc 2 -4a?— 1=0, 



of which the roots are cos — , cos — , cos—. Calculating then, 



by known methods*, to eight decimals, the positive root of this 

 equation, and thence deducing to seven decimals, by square 



IT 



roots, the sine and cosine of — , we find, without tables, the 

 values : 



cosy = a? =0-62348980; 



sin j =y/^ =0-4338837; 



cos 2" _ /L+f = 0-9009689; 



and these last agree so nearly with the values (A) of sin <f> and 

 cos</>, that at this stage a doubt may be felt, in which direction 

 does the construction err. In fact, Rober appears to have believed 

 that the construction above described was geometrically rigorous, 

 and had been known and prized as such from a very remote 

 antiquity, although preserved as a secret doctrine, entrusted 

 only to the initiated, and recorded only in stone. 



4. The following is an easier way, for a reader who may not 

 like so much arithmetic, to satisfy himself of .the extreme close- 

 ness of the approximation, by formulae adapted to logarithms, 

 but rigorously derived from the construction. It being evident 

 that 



r' = r sec — , and p — 2r tan ■=■} 

 5 1 5 



* Among these the best by far appears to be Horner's method, — for prac- 

 tically arranging the figures in the use of which method, a very compact and 

 convenient form or scheme was obligingly communicated to me by Professor 

 De Morgan, some time ago. We arrived independently at the following 

 value, to 22 decimals, of the positive root of the cubic mentioned above : 



cos^=0-G2348 98018 58733 53052 50. 



I had however found, by trials, before using Horner's method, the follow- 

 ing approximate value : 



cos ?ZT = 0-62348 98018 587; 



which was more than sufficiently exact for comparison with Rober's con- 

 struction. 



