Bx)yal Society. 461 



The absurdity of the construction may also be thus made appa- 

 rent geometricall)^ 



The radius O B being bi- 

 sected in F, CF joined and 

 produced to D, B D will, ac- 

 cording to this "mathematical 

 luminary," be aside of the in- 

 scribed octagon, and conse- 

 quently will subtend half a 

 right angle at the centre O. 

 The perpendicular D K, then, 

 will be equal to O K, and the 

 triangles COF and DFK 

 will be similar, so that 



CO : OF=DK : KF, or CO . KF=OF . DK. 

 But the radius being unity, we have clearly 



,-1 1 11 



C0=^-6, OF=K. DK= -7^, KF^-T--^; 



V2' 



\/2 



hence 



or squaring, 



U Voy 8 v'2 



and an incommensurable quantity is after all equal to a commensu- 

 rable one I 



The very first problem in the book contains a manifest blunder. 

 The construction fails in the generiil case, and only holds true for 

 circles, or for curves which arc symmetrical in relation to their ex- 

 tremities. 



Mr. Burchett, in the preface of ibc work, observes, " the 



present course it is believed, however, that many are new ....;" 

 now it happens, unfortunately, that the portions which are new 

 are not true, and those that are true are not new. We may observe, 

 by the way, that he has also committed a gross mathematical blunder 

 in his work on Perspective, where he states that the perspective pro- 

 jection of a circle is not exactly an ellipsis. 



LXXII. Proceedings of Learned Societies. 



ROYAL SOCIETY. 

 [Continued from p. 397-] 

 March 31, 1851). — Sir Benjamin C. Brodie, Bart., Pres., inthe Chair. 

 ''P^lIE following communications were read : — 

 -■- "On the Comparison of Uyperbolic Arcs." By C. W. Mer- 

 rifield, Esq. 



If in common trigonometry we take one arc equal to the sum of 



