50 



REPORTS ON THE STATE OF SCIENCE. — 1919. 



Lemniscate Trisection : r=30. 



8. L'hfc general formulas are given in p. 72, Keport 1911, as taken 

 from Phil. Trans. 1904, p. 261, 'The Ellipsotomic Problem,' and for the 



special case of the lemniscate, where b=2i/'d sin 75 = i?^?-^^' sinJ75^ 



(sin 45I'-! 

 the trigonometrical form of the division values can be written 



D{:-]0) 



^ (sin 75 )' 

 (sin 45)J 



B(:30): 



.(sin 45)3 (sin 60)^ 

 (sin 75).' 



C(30) + A^30) = i^ELi^, C(30)- A(30) = ( ^i^L^OjiJ^in J5)^ 

 , (sin 75)J ' ^ ' ^ ' (sin 45)J 



or otherwise, 



C(30)3 + A(30)'' -..= V 33V =: i^^*^-^^) \ C(30)3 - A(30)^^= ^ ^ + ^ = ^i^l^ , 



(sm 45) 2 sin 45 



E(30) = (^" ^?'^i^^', F(30)=E(60)=a-^ 

 (sm 60)t 



= (sin 45)i (sin 60)1 (sin 75) "^'^^ ^^)* ('^^ ^^P. 

 ^ ^ ^ ^ ^ ^^ (sin 60); 



Thus for D(30) 



sin 75 = al 1-98494 37781 

 sin 45=al 1-84948 50022 



(sin 75)5=al 1-99498 12594 



. (sin 45)?^ = al 1-97491 41670 



D(30) = al 0-02006 70924 



= 1 04729 03271 



(sin 75). i = al 1-94982 83340 



(sin 60)!= i-98438 26579 



1-93421 09919 



(sin 75)i'= 1.99749 06297 



B(30)=al 1-93672 03622 



= 0-86441 11542 



B(30)nX30)''* - sin 45" sin 60°. 



For C(30) and A(30) 



sin 60=al 1-93753 06317 



For B(30) 



sin 45 = al 1-84948 50022 

 sin 60=al 1-93753 06317 



sin75=al ] -98494 37781 



sin 75=al 1-98494 37781 

 sin 45=al 1-84948 50022 



sin 45=al 1-84948 50022 

 sin 75=al 1-98494 37781 



C(80) = 1-14189 38846, 



(sin 60)J = al 1-96876 53158 

 (sin 75): = al 1-99498 1259 3 



1-96374 65751 



(sin 45):;=a] 1-74914 16703 



C(30) f A(30)=al 0-21460 49048 



= 1-63909 79420 



(sin 45)5=al 1-79981 33829 



(sin 75)S = al 1-98996 25187 



C(3O)-A(30)=aH-80935 08142 



= 0-64468 98272 



A(30)= 0-49720 40572 



A(30)3 C(30)-^=sin 45° sin 15° 



