( 254 ) 
refraded Ray F G, and M K being taken to the Semi- 
diameter of the Circle in the Ratio of the Sine of In- 
cidence to the Sine of Refra^Stion ; let L M be joined, 
and laftly make the Angle under K M N equal to half 
the given Angle under E I H. This being done, 
if F G be produced to O, F O fhall be to K O as 
the Sine of the Angle of Incidence to the Sine of 
the refrad:ed Angle, that is as M K to K L ; in fb 
much that -K L being parallel to F O, and the Angle 
under M K L equal to that under F O K, the Angle 
under ML K iliall be equal to that under FK O, and 
the Angle under *K M L equal to that under K F O e- 
quat to that under FGK or half that under F GH, 
whence the Angle under KM'N being equal to half 
Ae Angle under F I H, the refiduary Angle under 
N'M L will be equal to half the Angle under I F G or 
to half that under M K L. Therefore L C being drawn, 
the Angle under L M N will be equal to that under 
M C L ; and in the laft place, if M C be divided into 
two equal Parts in P, and P Q^R be drawn parallel to 
CL, the Angle under Q^M R will be equal to that 
under R P M, and the Triangles QM R, M P R flmi- 
lar, fo that the Redtangle under P R ihall be equal 
to the Square of M R. Whence R L being equal to 
M R, the Point L Ihall be in an equilateral Hyperbola, 
touching the Line M N in the Point M, and having 
the Point P for its Center But this Hyperbola is gi- 
ven in Portion, and confequently the Point L, the 
Angle under M L K, and the equal Angle under C K F 
will be given, and therefore the Ray E F is given in 
Pofition. 
» Apoll. Conic, lib. i. prop. 37. compared with lib. 7. prop. 23. 
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