398 
PAPPUS OF ALEXANDRIA 
in length to the radius of the circle. In all other cases 
(Prop. 51 = Enel. Optics, 35) the diameters will appear unequal. 
Pappus’s other propositions carry farther Euclid’s remark 
that the circle seen under these conditions will appear 
deformed or distorted {Trapecnracrixevos:), proving (Prop. 53, 
pp. 588 -92) that the apparent form will be an ellipse with its 
centre not, ‘ as some think ’, at the centre of the circle hut 
at another point in it, determined in this way. Given a circle 
ABDE with centre 0, let the eye be at a point F above the 
plane of the circle such that FO is neither perpendicular 
to that plane nor equal to the radius of the circle. Draw P6r 
perpendicular to the plane of the circle and let ADG be the 
diameter through G. Join AF, JDF, and bisect the angle AFD 
by the straight line FG meeting A I) in G. Through C draw 
BE perpendicular to AD, and let the tangents at B, E meet 
A G produced in K. Then Pappus proves that C (not 0) is the 
centre of the apparent ellipse, that AD, BE are its major and 
minor axes respectively, that the ordinates to AD are parallel 
to BE both really and apparently, and that the ordinates to 
BE will pass through K but will appear to be parallel to AD. 
Thus in the figure, C being the centre of the apparent ellipse, 
F 
it is proved that, if LCM is any straight line through C, LG is 
apparently equal to GM (it is practically assumed—a proposi 
tion proved later in Book VII, Prop. 156—that, if LK meet 
the circle again in P, and if PM be drawn perpendicular to 
AD to meet the circle again in M, LM passes through G).