A history of Greek mathematics - Wilbourhall.org

398 PAPPUS OF ALEXANDRIA
in length to the radius of the circle. In all other cases
(Prop. 51 = Eucl. 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 (Trapecnracrfiii'os), 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 but
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 FG
perpendicular to the plane of the circle and let ADG be the
diameter through G. Join AF, DF, and bisect the angle AFD
by the straight line FG meeting AD in C. Through G draw
BE perpendicular to AD, and let the tangents at B, E meet
AG produced in K. Then Pappus proves that G (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, G being the centre of the apparent ellipse,
it is proved that, if LGM is
any straight line through G, LG is
apparently equal to CM (it is practically assumed— a proposition
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).