the Elliptic Representation of a Circle, 379 
perpendicularly in its centre, therefore (by perspective ) /z^ is 
perpendicular to that line, and vq parallel: whence is a 
right angle. But no two conjugate diameters are at right 
angles, except the axes. Therefore bf and pq\ always ^ con- 
jugate diameters, are now the axes. Q. E. D. 
Cor. 1. The perpendicular axis has (by perspective) its 
vanishing point in the centre of the vanishing line of the 
objective plane. 
Cor. 2. It is also directed to the centre of the picture. For 
the directing line is •f perpendicular to that plane, through the 
eye, which is perpendicular to the objective and directing 
planes. Therefore the central line is in that perpendicular 
plane: whence its indefinite representation is the common 
section of that plane with the picture; which common section 
passes through the centre of the picture. 
Cor. 3. If O be at the mean point M (which is always in 
BF), in which case the distance of A from the eye is equal to 
a tangent from A ; the representation is a circle, whose centre 
represents V. For, from the further extremity F of BF, draw 
FP; also FQ, produced to R in LR. Draw MR, OR, AQ. 
Then, if AOR, revolving on AR, bring O to the eye, AO and 
OR become the directors of fv and fq, therefore parallel to 
them respectively; whence the angle vfq is § equal to AOR, 
and the triangle vfq similar to AOR, wheresoever O is in AF. 
But, if O be at M, AO is equal to AR. For the angle AQR 
is equal || to FPQ, to FOP, to ARQ; whence AR is equal to 
• Cor. 3 to prop. III. + XI Eugl. prop. 19. 
t Cor. 2 to introd. prop. I. § XI Eucl. prop. lo. 
(] Its vertically opposite is so. Ill Eucl. prop 32. 
