Ihe Elliptic Representation of a Circle. 38«f 
sphere, and whose plane is perpendicular to a line * from O 
through E the centre of that circle to the centre of the sphere. 
If this line passed through the centre of the picture, such plane 
would be parallel to the picture, and the representation a circle. 
But, let it not so pass. Yet such plane is to be taken as the 
objective plane. And, since E is in every plane perpendicular 
to it through the eye, and the central •f' line is in one such 
plane, and both are in the objective plane, E is in the central 
line. Let FB,.the diameter in that line, have B its extremity 
nearer to its directing point A, and be produced to A. Draw 
OF, OB, OA ; and let the representation fb meet FB pro- 
duced in K. Then the angle OFB, being equal to OBF, is J 
greater than BbK or Obf. Therefore the axes are unequal, 
and the major axis is directed to the centre of the picture. § 
Q. E. D, 
Cor. 1. If the plane of a picture parallel to the former should 
cut off part of FB ; then the part viewed, of the circle or sphere, 
would have a representation || similar to its former one, and 
the major axis would still be directed to the centre of the 
picture. 
Cor. 2. The centre of the picture is not the vanishing point 
of the major axis. For FAO or FI^ is not a right angle. 
Lemma. Fig. 8. No. i and 2. If original lines, in one plane, 
converge to a point ; their representations will converge to a 
point, or be parallel. And, if the representations converge to 
a point ; the originals will so converge, or be parallel. 
• The intersection of this line with the picture is also its vanishing point. When 
this is given, the vanishing line of the perpendicular plane is found by perspective. 
Srook Taylor, Prob. XV. f In Cor. 2 to prop. V. 
J 1 Eucl. prop. 16. § Cor. 6 and 2 to prop. V. || Cor. to prop. I. 
3 D 2 
