S^o Dr. R. Key’s Propositions concerning 
AO, so to AM ; therefore vq to vf, and the axes are equal: 
but V remains the centroid. 
Cor. 4. If AO and AM be unequal, the axes are so. If AO 
be greater than AM, the axis perpendicular to the intersecting 
line is the major axis ; if less, the minor. 
Cor. 5. Fig. 4. If O be at the mean point, the section of the 
cone of rays is the subcontrary * section. For, let ABF be as 
before, but O now at the real place of the eye ; so that fig. 4 
shall be the plane in which are BF and its representation hf. 
Draw AO, OF. Then hf\s parallel to its director AO. And, 
since AO is equal to a tangent from A, its square is equal to 
the rectangle FAB ; whence the triangle FAQ is similar to 
OAB, and the angle OFA or OFB equal to BOA, so to Ohf. 
Cor. 6. Fig. 4. If OFB and Ohf be unequal, the axes are 
so. If OFB be greater than Obf, the axis perpendicular to 
the intersecting line is the major axis; if less, the minor. 
For, if A and B and F be fixed, whilst, by the motion of the 
eye directly from or towards A, the angle OFB and the length 
AO are varied ; these will increase or decrease together, and 
Obf will accordingly decrease or increase. 
Prop. VI. Prob. I. Fig. 3 and 5. A circular object being 
given in magnitude, and in position with respect to the eye and 
the plane of the picture ; to find the axes of the representing 
ellipse. 
Fig. 3. If the centre of the circle be in the central line of 
the plane, find (by perspective) bf,pq; which are J the axes. 
Fig. 5. If it be not so placed, let the circle GTH be the 
object, and E its centre. From the given position the 
* Known in conic sections. See T. Newton, prop. 82. 
f Cor. 4. t Prop. V. and its Dem. 
