the Elliptic Representation of a Circle. 38} 
^lirectlng plane is determined, and the directing line. Let this 
line be LR, and A its centre. Make a rig^it angle LAO, on 
the same side ol Lx\ as the circle is ; and make AO equal to 
the distance of A from the eye. Find V the * vertex to the 
base LR; also M the mean point. Draw OM; and bisect 
it perpendicularly in O, by OC cutting LR in C: and draw 
CO. In LR take CL and CR, each equal to CO; and, through 
V, draw, towards R and L, the chords GH and ST. Find gh 
and st, by perspective. These are the axes. 
Dem. With the centre C and radius CO, describe the semi- 
circle LOMR ; and draw LO, OR, LM, MR. Then, because 
V is J the centroid and LMR a right angle, gh and st are § 
conjugate diameters. And they are perpendicular to each 
other. For, if LOR, revolving on LR, bring O to the eye, 
OL and OR become the directors of st and gh, therefore pa- 
rallel to them respectively; whence the angle tvh is || equal 
to the right angle LOR. But no two conjugate diameters are 
perpendicular to each other, except the axes. Therefore gh 
and st are the axes. g. E. D. 
In practice, find (by trial) in the directing line a point C 
equidistant from O and M ; and, in the same line, take CL, 
CR, each equal to CO. And, if either L or R, suppose R, be 
at an inconvenient distance, the chord GH, directed to R, may 
be found as the tangent-chord of L. f 
Cor. 1. Fig. 6 ; the same as Fig. 5, so far as the letters are 
common. One mode of finding the axes, after GH and ST 
found, is this. Draw GT, and produce it cutting LR in B. 
Draw OB ; also BK, cutting OR perpendicularly in K. Find, 
♦ Cor. I to introd. prop. I. f Cor. 2. ib. % Prop. II. 
§ Cor. 2 to prop. III. |J XI Eucl. prop. 10. f Prop. III. 
MDCCCXIV. ' 3D 
