3^5 
the Elliptic Representation of a Circle. 
meet in a point P, of DG, making PD to PG as CD to FG. 
And, if CD and FG be equal, the three will either meet in a 
point equidistant from FH and CE, or be parallel, g. E. D. 
Cor. 1. If the originals neither converge to a point nor are 
parallel; the representations do not converge to a point. 
Cor. 2. It is impossible, tlierefore, that the major or minor 
axes should so converge, of ellipses representing any circles; 
unless the originals of the axes should in any case so converge 
or be parallel : which will only be, if the intervals between 
their intersecting points be proportional to those between the 
directing points.* 
Prop. VIII. Prob. II. Fig. g. The magnitude and positions 
being given, of any equal circles in one plane, having their 
centres in the central line ; to find a law which the axes, of 
the representing ellipses, observe in their directions. 
One axis is -f parallel to the intersecting line, the other-per- 
pendicular. Let AR, KI, AK, be the directing, intersecting, 
and central lines, indefinite ; AK passing through the centres 
of the circles z, zV, zzz, and any others such as supposed. Con- 
ceive O (not in the figure) at the real place of the eye. AO 
may be equal to a tangent from A to some one circle, or not. 
Let the zz be so placed. This alone will be ]J.' represented by 
a circle : those § between KI and zz by ellipses whose minor 
axes are parallel to KI, and major axes directed to the centre 
of the picture ; those beyond zz by ellipses whose major axes 
are parallel to KI, and minor axes directed to the centre of the 
picture. This, then, is a law observed. 
* Yet some advantage may be gained, by discovering any law or regularity observed 
in the directions of the axes. This is attempted in the remaining propositions. 
f Prop. V. t Cor. 3 and 4 to prop. V. § Cor. 4 and 2 to prop V« 
