‘37^ Dr, R. Key's Propositions concerning 
always through O, to be moved round, along the circumference 
ABC. A cone will thus be generated. And the section of this 
cone by FD is the perspective representation of the circle. But, 
since the circle is supposed to lie wholly beyond FD, from 0,the 
plane MR, through O, parallel to FD, neither cuts nor touches 
the circle. Therefore * the section is an ellipse, g. E. D. 
' Cor. If the eye remain fixed, and FD be moved, parallel to 
itself, till it cut ABC; then a part only of ABC is seen through 
FD : and its representation is part of an ellipse, and similar to 
the corresponding part of abc. For a given figure will have 
similar projections, formed by the same lines OA, OB, &c, on 
parallel planes.-f' 
Prop. II. Fig. 2 . The centroid of a- circle is the vertex to 
the directing line taken as a base. J 
Dem, Let GTH be the circle, LR its indefinite directing line, 
and V its vertex to the base LR. Then, whatever chord, as 
GH or ST, passes through V, its tangents meet § in a point 
of LR. Therefore they have the same director, and their 
I , 
representations are || parallel ; whence those of GH and ST 
are ^ diameters of the representing ellipse, and V is the cen- 
troid. 2* E. D, 
Cor.i. Hence, from the directing line, the centroid is found.** 
* By the known properties of the cone. Sec, in particular, prop. 8i, 82, 83, of 
Conic Sections by the Rev. T. Newton, Fellow of Jesus’ College, Cambridge; 1794: 
which is the treatise referred to in subsequent notes. 
f If the directing line touch or cut the circle, the representation is, accordingly, 
* a parabola or hyperbola. It is not the design of these propositions, to enter upon 
those cases. They may be seen elsewhere : particularly in Hamilton’s Stereogra- 
*phy; where also are various particulars of the elliptic representation. 
J See Def. 2, of introductory Propositions. § Introd. prop, I. 
|j Brook Taylor’s Perspective, 1719; Cor. i to Theo. V. 
4[[ By conic sections. A necessary converse of T. Newton’s prop. 28. 
** Cor. I to introd. prop. T. 
