384 Dr. R. Key’s Propositions concerning 
Dem. Let the indefinite original lines CF, DG, EH, meet 
in P; having C, D, E, their directing points, and F, G, H, 
their intersecting points. From A the centre of the directing 
line, and towards FH, draw AO perpendicular to AD and equal 
to the distance of A from the eye. Draw CO, DO, EO: which 
are the directors, brought down, into the original or objective 
plane, by the revolution of COE or the directing plane on CE. 
Parallel to these draw lines through F, G, H : which will be 
the indefinite representations, brought down by the revolution 
of the picture on FH. These indefinite representations will 
form, on FG and GH, triangles respectively similar to COD 
and DOE; and perpendiculars from their vertices to FH will 
bear to OA the ratios of FG to CD and GH to DE. But 
A 
these ratios are equal, because FH is parallel to CE. There- . 
fore the perpendiculars are the same line KN, and the vertices 
are one point K. If P be in the directing line, by which C, D, 
and E, coincide with P ; there is only one director, and the 
representations are * parallel. One part, then, is proved. 
Now, let F,"G, H, be the intersecting points, and FK, GK, 
HK, the indefinite representations, brought down as before: 
and let KN be a perpendicular on FH, and let AO be as be- 
fore. Draw OC, OD, OE, parallel respectively to KF, KG, 
KH, and cutting the directing line in C, D, E ; and draw, in- 
definitely, CF, DG, EH. These are the original lines repre- 
sented ; because C, D, E, are the directing points, and F, G, 
H, the intersecting points. From similar triangles, CD is to 
FG as CO to FK, as AO to NK, as EO to HK, as DE to GH; 
or CD is to FG as DE to GH. Therefore, if CD and FG be 
unequal, consequently DE and GH, then CF, DG, EH, will 
* Brook Taylor; Cor. i to Theo. Y. 
