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LINES OF THE SECOND ORDER. 
Section I. 
Referred to a Principal Diameter and its Tangent. Normals. 
1. From a point P of a conic section a normal is drawn to 
cut a principal axis (the major axis in the ellipse, the transverse 
axis in the hyperbola) in a point G: to prove that, of all 
straight lines drawn from G to the curve, GP is the least. 
The equation to the conic section is 
f = (1 + e) (2emx — (1 — e) x 2 j. 
The equation to the normal, at any point (A, A), is 
(1 + e) [em — (1 — e) A}. [y' — A) = — A [x — A). 
At the point G, y = 0, and therefore 
x = (1 + e) em + e*h. 
Let r represent the distance of G from any point (.r, y) in 
the curve: then 
r 2 = {x — (1 + e) em — e 2 A} 2 + y* 
= £C 2 — 2ex ((1 + e) m + eh] + e 2 {(1 + e) m + eh\ l 
+ (1 + e) (2emx — (1 — e) x] 
= e 2 [x — hf + e 2 (1 + e) 2 m 2 + 2e 8 (1 + e) mh — e 2 (1 — e 2 ) A 2 , 
= [x — Kf + (1 + ef m 2 + (1 + e) {2emh — (1 — e) Ji] 
= (x - A) 2 + (1 + ef m 2 + Jc\ 
Hence it appears that r is least when x = A, or when (x 1 y) 
coincides with (A, A). 
De la Hire : Sectiones Conicce, lib. vn. prop. 13.