of taking Lunar Ohservaiions. 291 



Let AHB be an ellipsis, and GH the normal or perpendicular 

 to the tangent KP which touches the curve in H. From F and 

 L, the foci, draw FP, 

 LK perpendicular to 

 KP; join FH, LH. 

 Now it is a known pro- 

 perty of the ellipsis, 

 that the angle FHP = 

 LHK, or that the tri- 

 angles PFH, KLH are 

 similar. Draw GQ,LR 

 parallel to KP, and the 

 triangles GLR, FGQ 

 will also be similar. 

 Whence PF - HG : 

 HG-KL::PH:HK 

 : : FP : KL, and 



H G = TT^-^^TT- = „p ^. • Rut when H and E do not coincide, 



FP+KL IP+KL ' 



KL + FP is greater than 2CD, because CD^ is known to be 

 equal KL x FP. Consequently, HG is less than CD the semi- 

 axis minor. 



Again : Since FP, GH and LK are parallel, and the triangles 

 FHP,LHK similar, we have FP + KL: FP-KL :: KP:HP-HK 

 : : FL : 2CG. Produce LR to meet FP in S, and HG to meet ED 

 in M. Then the triangles LFS,MGC will also be similar ; for 

 since MG is parallel to FS, the angle LFS = MGC, and the 

 angles FSL, GCM are likewise equal, being right angles. Hence 

 FP-KL : FL : : CG : GM. Consequently FP + KL : FL : : 



^^^^^¥^^> ^"^ MH = MG + GH = 



But when H and B do not coincide. 



iFL : GM = 



2CD« + 2CL2 



FP+KL 



2BC2 



FP + KL "~ FP+KL' 



FP + LK is less than FH + HL = AB = 2BC, because the side of 

 a right-angled triangle is less than the hypothenuse. MH is 

 therefore greater than BC. It is hence evident that GH : HM 

 : : CD^ : BC* : : DE' : AB% which is a very elegant property of 

 the ellipsis. 



When H comes to E, HG=CD, and HM = 7— ^ thesemipa- 



rameter of the axis minor, which is also the radius of curvature at 



CD* 

 E; and when H and B coincide, MH = BC, and GH = -;t77 the 



semiparameter of the axis major, which is likewise the radius of 

 curvature at B. 



It might likewise be shown conversely, that if a variable 



T 2 straight 



