Fig. 4. 



TG : GF : : PO : PC 



and GF : GE : : PQ^: PO by the ellipfe. 



whence TG : GE : : FQj PC and confequently the triangles TGE 



and PQC are equiangular, and the angle PCQ^= TEG ; therefore 



the angles at G and N are together equal two right angles, and fo 

 the angle at N a right angle. 



Now draw (^ parallel to TN, and 



CN : C(^: : CT : CV : : (becaufe CVXCP=CQI=: CFL= CGxCT) 



CP : CG : : CO : CF and therefore as C(^= CF, CN^iCO. Q^E.D 



Cor. When CO is a mean proportional between AC and CB, 

 CE = the femi-conjugate to CO, and alfo EN = AC— CB. 



Demonjlration. 



CNxCO = COi= (by hypothefis) ACxCB .-. CO = the femicon- 

 jugate to CE : and therefore AC^+CB'-aACxCB = CE'-CN' = EN^ 

 and therefore EN = AC— CB = Q^E. D. 



THEOREM. 



AEOBD is an ellipfe, the axis major of which is AD. Produce 

 any ordinate GE to meet the circumference of the circumfcribing cir- 

 cle in F. Draw COF and CT a femiconjugate to CO, aud alfo the 

 tangent TM meeting the perpendicular CM in M : then arc BT — 

 arc AE=TM. 



Demonjirat'wn, 



Draw fg indefinitely near to FG. Let C/ be femiconjugate to Co 

 and draw TR, tr parallel to the axis minor BC. Let alfo CR, Cr 

 meet the ellipfe in L,/. Draw LK, Ik parallel to the axis minor, join 

 K, M ; ^,C and draw the tang, im meeting k C, which interfefts TM 



in 



