1 ()0 The Construction of a Parabolic Trajectory. 



buoyancy o£ the liquid, towards the position with axis 

 vertical. In the course of the proof he quotes, as known^ 

 a property of the parabola which is an extension of that just 

 obtained. The source from which Archimedes derived it is 

 unknown. 



Using the lettering of fig. 2, which agrees with that 

 adopted in- Heath's edition of Archimedes, the theorem in 

 its most general form is as follows : — 



From a point B on a parabola ordinates BM, BV are 

 drawn to any two diameters AM, PV. Through any 

 point K of AM a line is drawn, parallel to the ordinate 

 BM, to meet PV in I. Then P V : PI = or > MK : KA. 



Archimedes refers only to the special case where A is the 

 vertex of the parabola. A proof of this case, on somewhat 

 algebraical lines, is given by Dr. Heath. The alternative 

 geometrical proof now given exhibits, perhaps, in a clearer 

 light the connexion of the theorem with the fundamental 

 properties of the curve. 



It has been shown above that the two ratios compared are 

 equal when I is the intersection of AB with the diameter 

 through P. It remains to prove the inequality in other 

 cases. 



Let K'F be another position lying, say, to the left of 

 KI, and let it meet AH in W. Join MH' meeting UB in 

 W, PW in W", and BH in H". We want to compare 

 I'W" with IW. We have 



WW" : BH" = AP : AH = AN : AK, 

 HH" : HH' = UM : UW' = AK : UW, 

 HH' :IP =AP:AN = UW:AN. 



Compounding these ratios 



ww" : ir = UW : UW', 

 WW" > IP and so r W" < IW. 



If I / K / is taken to the ri^-ht of IK it will be found in the 

 me ma 

 than IW. 



same manner that WW"<I 7 and so I'W" is again less 



/. in every case I'W" : I'P<IW : IT. 

 But IW = PV and I'W" : I'P = MK' : K'A, 

 MK' :K'A<PV : PI'. 



