Construction of a Parabolic Trajectory. 159 



the parabola, 56 is PI and 61 is AP. The line of collinearity 

 of intersections of opposite sides is HI and the tangent at A 

 meets this at infinity. 



To continue with the properties of the diagram fig. 2, 



Fis. 2. 



join HM meeting the diameter through P at W. Then 

 evidently XW = IP and BVVU is parallel to AP. Again 

 get the point V by joining HU. then VW = WX and 

 BV is parallel to the tangent at P. For, since NA = AT, 

 the figure BXWV is similar and parallel to PXAT. 

 From these results it follows that 



PV : PI = WI : IP = MK : KA. 



This brings us to a theorem used by Archimedes in the 

 course of his investigations of the positions of equilibrium of 

 a floating paraboloid of revolution, contained in the second 

 book of the work on Floating Bodies. In the sixth pro- 

 position of that book he proves that a paraboloid, the length 

 of whose axis has to the latus rectum a ratio lying between 



3 15 



the values ^ and -g-, if placed with a point on the circum- 

 ference of its base in the surface of the liquid and then 

 released, will turn, under the action of its weight and the 



