of floating Bodies , and the Stability of Ships, 105 
This construction of Archimedes * may be justly regarded as 
one of the most curious remains of the ancient geometrical 
synthesis, and is here inserted, in order that the agreement 
between the solutions by analytical investigation and geome- 
trical construction, may appear in the most satisfactory point 
-of view. 
Having given the parabola APBL, (fig. 27.) which is a 
section of a conoid passing through the axis BD, and having 
given the axis BD, which is to the parameter in a greater 
proportion than 15 to 8, it is required to express, by geome- 
trical construction, the two proportions which the specific gra- 
vity of the conoid must bear to that of the fluid, so that the 
solid may float permanently on the fluid when the surface 
passes through an extremity of the base. 
BD represents the axis of the conoid, DA is the greatest or- 
dinate to the axis ; join the points B and A, and bisect BA in 
T ; draw TH perpendicular to AD ; and with the axis TH, and 
ordinate AH, describe the parabola ATD ; in the axis BD set 
off DK = j of DB, and make KR = \ the parameter; also set 
offKC to DB in the prpportion of 4 to 15: consequently DB 
bears a greater proportion to KR than 15 to 4; and since 
KR is half the parameter, it follows that the axis is to 
the parameter in a greater proportion than that of 15 to 8. 
Through C draw CE parallel to DA intersecting BA in E, 
and draw EZ perpendicular to AD. With the ordinate AZ 
and axis ZE describe the parabola AEI, and through R draw 
the line RGY, intersecting the parabola AEI in the points G 
and Y ; through the points G and Y draw the lines ON, PQ, 
perpendicular to AD, intersecting the parabola ATD in the 
points X and F. 
* De iis quce in bumido vehuniur, Lib. ii. prop. x. 
MDCCXCVI. P 
