97 
of floating Bodies , and the Stability of Ships. 
If the specific gravity of the parabolic conoid should be less 
than the limit which has just been investigated, and if the axis 
should be to the parameter in a proportion greater than that of 
3 to fa and less than that of 15 to 8, it will float permanently 
on the fluid with the axis inclined to the horizon, and with the 
base wholly extant above the surface at some angle less than 
90°; which angle may be determined by the following geome- 
trical construction, subject to the limitation which will appear 
from the -construction itself, or rather from the computation 
founded upon it. 
Let ASBTD (Tab. VI. fig. 25.) represent a section of 
the parabolic conoid which passes through the axis ; which 
section will be a parabola. Let the axis BE be divided into 
three equal parts, one of which is EF. By the properties 
of this figure, F will be the centre of gravity of the solid. 
In the line FB take FH equal to half of the parameter, 
and through H draw the indefinite line rGZ perpendicular 
to BE, and in the line GZ take HK = FB ; in the line 
Hr take HI, which shall be to HK in the proportion of the 
specific gravity of the solid to that of the fluid ; and bisect 
IK in the point L ; with the centre L and radius LI describe 
the semicircle KOI, intersecting the axis BE in the point O ; 
through O draw OC parallel to KI, intersecting the parabola 
in the point C, and let PCN be drawn touching the parabola 
in the point C. Through C draw the indefinite line CR 
parallel to BE, intersecting the line KI in the point G ; in 
the revolution of a parabola, which is the section of any cone, whatever may be the 
angle at the vertex, half the parameter being substituted instead of the line, called by 
Archimedes “ea quae usque ad axem and it is a property of conics easily demon- 
strable, that any parabola being given, a similar and equal parabola may be formed 
from the section of any cone, whatever may be the angle at the vertex, the axis being 
of sufficient length. 
MDCCXCVI. O 
