98 Mr. Atwood's Propositions determining the Positions 
the line CR take GQ equal to half GC ; and through Q draw 
SQT parallel to PCN. When the conoid floats permanently 
and at rest, the surface of the fluid will coincide with the line 
SQT, and the axis will be inclined to the horizon at the 
angle ONC : through the points F and G draw the indefinite 
line FGM. 
The order of the demonstration will be as follows. First, 
to shew that, according to the construction, the volume of 
the immersed part SCBT is to the whole magnitude of the 
solid in the proportion which the specific gravity of the solid 
bears to that of the fluid : secondly, to shew that the centre 
of gravity of the solid and the centre of gravity of the part 
immersed are in the same vertical line ; and consequently the 
construction will place the solid in a position of equilibrium : 
thirdly, to demonstrate that the equilibrium so constituted is 
that of stability. 
Since by the properties of the circle, HI is to HK as 
the square of HO is to the square of HK ; and the square 
of HO is to the square of HK as the square of CO (= J- x HO) 
is to the square of BE (= J- BF) : therefore, since by the 
construction the specific gravity of the solid is to that of 
the fluid as HI to HK, it follows, that as the specific 
gravity of the solid is to the specific gravity of the fluid, 
so is the square of CO to the square of BE : but by the 
properties of the parabolic conoid the magnitude of the seg- 
ment SCBT is to the magnitude of the whole solid ACBTD 
as the square of CQ to the square of BE ; and consequently it 
is proved that when the solid floats according to the position 
described in the construction, the volume immersed SCPT will 
be to the whole magnitude as the specific gravity of the solid 
is to that of the fluid, which was in the first place to be de~ 
