99 
of floating Bodies, and the Stability of Ships. 
xnonstrated. Secondly, because CQ is the abscissa of the seg- 
ment SGT corresponding to the vertex C and ordinate SQ, 
and by the construction CG == 2 GQ, it follows from the 
properties of the solid that G is the centre of gravity of the 
segment or part immersed SCBT. By the properties of the 
parabola, as ON is to CO so is CO to half the parameter, that 
is, as ON : CO : : CO = GH : FH ; therefore since the triangles 
GHF, CON, have one right angle each, and the sides round the 
equal angles are proportional, the triangles will be similar; 
consequently the angle OCN = the angle NFG: the sum of 
the angles FNC, NFC, is therefore a right angle, and the 
line FGM is perpendicular to the horizontal line PCN ; and 
since F by construction is the centre of gravity of the para- 
bolic conoid, and G has been proved to be the centre of gra- 
vity of the part immersed, and the line FGM is vertical, it 
follows, that the centres of gravity of the entire solid and of 
the part immersed are in the same vertical line, and conse- 
quently the solid is in a position of equilibrium, according to 
the construction. Thirdly, this equilibrium is that of stabi- 
lity ; for let the solid be conceived to be turned round an axis 
passing through the centre of gravity, through a small angle, 
in such a direction as to depress the parts towards D, and to 
elevate those near to A ; in that case the lowest point of the 
curve will be situated between C and B ; suppose it to be at 
W, draw WX = CO, parallel to BE, and take W g = of 
2. . 2. 
WX. Then since* CQ is to BE as the specific gravity of the 
solid to that of the fluid, it is evident that however the axis 
BE is inclined to the horizon, CO and consequently CQ must 
* Page 98. 
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