343 
radius, as will, by its expansion, present a surface equal to the 
difference between the triangles, that is, the breadth must be 
to the radius as (the difference between the triangles divided 
by the expanded radius) is to the expansion of the radius. 
In balancing the semicylinders, less or more weight may 
be employed to produce the same effect, provided the centre 
of gravity is further from the axis, or nearer to it; for stance, 
if the balancing for a homogeneously balanced semicylinder 
be placed at its centre of gravity, 0°4244 from the axis, it will 
be half the specific gravity of water; but if the balance be 
placed at half that distance, the whole weight will equal that 
of an equal bulk of water, and if at one quarter the distance, 
it will be double the specific gravity of water. ‘Thus, the same 
effect is produced by bodies whose absolute weights are so 
different, that is, they will sink by the withdrawal of water 
below the surface level, or rise on any addition being made, 
though one may be much lighter than water, one equal to it, 
and one double its density. But if the weighting be made to 
act with a force greater than half the specific gravity of water, 
the power of such excess of weight acts as the whole weight, 
that is, as the sine of the angle of rotation ; and if the whole 
of the float be elevated to the fluid level, by the withdrawal 
of a quantity of the fluid, the float will commence to descend, 
and, in doing so, actually raise the level of the fluid surface, 
producing the paradox of raising the height of fluid in a vessel 
by withdrawing a part; but the fluid will continue to rise 
only while the float is descending through the first quadrant, 
for, as soon as the point B, in Fig. 1, falls on the line of sur- 
face mn, the level will fall, and continue to fall during the 
further descent of the float through the second quadrant ; 
the converse of this is also true. 
The form of the float ought to be that of a figure gene- 
rated by a plane revolving on its axis. If otherwise, let 
AEFC represent a parallelopiped, equal in weight to the semi- 
