A history of Greek mathematics - Wilbourhall.org

principal
ON FLOATING BODIES, I, II 95
parameter of the generating parabola, is a veritable
tour de force which must be read in full to be appreciated.
Prop. 1
is preliminary, to the effect that, if a solid lighter than
a fluid be at rest in it, the weight of the solid will be to that
of the same volume of the fluid as the immersed portion of
the solid is to the whole. The results of the propositions
about the segment of a paraboloid may be thus summarized.
Let h be the axis or height of the segment, p the principal
parameter of the generating parabola, s the ratio of the
specific gravity of the solid to that of the fluid (s always < 1 ).
The segment is supposed to be always placed so that its base
is either entirely above, or entirely below, the surface of the
fluid, and what Archimedes proves in each case is that, if
the segment is so placed with its axis inclined to the vertical
at any angle, it will not rest there but will return to the
position of stability.
I. If h is not greater than §p, the position of stability is with
the axis vertical, whether the curved surface is downwards or
upwards (Props. 2, 3).
II.
If h is greater than f p, then, in order that the position of
stability may be with the axis vertical, s must be not
t less
than (h — ^pY/h? if the curved surface is downwards, and not
greater than {h 2 — (h — %p)
2
}/h 2 if the curved surface is
upwards (Props. 4, 5).
III. If h>%p, but h/^p < 15/4, the segment, if placed with
one point of the base touching the surface, will never remain
there whether the curved surface be downwards or upwards
(Props. 6, 7). (The segment will move in the direction of
bringing the axis nearer to the vertical position.)
IV. If h>ip, but k/±p