A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

ON FLOATING BODIES, I, II 95
96 ARCHIMEDES
principal parameter of the generating parabola, is a veritable
where k is the axis of the segment of the paraboloid cut off by
tour de force which must be read in full to be appreciated.
the surface of the fluid.)
Prop. 1 is preliminary, to the effect that, if a solid lighter than
V. Prop. 10 investigates the positions of stability in the cases
a fluid be at rest in it, the weight of the solid will be to that
where h/±p> 15/4, the base is entirely above the surface, and
of the same volume of the fluid as the immersed portion of
« has values lying between five pairs of ratios respectively.
the solid is to the whole. The results of the propositions
Only in the case where s is not less than (h — ^pf/h
2 is the
about the segment of a paraboloid may be thus summarized.
position of stability that in which the axis is vertical.
Let h be the axis or height of the segment, p the principal
BAB 1
is a section of the paraboloid through the axis AM.
parameter of the generating parabola, s the ratio of the
G is a point on AM such that AG = 2 CM, K is a point on GA
specific gravity of the solid to that of the fluid (s always < 1 ).
such that AM-.CK =15:4. CO is measured along GA such
The segment is supposed to be always placed so that its base
that GO = %p, and R is a point on AM such that MR = §C0.
is either entirely above, or entirely below, the surface of the
A 2
is the point in which the perpendicular to AM from K
fluid, and what Archimedes proves in each case is that, if
meets AB, and A 3
is the middle point of AB. BA 2
B 2
, BA Z
M
the segment is so placed with its axis inclined to the vertical
are parabolic segments on A 2
M 2
, A 3
M 3 (parallel to AM) as axes
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 less
t
than (h — ^pY/h? if the curved surface is downwards, and not
greater than {h 2 — (h — 2
%p) }/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.)
and similar to the original segment. (The parabola BA.,B 2
is proved to pass through G by using the above relation
IV. If h>ip, but k/±p