ON FLOATING BODIES, I, II 
95 
y by measuring 
by three equal 
of gold, (3) an 
, as before, that 
Dntains weights 
¡n 
s fluid, V, say ; 
V v say, of the 
lume of 
W 
j, of the fluid; 
obtained by the 
al with the «ase 
I and immersed 
lurface is down- 
duid, or (2) the 
rely submerged, 
ent is in stable 
s expressed here 
>k II by saying 
)n that the base 
i, the figure will 
ight position 
ions of stability 
tion floating in 
ty and different 
igment and the 
principal 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 ip, 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 
than {h — ip) 2 /h 2 if the curved surface is downwards, and not 
greater than {Ti 2 — (h — |p) 2 ] / h 2 if the curved surface is 
upwards (Props. 4, 5). 
III. If h>ip, 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>%p, but h/^p< 15/4, and if s is less than 
(h — ^p) 2 /h 2 in the case where the curved surface is down 
wards, but greater than {h 2 ~Qi — \p) 2 ]/h 2 in the case where 
the curved surface is upwards, then the position of stability is 
one in which the axis is not vertical but inclined to the surface 
of the fluid at a certain angle (Props. 8, 9). (The angle is drawn 
in an auxiliary figure. The construction for it in Prop. 8 is 
equivalent to the solution of the following equation in 6, 
ip cot 2 6 = |{h—k)~ip,