A history of Greek mathematics - Wilbourhall.org

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94 ARCHIMEDES
method attributed to him by Vitruvius, 1
namely by measuring
successively the volumes of fluid displaced by three equal
weights, (1) the crown, (2) an equal weight of gold, (3) an
equal weight of silver respectively. Suppose, as before, that
the weight of the crown is W and that it contains weights
tu 1
and iv 2
of gold and silver respectively. Then
(1) the crown displaces a certain volume of the fluid, V, say
;
(2) the weight W of gold displaces a volume V v say, of the
fluid
therefore a weight w x
of gold displaces a volume yiy- V x
of
the fluid
(3) the weight W of silver displaces V 2
,
say, of the fluid;
w
therefore a weight w 2
of silver displaces —• V 2
.
It follows that V = ^
•
V 1
+ ^
• V
2
,
whence we derive (since W = w 1
+ w 2 )
y\ v 2
-v
w ~
2
V-V]'
the latter ratio being obviously equal to that obtained by the
other method.
The last propositions (8 and 9) of Book I deal with the case
of any segment of a sphere lighter than a fluid and immersed
in it in such a way that either (1) the curved surface is downwards
and the base is entirely outside the fluid, or (2) the
curved surface is upwards and the base is entirely submerged,
and it is proved that in either case the segment is in stable
equilibrium when the axis is vertical.
This is expressed here
and in the corresponding propositions of Book II by saying
that, ' if the figure be forced into such a position that the base
of the segment touches the fluid (at one point), the figure will
not remain inclined but will return to the upright position '.
Book II, which investigates fully the conditions of stability
of a right segment of a paraboloid of revolution floating in
a fluid for different values of the specific gravity and different
ratios between the axis or height of the segment and the
1
De architectural, ix. 3.