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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 
w x and w 2 of gold and silver respectively. Then 
(1) the crown displaces a certain volume of the fluid, F, 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 V 1 of 
the fluid; 
(3) the weight W of silver displaces V 2 , say, of the fluid; 
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 ) 
w x _ V 2 — V 
w 2 ~ V- V x 
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 ease 
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 down 
wards 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 architectura, ix. 3.