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

V
ON FLOATING BODIES, I 93
determining the proportions of gold and silver in a certain
crown.
Let W be the weight of the crown, w 1
and tv 2
the weights of
the gold and silver in it respectively, so that W = w x
+ w 2
.
(1) Take a weight IT of pure gold and weigh it in the fluid.
The apparent loss of weight is then equal to the weight of the
fluid displaced ; this is ascertained by weighing. Let it be F v
It follows that the weight of the fluid displaced by a weight
w i
°^ gold is -=^ . F
r
(2) Take a weight W of silver, and perform the same
operation. Let the weight of the fluid displaced be F 2
.
Then the weight of the fluid displaced by a weight w 2
of
silver is ^S> F .
(3) Lastly weigh the crown itself in the fluid, and let F be
loss of weight or the weight of the fluid displaced.
We have then ^ . F x + ^
.
F„ = F,
that is, w 1
F x
+ w 2
F 2
= (w x
+ w 2 ) F,
whence
—* = -=^—r^--
, w, F 2
-F
F-F x
w 2
According to the author of the poem de 'ponderibus et mensurls
(written probably about a.d. 500) Archimedes actually
used a method of this kind. We first take, says our authority,
two equal weights of gold and silver respectively and weigh
them against each other when both are immersed in water
this gives the relation between their weights in water, and
therefore between their losses of weight in water. Next we
take the mixture of gold and silver and an equal weight of
silver, and weigh them against each other in water in the
same way.
Nevertheless I do not think it probable that this was the
way in which the solution of the problem was discovered. As
we are told that Archimedes discovered it in his bath, and
that he noticed that, if the bath was full when he entered it,
so much water overflowed as was displaced by his body, he is
more likely to have discovered the solution by the alternative
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 = ^
•
1
+ ^
• V
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.
2
,