A history of Greek mathematics - Wilbourhall.org

;
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