ON FLOATING BODIES, I 93
l number. The
are that, of the
s, that which is
i is pressed the
3 fluid which is
:1 is shut up in
rne upwards in
ndicular drawn
a sphere, and
a the theorem
is a sphere the
th and in the
ways shown in
•of is similar to
y textbook, the
wing. A solid
fluid will, if let
. but not lower
3t down into it,
hat the weight
fluid displaced
will be driven
veen its weight
7 ) that a solid
3 the bottom of
in the fluid, be
t of the fluid
ag the famous
nat Archimedes
the problem of
determining the proportions of gold and silver in a certain
crown.
Let W be the weight of the crown, w x and iv 2 the weights of
the gold and silver in it respectively, so that W = w 1 + w 2 .
(1) Take a weight W 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 l .
It follows that the weight of the fluid displaced by a weight
w x of gold is ^ . F v .
(2) Take a weight If 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 m^-F 2 .
(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 l + ~ F = F,
If 1 If 2
that is, W \F X + w 2 F 2 = (w 1 +w 2 ) F,
, _ f 2 .
F
whence _ „
u' 2 F-J> x
According to the author of the poem de ponderibus et men
suris (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