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