# Archimedes' principle

**Archimedes' principle** is named after Archimedes of Syracuse, the first to discover this law. When he did, he ran naked out of his bathtub.<ref name=acottLaw>Template:Cite journal.</ref> Archimedes' principle relates buoyancy to displacement.<ref>Displacement here means the water pushed aside by the bottle (or other object).</ref>

## Principle

Archimedes' treatise, *On floating bodies*, proposition five states:

For more general objects, floating and sunken, and in gases as well as liquids (i.e. a fluid), Archimedes' principle may be stated thus in terms of forces:

For a sunken object, the volume of displaced fluid is the volume of the object, and for a floating object on a liquid, the weight of the displaced liquid is the weight of the object.

Briefly: **Buoyancy = weight of displaced fluid.**

## Gold

To use the principle to tell the difference between gold and another substance, the concept of **mass** (in practice, **weight**) must be added.

Apply this formula to a suitably pure piece of gold:

- [math] \frac { \text {density of object}} { \text{density of fluid} } = \frac { \text{weight}} { \text{weight} - \text{apparent immersed weight}}\,[/math]

That gives you the density of the gold sample. The only unknown is the *density of the (gold) object*; the density of water is 1.

Repeat for the experimental object (non-gold), when you get a different (and usually lesser) density.

Now you can tell what is gold and what is not, and collect your reward from the King of Syracuse. That is why Archimedes shouted **" Eureka!"**

### Second thoughts

We do not actually know if Archimedes used exactly this method. An alternative is to use a scale. On one side put the object to be tested (e.g. the crown). On the other side put gold of *equal weight*. Immerse the scales. The gold will go down, and the crown up (if it is not gold). That is because, being less dense than gold, it occupies a larger volume and receives more buoyancy.