Why do some things float and others sink?

That time, that place

Syracuse, in Sicily. Archimedes was a kinsman of King Hiero and an engineer who built machines for the city — while regarding pure mathematics as his real work.

Why this question

The king sets a problem: find out whether the goldsmith held back gold and mixed in silver — without damaging the crown.

Awkward. It cannot be melted, so the metal cannot be inspected. Weighing is useless: a thief would have matched the weight exactly. What remains is volume, since silver mixed in makes the same weight take up more space. But how do you measure the volume of a lumpy crown?

What was found

The famous story has him stepping into a bath, watching the water spill, and seeing it. Submerge an object and it pushes aside exactly its own volume of water; measure the overflow and you have the volume.

But his real achievement was what came next. He set out the principle: a submerged body is pushed upward by the weight of the water it displaces. So floating and sinking are not a matter of heavy and light but a contest between that upward push and the object's weight. This is why a ship made of iron floats.