Final answer:
Archimedes compared the volume of a sphere to a cylinder by using what is now known as Archimedes' Principle, where the buoyant force on an object is equal to the weight of the fluid it displaces. This allowed the determination of volume and density of objects based on their displacement of water.
Step-by-step explanation:
The story of how Archimedes compared the volume of a sphere to the volume of a cylinder is a famous one in both mathematics and physics, known as Archimedes' Principle. According to the tale, the ancient Greek mathematician Archimedes was tasked by the king to determine the purity of a gold crown without damaging it.
He realized that by submerging the crown in water and measuring the volume of water displaced, he could compare it to the known density and volume of pure gold. If the crown displaced more water than pure gold of the same weight would have, it meant that the crown was not made of pure gold.
This principle, which states that the buoyant force on an object is equal to the weight of the fluid it displaces, can be mathematically expressed as FB = Wfl, where FB is the buoyant force and Wfl is the weight of the fluid displaced.
In practice, this principle allows us to calculate the density of both fluids and solids, as Archimedes himself might have done to determine the purity of the gold crown. By submerging an object in a fluid and measuring the apparent loss in weight (ΔW), we can find the volume of the displaced fluid, which is directly proportional to the buoyant force.