# Is it true that Archimedes formulated his famous principle based on an observation he made as he immersed himself in a bath?

No. He did supposedly get an idea as he sat down in a bath, but it wasn’t about buoyancy which is the phenomenon on which his principle is based.

Archimedes’ principle states that any object, totally or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. But. it was not the discovery of the principle named after him that sent the Greek mathematician, physicist and inventor running naked through the street yelling “Eureka,” meaning “I have found it.” What he supposedly found was a solution to a problem posed to him by King Hiero of Syracuse who had suspected that a goldsmith tasked with making him a gold crown had substituted some of the gold with silver. How could one determine if the crown was pure gold, wondered the king?

As the story goes, the answer came to Archimedes as he noted that the level of the water in a tub rose as he immersed himself in it. This led him to conclude that a body immersed in water displaces a volume of water equal to its own volume, meaning that the volume of any irregularly shaped object that sinks in water can be determined from the volume of the water displaced. Then dividing the weight of that object by its volume allows its density to be calculated. When it came to the crown, Archimedes suggested that whether it had been adulterated with another metal could be determined by comparing its density to that of pure gold. At least so goes the story. How much reliance we can place on this account is questionable since we first hear of it from the Roman architect and engineer Vitruvius in the 1st century B.C., some two hundred years after the eureka moment supposedly happened. While that narrative may be somewhat fanciful, there is no doubt that Archimedes really did formulate the principle of buoyancy which explains, among other phenomena, why ships float.

Let’s begin with an imaginary snorkeling adventure. You see some pretty coral at a depth of some twenty feet and decide to dive down for a closer look. But the deeper you go, the greater the force you feel pushing you up towards the surface. This means that the downward force due to your weight plus the pressure of the water on top of you is less than the force exerted by the water on the bottom of your body pushing you up, which is referred to as the “buoyant force.”

Now imagine that we can hit the pause button on the dive at some point and somehow magically remove your body and replace it with water. This water has the same volume as your body had occupied, but not the same weight. However, this replacement water experiences the same pressure from below, that is, the same buoyant force, as your body did, since water pressure is dependent solely on depth. But the volume of water that has replaced your body has a certain weight, pushing downwards. Since you are now stationary, the downward force must equal the upward force, or the buoyancy. But since that downward force is exerted by the weight of the water occupying the same volume as your body had, the conclusion is that the buoyant force must equal the weight of the water that was displaced by your body. And here we have Archimedes principle, namely that the upward force on a body immersed in a fluid is equal to the weight of the fluid displaced.

How then does that explain why a ship floats? For flotation, the upward force exerted by the water, that is the buoyant force, must equal the downward force exerted by the ship’s weight. As a ship is lowered into the water it will displace more and more water until the weight of the displaced water equals the weight of the ship. At that point it floats. If more weight is piled on the ship, it will sink lower and lower until the weight of the water it displaces is less than the ship’s weight, at which point it will sink. But since ships are constructed in such a way as to have large volumes of empty space, the volume of water they displace will weigh more than the ship, since water is heavier than air. This also explains why a ship will float higher is salt water than in fresh water. Since salt water is heavier due to the salt content, a smaller volume has to be displaced to equal the ship’s weight. If you have wondered why it is almost impossible to sink in the Dead Sea, it is because of its astounding salt content of about 34%. You don’t need to be immersed too deeply to displace a volume of water equal to your weight. I hope all this talk of immersion hasn’t left you with a feeling that you are in over your head.

@JoeSchwarcz