Archimedes, Nudity, and Buoyancy

There is a story told to young scientists that is almost as old as science itself. It allegedly took place over two thousand years ago and involves the Greek aristocrat Archimedes (287-211 B.C.E), whom many consider to be the greatest scientific mind until Sir Isaac Newton. The story comes to us from Vitruvius, Roman author and engineer that was born sometime around 75 B.C.E. (Vitruvius was also the inspiration for Leonardo Da Vinci to draw the Vitruvian Man, a diagram of perfect geometric proportions in the human form.)

The story goes that King Hiero II of Syracuse paid a goldsmith to craft an ornate crown for display in a temple. The goldsmith was paid for his services, and was given the gold necessary for its construction from Hiero’s treasury. Though the crown he received was of the highest quality workmanship, Hiero suspected that he had been cheated. The suspected cheat was in the makeup of the crown itself; Hiero believed it was not made of the pure gold that the goldsmith had been supplied with. Hiero supposed that the goldsmith had taken a portion of the gold for himself and replaced it with an equal volume of some cheaper metal. In that case the crown would be made of a less valuable gold alloy.

Hiero was furious, despite not having a lick of proof. He did, however, have the crown in hand for testing. He summoned the greatest scientist in the land, Archimedes, to test his suspicion. Determining if the crown was made of pure gold or an alloy was simple… in theory.

The ancient Greeks were masters of engineering and had a very sophisticated understanding of geometry. They understood the concept of mass and volume (the amount of space an object takes up), and that if you divided the former by the latter you arrived at an object’s density – a measure of how compact its mass is. An object of high density has its mass contained in a small volume, while an object of low density has its mass spread out over a large volume. Alternatively, two objects of equal volume but different density must have different mass. Gold was one of the heaviest (most dense) metals known at the time. All Archimedes had to do was determine the volume of the crown, weigh it to determine its mass, then divide the latter by the former. If that number was the density of gold then the crown was pure gold. If the density of the crown were anything less than the density of gold then Archimedes would know that some volume of gold had been substituted for a less precious metal (perhaps silver or copper). Hiero could then do as he pleased with the offending goldsmith.

What was simple in theory proved daunting in practice. The crown, being an intricate work of art, was not a simple shape for which the ancient Greeks knew the volume. The determination of its volume by calculation required integral calculus, a mathematical discipline that would not exist until Sir Isaac Newton invented it 1,700 years in the future. He could, of course, make it a shape for which he could calculate the volume. We don’t know if Archimedes approached the king about smashing the crown into a ball so that he could calculate its volume. This would have undoubtedly worked, since Archimedes himself published a derivation of the volume and surface area of a sphere in 225 B.C.E. It’s not hard to imagine that Hiero would not approve of the destruction of the crown just to determine if he had been cheated. Nor is it a suitable solution in the long term, since it doesn’t guard against future cheating.

How, then, was Archimedes to calculate the crown’s volume?

As Vitruvius’s story goes, Archimedes toiled for some time to no avail, finally deciding to take a break at the public baths. Getting into the bath, Archimedes noticed the water steadily rise as he lowered himself into it. At this point he jumped from the tub and buck-ass nude and dripping wet through the streets shouting “Eureka!”, which translates to something along the lines of “I’ve figured it out!”

Archimedes had realized that when an object is totally submerged in water it must displace a volume of water exactly equal to its own volume. Once pointed out this isn’t hard to understand. If volume is the amount of space something takes up then when something is submerged it must displace exactly that volume of water. All Archimedes had to do was to fill a container to the top with water, then gently lower the crown into the water until it was entirely submerged while simultaneously catching all of the water that overflowed the edges of the container. Since one of the defining properties of a liquid is that it takes the shape of whatever container it is placed in, Archimedes could pour that water into any shape that he would calculate the volume of, such as a cylindrical glass jar. With the volume of the crown known it was a trivial matter of weighing it and calculating its density. As the story goes, Archimedes thus proved that the crown was indeed an alloy, and Hiero had the goldsmith executed.

Vitruvius published this account over 100 years after Archimedes lived, and I’ve always had a hard time believing it. Archimedes was one of the founding fathers of hydrostatics, the study of fluids and their forces. He had worked out far more complicated concepts than understanding that a submerged object displaces its volume of water. I’ll spare you the details of his derivation of the volume of a sphere, but suffice it to say that even with the use of modern mathematical notation it is a hard batch of equations to follow. I believe the true story involved Hiero setting the task before Archimedes, who immediately went home to calculate the volume of the crown. After some trying to estimate it using mathematics he decided that the most accurate thing to do was submerge it in water and measure the volume of the water displaced. I grant you that this is not an exciting story, but I find it entirely more believable. Of course Vetruvius’s version involves tension and intrigue, and has survived virtually unchanged for 2,000 years. For as much as Archimedes was a master mathematician and scientist, Vetruvius was clearly a master yarn spinner.

One reason I have trouble believing Vetruvius’s version of the tale is that Archimedes had such a great understanding of science that he described a concept that still gives aspiring scientists trouble 2,200 years later. The concept is all the more frustrating because it is something that we understand intuitively as children, but struggle to explain as adults. That concept is buoyancy, or more specifically, how boats float.

Anthropologists debate when the first boat was invented, with argument centered around the period from 10,000 – 8,000 B.C.E. With that in mind it is safe to say that boats – meaning man-made objects that float on the water and can carry at least one human – predated Archimedes by several thousand years. If buoyancy describes how an object floats, how was it possible for the civilizations before Archimedes to construct massive military ships?

One need not understand why a log floats in the water to observe that a log floats. It does not take mathematical understanding to guess that one could strap multiple logs together so that when one stood on them they wouldn’t sink. Once that was actually done the boat was invented in the form of the raft. More tools brought more sophisticated shapes of boats. I wonder if it was intuition or experience that led to the discovery that a boat full of water will sink. It was probably both, since boats were invented independently all over the world, and we can assume that each culture explored their construction in their own unique way.

There are countless examples throughout the history of science proving that humans will often do a thing before they understand a thing. Metallurgy is one such example. Blacksmiths worked out that mixing some metals with other metals would make the resulting metal (an alloy) stronger than the separate metals. This advancement in ability (but not understanding) was at the heart of the bronze age. Bronze is an alloy of copper with twelve percent tin that is stronger than copper alone. Understanding why requires an understanding of the crystalline properties of metals – and how atomic structure affects it – that would not come until a few thousand years after the first bronze age civilization appeared on Earth.

Archimedes’s discovery concerned not how to build a boat, but why the boat you built floats. This understanding allows boats to be designed by mathematical principle rather than experience, which allows for much more sophisticated boats to be built. His great insight begins with the simple understanding that a submerged object displaces water equal to its volume.

Archimedes’s Principle describes what is known as the “law of buoyancy.” A physical law is different from a theory in that a theory explains why something is true, whereas a law simply asserts that it is true based on repeated experimental evidence. Newton’s law of gravity states that the gravitational force between two objects is related to their masses, the distance between them, and a “Universal Gravitational Constant.” It does not explain why that constant is what it is, or why it is constant; it simple states it and that assertion is proven by experimental evidence. Einstein’s theory of relativity attempts to explain why gravity works the way it does, but it is possible that the description is wrong. (Indeed, quantum mechanics is not entirely compatible with relativity, so clearly something is wrong.)

Take any object that floats, for instance a wooden log, and force it underwater. If you let go of the log it will float back to the surface. It becomes harder to push the log the further under the surface you try to push it, and when released the log will come to the surface with greater speed. Clearly the log is experiencing some force trying to make it float. For clarity we will give this mysterious force a name: we will call it a buoyant force. (Lest you think this was Archimedes’s term, the first documented use of the word buoyant in this context dates to around 1570 A.D. and most likely comes from the Spanish boyante, meaning “to float.”)

Archimedes’s Principle states that “any object which is submerged experiences a buoyant force equal to the weight of the water it displaced.” Archimedes’s Principle doesn’t explain why this is the case, simple that it is and that it can be proven through experimentation. When Archimedes submerged Hiero’s crown, the crown experienced a buoyant force that tried to push it back to the surface of the water. This force must have been overcome by the crown’s own weight; if it hadn’t then the crown would float. Now we can begin to see how something might float. If the buoyant force is greater than the weight of the object, then the object will be pushed to the surface of the water.

But wait, that can’t be right. If the buoyant force is stronger than the weight of the object (exerted by the gravitational force) then at the surface of the water there is an upward force on the object which would send it flying out of the water. We can turn back to observation to solve this puzzle. When an object “floats” it does not sit on top of the water. Part of the object is under water, while part of it is above. We should properly define floating as to be when something sits partially in the water but is not entirely submerged. So for something to float, it must not sink any deeper into the water than it already has, and it must not be thrown out of the water and into the air, meaning that it is stable. Stability can only happen when all forces on an object are balanced. This means that something floats when the buoyant force is exactly balanced by the weight of the object.

Eureka! Archimedes’s Principle states that the buoyant force on an object is equal to the weight of the water the object has displaced. Meaning an object will float at the point that it has displaced its own weight in water! This is inherently tied to the shape of the object, or more specifically, the shape of the part of the object that is under water.

Think of a modern battleship, which weighs about 56,000,000 kilograms and is made mostly of steel, which has a density of 8,000 kilograms per cubic meter. Water has a density of 1,000 kilograms per cubic meter. One cubic meter of steel weighs more than one cubic meter of water, which should not surprise anyone. However, a battleship floats. (I will refrain from a “You sunk my battleship!” joke, and I’ll thank you to do the same.) The reason the battleship can float is that although steel has a density greater than water, the battleship does not. Battleships are full of rooms filled with air and people. Air, in particular, is much less dense than water (a dainty 1 kilogram per cubic meter, thank you very much). By spreading that 56,000,000 kilograms over a larger volume, which is then filled with air, the density of the battleship is much less than the density of the water it displaces. Thanks to Archimedes’s Principle we know for certain that the amount of water displaced by the submerged portion of the battleship has a mass of 56,000,000 kilograms. What would happen if we placed a 1,000 kilogram mass on our battleship? Well, the battleship would sink a little until an extra 1,000 kilograms of water was displaced. For this reason, boats are engineered with a maximum weight capacity which is directly related to how low they can sink before they start to have problems, like water coming in through windows on the side of the ship. (For anyone wondering, the ship I’ve used for reference is an American Iowa-class battleship, which is 270 meters long and 33 meters wide at the water line. When the ship has submerged 6.5 meters the displaced volume of water has a mass of 56,000,000 kilograms, meaning the ship will stably float at this point. Adding any weight to the ship will cause it to sink further, and knowing how tall the ship is – it’s “draft” – tells you how much weight it can support before sinking. The draft of an Iowa-class battleship is 11 meters, so she is floating quite comfortably with the load I’ve estimated.)

Remember that Archimedes’s Principe is a law, and as such doesn’t tell us why the buoyant force is equal to the weight of the water displaced, simply that it is. Understanding why would require a theory of buoyancy, which is perhaps a topic for another day.

About Andrew Porwitzky

Dr Andrew Porwitzky is a professional scientist, comic book junkie, and freelance writer. He is also on Twitter way too much.

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