Here is a story of how Archimedes supposedly arrived at his most famous discovery, the Archimedes Principle. Sit back and enjoy!
King Hiero II rises to power
King Hiero II was unlike any other king of the ancient Greek world. He started out as an illegitimate child of Hierocles, a Syracusan noble, who claimed descent from Gelon. At around 20 years old, he joined the army and quickly rose through the ranks, demonstrating exceptional leadership and tactical skills. He was appointed commander of the troops at 33 years of age and only five years later, Hiero led the Syracusan army to a critical victory against the Romans, firmly cementing his position as the commander of the army. Even though he did not come up from royal lineage, his military prowess as a general had won the hearts of the people of Syracuse, and in a rare turn of events, they made him their king.
King Hiero II, being a devout believer, sought to express his gratitude to the gods by making a crown made of pure gold, and having it placed in one of the temples in the city. Keen on pleasing the gods with his offering, Hiero devised a specific design for his gift to the gods: he wanted the crown-shaped like a laurel wreath – a popular symbol of victory, success, and achievement in ancient Greek mythology.
He proceeded to weigh out the precise amount of gold required for the task and sought out one of the best goldsmiths in Syracuse to carve out a near-flawless crown. Hiero’s instructions were simple: use all your wits and skills to carve out a golden wreath worthy of the gods. The King charged the goldsmith to deliver the job on time as he already had in mind the day when he will present his gift to the gods. As an incentive, he promised to reward the goldsmith handsomely if he completes the work before the deadline.
In principle, the goldsmith would melt the gold in a furnace and cast it into the exquisite laurel-shaped crown with as much accuracy as humanly possible. The goldsmith had to be careful to keep a near-perfect working environment, as any impurities would devalue the crown. In the end, he did an impressive job and pleased the King even more by completing the project within the allotted time. The King kept his word and showered the goldsmith with rewards, gifts, and praise. He was sure the gods would be pleased by his timely work … but not for long.
Whilst King Hiero II proceeded with the preparations for the ceremony, rumors began circulating around the city that the goldsmith had played a cruel trick on the king by mixing the pure gold with some silver (which is cheaper) and pocketed the difference. When the King caught wind of the rumors, he was very distressed. He called in the goldsmith who swore by the gods that he had used the exact amount of gold that the king had given him, pointing out that the weight of the crown was identical to the weight of gold that the king had given him. The king, however, was not convinced. If the goldsmith had mixed silver in it, he wanted to know for sure. And he knew just the right man to get to the bottom of this.
Archimedes and crown
Archimedes was a relative and a friend to King Hiero II. Although Archimedes was only 22 years old at the time, he had already built a reputation for himself with stunning inventions and his incredible aptitude in natural philosophy and mathematics. In fact, Archimedes was known to be so engrossed in his work that he would even skip baths and even food. The King reached out to Archimedes with the imminent task: determine the authenticity of the crown before the day of the ceremony. However, he did not want Archimedes to scratch it, break it apart, or melt it; if it turns out that the goldsmith was telling the truth, then the ceremony can proceed as planned, without the crown undergoing any further alterations. In modern terms, Archimedes had to do a non-destructive test on the crown. If there was any man in Greece capable of doing that, it had to be Archimedes.
Archimedes knew that silver and gold were fundamentally different, but if a small amount of silver is smelted together with gold, the resulting alloy can be impossible to distinguish from the actual gold by a mere inspection. Unless of course you melt them back again in which case the constituents will separate will clearly separate but that’s an option he didn’t have. Furthermore, he was dealing with a work of an accomplished goldsmith; Archimedes knew that if he dared cheat the king, he would be so careful to create a near-perfect silver-gold alloy. Archimedes pondered on the problem for days and it appeared he couldn’t find a way to get around it.
Archimedes has the “Eureka!” moment
One afternoon after yet another unproductive day at his lab, Archimedes went for a ritual bath in a public bathroom, as was the tradition back in the day. Archimedes may have been pondering on the king’s problem so much that he forgot to take notice of a filling tub. When he eventually submerged himself into the tub, it was already full of water (to the brim). Archimedes noticed that as he lowered himself into the tub, the water rose and spilled over the sides. This experience wasn’t all new, but today, he saw it in a different light:
Water has the ability to flow.
When you fish out a cup of water from a tub or a bucket, water flows back and fills up the hole. If you pour back the water, it flows out and evens out; returning as before. But what happens when you submerge an object into a body of water and leave it there?
See, when an object is immersed in water, it pushes water aside to create space for itself. The water, on the other hand, doesn’t like having holes poked on its surface so it’ll tend to flow back and even out the asymmetry. This will be felt like a soft continuous push on the object by the water, a force that physicists call the buoyant force or upthrust. The object in question will experience an apparent loss in weight as a result of this force; feeling lighter than in air. Anyone who’s ever dipped into a tub full of water (or a swimming pool) is aware of all these phenomena – as was Archimedes.
But on that particular day, Archimedes was able to make a remarkable connection by focusing on the water that was pushed away by the object.
Where does it go? Or rather,
Where does it flow to?
How much water is displaced?
Could the amount of water spilling over the edges be predicted?
If yes, how? And what does it depend on?
As a man who had taken his fair share of baths in a tub, Archimedes knew that the more you lower yourself into a tub of water, the more water keeps spilling over. You dip one leg into the tub, a little water overflows, dip a second leg, more water overflows, by the time you sit in the tub, the floor around the tub is now drenched.
Clearly, the bigger the volume of the object, the more water it displaces. Or put in another way, the amount of water displaced depends on the volume of the object in question.
Could the water displaced tell us something about the volume of the object? Oh, wait…
If an object pushes water aside to make room for itself, like poking a finger in wet clay, then it follows that the water displaced must be equal to the volume of the object immersed in it.
This was the epiphany, the fact that Archimedes could now deduce the volume of irregular objects by immersing them in water and measuring the volume of water displaced. Archimedes realized he had hit the nail on the head here; he had to get to his laboratory as soon as possible and test his idea. He was so overwhelmed with joy, excitement, and urgency that he leaped out of the tub, (skipping yet another bath) and ran through the streets of Syracuse naked shouting “Eureka! Eureka!” Which means, “I have found it” in Greek.
The concept of density
For a while now, Archimedes had been toying around with the concept of density. An important property of any physical object. It is not enough to say a certain object is heavy and another is light. A sack of cotton, for example, can be as heavy as a pack of stones, but not nearly as small. Similarly, a stack of papers can weigh the same as a block of concrete but it will occupy a much larger space! (As my college professor used to say, “how heavy is heavy”?)
Archimedes knew that gold is heavier than silver. That is, for the same volume, gold will weigh heavier. Or put in another way, for the same weight, gold will occupy a lesser volume than silver. So one way to check if the crown was made of pure gold was to compare it with a known sample of pure gold and see how it matches up.
For instance, Archimedes could get a lump of gold that weighs exactly as the crown and compare their volumes. If the crown is made of pure gold, then it had to occupy the same volume as the lump of gold in question.
Alternatively, he could get a lump of pure gold with the same volume as the crown and then compare their weights. If the crown was made of pure gold, then it should have the same exact volume as the lump of gold in question. (Think of it this way, a kilogram bag of sugar will occupy the same volume as any other kilogram bag of the same sugar, and a bag of wheat flour will weigh the same as another similar bag of wheat flour – unless of course, it’s not wheat flour!)
Either way, Archimedes had to determine the volume of the crown of gold. And it’s here where the trick was: Archimedes could only calculate the volumes of objects with regular shapes such as rectangular blocks, cylinders, and spheres. The shape of the crown wasn’t like anything regular and he had hit a dead-end there. But everything changed after that afternoon bath.
Back at his lab, Archimedes wasted no time testing his hypothesis. First, he wanted to be sure that the volume of an object is indeed related to the volume of water displaced when the object is immersed in water.
To test this, he took a series of rectangular slabs of which he knew how to calculate their volumes and began dipping them in water one after another while taking note of the displaced water. Afterwards, he compared the volumes of the rectangular blocks to the amount of water displaced and found a correlation. For example, a rectangular block of twice the volume of another displaced the amount of water, and so forth.
He repeated the procedure for other objects of known volumes such as cubes, cylinders, and spheres and found similar results.
Confident that he could now deduce the volume of objects by taking note of the displaced water, he set out to test the crown.
First, he took a lump of gold weighing exactly as the crown and dipped it in a vessel filled with water to the brim. Archimedes carefully collected the water that had spilled over and measured its volume.
Then he took the crown and repeated the above procedure. Archimedes found out that the crown displaced more water than the pure lump of gold of the same weight – giving him the impression that the crown was not all-pure gold. Archimedes repeated his experiments with more precision over the next few days and established that the crown is indeed an alloy of gold and silver, he then presented his findings to the king.
Do you know what happened to the goldsmith after that?
Me neither.
Archimedes Principle
Now with the pressure of the king’s task off his shoulders, Archimedes continued his research on buoyancy with full attention. He now turned his attention to what most swimmers usually take for granted, the buoyant force. Archimedes’ subsequent works dealt with investigating this property, and he came up with another important discovery – the Archimedes principle.
Archimedes discovered that “when a body is partially or totally immersed in a fluid, then it experiences a buoyant force which equals to the weight of the fluid displaced”.
This is the Archimedes Principle, stated mathematically as,
Buoyant force (Apparent loss in weight) = Weight of the fluid displaced
Disclaimer
This story originally came down from a Roman writer and architect Vitruvius, some 200 years after the event supposedly happened. So no one really knows whether this story is indeed true, in fact, there are holes in this story that indicates the contrary. For example, although Archimedes’ procedure works in theory, it is quite a different matter when it comes to practice. In fact, Galileo himself expressed skepticism about the veracity of the story in his tract La Bilancetta, or “The Little Balance,” published in 1586. The difference in the volume of water displaced between the crown and a similar sample of pure gold would have been so small for Archimedes to notice. In addition, other phenomena such as surface tension and the adhesive forces of water would spoil the experiment.
True or not, it’s a great story.
Interested in similar stories like this? Check out my piece on The falling apple story: How Sir Isaac Newton discovered gravity and the story behind Mpemba’s effect.
