Archimedes of Syracuse: the man that gave us the ‘Eureka moment’

By venturing into the abstract, Archimedes of Syracuse gave us a new way of thinking about the forces at play in the universe.

Illustration by Jeffrey Phillips

“Eureka!”, mathematician Archimedes is reputed to have exclaimed when, sitting in a bath, he discovered what is now called ‘Archimedes' principle’.

In ancient Greek, eureka simply means “I have found it!”, but it was Archimedes, a native of the city of Syracuse, who gave us today’s usage as a pronouncement of breakthrough or great discovery.

As well as being good with numbers, he was also an inventor – a device he came up with to lift water from a well or a ditch became known as the Archimedes Screw, and is still in use in many places today.

The principle that led to his famous outburst was also about the nature of fluids. It states that an object partially or fully submerged in any fluid is pushed upwards by a force equal to the weight of the fluid that the object displaces. So, for something to stay afloat, the weight of the object above the water level needs to be equal to the weight of the water the submerged part displaces.

This principle is one of the main reasons that Archimedes is credited as a pioneer of physics and mathematics.

His work is particularly striking because it brought the two disciplines together for one of the first times in history. Rather than reporting on the results of experiments or pointing to observations for his reasoning – techniques favoured by his predecessor, Aristotle – Archimedes’ works are entirely abstract, paving the way for the standardised approach to science that is essential for fields such as theoretical physics.

The situation that gave rise to his personal eureka moment, however, was rather more worldly in nature. Archimedes was ordered by the tyrant of Syracuse, Hiero, to discover whether a local crown-maker was ripping him off.

The tyrant suspected that the artisan, who had been commissioned to create a solid gold crown, had switched out some of the precious metal for less valuable gold.

Sitting in a bath, cogitating, Archimedes formulated his principle – the amount of water displaced by an object is directly proportional to the object’s weight. Therefore, silver being lighter than gold, a pure gold crown should displace more water than one made of the two metals combined. Eureka, indeed! (It wasn’t healthy to disappoint the tyrant.)

By using geometry and other mathematical abstractions in his work, Archimedes was able to isolate and describe the forces that govern our world. Mathematical reasoning is a necessary language for talking about force because we can only observe them indirectly, by measuring their effects. For example, we can see the apple fall from the tree, but not what’s causing it to fall.

Archimedes wrote many treatises, of which 11 remain. In one, On the Equilibrium of Planes, he dealt with the concept of weight. He considered it a mysterious “downward inclination” that affected even two dimensional shapes. By treating it in such an abstract way, Archimedes was able to pinpoint what later would be characterised as centres of gravity.

He also discovered the law of the lever: that weights are balanced when placed at distances proportional to them. For example, if weight A sits two centimetres from the centre of a scale while weight B sits one centimetre from the centre in the opposite direction, and it is balanced, then the ratio of A:B is 2:1.

This is a forerunner of Isaac Newton’s second law of motion: that change in the momentum of an object is proportional to the force applied. The law of the lever sees A and B as forces that push things down, rather than weights.

Archimedes’ most abstract success – and, evidence suggests, his proudest achievement – was discovering the relationship between spheres and cylinders. When a sphere and a cylinder have the same height and diameter, he showed, the sphere has two-thirds the volume and surface area of the cylinder.

How do we know that particular mathematical eureka moment was his proudest? The Roman writer Cicero reports visiting his tomb and seeing it topped by a large cylinder and a sphere.

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