Eureka!

Most Melburnians would be aware that our skyscraping Eureka Tower is named after the Eureka Stockade, the iconic battle from Ballarat’s gold rush days. But why “Eureka”?

The Greek word Eureka means “I have found (it)”, a very handy expression for a time and place where people were stumbling upon nuggets of gold. But why would an Australian miner exclaim his delight in Greek? It all stems from the granddaddy of Eureka moments, which also involved a quantity of gold.

Archimedes, the mathematical superstar from ancient Greece, was presented the task of determining whether a certain crown was solid gold. To do this, Archimedes only needed to calculate the volume of the crown, preferably without destroying it. While taking a bath, he suddenly realised that the crown’s volume could be determined by placing it in water and measuring the volume of water displaced. Thrilled with his discovery, Archimedes jumped up and ran naked through the street, shouting his famous cry.

It’s a great story, and possibly even true. However, we want to discuss a different Eureka moment, definitely true, and grand enough to have been commemorated on Archimedes’ tomb.

Not long ago, we wrote about the mathematics of ancient Egypt, and we raised the question of whether the ancient Egyptians knew of the formula for the surface area of a sphere. It is unlikely, but in any case Archimedes proved the formula, something the Egyptians most definitely did not do. A reader subsequently wrote to us, asking how Archimedes accomplished this. 

We endeavour to keep our readers happy, but this story will take some time to tell. To determine the area of a sphere, Archimedes first had to know the formula for the volume of a sphere. So, today we’ll discuss Archimedes’ ingenious study of volumes. Surface areas will be the Maths Masters’ homework for next week. 

We learn in school that a sphere of radius R has a volume of 4/3 π R³, but seldom does anyone ever hint at why this formula works. How could Archimedes prove such a formula? 

To begin, we have to discuss yet another instance of Archimedean brilliance: the law of the lever. This law states that two objects on a scale will balance if the weight multiplied by the distance to the fulcrum is the same for the two objects. For example, on the scale pictured below, the 100kg globe will exactly balance the 60kg Archimedes, because 100 x 3 = 60 x 5.

Now, what to do with this law? As well as the incredibly important practical applications, Archimedes had a new way of playing with geometric shapes. So, he went to his toy box and grabbed a sphere of radius R; it was the volume of this sphere that Archimedes wanted to find. Archimedes also fetched a cylinder and a cone, both of height 2R and circular base of radius 2R.

It is easy to calculate the volume of the cylinder (“base times height”), and Archimedes also knew that the volume of the cone was one third that of the cylinder: the mathematician Eudoxus had worked that out about a hundred years earlier. Archimedes’ brilliance was to recognise that his three solids could be balanced on the scale, as pictured:

From the law of the lever, Archimedes then knew that

2R x (volume of cone + volume of sphere) = R x (volume of cylinder)

So, knowing the volumes of the cone and the cylinder, some simple arithmetic gave Archimedes his desired result, the (now) well-known formula

volume of sphere = 4/3 π R³

But does balancing solids on a scale amount to a proof? The truly ingenious part of Archimedes’ balancing act is indicated by the pictured slices of the solids. The three solids have the same height, and the slices at corresponding levels are indicated by the colours.

Archimedes noted that any two slices of the same colour on the left are balanced by the slice of the same colour on the right. And, this is straight-forward to prove, because we are simply comparing the areas of three circles! Archimedes then argued that because the slices of the figures exactly balance, the whole solids must balance as well.

Archimedes’ genius here is reminiscent of the fundamental idea behind calculus, which was only properly developed 1900 years later. Indeed, many have argued that Archimedes (and, for similar reasons, Eudoxus) should be given at least some of the credit for the invention of calculus.

The top of the Eureka tower is intended to represent the gold rush, and the red line the blood that was shed during the Eureka stockade. But, without Archimedes there would be no Eureka, and so no Eureka Tower. Your Maths Masters prefer to regard the Eureka Tower as Melbourne’s monument to the great Archimedes. And the red line? It could signify Archimedes’ pointless death at the sword of a thuggish Roman soldier.

And, what about the surface area of a sphere? That is yet another Archimedean Eureka moment. As promised, we’ll write  about that next week.

Puzzle to Ponder: Can you provide the details for the last diagram, to show that the two left circles balance the right circle? (Another Greek hero, Pythagoras, may be of assistance, and you can find further clues here).

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator. 

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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