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.

Cosmos - - Cosmos Science Club - — JACK CONDIE IL­LUSTRATION 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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