Inspirational Thinkers : EUREKA ! Archimedes had the Answers

By

S. U Abdullahi

sheikhbeta@yahoo.co.uk

 

Arguably the best known story in scientific circles has to be the one about Archimedes in his bathtub – or rather running out of it. As the legend is told, a new golden crown had been sculpted for Hieron II, the reigning King of Syracuse at the time. The King, suspicious for an undisclosed reason, ordered Archimedes to determine if indeed the crown was of pure gold or merely some alloy of it. However, Archimedes was warned not to damage the crown in anyway during the investigation.

 

This last proviso had the sage baffled: how would one determine the composition of a crown without melting and testing even a tiny portion of it? Now, Hieron II as luck would have it, was a blood relation of Archimedes. Therefore it was not as though the life of the great man was in mortal danger should he fail to solve the problem. But there was the matter of professional pride. For Archimedes had already built a reputation for cracking just such puzzles. It was rather annoying that this particular one remained ……well, puzzling.

 

Then one day Archimedes entered his bathtub – the ancient Greeks, you would recall from history lessons, loved their small pleasures – and made an inspirational discovery. As he stepped into the tub, he noticed the water overflowing out of it, as those who could still afford the luxury of sluicing in bathtubs must have sometimes observed. This ordinary occurrence triggered an extraordinary train of thought in Archimedes, leading to what has now become known as Archimedes' Principle, and the realisation that he had splashed his way to a solution of the King's problem.

 

In his excitement Archimedes was said to have leapt out of the bathtub and sprinted to the palace stark naked shouting, ' Eureka !' ' Eureka !' along the way – ' Eureka ' being the Greek word for 'I've found it'. It was a spectacle that was bound to have drawn the curious glances of ordinary people walking along the streets of Syracuse , even in the rather permissive surroundings of ancient Greece . It certainly ensured that, thousands of years after the event, the story of Archimedes had lost none of its sensational appeal.

 

The law Archimedes discovered was that a partially or fully immersed body in a fluid experiences an upward force, or buoyancy, equal to the weight of the fluid it displaces. By a direct deduction from this law it can be shown that if bodies made of the same material and having equal weight are immersed in the same fluid, they should displace the same amount of that fluid: a unit weight of pure gold, for example, will displace the same quantity of water whether it comes in the shape of an uncut nugget, a ball bearing, a cup, a statuette or – yes – a crown.

 

Archimedes obtained a pure gold nugget of the same weight as the King's crown and immersed both separately in water only to observe that the crown displaced more volume of water than the gold nugget, a sure indication that the gold in the crown is mixed up with another material less dense or compact than gold – silver, in fact, as subsequent investigations confirmed. The King thus received the annoying verdict: the crown was of impure gold; while the poor goldsmith responsible for the adulteration received a rather harsh sentence: a summary execution.

 

Archimedes' Principle has widespread applications. It is the rule governing the operation of gadgets as small as the hydrometer, a device well known to our roadside electricians, who use it to measure the level of charge in the electrolytes of batteries. This is based on the theory that the density of electrolytes changes with their level of charges, so that differently charged electrolytes push up the indicator stick inside the hydrometer to different colour-coded bands of green, white and red. The principle also controls the workings of huge machines such as sailing ships. Indeed, nowhere is this more graphically illustrated than in submarines, where the buoyancy of the vessels could be adjusted through the use of ballast tanks to select the depths at which they navigate.

 

Another important contribution of Archimedes was the full mathematical development of the principle of the lever and compound pulleys. Archimedes found that a small force applied at a long distance from the fulcrum – the point about which the lever rotates – could lift a large load placed at a short distance from the fulcrum. He determined precisely that the force required to lift the load placed at a given distance on the short side of the lever was inversely proportional to the distance of the applied force on the long arm of the lever: the longer the lever arm on the long side, the larger the load that can be lifted for a given force applied on the short side. This is the principle on which the operation of crowbars and the common type of soft drink openers is based. It also governs the action of cranes – though these use a system of pulleys which operate in a slightly different way to confer similar 'mechanical advantages' as levers do.

In theory, levers and pulleys have virtually limitless possibilities. If a lever can be made sufficiently rigid and the long arm sufficiently long, there is virtually no size of load that cannot be lifted. Archimedes appreciated this fact, and was reported to have boasted “give me a place to stand on, and I can move the earth.” Heiron II promptly took up Archimedes on this boast, and challenged him to lift at least any sufficiently large object around them. Archimedes was then reported to have rigged a system of compound pulleys and, with little effort, towed a fully laden ship ashore.

In the field of Agriculture, Archimedes is credited with the design of the earliest type of a water pump. Known as the Archimedes Screw, the device employed a screw inside a close-fitting cylinder which, when rotated, raises water up for irrigation. It was possible that the great man might have got the idea for this pump from Egypt when he studied there. In any case, the pioneering ancient Egyptians were known to have had similar devices long before the time of Archimedes.

In mathematics, Archimedes calculated the value of 'pi' (π), the ratio of the circumference of a circle to its diameter, an important value in the evaluation of the areas and volumes of circles, spheres and cylinders – calculations which Archimedes also carried out. His method of calculating the value of 'pi' involved inscribing a circle inside one polygon and then drawing a second slightly smaller polygon inside the circle. As the polygons outside and inside the circle were given more and more sides their shapes approached that of the circle. Using this method Archimedes was able to narrow down the value of π to be between ²²³/₇₁ and ²²⁰/₇₀. In fact, the actual value π, which is ²²/_7, is exactly equal to the outer limit of ²²⁰/₇₀ specified by Archimedes.

 

Archimedes also famously attempted a calculation of the grains of sand contained in the whole universe. To many this might look like a pointless exercise, but in doing so Archimedes was precisely proving the point that any finite quantity can be measured. The sheer size of the numbers involved entailed Archimedes having to invent a method for expressing large numbers. He therefore devised a system similar to the 'scientific notation' – the shorthand used to write extremely large or small numbers – of today. If you had ever tried to write down, for example, a trillion trillion in full with all the eighteen zeros you would appreciate how expressing it as 10¹⁸  in the scientific notation has made life much easier.

 

Although Archimedes preferred that he be remembered as a natural philosopher or what will today be called a scientist, more at ease when tackling complex mathematical problems, in reality it was his heroic achievements in the design of practical, mostly military, devices that attracted much of his fame. Those were, after all, the turbulent days when one Kingdom would attack another just because a sovereign felt – like Dick Cheney and his neo-con colleagues did in the case of Iraq – that the adventure was 'doable’ and there were some advantages to be gained.

 

Archimedes built huge catapults that could be used to sink ships or destroy defensive walls; he designed a sophisticated arrangement of mirrors that could be focused on enemy ships to set them on fire; and his pulley systems were used to completely drag enemy vessels out of the sea, running them aground or even overturning them altogether. Indeed, when the Romans laid siege to Syracuse their fleet came up against the full range of such ingenious devices contrived by Archimedes.

 

For close to three years, it was reported, the Roman fleet fought against the ingenuity of Archimedes until the invaders came to utterly dread even approaching the walls of Syracuse for fear that Archimedes would unleash another deadly contrivance against them. But eventually Roman numbers and might prevailed and Syracuse fell in 212 B.C. As the victorious Roman soldiers were carrying out mopping up operations in the fallen town, one of them came across Archimedes deeply lost in, of all things, the study of figures he had drawn on the ground.

 

It may be that Archimedes did not hear the orders issued by the soldier to fall in line and follow him, or if he did the words never really registered in his consciousness; but he was said to have waved off the soldier, muttering something about " ….disturbing my diagrams." Unfortunately, conquering soldiers, in Rome , Iraq or anywhere else, are unlikely to tolerate such impertinence from, as they would say, a bloody civilian.

 

And so it happened that Archimedes, born of an astronomer father circa 287 B.C., and the greatest scientist and mathematician of ancient times, was regretfully killed by a common Roman soldier. Among the people who mourned his passing was the commander under whom the short-tempered soldier served: General Marcus Claudius Marcellus had looked forward to meeting the great man who had given his troops so much trouble. He had to content himself with the sad alternative of according Archimedes a state funeral with full military honours.

 

 

 

Engr. S.U. Abdullahi

N.N.D.C Quarters,

Kundila Kano.