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Primary Magazine - Issue 16: A little bit of history

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Created on 28 September 2009 by ncetm_administrator
Updated on 30 October 2009 by ncetm_administrator

Primary Magazine Issue 16

A little bit of history - Archimedes

In this issue’s article we look very briefly at some of the works and achievements of Archimedes, a world-renowned Greek mathematician, physicist, engineer and astronomer. He is considered by some to be the greatest mathematician of all time. His work is still used in maths and science education today, for example, the famous Archimedes spiral and use of pi for finding the area of circles.

map of Sicily showing SyracuseHe was born in 287BC in Syracuse, Sicily, and died in 212BC. He died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he shouldn’t be harmed. One of the stories of his death records that he was commanded to go with the soldier to meet the Roman General in charge, but he refused because he was too busy studying a drawing of circles and wanted to finish solving the related problem he was working on. The soldier was furious and killed Archimedes with his sword. It has been reported that the last words he said were ‘do not disturb my circles’, a reference to the drawing he was studying when he was killed.

His father was an astronomer and it was believed that Archimedes was related to King Hieron II, the ruler of Syracuse. Not much else is known about his life apart from his mathematical and scientific discoveries. A biography was written about him by his friend Herecleide but over time this has been lost.

cylinder and sphere

We do know however, that he thought his greatest mathematical achievement of all was to prove that a sphere has two thirds of the volume and surface area of a cylinder of the same height and diameter.

A sculpture of a circle and a cylinder was found on his tomb.


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Archimedes in the bathTo most people, Archimedes is probably best known for his ‘eureka’ moment and subsequent run through the streets naked, when he discovered the principal of buoyancy! According to the Roman writer, architect and engineer Vitruvius, a new crown in the shape of a laurel wreath had been made for King Hieron II. The king wanted to be sure that it was made of pure gold and not a cheaper version of gold mixed with silver. He gave it to Archimedes and told him to find out without damaging the crown. This was a bit of a problem for Archimedes because clearly he couldn’t melt it down to calculate its density and thereby its quality, so he had to come up with another idea. While he was having a bath he noticed that the level of the water in the tub rose as he got in and he realised that this could be used to determine the volume of the crown. He figured out that as water in this state is incompressible the submerged crown would displace an amount of water equal to its own volume and, by dividing the weight of the crown by the volume of water, the density of the crown could be found. If this density was lower than that of gold, the crown would be a cheap one.

He was so excited by his discovery that he leapt out of the bath and ran around the streets shouting "eureka!" ('I’ve found it'), but – he forgot to get dressed first! Great story, but sadly probably not true! The same principle would have been used, but Archimedes' method would possibly be to balance the crown on a balance scales with a piece of gold and immerse the lot in water. If the crown was less dense than the gold it would be more buoyant and the scales would tip accordingly.

A large part of Archimedes’ work in engineering arose form the needs of his home city of Syracuse. For example, he was asked by King Hieron II to design a ship that could be used for luxury travel, for carrying supplies and as a naval warship. It is said that this ship was the largest built at that time and for many years to come. Apparently it could carry 600 passengers, had a gym, garden and a temple for the goddess Aphrodite. A ship of this size in those days would have leaked a lot of water through the hull and so Archimedes invented his ‘screw’ which could remove a lot of the water that leaked in. Archimedes’ screw is still used today for pumping liquids and solids such as coal and grain. You can find details about how it worked from the Tiscali encyclopedia.

Archimedes Screw - by Silberwolf

Animation by Silberwolf. Used under the Creative Commons Attribution ShareAlike 2.5 License.

Why not let the children have a go at making one? Follow the instructions on NRICH. Archimedes invented various ways to defend his city from attack, for example, the ‘ship shaker’. It was a machine with a crane-like arm from which a large metal grappling hook was suspended, the claw was dropped onto an attacking ship, which then lifted it up and possibly sank it. It might be fun to try to design a version of this device that could be made into a model – it would be a great mathematics link to DT: scaling down, estimating and measuring. He also used mirrors to make a parabolic reflector in order to burn the ships attacking Syracuse.

Although he did not invent levers and catapults, he is responsible for their development through his invention of the odometer, a cart with a geared mechanism that dropped a ball into a container after each mile travelled – subsequent inventors used this idea to develop the milometer for measuring distances.

Some of his other inventions can be found on Wikipedia.

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pi - method of exhaustion by drawing circles within polygons
One of his major contributions to mathematics was pi (π), which we use today as part of a formula for finding the area of a circle. He worked this out using the ‘method of exhaustion’ for finding the circumference of a circle. This involved repeatedly drawing polygons outside a circle and smaller ones inside. As the number of sides of the polygons increased, the more like a circle they became and so more accurate. When they had 96 sides each, he calculated the lengths of their sides and from this was able to show that the value of π was between 31/7 and 310/71, giving a value of approximately 3.14.

From this, he was able to prove that the area of a circle was equal to π multiplied by the square of the radius of the circle. Try asking the children to find the areas of some circles that they draw using the formula \pi r^2 using a calculator.

This and his other contributions to geometry revolutionised the subject. More about his achievements and work can be found on the MacTutor History of Mathematics website.

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