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Archimedes (c. 287 c. 212 BCE):

a martyr for mathematics

The death of Archimedes by Edouard Vimont (1846-1930), from Wikipedia Commons.

The three best-known ancient Greek mathematicians are undoubtedly Pythagoras, Euclid and

Archimedes. Pythagoras is a semi-mythical figure, and Euclid owes his fame to his

Elements, the book which has set the pattern for mathematics ever since but about whose

author we know very little (we do not know how much of the material in the Elements wasEuclids own original mathematics). But we know a little more about Archimedes and his

original contributions to mathematics.

We know roughly when Archimedes died about 212 BCE and, as we shall see later,something of the manner of his death. We know that he was not young when he died, and

that his father was an astronomer. We know that he lived in Syracuse in Sicily, and since he

corresponded with mathematicians from Alexandria we deduce that he spent some time

studying there. We have imperfect copies of a number of his works, which show the

remarkable range and quality of his work, and we can deduce a little about his mathematical

life. And we have stories about his feats of engineering, and about his approach tomathematics, all of which however come from later sources.

Lets start with Archimedes mathematics. His Method of exhaustion was a precursor of

the integral calculus. By calculating the perimeter of regular polygons inside and outside a

circle, and increasing the number of sides of the polygons, he showed that is greater than 310/71 and less than 3 1/7. Similarly he approximated the area of the circle by that of

polygons, and used the method of exhaustion to show that the volume of a sphere is four

times that of a cone with base and height of the same radius. He also calculated the area of a

segment of a parabola, and presented theorems about the centres of gravity of plane figures.

In The Sand Reckoner Archimedes sets out to calculate the number of grains of sand that

could fill the universe: he has to provide notation for expressing very large numbers using asystem based on powers of a myriad of myriads (a myriad was 10,000), and his calculations

of the dimensions of the universe testify to his interest in astronomy. In his Stomachion he

analyses a dissection puzzle and (the text that has survived is fragmentary) he may calculate

in how many different ways the pieces can be assembled into a square perhaps the first

recorded work in combinatorics.

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This is one of a number of open mathematics resources. For information and more resources, including mp3 files, see

http://www.infj.ulst.ac.uk/~mmccart/hom.htm

There was great excitement a few years ago when the Archimedes Palimpsest was

rediscovered. This is a manuscript, created in the tenth century CE, which originally

contained copies of a number of the works of Archimedes. But in the twelfth century the

parchment was reused to make a prayer book. However some of the original text could still

be read. This manuscript disappeared after 1906 but its reappearance, together with very

sophisticated image processing techniques, are giving us improved copies of some ofArchimedes works and are helping us better understand his mathematical achievement. Anaward-winning book telling the story of the Palimpsest is listed amongst the further reading.

But Archimedes was also famous for his inventions. The story is well known of his bathtime

inspiration, when he apparently ran naked through the streets shouting Eureka! (I have

found it) after realising how to detect whether the kings new crown was solid gold,

measuring its volume by the quantity of water displaced when it was immersed. Histories

report that he used concave mirrors to set fire to enemy ships when the Romans attacked

Syracuse, and that he invented machines based on levers which plucked ships out of the sea

with giant claws. More pacifically, he is said to have created a working model of the

universe showing the motion of the planets. (It is possible that the Antikythera Mechanism google it! is a descendant of Archimedes machine.) It is suspected that some of these

stories are later exaggerations.

The historian Plutarch, writing three centuries after Archimedes lifetime, portrays him as

someone who was thinking about geometry even as he was being bathed by his slaves, a

theorist who valued pure mathematics above engineering and disdained mere use and

profit. This does not seem entirely consistent with his prodigious mechanical inventions and

military machines, and presumably reflects the then current ideal of a mathematician.

Archimedes died in the aftermath of the Roman capture of Syracuse, following a two-yearsiege in which the Romans were ultimately victorious despite Archimedes machines. The

Roman general Marcellus, appreciating the potential value of Archimedes engineering feats,

gave orders that he was not to be harmed, but nevertheless he was killed, while working on

mathematics, by an impatient Roman soldier.

So Archimedes was a martyr; a mathematician who made the ultimate sacrifice for his

subject. This may be part of his appeal to us today. But his delight in pure mathematics, his

willingness to apply mathematics for practical purposes, and the modern nature of his

mathematics, from integration to combinatorics, and his interest in large numbers, make it

entirely appropriate that his portrait appears on the Fields Medal, the ultimate mathematical

accolade.

Further Reading

The works of Archimedes were published by T.L. Heath in 1897. Reviel Netz is working on

a new critical edition, drawing on the Archimedes Palimpsest, of which Volume 1 has

recently appeared (Cambridge University Press, 2004).

For the story of the Palimpsest, see Reviel Netz and William Noel, The Archimedes Codex,

Weidenfeld and Nicolson, 2007.