35 May 2020 / www.theengineer.co.uk
optics dealing with the refl ection of
light from mirrors, plane or curved),
mechanics and pure mathematics.”
As Johnny Ball says in his book
Wonders Beyond Numbers, the most
famous legend concerning Archimedes
is the one in which he “jumped out of
the bath and ran down the street in the
nude” having shouted ‘Eureka!’ (which
is traditionally translated as ‘I have it’).
Ball explains that what Archimedes ‘had’
was a scientifi c method of verifying the
authenticity of the metallic composition
of a crown commissioned by King
Hiero II, subsequently produced by
two jewellers from a bar of pure gold.
According to Ball, when the king saw the
crown, “he was thrilled, until someone
whispered in his ear that the jewellers
were not completely honest and may
have substituted less valuable silver for
some of the gold, which they’d kept for
themselves. The problem was how he
could prove it without bashing the crown
back into its original gold bar shape?
Hiero turned to Archimedes for help,
but even he had no idea how to solve the
conundrum until one day, as he climbed
into the bath, water slopped over the
sides…”
The story continues that when the
crown was placed into a body of water,
it displaced more water by volume
than the original volume of the gold
bar, proving that the gold had been
adulterated with a lighter metal, silver.
Most tellings of the tale rejoice in
the supposition that the king, armed
with the evidence of malfeasance that
he required, then executed the two
jewellers, despite there being not one
shred of evidence for this. However,
the ‘incident’ led to Archimedes being
able to report in his treatise On Floating
Bodies what has now become known
as the Archimedes Principle, which
is that a solid denser than a fl uid will,
when immersed in that fl uid, be lighter
by the weight of the fl uid it displaces.
His other associated observations were
that an object immersed in water will
displace its own volume of water, while
an object less dense than water will sink
until it has displaced its own weight and
will then fl oat, eff ectively weightless,
on the surface. Which is why ships
fl oat. Archimedes used this knowledge
the second of these discoveries, Archimedes left instructions
for a sphere within a cylinder to be carved on his tomb, a fact
confi rmed by Roman statesman Cicero who rediscovered the
monument a century-and-a-half after Archimedes’ death. In
his Measurement of the Circle, that exists only as a fragment,
we see his approach to defi ning , which consists of inscribing
and circumscribing regular polygons with large numbers of
sides. His work on large numbers appears in a treatise called
The Sand-Reckoner in which he created a place-value system
of notation with the base 100,000,000, allowing him to express
how many grains of sand it would take to fi ll the entire universe.
Apart from the nine extant works, we can infer from later
authors that he wrote other treatises that have not survived,
as well as books of contested provenance on topics such as
touching circles and geometrical puzzles.
It wasn’t to be until the 8th and 9th centuries, when his
treatises were translated into Arabic, that the true signifi cance
of Archimedes as a mathematician was appreciated and his
work was developed by medieval Islamic mathematicians. The
invention of the printing press meant that his work (in Greek
this time) gained widespread popularity in Europe, infl uencing
the likes of Johannes Kepler and Galileo Galilei (who praised
Archimedes as ‘superhuman’). His infl uence extended into the
17th century via Latin translations that reached René Descartes
and Pierre de Fermat, and so had a profound infl uence on post-
Renaissance mathematics. Today, the Fields Medal awarded by
the International Congress of the International Mathematical
Union – one of the highest honours in mathematics – carries
a portrait of Archimedes along with the inscription Transire
suum pectus mundoque potiri (‘rise above oneself and grasp the
world’).
Give me a lever
and a place to stand
and I will move
the earth
Archimedes of Syracuse
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to design the Syracusia, which at 110m
presses a claim for being the largest
transport ship of antiquity. Capable
of carrying nearly 2000 people, it was
too big to dock at any port in Sicily and
so was sailed to Alexandria, where it
was presented to the king, Ptolemy III
Euergetes.
Although not famous in his lifetime
as a mathematician, from his treatises
we can deduce what breakthroughs
Archimedes made. In his On the Sphere
and Cylinder he tells us that the surface
area of any sphere of radius r is four
times that of its greatest circle (expressed
today as S = 4r2), while the volume of a
sphere is two-thirds that of the cylinder
in which it is inscribed, leading to the
formula for the volume V = 4/3r3. For
anyone doubting the signifi cance of
The Archimedes screw, also
known as the screw pump, was
originally used to move water from
low-lying sources such as rivers
into irrigation ditches
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