Some
of Archimedes' mathematical
proofs involve the use of infinitesimals
in a way that is similar to
modern integral calculus.
While he is often regarded
as a designer of mechanical
devices, Archimedes also made
contributions to the field
of mathematics.
Plutarch wrote: “He
placed his whole affection
and ambition in those purer
speculations where there can
be no reference to the vulgar
needs of life.”
By
assuming a proposition to
be true and showing that this
would lead to a contradiction,
Archimedes was able to give
answers to problems to an
arbitrary degree of accuracy,
while specifying the limits
within which the answer lay.
This technique is known as
the method of exhaustion, and
he employed it to approximate
the value of Pi.
He did this by drawing a
larger polygon outside a
circle, and a smaller polygon
inside the circle.
As the
number of sides of the polygon
increases, it becomes a more
accurate approximation of
a circle. When the polygons
had 96 sides each, he calculated
the lengths of their sides
and showed that the value of ¼ lay
between 3 + 1/7 (approximately
3.1429) and 3 + 10/71 (approximately
3.1408).
This was a remarkable achievement,
since the ancient Greek number
system was awkward and used
letters rather than the positional
notation system used today.
He also proved that the
area of a circle was equal
to Pi multiplied by the square
of the radius of the circle.
He used the method of exhaustion
to show that the value of the
square root of 3 lay between
265/153 (approximately 1.732)
and 1351/780 (approximately
1.7320512). |
The
modern value is around 1.7320508076,
making this a very accurate
estimate.
Another
noted mathematical work by
Archimedes is The Sand Reckoner
where he set out to calculate
the number of grains of sand
that the universe could contain.
Archimedes
suggested that there
are some who think that the
number of grains of sand
on earth is infinite,
not only that which exists
about Syracuse and the rest
of Sicily, but also throughout
the world.
He,
therefore, challenged the notion
that the number of grains of
sand was too large to be counted.
To solve the problem, Archimedes
devised a system of counting
based around the myriad.
This was a word used to mean
infinity, based on the Greek
word for uncountable, murious.
The word myriad was also used
to denote the number 10,000.
He proposed a number system
using powers of myriad myriads
Archimedes
was able to use infinitesimals in
a way that is similar to modern
integral
calculus .
In
his other works, Archimedes often
proves the equality of two areas
or volumes with his method of
double contradiction
assuming that the first is bigger
than the second leads to a contradiction,
as does the assumption that the
first be smaller than the second;
so the two must be equal.
These
proofs, still considered to
be rigorous and correct, used
what we might now consider secondary-school
geometry with
rare brilliance though later
writers often criticised Archimedes
for not explaining how he arrived
at his results in the first place. |
