From 0 to Infinity in 26 Centuries (6 page)

It is often said that many important examples of sculpture and architecture are made using the golden mean.

The Parthenon was designed according to the golden mean. Its length and height and the space between the columns were designed in perfect proportion to one other.

Aristotle

The son of the doctor at the court of the kings of Macedon, Aristotle was a nobleman who became a hugely influential philosopher. He was taught by Plato and later became a
teacher at Plato’s Academy, and he contributed ideas on a whole host of subjects, from politics and ethics to physics and zoology. So wide-ranging were his skills, it has been suggested
Aristotle knew
everything
it was possible to know. Indeed, his influence extended through to the philosophy of the modern world.

Aristotle is known chiefly for his logic, a series of works that comprised the earliest-known study of the theory of logic. His theories have since split into many different branches, some
highly mathematical, others more philosophical.

Aristotle’s work in mathematics and science focused on explaining the way things behave by describing them rather than
using numbers and equations. He was among the
first to explain the motion of objects (a subject we today call kinetics, from the Greek for ‘movement’). Aristotle’s descriptions acknowledged that time and space are not
arranged in indivisible chunks but are continuous, which allowed him to show that Zeno of Elea’s ideas were flawed and that Achilles would have been able to catch up with the tortoise!

E
UCLID
(
c.
325–
c.
265
BC
)

While little is known about the Greek mathematician Euclid, we do know that he was active in Alexandria in Egypt under Greek rule, and is notable for having penned a
groundbreaking book called
Elements
. Certainly one of the most important maths books of all time, Euclid’s
Elements
was considered essential reading for any scholar well into
the nineteenth century.

Elementary proof

Although Euclid drew on the ideas of others, he was one of the first mathematicians to produce work that used mathematical logic in order to prove theories. This idea of proof
is one of the foundations of mathematics.

Elements
covers much of geometry and ideas about numbers, including prime numbers and other number sequences, and all of Euclid’s geometrical constructions were made using only a
pair of compasses and a straight edge.

It is split into thirteen books, each of which starts with
definitions of words to help make it clear what Euclid means when he refers to words such as point, line,
straight, surface, etc. Euclid then sets out a list of axioms or statements that are evidently true, such as ‘all right angles are equal to each other’ and ‘if A=B and A=C, then
B=C’.

The next section of the
Elements
is called ‘Propositions’, in which Euclid proposes a method of how to carry out a mathematical task. For example, in Proposition 1 of Book 1
Euclid shows how to draw an equilateral triangle (all the sides are the same length and all the angles are equal to 60°), and he then goes on to prove that the triangle is, in fact,
equilateral.

E
RATOSTHENES
(276–195
BC
)

It would be wrong to talk too much about prime numbers without mentioning multi-disciplined mathematician Eratosthenes, who hailed from a Greek city in modern-day Libya. He was
responsible for many great intellectual endeavours, including calculating the earth’s circumference to a surprising degree of accuracy and coining the word ‘geography’, which
means ‘drawing the earth’ in Ancient Greek. Mathematically, Eratosthenes’ greatest contribution is the
Sieve of Eratosthenes
.

In their prime

Before we look at the sieve let us first contemplate
prime numbers
: numbers that have only two factors – themselves and 1. Hence 13 is a prime number because 1 and
13 are the only numbers that divide into it without leaving a remainder. 9 is not prime, because it can be divided by 1, 3 and 9, which means it has three factors. 1 is also not a prime number
because it has only one factor.

Prime numbers are important for two reasons:

There is only one way of doing this for each number so it seems to me, at least, that primes are the equivalent of DNA for numbers.

Eratosthenes’ sieve works using a very simple principle to help find prime numbers up to a certain limit. 2 is the first prime number. Anything that 2 goes into cannot be
prime, because it would then have 2 as a factor as well as itself and 1.

If we set ourselves a limit of 100, we could highlight 2 as a prime and then eliminate all the numbers that have 2 as a factor: 4, 6, 8, etc. up to 100. If we use a grid we can shade them in to
generate a pattern:

You don’t even need to be brilliant at your two-times table to do this – you could just count on 2 each time and shade in each square you land on.

After you’ve shaded in all the multiples of 2 you move on to the next unshaded number, which also happens to be the next prime number: 3. We highlight 3 as a prime and then eliminate all
the multiples of 3, some of which have already been eliminated in the first round. The next unshaded number is 5, which again is also prime. As before, highlight that and then eliminate the
multiples of 5.

As you move along, the next unshaded number must be prime because none of the prime numbers that went before it could go into it. If you keep on repeating this process eventually you’ll
have a completed sieve. Turn to page 55 to see what this looks like.

You can make the sieve as big as you like in order to work out higher and higher prime numbers. There are no hard calculations to do, but it is quite a tedious process –
something a Greek mathematician would probably have left to an educated slave.

A
RCHIMEDES
(287–212
BC
)

Archimedes was a friend of Eratosthenes and he hailed from the city of Syracuse in present-day Sicily. He was famous as a scientist and engineer: he invented the Archimedes
screw for pumping liquids and raising grain, which is still in use today. Archimedes is also said to have defended Syracuse from Roman warships by directing an intense ray of light from the sun
towards the approaching soldiers, setting their vessels alight.

From straight to circular

Archimedes’ contributions to mathematics are no less impressive, even if they are less well known. He worked out a value for π by noting that, as a polygon accrues more
sides, it gets closer and closer to becoming a circle. π is defined as a circle’s circumference divided by its diameter. It is hard to measure the curved edge of a circle accurately, but
easy to measure the straight sides of a polygon to find the perimeter. By approximating a circle as a polygon with a certain number of sides, Archimedes was able to find a value for π by
dividing the polygon’s perimeter by the distance across the polygon. Archimedes performed this calculation with a polygon that had up to 96 sides. During his investigations he came up with a
value of between 3.143 and 3.140 for π, which is pretty close to its actual value: 3.1415...