From 0 to Infinity in 26 Centuries (7 page)

Archimedes hit upon an important idea with this
method of exhaustion
– the idea that if an approximation is performed accurately enough it becomes indistinguishable from the true
answer. This idea has been used in many other areas of mathematics, perhaps most noticeably in the calculus of Newton (see
here
) and Leibniz (see
here
) almost 2000 years later.

Archimedes proved other important results, including that the area of a circle is π multiplied by the radius squared. He also proved that the volume of a sphere is 2/3 the volume of the
cylinder that it is able to fit into. Archimedes was so pleased with his discovery he had a sculpture of the sphere and cylinder erected on his tomb.

Lasting legacy

Archimedes died at the hands of a Roman soldier while working at his desk. Legend suggests Archimedes was so absorbed in his work he failed to respond to the soldier’s
orders that he come with him. Insulted, the soldier killed Archimedes, and presumably faced the wrath of his commanding officer, who would have regarded the slain intellectual as a highly valuable
scientific asset.

With the death of Archimedes we come to the end of Ancient Greece, when its territories were consolidated into the emerging Roman Empire. The mathematical legacy of the Greeks is long lasting
and most people today will have encountered the discoveries made by many of the mathematicians mentioned in this chapter. I think the Ancient Greeks’ greatest contribution was to invent
mathematics as a rich and diverse subject, moving it beyond the basic necessity of numeracy and arithmetic, the functional tools of an economy. They created a subject that would become the language
of science and which would eventually allow humanity to create scientific ideas from first principles, basing discoveries on a concept rather than from fudging equations and formulae to match
observations. Without this mode of thinking Sir Isaac Newton would have been unable to conduct much of his pioneering work.

The Romans

The Greek mainland was conquered by the Romans in 146
BC
, and the empire reached its zenith 200 years later, occupying a vast area that covered the
entirety of the Mediterranean on all sides.

A P
RACTICAL
P
EOPLE

Discipline was a central aspect of Roman life, which extended to its education system. The wealthier young Romans were taught basic arithmetic, most likely at home, but the main
thrust of their education was to understand the workings of their own society. Oration was seen as the pinnacle of education, along with physical training for boys, who would go on to do military
service, and home economics for girls, who were in charge of running their homes.

In terms of higher mathematics, it appears that very little was taught to the Romans when compared to their Greek predecessors. The Romans were a far more practical people,
focusing their attentions on developments in engineering and medicine; practicality is not the best mindset for exploring mathematics for its own sake.

A spanner in the works

The Roman number system, inherited from the Greeks, didn’t help matters. Roman numerals rely on the position they sit within a string of letters, which makes it very
difficult to use them in arithmetic.

The basic Roman numerals are:

I: one

V: five

X: ten

L: fifty

C: one hundred

D: five hundred

M: one thousand

The Romans wrote their numbers with the largest starting from the left. Therefore, in order to read a Roman numeral you have to add up the numbers from left to right. For
example:

MMMDCLXVII would be 1000+1000+1000+500+100 +50+10+5+1+1 = 3667

However, the Romans devised a useful shortcut for using when the value of a number was close to the value of the next letter. The method involved putting a
letter out of sequence, which indicated it should be subtracted from the next letter in sequence.

For example, in longhand the number 999 should be written DCCCCLXXXXVIIII, but with the shortcut it could be written as IM. However, there seemed to be no written rules, and the Romans, it
seems, didn’t like having an I before an M or a C if they could avoid it. Therefore, 999 would more likely have be written as CMXCIX which gives (1000 - 100) + (100 - 10) + (10 - 1) = 900 +
90 + 9 = 999. Needless to say, having more than one way to write a number did not make life easy!

Alexandria

The Roman Empire subsumed the old Greek Empire and, as such, the Greek mathematical tradition continued. It focused in Alexandria, Egypt, a remarkable centre of learning that
had been founded in 331
BC
by the leader of the Greeks, Alexander the Great, as he conquered his way East across Europe and Asia.

H
ERO
(10–70
AD
)

An Alexandrian scientist and mathematician, Hero is most famous for detailing a primitive steam engine, and for perhaps being the first person to harness wind power on land with
the aid of a windmill.

Hero also made two significant contributions to mathematics:

1.
He came up with a formula for working out the
area of a triangle
that only requires the lengths of the sides of the triangle.

2.
He devised a way of working out
square roots
: a number that when multiplied by itself gives a specific quantity.

Hero’s formula

There are many ways to work out the area of a triangle. Most of us were taught at school that:

area of triangle = ½ × base × height

For this formula you need to choose which side is the base and then work out the height of the triangle, which, if it’s non-right-angled, may not be one of the other two
sides:

Hero’s formula removed the need both to choose a base and to measure the height, although perhaps at the expense of simplicity:

area of triangle with sides of length a, b and c = ¼ × √[(a2 + b2 + c2)2 - 2(a4 + b4 + c4)]

The root of the problem

Hero’s method for working out square roots involved using a formula to generate a new value; this new value would then be put back into the formula and the process would
be repeated a number of times with the answer getting closer to the true value.

This technique is called
iteration
– another important development in mathematics. For example, if you wanted to work out the square root of 2, which, as we saw earlier, is an
irrational number – one which cannot be written as a fraction and whose decimal goes on for ever without repeating (see
here
) – Hero’s method would work like this:

new value = ½ × (old value + R ÷ old value)

where R is the number you want to know the square root of. The first time you use the formula there is no ‘old value’, so you have to take a guess. The square root
of 2 must be between 1 and 2, because 1 × 1 = 1 and 2 × 2 = 4 and 2 lies between 1 and 4. Let’s
opt for the middle value, 1.5, and see what happens:

new value = ½ × (1.5 + 2 / 1.5) = 1.41666666...

You can now repeat this process using 1.41666 as your old value:

new value = ½ × (1.41666 + 2 / 1.41666) = 1.414215686

new value = ½ × (1.414215686 + 2 / 1.414215686) = 1.41423562

new value = ½ × (1.41423562 + 2 / 1.41423562) = 1.41423562

At this point you should notice that the old value and new value are the same, so our work here is done – and this is indeed the square root of 2, accurate to 8 decimal
places.

If you wanted to work out the square root of another number you would start with a different R. It’s important to note that if you make R a negative number the formula does not work. For
example, if you make R = -2 and have 1 as your first guess you get:

new value = ½ × (1 - 2/1) = -0.5

If you repeat as before you get:
    

1.75

 

0.3035714286

 

-3.142331933

 

-1.252930967

 

0.1716630854

This process continues for ever without ever settling on a value. Why? Because negative numbers cannot have a square root – a negative number multiplied by a negative
number always gives a positive answer. Hence the formula is searching for something that does not exist!

Hero did, however, postulate that it could be possible for a negative number to have a square root, if you use a bit of imagination (see
here
).

D
IOPHANTUS
(
c.
200–
c.
284
AD
)

A resident of Alexandria from
c.
250
AD
, Diophantus is sometimes referred to as the ‘Father of Algebra’ because of his contribution to
solving equations. While today thoughts of algebra conjure up a process of replacing numbers with letters, Diophantus did not adhere to this principle. Before true
symbolic
algebra was invented, mathematicians were forced to write out equations longhand.

These days it’s very easy to write a simple algebraic equations, such as: 3a + 4a
2
. However, Diophantus’ method would have been far more laborious, involving something
along the lines of: ‘three multiplied by the unknown number added to four times the unknown number multiplied by itself.’ This made solving equations a tricky process, both in terms of
writing and reading them.

An imaginary triangle

Diophantus was interested in Pythagoras’ theorem. He noticed something strange when he tried to work out the sides of a right-angled triangle with a perimeter of 12 and an
area of 7. It produced an equation that could not be solved, indicating a triangle with those specific dimensions cannot exist. However, Diophantus remarked that if negative numbers could have
square roots he would be able to solve the equation – which would mean the triangle would then exist. Much later, these numbers were called
imaginary numbers
(see box
here
),
because in order to get round the problem you have to imagine that there is a number, represented by the symbol ‘i’, that is the square root of -1.

Triple the fun

His interest in Pythagoras’ theorem also sparked another mathematical mystery that would take hundreds of years to
solve. Diophantus was interested
in
Pythagorean triples
, which are solutions to the theorem that are whole numbers. For example:

3
2
+ 4
2
= 5
2

5
2
+ 12
2
= 13
2

8
2
+ 15
2
= 17
2

In his great work
Arithmetica
, Diophantus included instructions on how to find such numbers. In 1637 French mathematician Pierre de Fermat wrote in the margin of his copy
of
Arithmetica
that it was not possible to find the Pythagorean triples where the numbers were raised to any power other than 2. He finished with a tantalizing comment that was to tease
mathematicians for years to come: ‘I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.’

These innocuous words started a 350-year challenge to solve what became known as
Fermat’s last theorem
.

Unravel the Riddle

Although we know very little about Diophantus’ life, a charming riddle, sometimes known as ‘Diophantus’ Epitaph’, associated with
him provides a brief overview of his days on this earth. The riddle was first noticed in a puzzle book by the Greek philosopher Metrodorus some time in the sixth century
AD
.

‘Here lies Diophantus,’ the wonder behold.

Through art algebraic, the stone tells how old:

‘God gave him his boyhood one-sixth of his life,

One twelfth more as youth while whiskers grew rife;

And then yet one-seventh ere marriage begun;

In five years there came a bouncing new son.

Alas, the dear child of master and sage

After attaining half the measure of his father’s life

chill fate took him. After consoling his fate by the

science of numbers for four years, he ended his life.’

Can you work out how old Diophantus was when he died?

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