**HERE’S
LOOKING AT EUCLID**

**Introduction**

This is the second of three lectures on the multi-cultural
origins of mathematics, shown here on this time-line. In my last lecture
I introduced you to the world of papyruses and clay tablets, the mathematics
of Ancient Egypt and Mesopotamia. Today I’ll move forward more
than a thousand years to the world of Greek mathematics, before spending
the next lecture on the cultures of China, India and Central America.

The period of Greek mathematics lasted for about
a thousand years, starting around 600 BC, and falls into three main
sections. The first of these concerns the semi-legendary figures of
Thales and Pythagoras; the second deals mainly with Athens and Plato’s
Academy; and for the third we’ll visit Alexandria, starting with
Euclid and extending over seven hundred years. We’ll also make
a brief excursion to the island of Sicily to meet Archimedes.

In which parts of the Greek empire was the story
acted out? Thales came from Miletus in modern Turkey, while Pythagoras
was based in Crotona, now in Italy. Plato’s Academy was in Athens,
while Alexandria is now situated in Egypt. Archimedes was based in
Syracuse in Sicily.

Before proceeding, I’d like to mention the available
sources. Unlike Ancient Egypt, where there are a few well-preserved
papyruses, and Mesopotamia where there are many thousands of surviving
clay tablets, we have hardly any Greek primary sources. As in Egypt,
the Greeks wrote on papyrus which didn’t last the centuries,
although a few fragments have survived from the later period. There
were also disasters, such as the fire in the library at Alexandria,
in which many primary sources perished.

So we have to rely entirely on commentaries and later
versions. The best-known commentator on Greek mathematics was Proclus
from the fifth century AD, who supposedly derived much of his material
from earlier commentaries (now lost) by Eudemus of Rhodes, a fourth
century BC student of Aristotle. But Proclus lived some 800 years
after Euclid – it’s rather like us trying to comment on
contemporary oral and written accounts of Robin Hood – so we
have to treat his commentaries with scepticism, while recognising
that they’re all we’ve got.

We also have translations and commentaries by Islamic
scholars, which are most useful – but the fact remains that our
earliest surviving copy of Euclid’s *Elements* is an Arabic
translation in the Bodleian Library in Oxford, dating from the year
888, which is as far removed from Euclid’s time as it is from
our own.

**The first period**

Moving on to our first period, that of Thales and
Pythagoras, we have to admit that we don’t know much about either
of them. According to legend, Thales came from Miletus, he brought
geometry to Greece from Egypt , he predicted a solar eclipse in 585
BC, and he showed how rubbing feathers with a stone produced electricity.

Further, as Proclus commented while writing about Euclid’s Elements:

*The famous Thales is said to have been the first
to demonstrate that the circle is bisected by the diameter.*

*If you wish to demonstrate this mathematically,
imagine the diameter drawn and one part of the circle fitted upon
the other.*

*If it is not equal to the other, it will fall
either inside or outside it, and in either case it will follow that
a shorter line is equal to a longer.*

*For all the lines from the centre to the circumference
are equal, and hence the line that extends beyond will be equal to
the line that falls short, which is impossible.*

You’ll notice that the Thales extract is concerned
with *mathematical proof*. Of all the many mathematical contributions
by the Greeks – and they include the introduction and study of
*conics* and the study of the *three classical problems*,
as you’ll see – the idea of deductive reasoning and mathematical
proof is the most fundamental. Starting with some initial assumptions
– or *axioms* – we derive some simple results, and
then more complicated ones, and so on, eventually creating a great
hierarchy of results, each depending on previous ones.

The Greeks adopted several methods of proof –
the Thales extract used a proof by contradiction (or *reductio ad
absurdum*), where we first assume that the result we want to prove
is incorrect and then deduce a result that contradicts our assumptions.
You’ll see other examples of this later on.

These ideas were developed around 550 BC by Pythagoras
and the Pythagoreans – the brotherhood that supposedly gathered
around him in the Greek seaport of Crotona, now in Italy. There’s
a representation of Pythagoras in Raphael’s fresco *The School
of Athens*, even though this School (Plato’s Academy) was
two hundred years after Pythagoras.

The Pythagoreans believed that *all is number*
– that everything worthy of study can be quantified – and
they then divided the mathematical sciences into four parts. Two of
these are *arithmetic*, which deals with numerical quantities,
and *music*, which deals with intervals regarded as simple ratios
between these quantities – for example, an octave corresponds
to a ratio of 2 to 1 and a perfect fourth gives a ratio of 3 to 2.
The other two are *geometry*, which deals with magnitudes at
rest; and *astronomy*, which deals with magnitudes in motion.
These four mathematical areas were later combined into the *quadrivium*
which, together with the *trivium* of grammar, rhetoric and logic,
formed the seven liberal arts – the subjects studied in academies
and universities for the next 2000 years or so.

The Pythagoreans were concerned with many areas of
mathematics. For them, *arithmetic* meant both ‘arithmos’
– ordinary arithmetic, calculating with whole numbers, and also
what we now call number theory. For example, they knew how to find
the triangular numbers by adding consecutive integers (for example,
15 = 1 + 2 + 3 + 4 = 5), and they realised that square numbers are
obtained by adding consecutive odd numbers – for example, 16
= 1 + 3 + 5 + 7. They also studied prime numbers, which we’ll
meet later.

The Pythagoreans were also very interested in *commensurable* and
*incommensurable* numbers, which came to play an important role in Euclid’s
*Elements*. We say that the numbers 8 and 12 are *commensurable* because they
can both be ‘measured’ an exact number of times by a ruler of length 4, and the
numbers 3*π* and 5*π* are commensurable because they can both be measured
a whole number of times by a ruler of length *π*. Essentially, two numbers are
commensurable if their ratio can be written as a fraction (a ratio of whole numbers) –
so 5*π* divided by 3*π* is just 5/3, which is a fraction. However, as they
discovered, the diagonal and side of a square are not commensurable.

The current proof of this last fact is well known
and is typical of the Greek approach, except that here I’ll use
algebra, whereas the Greeks would have couched everything in geometrical
terms. The proof is by contradiction.

By Pythagoras’s theorem the ratio of the diagonal and side of a square
is √2, and we must prove that this number cannot be written as a fraction – as
*a*/*b*, where *a* and *b* are whole numbers. So, to obtain a
contradiction *we assume that √2 can be written as a fraction, and we can assume that
this fraction is written in its lowest terms, so a and b have no common factor*. By
squaring, we can rewrite this as *a* 2 = 2*b* 2, which means that *a* 2
must be an even number. But if *a* 2 is even, then *a* must also be even
(because otherwise, *a* is odd, so *a* 2 is odd). Since *a* is even, we
can write *a* = 2*k*, for some whole number *k*. So 2*b* 2 =
4*k* 2, which gives *b* 2 = 2*k* 2, so *b* 2 is even, and
*b* is even. But this gives us a contradiction – *a* and *b* are
both even, so both are divisible by 2, contradicting the fact that *a* and *b*
had no common factor. This contradiction arises from our original assumption – that
√2 can be written as a fraction – so this assumption is wrong: √2
*cannot* be written as a fraction, and so the diagonal and side of a square are
incommensurable.

For no apparent reason, Pythagoras’s name is
associated with *Pythagoras’s Theorem*, even though the
Mesopotamians knew the result a thousand years earlier, as we saw
last time. But it was in Greek times that the theorem was first proved.

The theorem states that in any right-angled triangle,
the area of the square on the hypotenuse (the longest side) is equal
to the sum of the areas of the squares on the other two sides. So
it’s a geometrical result about areas – there’s no
mention of an algebraic equation such as a 2 + b 2 = c 2.

A dissection proof of Pythagoras’s theorem is
typical of the Pythagorean school. We can draw two different dissections
of a square of side a + b. Removing the four triangles in each case,
and comparing the red squares, we see that the largest area must be
the sum of the two smaller ones. It’s a far cry from the axiomatic
proof that would appear later in Book I of Euclid’s *Elements*.

Charles Dodgson, better known as Lewis Carroll, was
a great enthusiast for Pythagoras’s theorem:

It is as dazzlingly beautiful now as it was in the
day when Pythagoras first discovered it, and celebrated the event,
it is said, by sacrificing a hecatomb of oxen [that’s a whole
oxen] – a method of doing honour to Science that has always seemed
to me slightly exaggerated and uncalled-for. One can imagine oneself,
even in these degenerate days, marking the epoch of some brilliant
scientific discovery by inviting a convivial friend or two, to join
one in a beefsteak and a bottle of wine. But a hecatomb of oxen! It
would produce a quite inconvenient supply of beef.

**The second period**

The second great period of Greek mathematics took
place in Athens, with the founding of Plato’s Academy around
387 BC in a suburb of Athens called ‘Academy’ – that’s
where the word Academy comes from. Plato’s Academy is featured
in Raphael’s fresco *The
School of Athens* with Plato and Aristotle at the top of the
steps, and with Pythagoras on the left and Euclid on the right.

Plato’s Academy soon became the focal point
for mathematical study and philosophical research, and it is said
that over its entrance were the words Let no-one ignorant of geometry
enter these doors.

Plato wrote a short dialogue called Meno in which
Socrates asks a slave boy how to double the size of a square. The
boy first suggests doubling the side of the square, but that gives
four times the area. Eventually he settles on the square based on
the diagonal of the original square. It’s a wonderful example
of teaching by experiment and it’s far removed from anything
we saw in Egypt and Mesopotamia.

Plato believed that the study of mathematics and
philosophy provided the finest training for those who were to hold
positions of responsibility in the state, and in his Republic he discussed
at length the importance of each of the mathematical arts for the
so-called ‘philosopher-ruler’: arithmetic, geometry (which
he divided into plane geometry and solid geometry), astronomy and
harmonics.

His book *Timaeus* includes a discussion of
the five regular, or ‘Platonic’, solids – the tetrahedron,
cube, octahedron, dodecahedron and icosahedron – in which the
faces are all regular polygons of the same type and the arrangement
of polygons at each corner is the same: for example, the cube has
six square faces, with three meeting at each corner. He also linked
four of the polyhedra with the Greek elements of earth, air, fire
and water, and assigned the cosmos to the dodecahedron, which had
only recently become known. One hundred years later, Archimedes would
find all the semi-regular solids, in which the faces are all regular
polygons, but they’re not all the same – for example, the
truncated cube, obtained from a cube by chopping off the corners,
is made up of triangles and octagons.

One of the students at the Academy was Aristotle,
who remained there for some twenty years. He was fascinated by logical
questions and systematised the study of logic and deductive reasoning.
In particular, he studied the nature of mathematical proof, and considered
syllogisms such as *All men are mortal, Socrates in a man; therefore
Socrates is mortal*. Another early student there was the mathematician
and astronomer Eudoxus of Cnidus, who advanced the hypothesis that
the sun, moon and planets move around the earth on rotating concentric
spheres, a hypothesis later adopted in modified form by Aristotle.
Eudoxus is often credited with developing the theory behind two of
the books in Euclid’s *Elements* – *Book V* on
proportion and *Book XII* on the so-called method of exhaustion.

It was around this time that the Greeks adopted a
new counting system. This was a decimal system in which separate Greek
letters were used for 1, 2, 3, … up to 9; then new letters for
10, 20, … up to 90; and then nine further letters taking them
to 900.

It was also around this time that the three classical
problems emerged. Each asked for a construction that uses only a straight
edge and a pair of compasses – no measuring was to be allowed.
The first problem was that of *doubling the cube*, said to be
based on the need to double the size of an altar at Delos to appease
the Gods: given a cube, construct another cube with twice the volume
– this effectively involved constructing the cube root of 2.
The second is *trisecting the angle*: given any angle, construct
two lines that divide it into three equal parts. The third is the
best known – *squaring the circle*: given a circle, construct
a square with the same area. These problems fascinated the Greeks,
but in the event it would take two thousand years until all three
constructions were proved to be impossible.

**The third period**

Around 300 BC, with the rise to power of Ptolemy
I, mathematical activity moved to the Egyptian part of the Greek empire.
In Alexandria Ptolemy founded a university that became the intellectual
centre for Greek scholarship for the next 800 years – our third
period of Greek mathematics. Ptolemy also started its famous library,
which held over half-a-million manuscripts before eventually being
destroyed by fire.

A number of important mathematicians were associated
with Alexandria – Apollonius (who wrote the standard work on
conics), the great astronomer Ptolemy (after whom the Ptolemaic system
of planetary motion is named), and the Neo-platonists such as the
geometers Pappus and Hypatia (one of the most important women mathematicians
of all time). But the greatest of all, and the first important mathematician
to be associated with Alexandria , was Euclid, who lived and taught
there around 300 BC.

We know virtually nothing about Euclid’s life.
As well as the *Elements*, he’s credited with writing many
other books, including several in geometry (the *Data* and *On
Divisions of Figures*), the *Porisms* (a three-book work on
problems which hasn’t survived), a four-book work on *Conics*
(which also hasn’t survived), and books on *Astronomy* and
*Optics*.

Euclid’s *Elements* was a compilation of
results known at the time, organised in a systematic way. It wasn’t
the earliest such work – Hippocrates of Chios and others had
written *Elements* earlier, though these haven’t survived.
However, Euclid’s was the most important. As the commentator
Proclus observed:

*It is a difficult task in any science to select
and arrange properly the elements out of which all other matters are
produced and into which they can be resolved.*

*Such a treatise ought to be free of everything
superfluous, for that is a hindrance to learning; the selections chosen
must all be coherent and conducive to the end proposed, in order to
be of the greatest usefulness for knowledge; it must devote great
attention both to clarity and to conciseness, for what lacks these
qualities confuses our understanding.*

*Judged by all these criteria, you will find Euclid’s
introduction superior to others*.

As a result, Euclid’s Elements is the best-selling
mathematics book of all time, over more than 2000 years – indeed,
it’s possibly the most printed book ever, apart than the Bible.
Much has been written about it: the philosopher Immanuel Kant observed
that:

*There is no book in metaphysics such as we have
in mathematics. If you want to know what mathematics is, just look
at Euclid’s Elements*.

and the Victorian mathematician Augustus De Morgan observed:

*The thirteen books of Euclid must have been a
tremendous advance, probably even greater than that contained in the
‘Principia’ of Newton. The sacred writings excepted, no
Greek has been so much read and so variously translated as Euclid.*

Euclid’s Elements consists of thirteen books,
traditionally divided into three main parts – on plane geometry,
arithmetic and solid geometry. Here’s a quick overview.

*Books I* and *II* deal with the foundations
of plane geometry and the geometry of rectangles; *Books III*
and *IV* then proceed to the geometry of circles. Book V is on
proportion, which is then applied to the geometry of similar figures
in *Book VI*. *Books VII*, *VIII* and *IX* are
on arithmetic, and include basic properties such as the divisibility
of integers and the so-called Euclidean algorithm, as well as a discussion
of prime numbers and perfect numbers. *Book X*, the longest and
most difficult book amounting to a quarter of the whole work, is on
incommensurable line segments. The final three books are on solid
geometry, and conclude with the construction and classification of
the five Platonic solids. I won’t go through all these systematically,
but I’ve chosen a few topics that interest me.

*Book I* starts with twenty-three definitions
of basic terms such as point, line and circle. There are then five
geometrical Postulates, beginning with three allowable constructions:

He then observes:

*That all right angles are equal to one other*.
The last postulate is much longer:

*That, if a straight line falling on two straight lines makes
the interior angles on the same side less than two right angles, the
two straight lines, if produced indefinitely, meet on that side on
which are the angles less than two right angles*.
This fifth postulate seems to be of a different style
from the others – in fact, as we’ll see in a later lecture,
it caused no end of problems over the next 2000 years.

Finally, we get down to our first proposition:

*Proposition 1. On a given straight line to construct an equilateral
triangle*.

To do this, we draw the line AB and trace out circles
with centre A and radius AB and with centre B and radius AB. These
meet at a point C, and the triangle ABC is then the required equilateral
triangle. Euclid then proceeded to prove that the construction actually
works – that the resulting triangle is equilateral. At each stage
there is a reference to a definition or a postulate, and in later
propositions there are frequent references to earlier propositions.

Here are some other Propositions from *Book I*.
Proposition 5 is the famous *Pons Asinorum*, the bridge of asses,
that *in an isosceles triangle, the angles at the base are equal
to each other*. This result is credited to Thales, and in medieval
universities it was often as far as students of Euclid ever reached:
if you could cross the bridge of asses, you could then go on to all
the treasures that lay ahead! Propositions 1 to 26 are all basic results
and constructions in plane geometry, such as the bisection of an angle
and congruence theorems for triangles, and these are followed by nineteen
propositions on parallel lines and parallelograms. These include the
results that *the angles of a triangle add up to two right angles,
and that given any triangle, we can construct a rectangle with the
same area*. Since any polygon, such as a pentagon, can be split
into triangles, we can construct a rectangle with the same area as
any given polygon – when combined with other results, this shows
that we can square any polygon, even though we can’t square the
circle.

*Book III* introduces the properties of circles
– I’ll mention just a couple of results here:

*Proposition 20. In a circle the angle at the centre
is double the angle at the circumference when the angles have the
same arc as base*.

A special case is:

*Proposition 31. In a circle the angle in a semicircle is right*.

and a related result is:

*Proposition 22. The opposite angles of quadrilaterals in circles
are equal to two right angles*.

Books VII to IX take us into arithmetic – but
the descriptions are still given in geometrical terms – using
lengths of lines to represent numbers, rather than the numbers themselves.
A good example of this is Euclid’s proof in Book IX that *there
are infinitely many prime numbers* – one of the most famous
proofs in the whole of mathematics.

Here’s the modern way of writing out the proof:
recall that a *prime number* is a number whose only factors are
itself and 1 – for example, 11, 13, 17 and 19 are all prime numbers,
whereas 15 isn’t.

*Suppose, for a contradiction, that the only prime
numbers are p 1, p 2, … , p n and form the number N = p 1 p 2
… p n + 1. Since each of the primes p 1, p 2, … , p n divides
their product p 1p 2 … p n, none of them can divide N. So N is
itself a prime number, or else is divisible by some prime number different
from p 1, p 2, … , p n. In either case, there is a prime number
different from the given ones, giving the required contradiction.
So there are infinitely many primes*.

Euclid’s proof involves the lengths of lines,
and starts with only three lines of prime length, representing the
general case. Apart from this, the proof is essentially the same.

The final three books deal with aspects of three-dimensional
geometry. Of these, *Book XIII* is the most remarkable. It introduces
the five regular solids that we saw earlier, and then shows how they
can be constructed. For example, to construct a dodecahedron, we take
a cube and add to each face a ‘roof’ whose proportions are
such that the faces all become pentagons; these proportions involve
the so-called ‘golden ratio’, whose geometrical properties
are worked out in detail at the beginning of *Book XIII*.

In this book, Euclid also proved such remarkable results as the following:

*If an equilateral pentagon is inscribed in a circle,
the side of the pentagon is equal in square to that of the hexagon
and that of the decagon inscribed in the same circle*.

In other words, if we calculate the lengths of the
sides of a pentagon, a hexagon and a decagon inscribed in a circle,
then these turn out to be the lengths of the sides of a right-angled
triangle. He also presented this complicated diagram in his construction
of an icosahedron.

Euclid concluded the *Elements* by proving that
*the only possible regular solids are the tetrahedron, cube, octahedron,
dodecahedron and icosahedron* – there can be no more. This
is the first ever ‘classification theorem’ in mathematics,
and forms a fitting climax to this great work.

Euclid’s *Elements* was warmly received,
and quickly replaced all its predecessors and competitors. After the
invention of printing, an enormous number of printed versions appeared.
Two I particularly like – the first English edition, produced
by Henry Billingsley in 1570, and Oliver Byrne’s colourful edition
of the first six books in 1847 in which all the symbols are replaced
by coloured diagrams.

One major figure who seems not to have been associated
in any substantial way with Alexandria was Archimedes, a native of
Syracuse on the island of Sicily. Here’s looking at Archimedes
– notice how different artists and sculptors had different ideas
on what he might have looked like – this is Ribera’s portrait
of him in the Prado Museum in Madrid.

One of the greatest mathematicians of all time, Archimedes
is mainly remembered for running naked through the street shouting
*Eureka* (I have found it). The reason for this outburst is apparently
that his friend King Hiero wanted to ascertain whether his crown was
of pure gold or whether it was partially made of silver. To solve
this problem, Archimedes immersed the crown in water and observed
that the weight was reduced by an amount equal to the weight of water
displaced, from which he could make the necessary calculations.

Another result of his was the *law of the lever*
– that if unequal weights are placed at opposite ends of a scale,
they balance at distances that are inversely proportional to the weights.
This is sometimes called the *law ofmoments*: W 1a = W 2b, or
a/b = 1/W 1 divided by 1/W 2.

But Archimedes didn’t only work in applied mathematics.
A list of his works includes studies on the measurement of a circle,
on the so-called *Archimedeanspiral*, on properties of the sphere
and cylinder and other solids, and *The sandreckoner* on enormous
numbers in which he estimates the number of grains of sand in the
universe.

Among his geometrical results are calculations of
centres of gravity for a triangle, hemisphere and parallelogram, his
impressive calculations of volumes and surface areas such as the sphere,
and his celebrated result (which he apparently wanted engraved on
his tomb) that the surface area of any horizontal section of a sphere
is the same as that of the surrounding cylinder. And although we can’t
square the circle, Archimedes showed that parabolas can be squared
– the area of any parabolic segment is 4/3 times the area of
the enclosed triangle, and the corresponding square can then be obtained.

Some of his most profound results can be found in
*The Method*, and there was much excitement recently when a palimpsest
turned up (and sold for two million dollars) on which a monk had overwritten
what was already known to be an Arabic version of *The Method*.
This is still being worked on and interpreted, and is due to go on
exhibit around the world in 2007.

One of Archimedes’ most well-known results is his invention of a method
for estimating π – by drawing hexagons inside and outside a circle and comparing
their perimeters with the circumference of the circle. This wasn’t very accurate, so he
replaced the hexagon by a 12-sided polygon, then 24, 48 and 96 sides, obtaining successively
better estimates. He found that π is a little bit less than 22/ 7, the value we learned at
school, and a little bit more than 3 10/ 71 – this gives a value of about 3.14, correct
to two decimal places.

Back in Alexandria, Apollonius was writing his celebrated
treatise on *conics*. These curves can be traced back to Menaechmus
in the fourth century BC, and arise from slicing a cone in various
ways. There are three different types – the *ellipse* (with
the circle as a special case), the *parabola*, and the *hyperbola*.

The frontispiece of Edmond Halley’s 1710 edition
of Apollonius’s celebrated treatise on conics shows the philosopher
Aristippus, shipwrecked on the island of Rhodes, noticing some conics
that had been drawn in the sand and claiming that the inhabitants
must thus surely be civilised.

In astronomy, too, there was much activity. Eudoxus, whom I mentioned
earlier in connection with the theory of proportion in Euclid’s
*Elements*, advanced the hypothesis that the sun, moon and planets
move around the earth on rotating concentric spheres, a hypothesis
later adopted in modified form by Aristotle.

Aristarchus advanced an alternative hypothesis –
that the fixed stars and the sun remain unmoved while the earth revolves
about the sun in a circle – anticipating by 1700 years the revolutionary
work of Copernicus, but his hypothesis found few supporters at the
time.

Trigonometry made its first appearance around 150
BC by Hipparchus, possibly the greatest astronomical observer of antiquity.
Sometimes called ‘the father of trigonometry’, he constructed
a ‘table of chords’ giving essentially the sines of certain
angles. He also constructed a star catalogue using a coordinate system
for the stars.

The earth-centred hypothesis was developed by Ptolemy
of Alexandria around 150 AD, and it eventually became known as the
*Ptolemaic system*. Ptolemy wrote an important 13-volume work,
usually called by its Arabic name *Almagest*, containing a mathematical
description of the motion of the sun, moon and planets, and including
a table of chords listing the sines of angles from 0° to 180°
in steps of ½°. He also published a work on map-making
called *Geographia*, discussing various types of map projection
and containing the latitude and longitude of 8000 places in the known
world: his maps were used by navigators for many centuries.

I’d like to conclude with three mathematicians
from later times. Diophantus of Alexandria wrote an important *Arithmetic*,
in which he solved a number of problems whose answers were to be given
as whole numbers, or as fractions. A typical example is: find three
numbers such that the product of any two added to the third is a square.
In the seventeenth-century French translation of Diophantus’s
*Arithmetic* by Bachet, Fermat annotated the margin claiming
to have a proof of what is now known as ‘Fermat’s last theorem’.
We don’t know when Diophantus lived, but it may have been around
250 AD. All we have is a puzzle from the Greek anthology which states
that Diophantus spent 1/ 6 of his life in childhood, 1/ 12 in youth,
and 1/ 7 more as a bachelor. Five years after his marriage there was
a son who died four years before his father, at ½ his father’s
final age. You can work out from this that Diophantus lived to be
84.

Another later Greek mathematicians is Pappus
of Alexandria, from the early fourth century AD, and I’d like
to mention two contrasting results of his. In his treatise *On the
sagacity of bees*, he credited bees with a certain geometrical
forethought in planning their honeycombs. After showing that there
can only be three regular arrangements – with triangles, squares
and hexagons – he noted that the bees in their wisdom chose that
which has the most angles, perceiving that it would hold more honey
than the other two.

Pappus’s
other result is one of the great theorems of mathematics, and you
might like to try it out for yourself. Draw two lines, and mark points
A, B, C on one and a, b, c on the other. Then join A with b and c,
B with a and c, C with a and b – this provides three new points
of intersection. Amazingly, whatever points you originally chose,
these new points always lie on a straight line.

The last person in today’s story was the first
great woman mathematician. Women scholars had always been acceptable
to the Greeks – for example, they were welcomed at Plato’s
Academy – and around 400 AD we meet Hypatia, daughter and pupil
of the geometer Theon of Alexandria. A fine geometer herself, she
became Head of the Neoplatonic school in Alexandria, and was such
a renowned expositor and lecturer that people came many miles to hear
her. She is credited with impressive commentaries on many classic
texts, such as Apollonius’s *Conics*, Ptolemy’s *Almagest*,
and Diophantus’s *Arithmetic*, and she showed how to construct
various astronomical and navigational instruments, such as the astrolabe.
Tragically, her life was cruelly and savagely cut short when she was
murdered by a mob of fanatical Christians in 415 AD. After her death,
little further was achieved in Alexandria, and the city fell to the
Arabs in 641.

This brings to an end one thousand years of Greek
mathematics, and I hope that you found the journey worthwhile. If
not, may I conclude by recalling that a student in Alexandria who
had begun to read geometry with Euclid asked him ‘What advantage
shall I get by learning these things?’ Euclid called his slave
and said ‘Give him threepence, for he must needs make profit
out of what he learns.’ I do hope that you don’t *all
want* to claim your threepences from me.

**Robin Wilson**

Professor of Mathematics at the Open University and a fellow of Keble
College, Oxford.

27 October 2004