Chapter 1. Classical Greek Mathematics

Greek science and mathematics is distinguished from that of earlier cultures by its desire

to know, in contrast to a need to make purely utilitarian advances or improvements. Greek

geometry displays abstract and deductive elements which were largely lost during the Dark

Ages, following the collapse of the Roman Empire, and only gradually recovered in the 16th

and 17th centuries. It must be understood that many of the great discoveries in geometry

were made about two and half thousand years ago. Given the difficulty of preserving

fragile manuscripts, written on parchment or papyrus, over centuries when warfare could

wipe out civilizations, it is not too surprising to find that we do not have many reliable

records about the origin of Greek geometry or of its practitioners. We may count ourselves

lucky that a few commentaries on Greek geometry, written in the fourth or fifth centuries

of the present era, have survived to provide us with what details we have.

Greeks from Ionia had settled in Asia Minor and there they had contact with two

ancient civilizations, those of Babylon and Egypt. Although knowledge of science was

elementary among the Babylonians and Egyptians, nonetheless they supplied the initial

impetus that directed the Greeks towards the pursuit of systematic science. All authorities

trace the beginnings of Greek geometry to Egypt. The rudimentary study of geometry

in Egypt arose out of practical needs. Revenue was raised by taxation of landed prop-

erty, and its assessment depended on the accurate fixing of the boundaries of fields in the

possession of the landowners. The landmarks were constantly removed by the periodic

flooding of the Nile and it became necessary to determine the taxable area by a technique

of land surveying. (The Greek historian Herodotus believed that basic knowledge of ge-

ometry originated from the recurrent need to measure land after inundation by the Nile.

Aristotle, on the other hand, believed that mathematics was the invention of Egyptian

priests with the time and leisure to speculate on abstract things.) Egyptian geometry was

limited mainly to mensuration of areas and volumes, in addition to what was required to

construct pyramids. The written records show a collection of practical rules for measuring

the areas of squares, triangles, trapeziums and circles, these rules often being imprecise,

and for estimating the volume of various measures of corn in different shapes. The Egyp-

tians also required familiarity with the notion of similarity of triangles, to enable them

to construct pyramids of different slopes. We have some knowledge of ancient Egyptian

mathematics, thanks to the survival of a few papyri. The most complete is the Rhind

Mathematical Papyrus, now in the British Museum. It was copied by the scribe Ahmes

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around 1650 BCE, and was itself copied from documents written before 2000 BCE. It

contains 85 mathematical problems and solutions. Another important papyrus relating to

mathematics is in Moscow.

We now present a little history of those considered to be the founders of Greek

mathematics or logical method.

• Thales (c. 624-546 BCE)

Thales is considered to be the founder of Greek geometry. He was born in Miletus, a town

now in modern Turkey (Asia Minor). He was also an astronomer and philosopher. He was

held in high regard by the ancient Greeks, and named as one of the seven ‘wise men’ of

Greece. (The seven wise men or sages of ancient Greece were held to be: Thales, Solon,

Periander, Cleobulus, Chilon, Bias and Pittacus. Various legends grew up about them,

and they are mentioned in the writing of Plato. They seem to have been astute politicians

and businessmen, rather than great thinkers.) He is said to have made a prediction of a

solar eclipse which, according to the famous historian Herodotus, occurred during a battle

of the Medes and the Lydians. Modern astronomers have dated this eclipse to 28 May,

585 BCE, which serves to give us some idea of the dates of Thales. While it is doubted if

someone could have predicted an eclipse so accurately at the given date, the story of its

happening assured his fame. (The Babylonians already knew that the eclipses followed a

cycle of 223 lunations, and Thales may have learnt this at some stage in his journeys.)

Various stories about Thales have come down to us from historians, especially Dio-

genes Laertius. One story relates that he travelled to Egypt, where he became acquainted

with Egyptian geometry. While, as we noted above, the Egyptian approach to geome-

try was essentially practical, Thales’s work was the start of an abstract investigation of

geometry. The following discoveries of elementary geometry are attributed to Thales.

• A circle is bisected by any of its diameters.

• The angles at the base of an isosceles triangle are equal.

• When two straight lines cut each other, the vertically opposite angles are equal.

• The angle in a semicircle is a right angle.

• Two triangles are equal in all respects if they have two angles and one side respec-

tively equal.

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He is also credited with a method for finding the distance to a ship at sea, and a method to

determine the height of a pyramid by means of the length of its shadow. It is not certain

whether this implies that he understood the theory of similar (equiangular) triangles.

Thales may be considered to have originated the geometry of lines, which forms a

basic part of elementary geometry. We also see in the theorems of Thales the first attempt

to find order and constancy in the midst of geometric data. It seems that he passed on

no written work to later generations, so we must rely on traditional stories, not all likely

to be true, for our information about him.

The commentator Proclus (whom we will discuss in more detail later), writing almost

one thousand years after the time in which Thales flourished, says that Thales first brought

knowledge of geometry into Greece after his time spent in Egypt. There is controversy

among modern historians of mathematics about the extent of Thales’s discoveries, for as

we observed, Egyptian geometry was rudimentary, had no theoretical basis, and consisted

mainly of a few techniques of mensuration. It is also considered unlikely that Thales could

have obtained theoretical proofs of the theorems attributed to him, but he may guessed

the truth of the results on the basis of measurements in particular cases.

• Pythagoras (c. 582–c. 500 BCE)

It is believed that Pythagoras was born around 582 BCE, in Samos, one of the Greek

islands. He had a reputation of being a highly learned man, a reputation that endured

for many centuries. He is said to have visited Egypt and possibly Babylon, where he

may have learnt astronomical and mathematical information, as well as religious lore. He

emigrated around 529 BCE to Croton in the south of Italy, where a Greek colony had

earlier been founded. He became the leader there of a quasi-religious brotherhood, who

aimed to improve the moral basis of society. After opposition developed to the influence

of his followers, he moved to Metapontum, also in the south of Italy, where he is thought

to have died around 500.

While geometry was introduced to Greece by Thales, Pythagoras is held to be the

first to establish geometry as a true science. It is difficult to distinguish the work of the

followers of Pythagoras (the Pythagoreans, as they are called) from that of Pythagoras

himself. Part of the problem is that later Pythagoreans tended to refer everything to the

master, remarking that he himself has said it. Furthermore, the Pythagorean school only

transmitted their knowledge orally and left no written record of their work. By the time

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of Aristotle (4th cent BCE), precise knowledge of any aspects of Pythagorean ethical or

physical theories was lacking.

The Pythagorean school was effectively divided into two groups. There was an inner

(esoteric) circle, whose followers had learnt the Pythagorean theory of knowledge in its

entirety. They were called mathematikoi (mathematicians). The outer (exoteric) circle,

who knew only the Pythagorean rules of conduct, were called akousmatikoi (hearers). The

word mathematics derives from the Greek mathema (µαθηµα), which means that which

is learnt. In Plato’s writing, this word is used for any subject of study or instruction,

although with a tendency to restrict it to the studies now called mathematics.

A number of statements regarding the Pythagoreans have been transmitted to us,

among which are the following.

• Aristotle says “the Pythagoreans first applied themselves to mathematics, a science

which they improved; and penetrated with it, they fancied that the principles of

mathematics were the principles of all things.”

• Eudemus, a pupil of Aristotle, and a writer of a now lost history of mathematics,

states that “Pythagoras changed geometry into the form of a liberal science, regard-

ing its principles in a purely abstract manner, and investigated its theorems from

the immaterial and intellectual point of view.”

• Aristoxenus, who was a musical theorist, claimed that Pythagoras esteemed arith-

metic above everything else. (“All is number” is a motto attributed to Pythagoras.)

• Pythagoras is said to have discovered the numerical relations of the musical scale.

Concerning the geometric work of the Pythagoreans we have the following testimony.

• Eudemus states that the theorem that the sum of the angles in a triangle is two right

angles is due to the Pythagoreans and their proof is similar to that given in Book 1

of Euclid’s Elements.

• According to Proclus, they showed that space may be uniformly tesselated by equi-

lateral triangles, squares, or regular hexagons.

• Eudemus states that the Pythagoreans discovered the five regular solids.

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• Heron of Alexandria and Proclus ascribe to Pythagoras a method of constructing

right–angled triangles whose sides have integer length.

• Eudemus ascribes the discovery of irrational quantities to Pythagoras.

We should now address some of the issues that arise from these claims, as they are not

accepted by all commentators and, indeed, some are implausible. The theorem that the

sum of the angles in a triangle is two right angles is not provable without recourse to

Euclid’s fifth or parallel postulate. This is a highly subtle point and any proof by the

Pythagoreans that the sum is constant must have had some implied appeal to the parallel

postulate. The Greek historian Plutarch tells us that the Egyptians knew of the right–

angled triangle whose sides have lengths equal to 3, 4, and 5 units, and that in this case

they observed that the square of the hypotenuse equals the sum of the squares of the other

two sides. Other versions of this arithmetical construction seem to have been known earlier

in Babylon. More generally, positive integers a, b and c are said to form a Pythagorean

triple (a, b, c) if a2 + b2 = c2. It seems to have become known at some time that such

Pythagorean triples may be used to form the sides of a right angled triangle, where the

hypotenuse has length c units, and so on. Proclus has described a method of finding such

Pythagorean triangles using an odd integer m, which he attributes to Pythagoras. We

take an odd integer m and set

a = m, b =m2 − 1

2, c =

m2 + 12

.

Note that both b and c are integers, because m is odd. It is straightforward to verify that

(a, b, c) is a Pythagorean triple, and this is the method used by Pythagoras to generate such

triples. There seems to be agreement that what we know as the theorem of Pythagoras

concerning right-angled triangles is not due to Pythagoras or the Pythagoreans. A proof

of the general theorem is found as Proposition 47 in Book 1 of Euclid’s Elements, but

is more complicated than the proof that would be given nowadays, using the theory of

similar triangles. Regarding this so–called theorem of Pythagoras, Proclus says: “If we

listen to those who wish to recount ancient history, we may find some of them referring

this theorem to Pythagoras and saying he sacrificed an ox in honour of his discovery.”

Proclus seems to suggest that we owe to Euclid the first rigorous proof.

One of great discoveries made by Pythagoras is the dependence of musical intervals

upon numerical ratios. He found that, with strings in the same tension, a difference of

length in the ratio of 2 to 1, gives the octave, the ratio of 3 to 2 gives the fifth, and that

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of 4 to 3 the fourth. In view of connections such as this, between numbers and natural

phenomena, the tendency of the Pythagoreans to liken all things to numbers, and find in

the principles of numbers the principles of all things, is perhaps explainable.

Concerning irrational quantities, we encounter some problems about the Greek con-

cepts of magnitude and number. Magnitudes are what we would call continuous quantities,

such as lengths of lines or areas of plane figures. Number is a discrete quantity, such as an

integer. Aristotle made a distinction between these two quantities. A magnitude is that

which is divisible into divisibles that are infinitely divisible, while the basis of number is

the indivisible unit. The Pythagoreans did not make such a distinction, as they considered

number to be the basis of everything and believed that everything can be counted. To

count a length, one needed a unit of measure. Once this unit was chosen, it was indivisible.

They then assumed that it is possible to choose a unit so that the diagonal and side of a

square can both be counted. This was eventually shown to be untrue–the precise time is

uncertain. As was said in Greek mathematics, the lengths of the diagonal and the side of

a square are incommensurable–they do not possess a common unit of measure. Nowadays

we would say that, in its initial stages, the Greek theory of numbers essentially held that

all numbers are rational. The present consensus is that the discovery of incommensurable

magnitudes, or equivalently, of irrational quantities, is not due to Pythagoras or the group

associated with him, but to later Pythagoreans, around 420 BCE.

The modern approach to the question is quite straightforward. Suppose that we

have a unit square. Then by the Pythagoras theorem, if c is the length of the diagonal,

c2 = 2. Now we claim that c cannot be a rational number, that is, it cannot be expressed

as a quotient of two integers. For suppose that c =r

s, where r and s are integers. We can

assume that r and s have no common factors. Then, on squaring, we obtain

2s2 = r2.

Since 2s2 is an even integer, r2 is even and thus r is even. Thus we can write r = 2t, where

t is an integer. Substituting, we obtain s2 = 2t2, and hence s is also even. This contradicts

the assumption that r and s have no common factor. Rather similar arguments can be

used to prove that several other square roots of integers are irrational. This was already

known in the circle of scholars around Plato. Indeed, in his dialogue Theaetetus, Plato

says that his teacher, Theodorus of Cyrene, also the teacher of Theaetetus, had proved

that the square root of any non-square integer between 3 and 17 is irrational. From the

modern point of view, this is easy to prove, as the the square root of any non-square integer

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is irrational. Presumably the method used by Theodorus involved specific arguments that

missed the full mathematical generality.

• Zeno of Elea (c. 490–c. 425 BCE).

Although a philosopher rather than a mathematician, Zeno is famous for his paradoxes

which demonstrate the danger of using infinite processes in logical arguments.

The best known paradox is that of Achilles and the tortoise. In a race, Achilles

travels 100 times faster than the tortoise, but it has the advantage of starting ahead of

Achilles. By the time Achilles has reached the tortoise’s starting point, the tortoise has

travelled forward 1/100 times the original distance. When Achilles covers this distance,

the tortoise has again forward moved 1/100 times that distance, and so on. In this way,

Achilles can never overtake the tortoise.

A second paradox denies the existence of motion by considering a flying arrow. At

any given instance, the arrow occupies a space equal to its own size. It can neither occupy

a larger space nor be in two different places at the same time. Since there is nothing

between one instant and the next, and the arrow cannot move in an instant, it cannot

move at all. Philosophers have studied Zeno’s paradoxes for many centuries. Aristotle,

in particular, sought to refute the arguments by denying that time is composed of ‘nows’

or instants. From the modern point of view, the paradox of Achilles and the tortoise is

explained by noting that the sum of the infinite geometric series

12

+14

+18

+ · · ·

is equal to 1. Nonetheless, it is still problematic whether one can speak of the infinite

divisibility of time, or postulate the existence of arbitrarily small lapses of time.

It is certainly true that Greek mathematicians tended to avoid arguments that in-

volved limiting processes, although these seem the appropriate techniques to use in con-

junction with the method of exhaustion when computing lengths, areas and volumes.

• Plato (427-347 BCE).

The personality and thought of Plato play a large role in the history of Greek mathemat-

ics, so we should say something about his life and influence. Plato is known primarily as

a philosopher but he was an important promoter of mathematics, especially geometry. He

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founded the famous Academy in Athens, around 380 BCE, which became a centre where

specialists met and discussed intellectual topics. Innovative mathematicians, including

Theodorus of Cyrene, Eudoxus of Cnidus, Theaetetus and Menaechmus, are closely as-

sociated with the Academy. Plato himself made no significant contribution to creative

mathematics, but he inspired others to ground-breaking work and guided their activity.

It is said that over the doors of his school the motto “Let no one ignorant of geometry

enter” was written. The authenticity of this claim is doubtful, as the earliest reference to it

occurs in the sixth century CE, but nonetheless it encapsulates the spirit of his Academy.

We know much detail about Plato’s life and career, and virtually all his writings

have survived. The source for much of our information about Plato, and indeed about

many other philosophers, is “Lives of the Philosophers” by Diogenes Laertius (3rd century

CE?).

Plato became associated with Socrates, close to the trial and execution of the latter

for impiety in 399. Plato was impressed by Socrates’s use of the art of argument and his

search for truth, but we should note that Socrates was himself no enthusiast for mathemat-

ics. Plato felt that it was his duty to defend Socratic ideas and methods, and conceived

the notion of training the young men of Athens in the discipline of mathematics and then,

when mentally ready, in Socratic interrogation. This was to counteract what he saw as

the problem of young people bewildering themselves in philosophical enquiry at too early

an age.

Around the year 390, Plato visited Sicily, where he came under the influence of

Archytas of Tarentum, a follower of the Pythagoreans. Archytas studied, among other

mathematical topics, the theory of those means that are associated with Greek math-

ematics: the arithmetic, geometric and harmonic means. Plato returned to Athens in

388, and in the next twenty years, his Academy came into existence. The purpose of the

Academy was to train young people in the sciences (mathematics, music and astronomy)

before they undertook careers as legislators and administrators. The two main interests

of the Academy were mathematics and dialectic (the Socratic examination of the assump-

tions made in reasoning). While Plato regarded the study of mathematics as preparatory

to the study of dialectic, he nonetheless believed that the study of arithmetic and plane

geometry, as well as the geometry of solids, must form the basis of an education leading

to knowledge, as opposed to opinion. Plato’s teaching at the Academy was assisted by

Theaetetus. Eudoxus of Cnidus, a pupil of Archytas and an important contributor to the

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emerging Greek theory of magnitude and number, also taught from time to time at the

Academy. Plato’s role in the teaching at the Academy was probably that of an organizer

and systematizer, and he may have left the specialist teaching to others. We may view the

Academy as a place where selected sciences were taught and their foundations examined

as a mental discipline, the goal being practical wisdom and legislative skill. Clearly, this

has relevance to the nature of university learning nowadays, especially as it relates to the

conflict between a liberal education, as espoused by Plato, and vocational education with

some special aim or skill in mind.

Plato’s opinion of the value and difficulty of mathematics is expressed in the following

dialogue, taken from his Republic.

• Have you ever noticed that those who are by nature apt at calculation are–not to

make a short matter long– naturally sharp at all studies, and that the slower–witted,

if they be trained and exercised in this discipline, even supposing they derive no other

advantage from it, at any rate progress so far as to become sharper than they were

before?

Yes, that is true, he said.

And I am of the opinion, also, that you would not easily find many sciences which

give the learner and the student greater trouble than this.

No, indeed.

For all these reasons, then, this study must not be rejected, but all the finest spirits

must be educated in it.

A further dialogue from the Republic gives some idea of the nature of mathematical thought

and method, and its abstraction from the particular to the general.

• I think you know that those who deal with geometrics and calculations and such

matters take for granted the odd and the even, figures, three kinds of angle and

other things cognate to these in each field of inquiry; assuming these things to be

known, they make them hypotheses, and henceforward regard it as unnecessary to

give any explanation of them either to themselves or to others, treating them as if

they were manifest to all; setting out from these hypotheses, they go at once through

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the remainder of the argument until they arrive with perfect consistency at the goal

to which their inquiry was directed.

Yes, he said, I am aware of that.

Therefore I think you also know that although they use visible figures and argue

about them, they are not thinking about these figures but those things which the

figures represent; thus it is the square in itself and the diameter in itself which are

the matter of their arguments, not that which they draw; similarly, when they model

or draw objects, which may themselves have images in shadows or in water, they

use them in turn as images, endeavouring to see absolute objects which cannot be

seen otherwise than by thought.

The regular solids were a major feature of Plato’s view of the cosmos. We mentioned that

Pythagoras is credited with the discovery of these solids, but modern opinion suggests that

this is unlikely. There are five regular solids, the cube, regular tetrahedron and octahedron,

dodecahedron and icosahedron. While the first three regular solids were probably known

long before the time of Pythagoras, this seems less likely for the other two, which are much

harder to visualize and to construct. It seems instead that Theaetetus, an Athenian who

died in 369 BCE, discovered the other two regular solids and wrote a study of all five solids.

It is possible that he proved that only five different types of regular solid exist (this is a

theorem in Euclid’s Elements). Theaetetus was associated with Plato and his Academy

in Athens, and his death was commemorated by Plato’s dialogue entitled Theaetetus.

This dialogue also contains information on irrational numbers, which had recently been

discovered and had caused a furore in mathematical and philosophical circles of the time.

Theaetetus was associated with some of this work on irrationals. The regular solids are

also called Platonic solids, because of the importance they held in the teaching of Plato.

He used the solids to explain various scientific phenomena. Indeed, the four elements

(earth, air, fire and water) were associated with the five regular solids in a cosmic scheme

that fascinated thinkers well into Renaissance times.

Plato’s enthusiasm for mathematics is described by Eudemus, writing some time

after the death of Plato:

• Plato . . . caused the other branches of knowledge to make a very great advance

through his earnest zeal about them, and especially geometry: it is very remarkable

how he crams his essays throughout with mathematical terms and illustrations, and

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everywhere tries to arouse an admiration for them in those who embrace the study

of philosophy.

• Aristotle (384-322 BCE).

Aristotle, the famous philosopher and logician, came to Athens in 367 and became a

member of Plato’s Academy. He remained there for twenty years, until Plato’s death in

347.

Here are two of his many sayings:

• As sight takes in light from the surrounding air, so does the soul from mathematics,

and

• The roots of education are bitter, but the fruit is sweet.

As we noted above, in Plato’s time, dialectic was of primary importance at the Academy,

with mathematics an important prerequisite. Aristotle held that the mathematical method

then being developed was to be a model for any properly organized science. Greek mathe-

matics at the time was distinguished by its axiomatic method, and sequence of reasoning,

from which irrefutable theorems are derived. Aristotle required that any science should

proceed as mathematics does, and the mathematical method should be applied to all

sciences.

Aristotle is important for laying down the working method for each demonstrative

science. Writing in his Posterior Analytics, he says:

• By first principles in each genus I mean those the truth of which it is not possible to

prove. What is denoted by the first terms and those derived from them is assumed;

but, as regards their existence, this must be assumed for the principles but proved for

the rest. Thus what a unit is, what the straight line is, or what a triangle is must be

assumed, but the rest must be proved. Now of the premises used in demonstrative

sciences some are peculiar to each science and others are common to all . . .Now

the things peculiar to the science, the existence of which must be assumed, are the

things with reference to which the science investigates the essential attributes, e.g.

arithmetic with reference to units, and geometry with reference to points and lines.

With these things it is assumed that they exist and that they are of such and such

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a nature. But with regard to their essential properties, what is assumed is only the

meaning of each term employed: thus arithmetic assumes the answer to the question

what is meant by ‘odd’ or ‘even’, ‘a square’ or ‘a cube’, and geometry to the question

what is meant by ‘the irrational’ or ‘deflection’ or the so-called ‘verging’ to a point.

Aristotle notes that every demonstrative science must proceed from indemonstrable prin-

ciples; otherwise, the steps of demonstration would be endless. This is especially apparent

in mathematics. He discusses the nature of what is an axiom, a definition, a postulate and

a hypothesis. It is quite difficult to distinguish between a postulate and a hypothesis. All

these terms play a leading role in Euclid’s Elements.

Aristotle’s influence on later European thought was immense. For many centuries,

virtually all Greek learning, except that of Aristotle, fell into oblivion. Aristotle was held

to be the basis of all knowledge. Universities and grammar schools were founded with the

study of Aristotle as their main intellectual activity. We see the extent of his influence even

now by noting how many Aristotelian words have survived in modern use, for example:

principle, maxim, matter, form, energy, quintessence, category, and so on. It was really

only in the Renaissance that the authority of Aristotle was questioned and supplanted.

• Euclid (active around 300 BCE).

We have little reliable knowledge about the lives of the early Greek geometers, and our

best sources are the Alexandrian mathematician Pappus (exact dates unknown, probably

third century CE) and the Byzantine Greek mathematician Proclus (410-485 CE), who

both lived many centuries after the golden age of Greek geometry had ended. Proclus,

who wrote a commentary on the first book of Euclid’s Elements, is our main authority on

Euclid. He states that Euclid lived in the time of Ptolemy I, king of Egypt, who reigned

323-285 BCE, and that Euclid was younger than the associates of Plato (active around 350

BCE) but older than Eratosthenes (276-196 BCE) and Archimedes (287-212 BCE). Euclid

is said to have founded the school of mathematics in Alexandria, a city that was becoming

a centre of commerce, and of learning, following its foundation around 330 BCE. Proclus

has preserved a famous incident relating to Euclid. On being asked by Ptolemy whether

he might learn geometry more easily than by studying the Elements, Euclid replied that

“there is no royal road to geometry”. The exact dates of Euclid, his place of birth, and

details of his life are not known, but we can say that he flourished around 300 BCE.

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The most famous of Euclid’s works is his Elements (Στoιχεια). It consists of thirteen

books (a book here means a long chapter). The first book contains:

(a) the definitions, or explanations of the terms used in the text.

(b) the postulates, which limit the instruments to be used in constructions to the ruler

and compass.

(c) the common notions or axioms, the fundamental principles from which the theorems

or propositions are deduced. The axioms are taken as given, and are not provable.

The propositions of Book 1 deal with rectilineal figures (figures bounded by straight line

segments), mainly the triangle and parallelogram. Book 1 concludes with the theorem of

Pythagoras (Proposition 47) and its converse (Proposition 48). Proposition 5 is the proof

that the angles at the base of an isosceles triangle are equal. Various constructions are

described and justified, for example, how to bisect an angle or bisect a line segment.

Book 2 is concerned with rectangular parallelograms contained by segments of straight

lines, and their relation to certain squares. It gives geometric explanations of various ele-

mentary results of algebra, and may be said to constitute a theory of geometrical algebra,

written long before algebraic symbolism became prevalent. The book is short, containing

only 14 propositions.

Book 3 discusses properties of circles. It contains many of the famous theorems

concerning circles that are used in elementary geometry. There are 37 propositions.

Book 4 contains no theorems. It is concerned with constructions, such as how to in-

scribe a circle inside a triangle, or circumscribe a circle about a triangle. There is a method

for constructing a regular pentagon (important to the Pythagoreans), for circumscribing

a regular pentagon about a circle, and for inscribing a circle in a regular pentagon.

The first four books build successively on the earlier books, and may be considered

a thorough introduction to the study of plane figures. Books 1 and 3, together with parts

of Book 4, constitute a traditional course of geometry for beginners.

Book 5 is unusual, as it is independent of the first four books. It contains the theory

of proportion, applied not only to geometrical quantities, such as lines, angles, areas, etc,

but also to arithmetical quantities. The doctrine of proportion in Euclid has been much

investigated, especially as it relates to the concepts of magnitude and number. Roughly

13

speaking, as we observed earlier, early Greek mathematicians had considered all numbers

to be rational (i.e. fractions) and could then compare them in size by simple proportion.

The Pythagoreans discovered through the theorem of Pythagoras that not all numbers

are rational, for example,√

2 is not rational. This led to a knowledge of the existence

of incommensurable magnitudes, causing a crisis in mathematics. Euclid’s fifth book is

considered to be difficult to follow, because of the inherent complexity of the subject.

Book 6 contains applications of the theory of proportion, mainly to rectilineal figures.

The theory of similar triangles is developed and also conditions for the congruence of

triangles. There are 33 propositions. It is surprising that the theory of similar triangles

does not appear earlier in the work, as it can simplify many of the earlier proofs. This is

related to the need to have an adequate theory of magnitudes and their ratios, as presented

in Book 5.

Books 7, 8 and 9 are devoted to arithmetic and properties of numbers. Proposition

2 of Book 7 is the famous Euclid’s algorithm of arithmetic. The last proposition of Book

9 gives a construction of even perfect numbers, a construction that Euler showed in the

18th century to give all even perfect numbers.

Book 10 is the longest book in the Elements, containing 115 propositions. It is

concerned with incommensurable line segments of the form a ±√

b,√

a +√

b,√

a±√

b

and√√

a +√

b. Rules are given which are the counterparts for rationalizing fractions of

the form a/(b +√

c). They are expressed in geometrical form, of course, not in algebraic

form.

Books 11, 12 and 13 are devoted to solid geometry. Book 11 is mainly concerned

with planes, lines and solid angles. Propositions 20 and 21 are particularly important as

they are needed to prove that there are at most five regular solids. In Book 12, the volumes

of such solids as cones, cylinders, pyramids and spheres are compared. Euclid employs the

method of exhaustion, used to approximate area and volume by successive subdivision, to

obtain his conclusions. The method is due to Eudoxus, whom we mentioned in connection

with Plato’s Academy, and it was much used later by Archimedes in volume calculation.

It is a forerunner of the methods of the integral calculus, developed in the 17th century.

Book 13 finishes with the theory of the five regular or Platonic solids, and considers

in particular how these solids may be constructed inside spheres. Some commentators have

14

suggested that the whole of the Elements was designed to provide the theory necessary

for the classification of such solids.

We do not know for certain how much of the Elements was original with Euclid, or

how much derived from earlier work. We have to rely on the evidence offered by Pappus

and Proclus. Proclus cites earlier work of Hippocrates of Chios, Eudoxus, Theaetetus, and

others, and this may have been influential. It is claimed that the theorem of Pythagoras on

right angled triangles was already known (although not necessarily proved by Pythagoras),

that the subject matter of plane geometry was already restricted to the straight line and

circle, and that the regular solids and their properties were already known (around the

time of Plato).

Euclid’s Elements was much used as a school and university textbook, especially the

first six books, and books eleven and twelve. In the latter half of the nineteenth century,

attempts were made to revise the way in which geometry was taught. In particular, it

was considered necessary to depart from the order in which Euclid presented his material,

as this was held to be arbitrary and unsystematic. People also objected to the frequent

use of proof by reductio ad absurdum–assuming the opposite of what was required to be

proved.

Let us now examine the opening of Book 1 and comment on Euclid’s working method.

There are 23 definitions.

1. A point is that which has no part.

2. A line is breadthless length.

3. The extremities of a line are points.

4. A straight line is a line which lies evenly with the points on itself.

5. A surface is that which has length and breadth only.

6. The extremities of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on itself.

8. A plane angle is the inclination to one another of two lines in a plane which meet

one another and do not lie in a straight line.

15

9. And when the lines containing the angle are straight, the angle is called rectilineal.

10. When a straight line set up on a straight line makes the adjacent angles equal to

one another, each of the equal angles is right, and the straight line standing on the

other is called a perpendicular to that on which it stands.

11. An obtuse angle is an angle greater than a right angle.

12. An acute angle is an angle less than a right angle.

13. A boundary is that which is an extremity of anything.

14. A figure is that which is contained by any boundary or boundaries.

15. A circle is a plane figure contained by one line such that all the straight lines falling

upon it from one point among those lying within the figure are equal to one another;

16. and the point is called the centre of the circle.

17. A diameter of the circle is any straight line drawn through the centre and termi-

nated in both directions by the circumference of the circle, and such a straight line

also bisects the circle.

18. A semicircle is the figure contained by the diameter and the circumference cut off

by it. And the centre of the semicircle is the same as that of the circle.

19. Rectilinear figures are those which are contained by straight lines, trilateral

figures being those contained by three, quadrilateral those contained by four, and

multilateral those contained by more than four straight lines.

20. Of trilateral figures, an equilateral triangle is that which has its three sides equal,

an isosceles triangle that which has two of its sides alone equal, and a scalene

triangle that which has its three sides unequal.

21. Further, of trilateral figures, a right-angled triangle is that which has a right angle,

an obtuse-angled triangle that which has an obtuse angle, and an acute-angled

triangle that which has its three angles acute.

22. Of quadrilateral figures, a square is that which is both equilateral and right-angled;

an oblong that which is right-angled but not equilateral; a rhombus that which is

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equilateral but not right-angled; and a rhomboid that which has its opposite sides

and angles equal to one another but is neither equilateral nor right-angled. And let

quadrilaterals other than these be called trapezia.

23. Parallel straight lines are straight lines which being in the same plane and being

produced indefinitely in both directions, do not meet one another in either direction.

The definitions are followed by five postulates.

Let the following be postulated.

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any centre and distance.

4. That all right angles are equal to one another.

5. That, if a straight line falling on two straight lines make 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 the two right angles.

Finally, there are five common notions (or axioms).

1. Things which are equal to the same thing are also equal to one another.

2. If equals be added to equals, the wholes are equal.

3. If equals be subtracted from equals, the remainders are equal.

4. Things that coincide with one another are equal to one another.

5. The whole is greater than the part.

Euclid’s definitions, postulates and common notions show the influence of Aristotle’s

discussion of these topics. On the other hand, Aristotle shows no awareness of Euclid, and

in one of his works he even gives a pre-Euclidean proof of the theorem that the base angles

of an isosceles triangle are equal. This strongly suggests that Euclid was active after the

death of Aristotle (recall that we do not know Euclid’s exact dates).

17

Regarding the definitions, various commentators have observed that certain are inad-

equate. For example, the definition of a plane surface is inadequate, and further postulates

must be added to obtain its basic properties. This is important, as much of Euclid’s ge-

ometry is studied in a plane. Modern editions of the Elements have expanded the number

of definitions to 35, reduced the postulates to three, and given as many as 12 axioms.

The most notorious of the definitions and postulates is the fifth postulate, also

known as the parallel postulate. The postulate asserts that, if a straight line cuts two

other straight lines AB and CD in points P and Q so that the sum of the angles BPQ

(β) and DQP (α) is less than two right angles, AB and CD will meet on the same side

of PQ as α and β, that is, they will meet if produced beyond B and D.

There was a feeling in antiquity that the fifth postulate could be proved, and erro-

neous proofs were offered, resting on some assumption or other that could not be justified.

In the 18th century, the Italian mathematician Saccheri and the German mathematician

Lambert looked at what could be achieved in geometry if the parallel postulate is not

assumed. They argued that if the parallel postulate, or some equivalent, is not assumed,

we cannot prove that the sum of the angles in a triangle is two right angles (this is Propo-

sition 32 of Book 1 of Euclid, the theorem attributed to the Pythagoreans). Furthermore,

if a quadrilateral is drawn, in which three of its four angles are right angles, we cannot

conclude that the fourth angle is also right. The fourth angle may be acute, or right, or

obtuse, and they proved that whatever occurs in one case is true in all cases. They then

inferred that there are three conceivable types of geometry, in two of which the parallel

postulate is not true, and in the third it is true. Thanks to the work of the 19th century

mathematicians Bolyai, Gauss and Lobachevski, we now know that all three possibilities

can occur in suitable models of space. Thus the fifth postulate is an unprovable assump-

tion defining a particular type of geometry of space–the Euclidean geometry. The other

two models of geometry, in which the parallel postulate is abandoned, define so-called

non-Euclidean geometries.

Sir Thomas Heath, the editor of the most authoritative edition of Euclid’s Elements

in English, has written:

This postulate must ever be regarded as among the most epoch–making achievements

in the domain of geometry. . . .When we consider the countless successive attempts

made through more than twenty centuries to prove the postulate, many of them by

18

geometers of ability, we cannot but admire the genius of the man who concluded

that such a hypothesis, which he found necessary to the validity of his whole system

of geometry, was really indemonstrable.

It may be of interest to examine Euclid’s methods of proof. In a proposition of Euclid,

there were definite subdivisions, called by different names, which were arranged in strictly

logical order. Firstly, there was the enunciation; secondly, the setting–out , which was the

statement of the precise data of the problem; thirdly, the definition or declamation, which

is a statement of what we are required to prove, with reference to the particular data of the

setting–out; fourthly, there was the construction, which might involve drawing additional

lines on the figure; fifthly, the proof and finally the conclusion, stating what had actually

been done, following the wording of the enunciation. This procedure had evolved over

several centuries, and was not unique to Euclid. As the Elements were used more and

more for pedagogical purposes, even into the modern era, it was expected that students

would follow these subdivisions, otherwise proofs would be considered incomplete.

• The five regular solids.

We have mentioned the importance of the five regular solids in Greek mathematics, espe-

cially in Plato’s cosmology. Euclid investigates the regular solids in Books 11 and 13 of

the Elements. The notion of a solid angle plays an important role in his analysis. We will

briefly discuss solid angles and state without proof one of the main theorems relating to

them. We will then show how to deduce that there are at most five regular solids.

When three or more planes taken in order intersect, each with the next, in lines

which meet at a point, they are said to form a solid angle. The point of concurrence is

called the vertex; the lines of intersection of consecutive planes are called the edges of the

solid angle; the angles between consecutive planes are called its dihedral angles; the plane

angles formed by consecutive edges are called its face angles.

A solid angle formed by three concurrent planes is said to be trihedral; if it is formed

by more than three concurrent planes, it is said to be polyhedral.

The following two important results are proved in Book 11 of the Elements.

Proposition 20. If a solid angle be contained by three plane angles, any two, taken

together in any manner, are greater than the remaining one. (In more modern

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language, in a trihedral angle, the sum of any two face angles is greater than the

third.)

Proposition 21. Any solid angle is contained by plane angles less than four right

angles. (Modern form: in a convex solid angle, the sum of the face angles is less

than four right angles.) A solid angle is convex if any point lying in the interior of

the angle can be joined to any other point lying in the interior of the angle by a line

segment also lying in the interior of the angle.

It should be noted that, although Euclid enunciates Proposition 21 for any solid angle, he

only proves it for trihedral angles. He also neglects to include any mention of restriction to

convex solid angles. Modern proofs of Proposition 21 use Proposition 20 as a preliminary

result.

A polyhedron is a solid bounded by plane faces. A regular polyhedron has faces

which are equal regular polygons. Suppose a regular polygon has n ≥ 3 edges. It is

straightforward to see that each edge subtends an angle equal to4n

right angles (we

measure angles, as the Greeks did, in terms of right angles). It follows easily that the

angle formed at each vertex of the polygon by any two adjacent edges is

2− 4n

right angles.

We can now desribe the possibilities for the faces of a regular polyhedron.

Let P be a regular polyhedron whose faces are regular polygons of n sides. Suppose

that at each vertex of P , r faces meet (r is the same for each vertex, by regularity). At

each vertex, a convex solid angle is thus formed by r concurrent planes (where we must

have r ≥ 3). The face angles of each of these solid angles are all equal to 2 − 4n

right

angles and the sum of the face angles of the solid angle is thus

r

(2− 4

n

)right angles.

By Proposition 21 above, we obtain

r

(2− 4

n

)< 4,

and hence

n <2r

r − 2.

20

Since necessarily 3 ≤ n, we obtain 3(r − 2) < 2r and hence r < 6. It follows therefore

that r = 3, r = 4 or r = 5 and we go through each value in turn. If r = 3, we obtain

n < 6. Thus we can have n = 3, n = 4 or n = 5 in this case. These three solutions

correspond to three faces meeting at each vertex, the faces being equilateral triangles,

squares, or regular pentagons. The corresponding solids are the tetrahedron, cube, and

dodecahedron. If r = 4, we obtain n < 4. It follows that n = 3 is the only solution. This

corresponds to four faces meeting at each vertex, the faces being equilateral triangles, and

the solid is an octahedron. If r = 5, we obtain n < 103 . This leaves only the solution n = 3.

Here, five faces meet at each vertex, the faces being equilateral triangles, and the solid

an icosahedron. Hence there are exactly five solutions and each corresponds to a different

solid.

• Three classical problems of antiquity

Euclid’s Elements describes the achievements of Greek geometry over the three hundred

years or so since Thales began studying the subject. There were, however, three famous

problems that exercised the minds of Greek mathematicians over many centuries and never

admitted any satisfactory solution. These problems were:

• to square the circle

• to construct a cube whose volume is twice that of a given cube

• to trisect any given angle.

It was assumed that, ideally, any solution of a given problem would be accomplished within

the framework of Euclidean geometry, namely, by using a straight edge and compass. The

first problem, to square the circle, is as follows: to construct a square, by geometrical

methods, whose area equals that of a given circle. Thus, if we take a circle of radius 1

unit, we know now that its area is π square units and we therefore need to construct a

square whose sides have length√

π units. At first, the value of π was not known accurately

to Greek mathematicians. Later, Archimedes realized that π could be approximated to

any degree of accuracy, by a process of subdivision similar to the methods of integral

calculus. He obtained the following estimate for π:

31071

< π <227

.

21

This did not solve the squaring the circle problem, as there was still no geometric method

for constructing a line of length√

π units. In 1882, the German mathematician Lindemann

proved that the number π is transcendental, meaning that it is not a root of any non-zero

polynomial with rational coefficients. This finally laid the Greek problem to rest, as the

only numbers that can be constructed by the methods of Greek geometry are algebraic

numbers of a special type (lying in iterated quadratic extensions of the rational numbers),

and these numbers are certainly not transcendental.

The second problem, also known as the problem of duplicating the cube, involves

constructing a line of length equal to the cube root of two, 3√

2, since a cube whose sides

have length 3√

2 will have volume equal to 2 cubic units. Work of Gauss in the early

19th century showed that a length 3√

2 cannot be obtained by straight edge and compass,

because 3√

2 determines a degree 3 extension of the rationals, and hence this number

does not lie in an iterated quadratic extension. Already Archytas, whom we mentioned in

connection with Plato, proposed a solution to the duplication problem using a cylinder, but

Plato held that such methods were contrary to the spirit of geometry. Later, Menaechmus

devised another method, using special curves that cannot be constructed by Euclidean

means.

Concerning the third problem, it is well known that one can bisect an angle using

a straight edge and compass. One can then divide an angle into four, eight, etc, equal

parts. However, no construction for trisecting an angle using a straight edge and compass

was ever discovered. It was later shown as a consequence of the work of Gauss that it is

impossible to trisect a general angle using only a straight edge and compass. The proof

lies within the theory of abstract fields, as do the first and second problems previously

described. It is possible to trisect an angle using an infinite sequence of bisections of

related angles, but this is inadmissible as a solution. Nonetheless, the idea allows the

possibility of obtaining an approximation to the trisection of an angle to any degree of

accuracy using only a finite number of bisections.

• The transmission and later history of the Elements

It is of some interest to know how the Elements were transmitted to later generations. In

the 4th century CE, Theon of Alexandria edited a version of the work which has served

as the basis for virtually all subsequent editions of the Elements. Theon’s daughter,

Hypatia, was herself a celebrated geometer, but, as a pagan, she suffered the indignity of

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being murdered by a Christian mob in Alexandria in 415. The earliest extant manuscript

of a Greek version of the Elements dates from 888, and is located in Oxford. In the

early 7th century, around 620-630, the creation of Islam and the subsequent expansion

of Arab conquerors into much of North Africa and Asia led to the demise of the greater

part of the Byzantine Empire, which was the inheritor of the Greek tradition, and of the

remains of the Western Roman Empire. The Arabs came into contact with Greek learning,

especially in Alexandria. Initially, they were without intellectual interest in this unfamiliar

knowledge, rather as the Romans had felt with regard to their Greek predecessors, but by

750, the Arabs began to absorb the Greek legacy of mathematics and astronomy. A centre

of learning was established in Baghdad in the early 9th century, under the support of a

series of enlightened caliphs. Here, translations into Arabic were made of Greek classics,

such as Euclid’s Elements. Without these translations, many of the most famous and

original of the Greek treatises might have been lost during periods of political confusion,

barbarian invasion and intellectual sterility in Europe.

Knowledge of the Greek version of Euclid was lost in Western Europe for many

centuries. Eventually, in the early years of the twelth century, an English monk, Athelhard

of Bath (also known as Adelhard) travelled in Spain, Greece, Asia Minor and Egypt, where

he gained a knowledge of Arabic. Probably working from a Spanish source, he translated

the Elements from Arabic into Latin. His translation served as the basis for virtually

all 15th and 16th century translations of Euclid’s work into the vernacular languages of

Europe. The earliest printed edition of the Elements was produced in Venice in 1482,

and in the next century, printed editions in most modern European languages appeared.

Scholarly attempts to obtain as accurate a version as possible of the Elements began

in the 18th century, using manuscript copies of the work available in European libraries

(such as the Oxford manuscript). This process culminated in the work of the Danish

mathematician Johan Heiberg, who produced a definitive version of the text in parallel

Greek and Latin forms, in a book published in 1883.

We mentioned earlier that much of our knowledge of ancient Greek geometry derived

from the writing of Proclus. As he plays a significant role in the history of mathematics, we

will provide some information about him. Proclus (410-485 CE) was born in Constantino-

ple. He studied in Alexandria and then decided to devote his life to philosophy. He

therefore moved to Athens, where he became a member of the revived Platonic Academy

there. He eventually became a teacher and then head of the Academy.

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Proclus was the last great representative of the philosophical movement known as

Neoplatonism, which had begun about two centuries earlier under the influence of Plotinus.

Proclus aimed to perfect Neoplatonism by systematizing and extending the views of his

predecessors, and showing how all was derived from the teaching of Plato. He collected

information about a variety of Greek cultural activities, such as religion, literature, science

and philosophy. Many of his writings are lost, but what has survived gives us one of

our best sources of information on this last stage of Greek culture, and also on earlier

achievements.

Proclus believed that one needed a thorough grounding in logic, mathematics and

natural science to understand philosophy. His Commentary on the first book of Euclid’s

Elements has been transmitted to us. It was probably written as part of his work in the

Academy. He had a detailed knowledge of Greek mathematics from Thales to his own

time, and he provides us with one of the best references to Euclid and his predecessors

and successors. He supposedly was able to use a history of geometry due to Eudemus of

Rhodes. Eudemus lived a few years before Euclid, and he could have collected information

about the predecessors of Euclid. Eudemus’s work is now lost, and we must take it on

trust that Proclus had access to this source, seven hundred years after it was written.

24