Euclid of Alexandria is the most prominent
mathematician of antiquity best known for his treatise on mathematics The
Elements. The long lasting nature of The Elements must make Euclid the leading
mathematics teacher of all time. However little is known of Euclid's life except
that he taught at Alexandria in Egypt. Proclus, the last major Greek
philosopher, who lived around 450 AD wrote (see or or many other sources):-

Not much younger than these [pupils of Plato] is
Euclid, who put together the "Elements", arranging in order many of
Eudoxus's theorems, perfecting many of Theaetetus's, and also bringing to
irrefutable demonstration the things which had been only loosely proved by his
predecessors. This man lived in the time of the first Ptolemy; for Archimedes,
who followed closely upon the first Ptolemy makes mention of Euclid, and
further they say that Ptolemy once asked him if there were a shorted way to
study geometry than the Elements, to which he replied that there was no royal
road to geometry. He is therefore younger than Plato's circle, but older than
Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes
somewhere says. In his aim he was a Platonist, being in sympathy with this
philosophy, whence he made the end of the whole "Elements" the
construction of the so-called Platonic figures.

There is other information about Euclid given by
certain authors but it is not thought to be reliable. Two different types of
this extra information exists. The first type of extra information is that
given by Arabian authors who state that Euclid was the son of Naucrates and
that he was born in Tyre. It is believed by historians of mathematics that this
is entirely fictitious and was merely invented by the authors.

The second type of information is that Euclid was born
at Megara. This is due to an error on the part of the authors who first gave
this information. In fact there was a Euclid of Megara, who was a philosopher
who lived about 100 years before the mathematician Euclid of Alexandria. It is
not quite the coincidence that it might seem that there were two learned men
called Euclid. In fact Euclid was a very common name around this period and
this is one further complication that makes it difficult to discover
information concerning Euclid of Alexandria since there are references to
numerous men called Euclid in the literature of this period.

Returning to the quotation from Proclus given above,
the first point to make is that there is nothing inconsistent in the dating
given. However, although we do not know for certain exactly what reference to
Euclid in Archimedes' work Proclus is referring to, in what has come down to us
there is only one reference to Euclid and this occurs in On the sphere and the
cylinder. The obvious conclusion, therefore, is that all is well with the
argument of Proclus and this was assumed until challenged by Hjelmslev in. He
argued that the reference to Euclid was added to Archimedes book at a later
stage, and indeed it is a rather surprising reference. It was not the tradition
of the time to give such references, moreover there are many other places in
Archimedes where it would be appropriate to refer to Euclid and there is no
such reference. Despite Hjelmslev's claims that the passage has been added
later, Bulmer-Thomas writes in:-

Although it is no longer possible to rely on this
reference, a general consideration of Euclid's works ... still shows that he
must have written after such pupils of Plato as Eudoxus and before Archimedes.

For further discussion on dating Euclid, see for example.
This is far from an end to the arguments about Euclid the mathematician. The
situation is best summed up by Itard
who gives three possible hypotheses.

(i) Euclid was an historical character who wrote the
Elements and the other works attributed to him.

(ii) Euclid was the leader of a team of mathematicians
working at Alexandria. They all contributed to writing the 'complete works of
Euclid', even continuing to write books under Euclid's name after his death.

(iii) Euclid was not an historical character. The
'complete works of Euclid' were written by a team of mathematicians at
Alexandria who took the name Euclid from the historical character Euclid of
Megara who had lived about 100 years earlier.

It is worth remarking that Itard, who accepts Hjelmslev's
claims that the passage about Euclid was added to Archimedes, favours the
second of the three possibilities that we listed above. We should, however,
make some comments on the three possibilities which, it is fair to say, sum up
pretty well all possible current theories.

There is some strong evidence to accept (i). It was
accepted without question by everyone for over 2000 years and there is little
evidence which is inconsistent with this hypothesis. It is true that there are
differences in style between some of the books of the Elements yet many authors
vary their style. Again the fact that Euclid undoubtedly based the Elements on
previous works means that it would be rather remarkable if no trace of the
style of the original author remained.

Even if we accept (i) then there is little doubt that
Euclid built up a vigorous school of mathematics at Alexandria. He therefore
would have had some able pupils who may have helped out in writing the books.
However hypothesis (ii) goes much further than this and would suggest that
different books were written by different mathematicians. Other than the
differences in style referred to above, there is little direct evidence of
this.

Although on the face of it (iii) might seem the most
fanciful of the three suggestions, nevertheless the 20th century example of
Bourbaki shows that it is far from impossible. Henri Cartan, André Weil,
Jean Dieudonné, Claude Chevalley, and Alexander Grothendieck wrote
collectively under the name of Bourbaki and Bourbaki's Eléments de mathématique
contains more than 30 volumes. Of course if (iii) were the correct hypothesis
then Apollonius, who studied with the pupils of Euclid in Alexandria, must have
known there was no person 'Euclid' but the fact that he wrote:-

.... Euclid did not work out the syntheses of the
locus with respect to three and four lines, but only a chance portion of it ...

certainly does not prove that Euclid was an historical
character since there are many similar references to Bourbaki by mathematicians
who knew perfectly well that Bourbaki was fictitious. Nevertheless the
mathematicians who made up the Bourbaki team are all well known in their own
right and this may be the greatest argument against hypothesis (iii) in that
the 'Euclid team' would have to have consisted of outstanding mathematicians.
So who were they?

We shall assume in this article that hypothesis (i) is
true but, having no knowledge of Euclid, we must concentrate on his works after
making a few comments on possible historical events. Euclid must have studied
in Plato's Academy in Athens to have learnt of the geometry of Eudoxus and
Theaetetus of which he was so familiar.

None of Euclid's works have a preface, at least none
has come down to us so it is highly unlikely that any ever existed, so we cannot
see any of his character, as we can of some other Greek mathematicians, from
the nature of their prefaces. Pappus writes (see for example) that Euclid was:-

... most fair and well disposed towards all who were
able in any measure to advance mathematics, careful in no way to give offence,
and although an exact scholar not vaunting himself.

Some claim these words have been added to Pappus, and
certainly the point of the passage (in a continuation which we have not quoted)
is to speak harshly (and almost certainly unfairly) of Apollonius. The picture
of Euclid drawn by Pappus is, however, certainly in line with the evidence from
his mathematical texts. Another story told by Stobaeus is the following:-

... someone who had begun to learn geometry with Euclid,
when he had learnt the first theorem, asked Euclid "What shall I get by
learning these things?" Euclid called his slave and said "Give him
threepence since he must make gain out of what he learns".

Euclid's most famous work is his treatise on mathematics
The Elements. The book was a compilation of knowledge that became the centre of
mathematical teaching for 2000 years. Probably no results in The Elements were
first proved by Euclid but the organisation of the material and its exposition
are certainly due to him. In fact there is ample evidence that Euclid is using
earlier textbooks as he writes the Elements since he introduces quite a number
of definitions which are never used such as that of an oblong, a rhombus, and a
rhomboid.

The Elements begins with definitions and five
postulates. The first three postulates are postulates of construction, for
example the first postulate states that it is possible to draw a straight line
between any two points. These postulates also implicitly assume the existence
of points, lines and circles and then the existence of other geometric objects
are deduced from the fact that these exist. There are other assumptions in the
postulates which are not explicit. For example it is assumed that there is a
unique line joining any two points. Similarly postulates two and three, on
producing straight lines and drawing circles, respectively, assume the
uniqueness of the objects the possibility of whose construction is being
postulated.

The fourth and fifth postulates are of a different
nature. Postulate four states that all right angles are equal. This may seem
"obvious" but it actually assumes that space in homogeneous - by this
we mean that a figure will be independent of the position in space in which it
is placed. The famous fifth, or parallel, postulate states that one and only
one line can be drawn through a point parallel to a given line. Euclid's
decision to make this a postulate led to Euclidean geometry. It was not until
the 19th century that this postulate was dropped and non-euclidean geometries
were studied.

There are also axioms which Euclid calls 'common
notions'. These are not specific geometrical properties but rather general
assumptions which allow mathematics to proceed as a deductive science. For
example:-

Things which are equal to the same thing are equal to
each other.

Zeno of Sidon,
about 250 years after Euclid wrote the Elements, seems to have been the first
to show that Euclid's propositions were not deduced from the postulates and
axioms alone, and Euclid does make other subtle assumptions.

The Elements is divided into 13 books. Books one to
six deal with plane geometry. In particular books one and two set out basic
properties of triangles, parallels, parallelograms, rectangles and squares.
Book three studies properties of the circle while book four deals with problems
about circles and is thought largely to set out work of the followers of
Pythagoras. Book five lays out the work of Eudoxus on proportion applied to
commensurable and incommensurable magnitudes. Heath says:-

Greek mathematics can boast no finer discovery than
this theory, which put on a sound footing so much of geometry as depended on
the use of proportion.

Book six looks at applications of the results of book
five to plane geometry.

Books seven to nine deal with number theory. In
particular book seven is a self-contained introduction to number theory and
contains the Euclidean algorithm for finding the greatest common divisor of two
numbers. Book eight looks at numbers in geometrical progression but van der
Waerden writes in that it contains:-

... cumbersome enunciations, needless repetitions, and
even logical fallacies. Apparently Euclid's exposition excelled only in those
parts in which he had excellent sources at his disposal.

Book ten deals with the theory of irrational numbers
and is mainly the work of Theaetetus. Euclid changed the proofs of several
theorems in this book so that they fitted the new definition of proportion
given by Eudoxus.

Books eleven to thirteen deal with three-dimensional
geometry. In book thirteen the basic definitions needed for the three books
together are given. The theorems then follow a fairly similar pattern to the
two-dimensional analogues previously given in books one and four. The main
results of book twelve are that circles are to one another as the squares of
their diameters and that spheres are to each other as the cubes of their
diameters. These results are certainly due to Eudoxus. Euclid proves these
theorems using the " method of exhaustion" as invented by Eudoxus.
The Elements ends with book thirteen which discusses the properties of the five
regular polyhedra and gives a proof that there are precisely five. This book
appears to be based largely on an earlier treatise by Theaetetus.

Euclid's Elements is remarkable for the clarity with
which the theorems are stated and proved. The standard of rigour was to become
a goal for the inventors of the calculus centuries later. As Heath writes in:-

This wonderful book, with all its imperfections, which
are indeed slight enough when account is taken of the date it appeared, is and
will doubtless remain the greatest mathematical textbook of all time. ... Even
in Greek times the most accomplished mathematicians occupied themselves with
it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon
of Alexandria re-edited it, altering the language here and there, mostly with a
view to greater clearness and consistency...

It is a fascinating story how the Elements has
survived from Euclid's time and this is told well by Fowler in. He describes
the earliest material relating to the Elements which has survived:-

Our earliest glimpse of Euclidean material will be the
most remarkable for a thousand years, six fragmentary ostraca containing text and
a figure ... found on Elephantine Island in 1906/07 and 1907/08... These texts
are early, though still more than 100 years after the death of Plato (they are
dated on palaeographic grounds to the third quarter of the third century BC);
advanced (they deal with the results found in the "Elements" [book
thirteen] ... on the pentagon, hexagon, decagon, and icosahedron); and they do
not follow the text of the Elements. ... So they give evidence of someone in
the third century BC, located more than 500 miles south of Alexandria, working
through this difficult material... this may be an attempt to understand the
mathematics, and not a slavish copying ...

The next fragment that we have dates from 75 - 125 AD
and again appears to be notes by someone trying to understand the material of
the Elements.

More than one thousand editions of The Elements have
been published since it was first printed in 1482. Heath discusses many of the
editions and describes the likely changes to the text over the years.

B L van der Waerden assesses the importance of the
Elements in:-

Almost from the time of its writing and lasting almost
to the present, the Elements has exerted a continuous and major influence on
human affairs. It was the primary source of geometric reasoning, theorems, and
methods at least until the advent of non-Euclidean geometry in the 19th
century. It is sometimes said that, next to the Bible, the "Elements"
may be the most translated, published, and studied of all the books produced in
the Western world.

Euclid also wrote the following books which have
survived: Data (with 94 propositions), which looks at what properties of
figures can be deduced when other properties are given; On Divisions which
looks at constructions to divide a figure into two parts with areas of given
ratio; Optics which is the first Greek work on perspective; and Phaenomena
which is an elementary introduction to mathematical astronomy and gives results
on the times stars in certain positions will rise and set. Euclid's following
books have all been lost: Surface Loci (two books), Porisms (a three book work
with, according to Pappus, 171 theorems and 38 lemmas), Conics (four books),
Book of Fallacies and Elements of Music. The Book of Fallacies is described by
Proclus:-

Since many things seem to conform with the truth and
to follow from scientific principles, but lead astray from the principles and
deceive the more superficial, [Euclid] has handed down methods for the
clear-sighted understanding of these matters also ... The treatise in which he
gave this machinery to us is entitled Fallacies, enumerating in order the
various kinds, exercising our intelligence in each case by theorems of all
sorts, setting the true side by side with the false, and combining the
refutation of the error with practical illustration.

Elements of Music is a work which is attributed to
Euclid by Proclus. We have two treatises on music which have survived, and have
by some authors attributed to Euclid, but it is now thought that they are not
the work on music referred to by Proclus.

Euclid may not have been a first class mathematician
but the long lasting nature of The Elements must make him the leading
mathematics teacher of antiquity or perhaps of all time. As a final personal
note let me add that my [EFR] own introduction to mathematics at school in the
1950s was from an edition of part of Euclid's Elements and the work provided a
logical basis for mathematics and the concept of proof which seem to be lacking
in school mathematics today.

J J O'Connor and E F Robertson

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