Euclid

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 [1]
or [9] 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 [48]. 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 [1]:-

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 [8]. This is far from an end
to the arguments about Euclid the mathematician. The situation is best summed up
by Itard [11] 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, Andr� Weil, Jean Dieudonn�, Claude Chevalley,
and Alexander Grothendieck wrote collectively under the name of Bourbaki and
Bourbaki's El�ments de math�matique 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 [1]) 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 [9] 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
[9]:-

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 [2] 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 [9]:-

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 [7]. 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 [9] 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 [2]:-

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 [1]:-

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.

Article by: J J O'Connor and E F Robertson