Euclid and the Elements
A Center of Learning:
The Museum
The Alexandrian School: Euclid
It is the glory of geometry that from so few principles, fetched from without, it is able to accomplish
so much.
ISAAC NEWTON
4.1 Euclid and the Elements
A Center of Learning:
The Museum
Toward the end of the fourth century B.C., the scene
of mathematical activity shifted from Greece to Egypt.
The battle of Chaeronea, won by Philip of Macedon in
338 B.C., saw the extinction of Greek freedom as well
as the decay of productive genius on its native soil. Two
years later, Philip was murdered by a discontented noble
and was succeeded by his 20-year-old son, Alexander the Great. Alexander conquered a
great part of the known world within 12 years, from 334 B.C. to his death in 323 B.C.,
at the age of 33. Because his armies were mainly Greek, he spread Greek culture over
wide sections of the Near East. What followed was a new chapter of history, known as
the Hellenistic (or Greek-like) Age, which lasted for three centuries, until the Roman
Empire was established.
Alexander’s great monument in Egypt was the city that still bears his name, Alexan-
dria. Having taken and destroyed the Phoenician seaports in a victorious march down
the Eastern Mediterranean, Alexander was quick to see the potential for a new maritime
city (a sort of Macedonian Tyre) near the westernmost mouth of the Nile. But he could
do little more than lay out the site, because he departed for the conquest of Persia soon
afterward. The usual story is that Alexander, with no chalk at hand to mark off the streets,
used barley from the commissary instead. This seemed like a good idea until clouds of
birds arrived from the delta and ate the grain as fast as it was thrown. Disturbed that this
might be a bad omen, Alexander consulted a soothsayer, who concluded that the gods
were actually showing that the new city would prosper and give abundant riches.
At Alexander’s death, one of his leading generals, Ptolemy, became governor of
Egypt and completed the foundation of Alexandria. The city had the advantage of a superb
harbor and docking facilities for 1200 ships, so it became with the shortest possible delay
the trading center of the world, the commercial junction point of Asia, Africa, and Europe.
Alexandria soon outshone and eclipsed Athens, which was reduced to the status of an
impoverished provincial town. For nearly a thousand years, it was the center of Hellenistic
culture, growing in the later years of the Ptolemaic dynasty to an immense city of a million
people. Following its sacking by the Arabs in A.D. 641, the building of Cairo in 969, and
the discovery of a shipping route around the Cape of Good Hope, Alexandria withered
away, and by the time of the Napoleonic expedition its population had dwindled to a mere
4000.
The early Ptolemies devoted themselves to making Alexandria the center of intel-
lectual life for the whole eastern Mediterranean area. Here they built a great center of
learning in the so-called Museum (seat of the Muses), a forerunner of the modern uni-
versity. The leading scholars of the times—scientists, poets, artists, and writers—came to
Alexandria by special invitation of the Ptolemies, who offered them hospitality as long
as they wished to stay. At the Museum, they had leisure to pursue their studies, access
to the nest libraries, and the opportunity of discussing matters with other resident spe-
cialists. Besides free board and exemption from taxes, the members were granted salary
stipends, the only demand being that they give regular lectures in return. These fellows
of the Museum lived at the king’s expense in luxurious conditions, with lecture rooms
for their discussions, a colonnaded walkway in which to stroll, and a vast dining hall,
where they took their meals together. The poet Theocritus, enjoying the bounty, hailed
Ptolemy as “the best paymaster a free man can have.” And another sage, Ctesibius of
Chalcis, when asked what he gained from philosophy, candidly replied, “Free dinners.”
Built as a monument to the splendor of the Ptolemies, the Museum was nonetheless a
milestone in the history of science, not to mention royal patronage. It was intended as an
institution for research and the pursuit of learning, rather than for education; and for two
centuries scholars and scientists ocked to Egypt. At its height, this center must have had
several hundred specialists, whose presence subsequently attracted many pupils eager to
develop their own talents. Although one poet of the time contemptuously referred to the
Museum as a birdcage in which scholars fattened themselves while engaging in trivial
argumentation, science and mathematics ourished with remarkable success. Indeed, it
is frequently observed that in the history of mathematics there is only one other span
of about 200 years that can be compared for productivity to the period 300–100 B.C.,
namely the period from Kepler to Gauss (1600–1850).
Scholars could not get along without books, so the rst need was to collect
manuscripts; when these were suf ciently abundant, a building was required to hold
them. Established almost simultaneously with the Museum and adjacent to it was the
great Alexandrian library, housing the largest collection of Greek works in existence.
There had of course been libraries before it, but not one possessed the resources that
belonged to the Ptolemies. Manuscripts were of cially sought throughout the world, and
their acquisition was vigorously pressed by agents who were commissioned to borrow
old works for copying if they could not otherwise be obtained; travelers to Alexandria
were required to surrender any books that were not already in the library. Many stories
are told of the high-handed methods by which the priceless manuscripts were acquired.
One legend has it that Ptolemy III borrowed from Athens the rolls kept by the state con-
taining the authorized texts of the writers Aeschylus, Sophocles, and Euripides. Although
he had to make a deposit as a guarantee that the precious volumes would be returned,
Ptolemy kept the original rolls and sent back the copies (needless to say, he forfeited the
deposit). A staff of trained scribes catalogued the books, edited the texts that were not
in good condition, and explained those works of the past that were not easily understood
by a new generation of Greeks.
The Alexandrian library was not entirely without rivals in the ancient world. The most
prominent rival was in Pergamon, a city in western Asia Minor. To prevent Pergamon
from acquiring copies of their literary treasures, the jealous Ptolemies, it is said, prohibited the export of papyrus from Egypt. Early writers were careless with numbers and often
exaggerated the size of the library. Some accounts speak of the main collection at the
library as having grown to 300,000 or even 500,000 scrolls in Caesar’s time (48 B.C.),
with an additional 200,000 placed in the annex called the Serapeum. The collection
had been built partly by the purchase of private libraries, one of which, according to
tradition, was Aristotle’s. After the death of Aristotle, his personal papers passed into
the hands of a collector who, fearing that they would be con scated for the library at
Pergamon, hid all the manuscripts in a cave. The scrolls were badly damaged by insects
and moisture, and the Alexandrian copyists made so many errors when restoring the texts
that they no longer agreed with the versions of Aristotle’s works already housed in the
library.
Euclid’s Life and Writings
Before the Museum passed into oblivion in A.D. 641, it produced many distinguished
scholars who were to determine the course of mathematics for many centuries: Euclid,
Archimedes, Eratosthenes, Apollonius, Pappus, Claudius Ptolemy, and Diophantus. Of
these, Euclid (circa 300 B.C.) is in a special class. Posterity has come to know him as the
author of the Elements of Geometry, the oldest Greek treatise on mathematics to reach us
in its entirety. The Elements is a compilation of the most important mathematical facts
available at that time, organized into 13 parts, or books, as they were called. (Systematic
expositions of geometry had appeared in Greece as far back as the fth century B.C., but
none have been preserved, for the obvious reason that all were supplanted by Euclid’s
Elements.) Although much of the material was drawn from earlier sources, the superbly
logical arrangement of the theorems and the development of proofs displays the genius
of the author. Euclid uni ed a collection of isolated discoveries into a single deductive
system based on a set of initial postulates, de nitions, and axioms.
Few books have been more important to the thought and education of the Western
world than Euclid’s Elements. Scarcely any other book save the Bible has been more
widely circulated or studied; for 20 centuries, the rst six books were the student’s usual
introduction to geometry. Over a thousand editions of the Elements have appeared since
the rst printed version in 1482; and before that, manuscript copies dominated much of
the teaching of mathematics in Europe. Unfortunately, no copy of the work has been
found that actually dates from Euclid’s own time. Until the 1800s, most of the Latin
and English editions were based ultimately on a Greek revision prepared by Theon of
Alexandria (circa 365) some 700 years after the original work had been written. But in
1808, it was discovered that a Vatican manuscript that Napoleon had appropriated for
Paris represented a more ancient version than Theon’s; from this, scholars were able to
reconstruct what appears to be the de nitive text.
Although the fame of Euclid, both in antiquity and in modern times, rests almost
exclusively on the Elements, he was the author of at least 10 other works covering a
wide variety of topics. The Greek text of his Data, a collection of 95 exercises prob-
ably intended for students who had completed the Elements, is the only other text by
Euclid on pure geometry to have survived. A treatise, Conic Sections, which formed
the foundation of the rst four books of Apollonius’s work on the same subject, has
been irretrievably lost, and so has a three-volume work called Porisms (the term porism
in Greek mathematics means “a corollary”). The latter is the most grievous loss, for it
apparently was a book on advanced geometry, perhaps an ancient counterpart to analytic
geometry.
As with the other great mathematicians of ancient Greece, we know remarkably little
about the personal life of Euclid. That Euclid founded a school and taught in Alexandria
is certain, but nothing more is known save that, the commentator Proclus has told us,
he lived during the reign of Ptolemy I. This would indicate that he was active in the
rst half of the third century B.C. It is probable that he received his own mathematical
training in Athens from the pupils of Plato. Two anecdotes that throw some light on the
personality of the man have ltered down to us. Proclus, who wrote a commentary to the
Elements, related that King Ptolemy once asked him if there was not a shorter way to
learning geometry than through the Elements, to which he replied that there is “no royal
road to geometry”—implying thereby that mathematics is no respecter of persons. The
other story concerns a youth who began to study geometry with Euclid and inquired, after
going through the rst theorem, “But what shall I get by learning these things?” After
insisting that knowledge was worth acquiring for its own sake, Euclid called his servant
and said, “Give this man a coin, since he must make a pro t from what he learns.”
The rebuke was probably adapted from a maxim of the Pythagorean brotherhood that
translates roughly as, “A diagram and a step (in knowledge), not a diagram and a coin.”
4.2 Euclidean Geometry
Euclid’s Foundation
for Geometry
For more than two thousand years Euclid has been the hon-
ored spokesman of Greek geometry, that most splendid cre-
ation of the Greek mind. Since his time, the study of the
Elements, or parts thereof, has been essential to a liberal ed-
ucation. Generation after generation has regarded this work
as the summit and crown of logic, and its study as the best
way of developing facility in exact reasoning. Abraham Lincoln at the age of 40, while
still a struggling lawyer, mastered the rst six books of Euclid, solely as training for
his mind. Only within the last hundred years has the Elements begun to be supplanted
by modern textbooks, which differ from it in logical order, proofs of propositions, and applications, but little in actual content. (The rst real pedagogical improvement was
by Adrien-Marie Legendre, who in his popular El´ements de G´eom´etrie, rearranged and
simpli ed the propositions of Euclid. His book ran from an initial edition in 1794 to a
twelfth in 1823.) Nevertheless, Euclid’s work largely remains the supreme model of a
book in pure mathematics.
Anyone familiar with the intellectual process realizes that the content of the Elements
could not be the effort of a single individual. Unfortunately, Euclid’s achievement has
so dimmed our view of those who preceded him that it is not possible to say how far
he advanced beyond their preparatory work. Few, if any, of the theorems established
in the Elements are of his own discovery; Euclid’s greatness lies not so much in the
contribution of original material as in the consummate skill with which he organized
a vast body of independent facts into the de nitive treatment of Greek geometry and
number theory. The particular choice of axioms, the arrangement of the propositions,
and the rigor of demonstration are personally his own. One result follows another in
strict logical order, with a minimum of assumptions and very little that is super uous.
So vast was the prestige of the Elements in the ancient world that its author was seldom
referred by name but rather by the title “The Writer of the Elements” or sometimes
simply “The Geometer.”
Euclid was aware that to avoid circularity and provide a starting point, certain facts
about the nature of the subject had to be assumed without proof. These assumed state-
ments, from which all others are to be deduced as logical consequences, are called
the “axioms” or “postulates.” In the traditional usage, a postulate was viewed as a “self-
evident truth”; the current, more skeptical view is that postulates are arbitrary statements,
formulated abstractly with no appeal to their “truth” but accepted without further justi -
cation as a foundation for reasoning. They are in a sense the “rules of the game” from
which all deductions may proceed—the foundation on which the whole body of theorems
rests.
Euclid tried to build the whole edi ce of Greek geometrical knowledge, amassed
since the time of Thales, on ve postulates of a speci cally geometric nature and ve
axioms that were meant to hold for all mathematics; the latter he called common no-
tions. (The rst three postulates are postulates of construction, which assert what we
are permitted to draw.) He then deduced from these 10 assumptions a logical chain of
465 propositions, using them like stepping-stones in an orderly procession from one
proved proposition to another. The marvel is that so much could be obtained from so
few sagaciously chosen axioms.
Abruptly and without introductory comment, the rst book of the Elements opens
with a list of 23 de nitions. These include, for instance, what a point is (“that which has
no parts”) and what a line is (“being without breadth”). The list of de nitions concludes:
“Parallel lines are straight lines which, being in the same plane and being produced
inde nitely in both directions, do not meet one another in either direction.” These would
not be taken as de nitions in a modern sense of the word but rather as naive descriptions
of the notions used in the discourse. Although obscure and unhelpful in some respects,
they nevertheless suf ce to create certain intuitive pictures. Some technical terms that
are used, such as circumference of a circle, are not de ned at all, whereas other terms,
like rhombus, are included among the de nitions but nowhere used in the work. It is
curious that Euclid, having de ned parallel lines, did not give a formal de nition of
parallelogram. Euclid then set forth the 10 principles of reasoning on which the proofs in the
Elements were based, introducing them in the following way:
Postulates
Let the following be postulated:
1. A straight line can be drawn from any point to any other point.
2. A nite straight line can be produced continuously in a line.
3. A circle may be described with any center and distance.
4. All right angles are equal to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same
side less than two right angles, then the two straight lines, if produced inde nitely
meet on that side on which the angles are less than two right angles.
Common Notions
1. Things that are equal to the same thing are also equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are 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.
Postulate 5, better known as Euclid’s parallel postulate, has become one of the most
famous and controversial statements in mathematical history. It asserts that if two lines l
and l
0 are cut by a transversal t so that the angles a and b add up to less than two right
angles, then l and l
0 will meet on that side of t on which these angles lie. The remarkable
feature of this postulate is that it makes a positive statement about the whole extent of a
straight line, a region for which we have no experience and that is beyond the reach of
possible observation.
t
l'
l
b
a
Those geometers who were disturbed by the parallel postulate did not question that
its content was a mathematical fact. They questioned only that it was not brief, simple,
and self-evident, as postulates were supposed to be; its complexity suggested that it
should be a theorem instead of an assumption. The parallel postulate is actually the
converse of Euclid’s Proposition 27, Book I, the thinking ran, so it should be provable.
It was thought impossible for a geometric statement not to be provable if its converse
was provable. There is even some suggestion that Euclid was not wholly satis ed with his fth postulate; he delayed its application until he could advance no further without
it, though its earlier use would have simpli ed some proofs.
Almost from the moment the Elements appeared and continuing into the nineteenth
century, mathematicians have tried to derive the parallel postulate from the rst four
postulates, believing that these other axioms were adequate for a complete development
of Euclidean geometry. All these attempts to change the status of the famous asser-
tion from “postulate” to “theorem” ended in failure, for each attempt rested on some
hidden assumption that was equivalent to the postulate itself. Futile so far as the main
objective was concerned, these efforts led nevertheless to the discovery of non-Euclidean
geometries, in which Euclid’s axioms except the parallel postulate all hold and in which
Euclid’s theorems except those based on the parallel postulate all are true. The mark
of Euclid’s mathematical genius is that he recognized that the fth postulate demanded
explicit statement as an assumption, without a formal proof.
Detailed scrutiny for over 2000 years has revealed numerous aws in Euclid’s treat-
ment of geometry. Most of his de nitions are open to criticism on one ground or another.
It is curious that while Euclid recognized the necessity for a set of statements to be as-
sumed at the outset of the discourse, he failed to realize the necessity of unde ned terms.
A de nition, after all, merely gives the meaning of a word in terms of other, simpler
words—or words whose meaning is already clear. These words are in their turn de ned
by even simpler words. Clearly the process of de nition in a logical system cannot be
continued backward without an end. The only way to avoid the completion of a vicious
circle is to allow certain terms to remain unde ned.
Euclid mistakenly tried to de ne the entire technical vocabulary that he used.
Inevitably this led him into some curious and unsatisfactory de nitions. We are told
not what a point and a line are but rather what they are not: “A point is that which
has not parts.” “A line is without breadth.” (What, then, is part or breadth?) Ideas of
“point” and “line” are the most elementary notions in geometry. They can be described
and explained but cannot satisfactorily be de ned by concepts simpler than themselves.
There must be a start somewhere in a self-contained system, so they should be accepted
without rigorous de nition.
Perhaps the greatest objection that has been raised against the author of the Elements
is the woeful inadequacy of his axioms. He formally postulated some things, yet omitted
any mention of others that are equally necessary for his work. Aside from the obvious
failure to state that points and lines exist or that the line segment joining two points is
unique, Euclid made certain tacit assumptions that were used later in the deductions but
not granted by the postulates and not derivable from them. Quite a few of Euclid’s proofs
were based on reasoning from diagrams, and he was often misled by visual evidence.
This is exempli ed by the argument used in his very rst proposition (more a problem
than a theorem). It involved the familiar construction of an equilateral triangle on a given
line segment as base.
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