Thursday, December 8, 2022

Euclid and the Elements A Center of Learning: The Museum

 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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