The Beginnings of Greek Mathematics He is unworthy of the name of man who does not know that the diagonal of a square is incommensurable with its side.

The Beginnings of Greek Mathematics

He is unworthy of the name of man who does not know that the diagonal of a square is incommensurable

with its side.

PLATO

3.1 The Geometrical Discoveries of Thales

Greece and the Aegean Area

The Greeks made mathematics into one dis-

cipline, transforming a varied collection of

empirical rules of calculation into an orderly

and systematic unity. Although they were

plainly heirs to an accumulation of Eastern

knowledge, the Greeks fashioned through their own efforts a mathematics more profound,

more abstract (in the sense of being more remote from the uses of everyday life), and

more rational than any that preceded it. In ancient Babylonia and Egypt, mathematics

had been cultivated chie y as a tool, either for immediate practical application or as part

of the special knowledge be tting a privileged class of scribes. Greek mathematics, on

the other hand, seems to have been a detached intellectual subject for the connoisseur.

The Greeks’ habits of abstract thought distinguished them from previous thinkers; their

concern was not with, say, triangular elds of grain but with “triangles” and the char-

acteristics that must accompany “triangularity.” This preference for the abstract concept

can be seen in the attitude of the different cultures toward the number p

2; the Babylo-

nians had computed its approximations to a high accuracy, but the Greeks proved that it

was irrational. The notion of seeking after knowledge for its own sake was almost com-

pletely alien to the older Eastern civilizations, so that in the application of reasoning to

mathematics, the Greeks completely changed the nature of the subject. Plato’s inscription

over the door of his Academy, “Let no man ignorant of geometry enter here,” was not

the admonition of an eccentric but rather a tribute to the Greek conviction that through

the spirit of inquiry and strict logic one could understand a person’s place in an orderly

universe.

All history is based on written documents. Although documentation concerning

Egyptian and Babylonian mathematics is often very precise, the primary sources that can

give us a clear picture of the early development of Greek mathematics are meager. In

Greece, there was no papyrus such as was available in Egypt, no clays as in Babylonia.

Such “books” as were written must have been very few; and with the passing of time and

ravages of the elements, little original material has survived. Consequently, early Greek

history is a morass of myths, legends, and dubious anecdotes, preserved by writers who lived centuries later than the events under consideration. We depend on fragments and

copies of copies many times removed from the original document. However scrupulous

the copyist may have been in lling in obscure passages in an earlier text, we can never

be sure how much the copyist had to call on his or her own imagination or indeed, how

well the copyist understood the original.

The Greeks were not always con ned to the southeastern corner of Europe, their

location in modern times. Although the Egyptians had kept to themselves, the Greeks were

great travelers. Their colonization of the coasts and the offshore islands of Asia Minor

from the eleventh to the ninth century B.C. was a prologue to later large-scale movements

from mainland Greece. About the middle of the eighth century B.C., a network of Greek

cities was founded on the coastal reaches of the Mediterranean, with scattered settlements

as far a eld as the eastern end of the Black Sea. Down to 650 B.C., the main vent for

Greek expansion was lower Italy and Sicily; the many ourishing colonies there caused

the whole area to be given the name Greece-in-the-West. Although the earlier migration to

Asia Minor was probably the result of the Dorian conquest of large portions of the Balkan

Peninsula, economic distress and political unrest in the homeland were the new incentives

to spread overseas. An increase in population caused a crisis in land ownership, as well as

a serious shortage of food. All these migrations not only provided an outlet for dissatis ed

elements of the population at home but also served to establish foreign markets and to

lay the material foundations of art, literature, and science. Although Hellenic culture had

its beginnings in an expanded Greece, in due course peninsular Greece became only a

part of “Greater Greece.” By 800 B.C., there was, broadly speaking, a unity of language

and custom throughout the ancient Mediterranean world.

The wave of colonization that took place outside of the Aegean from the eighth to

the sixth century B.C. paved the way for an extraordinary breakthrough of reason and

the attendant cultural advancement. Historians have called this phenomenon the Greek

miracle. The miracle of Greece was not single but twofold— rst the unrivaled rapidity

and variety and quality of its achievement; then its success in permeating and imposing

its values on alien civilizations. For this, the colonies were like conduits through which

Greek culture owed to the world of the “Barbarians,” and the older Egyptian and

Babylonian cultures streamed to the Greeks. It is remarkable that all the early Greek

mathematics came from the outposts in Asia Minor, southern Italy, and Africa, and not

from mainland Greece. It is as if the scanty Greek populations living next door to the more

developed societies had their wits sharpened by this contact, as well as having access to

the knowledge gathered by them. The most decisive of all Greek borrowings was the art of

writing with the convenient Phoenician alphabet. Each of the symbols of the Phoenician

alphabet stood for a consonant; there were no signs for vowels. The Phoenician alphabet

had more consonant symbols than the Greek language required, so the Greeks set out

by selecting and adapting the consonant symbols they needed. Thereafter, they assigned

vowel values to the remaining symbols, adding only such new signs as they needed

(for instance Þ, which had a consonantal value in the Phoenician alphabet, became the

symbol for the vowel A in the Greek alphabet). As in other matters, the Greek city-states

vied with each other in the elaboration of the alphabet, with as many as 10 different

versions getting under way. Gradually, one of these local alphabets, the Ionian, gained

the ascendancy; and after its of cial adoption by Athens in 403 B.C., it spread rapidly

through the rest of Greece. Although the acceptance of alphabetic writing did not initiate

anything like popular education, the ease with which it could be learned made possible a wider distribution of learning than had prevailed in the older cultures, where reading and

writing were the property of a priestly class. (Although the Phoenician traders eventually

spread the new device throughout the Mediterranean world, the intelligentsia of Egypt

and Babylonia disdained the alphabet—possibly because they had invested lifetimes in

learning the elaborate ideograms that were the mysterious delight of specialists.)

Coinage in precious metals was invented in the Greek cities of Asia Minor about

700 B.C., stimulating trade and giving rise to a money economy based not only on

agriculture but also on movable goods. In rendering possible the accumulation of wealth,

this new money economy permitted the formation of a leisure class from which an

intellectual aristocracy could emerge. Aristotle recognized how important nonpractical

activity is in the advancement of knowledge when he wrote in his Metaphysics:

When all the inventions had been discovered, the sciences which are not concerned with the

pleasures and necessities of life were developed rst in the lands where men began to have

leisure. This is the reason why mathematics originated in Egypt, for there the priestly class

was able to enjoy leisure.

In most ancient civilized societies, an educated elite, usually priests, directed the

activities of the community. Whether the priests were themselves the government (as

in early Babylonia) or merely its servants (as in Egypt), pro ciency in writing and

mathematics was considered part of their special skills. In the structure of Egyptian

bureaucracy, the man of learning held a position of great privilege and potential power.

The Greek historian Polybius remarked that “the Egyptian priests obtained positions

of leadership and respect because they surpassed their fellows in knowledge.” Eastern

learning was a mystery shared only by the specialists and not destined for the citizenry.

Although an able and ambitious man had some opportunity to improve his lot through

education, these hopes were seldom realized—just as very few of Napoleon’s soldiers

ever became eld marshals. By contrast, Greek education was far more broadly based

and designed to produce gentleman amateurs. Perhaps the difference was that the Greeks

had no powerful priesthood that could monopolize learning as its own preserve; no

sacred writings or rigid dogmas that required the mind’s subservience. In any event, the

rst Greek intellectuals came not from the class of governmental managers but from

people of affairs, for whom business was a profession and learning a pastime.

Geography shaped the pattern of Greek political life. In Egypt and Babylonia, it was

easy to subject a large population to a single ruler, but in Greece, where every district was

separated from the next by mountains or the sea, central control by an absolute monarch

was impossible. Mountainous barriers were not enough to prevent invasion, but they

were enough to prevent one state from being merged with another. Patriotic loyalty was

to the native city—Athens, Corinth, Thebes, or Sparta—and not to Greece as a whole. In

great emergencies the Greek states acted collectively, seeing that they must unite or be

destroyed. During the Persian invasions of the later sixth and early fth centuries B.C.,

they pooled their ghting forces to defeat Darius at Marathon (490 B.C.) and Xerxes at

Salamis (480 B.C.), after a rearguard action by 300 Spartans at Thermopylae. On none

of these occasions was the union successful or long-lasting, because with each victory

the city-states would promptly fall out and exhaust themselves in long local wars. The

lack of political unity made the outcome inevitable. The end came when Philip II of

Macedonia overpowered the mixed Greek forces at the battle of Charonea in 338 B.C.

and established himself as the head of all the Greek states except Sparta. Philip died two years later, and the power passed into the hands of his son, Alexander the Great,

who achieved what no leader before had done. He uni ed Greece and carried Greek

civilization to the limits of the known world. In 323 B.C., when Alexander died at age

32, he ruled over conquests of more than 2 million square miles. But neither Greeks

nor Greek culture vanished with the change of masters. The years that followed—from

the time of Alexander the Great into the rst century B.C.—formed a brilliant period of

history known to scholars as the Hellenistic Age.

The Dawn of Demonstrative Geometry: Thales of Miletos

The rise of Greek mathematics coincides in time with the general owering of Greek

civilization in the sixth century B.C. (“Greek civilization” usually indicates a culture be-

ginning in the Iron Age and ourishing most brilliantly in the fth and fourth centuries

B.C.) From the modest beginnings with the Pythagoreans, number theory and geome-

try developed rapidly, so that early Greek mathematics reached its zenith in the work

of the great geometers of antiquity—Euclid, Archimedes, and Apollonius. Thereafter,

the discoveries were less striking, although great names such as Ptolemy, Pappus, and

Diophantus testify to memorable accomplishments from time to time. These pioneering

contributors exhausted the possibilities of elementary mathematics to the extent that little

signi cant progress was made, beyond what we call Greek mathematics, until the six-

teenth century. What is more striking still is that almost all the really productive work

was done in the relatively short interval from 350 to 200 B.C., and not in the old Aegean

world but by Greek immigrants in Alexandria under the Ptolemies.

The rst individuals with whom speci c mathematical discoveries are traditionally

associated are Thales of Miletus (circa 625–547 B.C.) and Pythagoras of Samos (circa

580–500 B.C.). Thales was of Phoenician descent, born in Miletus, a city of Ionia, at

a time when a Greek colony ourished on the coast of Asia Minor. He seems to have

spent his early years engaged in commercial ventures, and it is said that in his travels

he learned geometry from the Egyptians and astronomy from the Babylonians. To his

admiring countrymen of later generations, Thales was known as the rst of the Seven

Sages of Greece, the only mathematician so honored. In general, these men earned the

title not so much as scholars as through statesmanship and philosophical and ethical

wisdom. Thales is supposed to have coined the maxim “Know thyself,” and when asked

what was the strangest thing he had ever seen, he answered “An aged tyrant.”

Ancient opinion is unanimous in regarding Thales as unusually shrewd in politics

and commerce no less than in science, and many interesting anecdotes, some serious and

some fanciful, are told about his cleverness. On one occasion, according to Aristotle,

after several years in which the olive trees failed to produce, Thales used his skill in

astronomy to calculate that favorable weather conditions were due the next season.

Anticipating an unexpectedly abundant crop he bought up all of the olive presses around

Miletus. When the season came, having secured control of the presses, he was able to

make his own terms for renting them out and thus realized a large sum. Others say that

Thales, having proved the point that it was easy for philosophers to become rich if they

wished, sold his olive oil at a reasonable price.

Another favorite story is related by Aesop. It appears that once one of Thales’ mules,

loaded with salt for trade, accidentally discovered that if it rolled over in a stream, the

contents of its load would dissolve; on every trip thereafter, the beast deliberately repeated two years later, and the power passed into the hands of his son, Alexander the Great,