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Mathematics in Ancient Greece
By Tobias Dantzig
Dover Publications, Inc.Copyright © 1983 George Dantzig
All rights reserved.
ON GREEKS AND GRECIANS
A dazzling light, a fearful storm, then unpenetrable darkness.
The stage on which were enacted the early episodes of the drama which I am about to unfold was Ancient Greece; the cast bore names unmistakably Greek; and the medium through which they conveyed their thoughts and deeds to their peers in culture was Greek, even though some of the records of these thoughts and deeds have passed through Latin and Arabic translations before reaching us. In these records we find the germs of theories and problems which have agitated the mathematical world ever since, and of which some remain unsolved to this day. We are told, indeed, that in mathematics most roads lead back to Hellas, and thus a book which makes any historical pretensions at all must needs begin with the question: Who were these Ancient Greeks?
The term conjures up in our minds a group of Aryan tribes, which had originally settled on the southern part of the Balkan Peninsula and adjacent islands of the Aegean Sea; then, spreading out in all directions, eventually reached the shores of Asia Minor, Lower Italy and the African littoral. The insular character of the land and its maritime activities encouraged independence and local rule; yet, dwelling as these people did at the very gateway of Europe, and menaced as they continually were by Oriental encroachment, they were often driven to "totalitarianism" as a means of self-preservation. Thus Ancient Greece became the proving ground of that struggle between oligarchy and democracy which has prevailed to this day; and as such we know it best.
But this is just one aspect of the complex pattern which the term evokes in our minds. In Ancient Greece stood the cradle of our culture: literature and philosophy, architecture and sculpture, in the various forms in which these arts are cultivated today, all had their origin in Greece. The songs of her poets, the works of her sculptors and the tracts of her philosophers are not mere monuments of a glory that was, but sources of study and inspiration today and, probably, for many centuries to come. Nor was the genius of these people limited to the arts and letters; their penetrating insight into the mysteries of number, form and extension had led them to develop to a high degree of perfection a discipline which they named mathematics, and which was destined to become both the model and the foundation of all sciences called exact.
This pattern becomes even more amazing when we contemplate that all this magnificent culture was erected in a few short centuries. We are told, indeed, that this great intellectual upheaval had reached its peak in the fifth century B.C.; that soon afterwards a general and rapid decline had set in, as though a fatal blow had been struck at the very roots of the mighty tree, a blow from which it never recovered; that after lingering on for a few more centuries, vainly endeavouring to live up to the grandeur of its past, it had finally succumbed to the coup de grâce administered by rising Rome.
Such is the picture of Ancient Greece as we perceive it through the thick historical fog of two thousand years. It is a perplexing picture, to say the least, for, as far as we know, nothing that ever happened before, or since, has even remotely resembled it. It is a picture of a people numerically insignificant, even when measured by standards of the Ancient World, which in the course of a few centuries erected a civilization of unprecedented magnitude, bequeathing to mankind for all time to come immortal treasures of literature, philosophy and mathematics. And the mystery becomes even more profound when we attempt —as is indeed our duty—to appraise this past in the light of the present. For the modern representatives of this ethnic group, far from exhibiting the acumen and finesse of their illustrious forebears, have contributed so little to the intellectual and artistic life of our time that it is difficult to conceive of any kinship between this Balkan people and the intellectual giants to whom our culture owes so much.
Is there anything wrong with this picture? Could it be that it is but another cliché, one of the many synthetic products of the diversified industry which passes today for liberal education? Well, this much is certain: in so far as the history of mathematics is concerned, this conception of Ancient Greece calls for a wholesale and drastic revision.
To begin with, the mathematical activity of Ancient Greece reached its peak during the glorious era of Euclid, Eratosthenes, Archimedes and Apollonius, a time when Greek letters, art and philosophy were already on the decline. There is a modern counterpart to this singular phenomenon. It was in the sixteenth century, the age of Cavalieri, Cardano, Galileo and Vieta, that mathematics was reborn; and the resurgence took place when the renaissance in arts and letters had already run its course, and the very names of Dante and da Vinci had become memories. Galileo was the central figure of that era, and the fact that he died one week before Newton's birth has been the subject of much historical comment. It is as significant, perhaps, that Galileo was born in 1564, within a few months of the death of the last great representative of Italian renaissance, Michelangelo.
In the second place, while Roman contributions to mathematics were less than negligible, there is no evidence whatsoever that either the Republic or the early Empire had in any way hampered its progress. The eclipse of mathematics began with the Dark Ages, and the blackout did not end until the last Schoolman was shorn of the power to sway the mind of man.
Lastly, it was not Greece proper but its outposts in Asia Minor, in Lower Italy, in Africa that had contributed most to the development of mathematics. Some of these outposts were Greek conquests, others had come under Greek domination through alliance or trade. Moreover, since the Greeks had no Gestapo to protect their Aryan blood from pollution, racial intermingling was widespread, and there is no evidence that these misalliances were frowned upon by Greeks of pure strain.
To be sure, the Greeks did divide mankind into barbarians and Hellenes; yet, scions of barbarian families who had adopted Greek names and customs were viewed by Greeks as Hellenes. Thales of Miletus is a case in point. From all accounts he was of Phoenician origin, as was, indeed, Pythagoras; still, not only was Thales classed by his contemporaries as a Greek but proudly hailed by them as among the seven wisest Greeks. As to his own attitude, listen to the words which one biographer puts in his mouth: "For these three blessings I am grateful to Fortune: that I was born human and not a brute, a man and not a woman, a Greek and not a barbarian."
In the centuries of Euclid and Archimedes Greek was the language of most educated men, whether they hailed from Athens, Syracuse, Alexandria or Perga. The significance of this will not escape the American observer familiar with the many melting pots strewn over this wide land. Could one assert that a man was of Anglo-Saxon blood because he was named Archibald or Percival and enjoyed a good command of English? Is it not equally naive to contend that a man who lived in the third century B.C. was racially a Greek because he called himself Apollonius and wrote in Greek?
The shores of the Mediterranean harboured many a melting pot into which Greeks and Etruscans, Phoenicians and Assyrians, Jews and Arabs were promiscuously cast. Who can tell today how the Aryans and Semites had been apportioned within these seething brews, or the Hamites, or the Ethiopians, for that matter? The motley mash passed into the sewers of history without leaving a trace of its composition behind it. The distilled essence alone remains, bottled in vessels which bear Greek inscriptions.
"A dazzling light, a fearful storm, then unpenetrable darkness." So wrote Galois on the eve of his fatal duel; and if we did not know that he intended these words as a summary of his own short span of nineteen eventful years, we could take it as a description of the era of Hellenic mathematics.
What brought about this brilliant progress, and what caused the subsequent eclipse? I shall not add my own speculations to those of Taine and Comte, and Spencer, and Spengler and countless other historians of culture. This much is clear: mathematics flourished as long as freedom of thought prevailed; it decayed when creative joy gave way to blind faith and fanatical frenzy.CHAPTER 2
To Thales ... the primary question was not What do we know, but How do we know it, what evidence can we adduce in support of an explanation offered.
Six centuries before the zero hour of history struck, there thrived on an Aegean shore of Asia Minor, not far from what today exists as Smyrna, a group of Greek settlements which went under the collective name of Ionia. It consisted of a dozen or so towns on the mainland, of which Miletus was the most prosperous, and of about as many islands, of which Samos and Chios were the largest. When measured by present-day standards, the territory was so small that if modern Smyrna were run by American realtors all that was once Ionia would be reduced to mere suburban "additions" to the "greater city."
Here, within the span of about fifty years, were born the two "Founders" of mathematics, Thales of Miletus and Pythagoras of Samos. According to some of their biographers both were of Phoenician descent, which seems plausible enough since most of the coast of Asia Minor was at that time honeycombed with Phoenician colonies.
Who were these Phoenicians? We remember them chiefly today as the inventors of the phonetic script which was so vast an improvement over all the previous methods of recording experience that, in principle at least, it has undergone no significant change in the twenty-five hundred years which followed. The Greek alpha, beta, gamma, as well as the Hebrew aleph, beth, gimel are but adaptations of the Phoenician symbols for these letters.
And yet, having bestowed upon mankind this marvellous method of recording events, the Phoenicians left practically no records of their own, and what little we know of them today we owe to Greek or Hebrew sources. They were a Semitic tribe, and their homeland was what we call today Syria. As Canaanites, Moabites, Sidonians, they fill many a page of the Bible. Apparently, whenever they did not engage the Hebrews in mortal combat they fought them with subtle propaganda, inducing the fickle sons of Israel to abandon Jehovah for Baal and other more tangible gods.
The Greeks knew the Phoenicians under a different guise. They spoke of them as crafty merchants and skilled navigators, and called them "Phoenixes," i.e., red, because of the ruddy complexions which the Mediterranean sun and winds had imparted to these ancient mariners. For, the Phoenicians roved that "landlocked ocean of yore" from end to end, exchanging wares and founding colonies, such as ill-fated Carthage, or Syracuse, the birthplace of Archimedes, who, reputedly, was also of Phoenician descent.
I said that Thales was classed by the Greeks as one of the Seven Sages. Indeed, he was the only mathematician so honoured, and it was his reputed political sagacity and not his mathematical achievements that had earned him the title. Because of this distinction, Thales was the subject of many historical studies, with the result that much had been written on his life and deeds. Of what value are these biographical accounts? Here are some highlights from which you can draw your own conclusion.
We are told by one of these commentators that Thales was so keen an observer that nothing would escape his alert attention; yet, according to another, he was so absent-minded that even as a grown-up man he had to be followed on his walks by his nurse lest he land in a ditch. We are informed by one that he was a seasoned merchant in salts and oils, and that it was the pursuit of this trade that had taken him to Egypt; but another tells us that he had come to Egypt as a very young man, and that, struck by the learning of the priests, he had tarried among them for more than a quarter of a century, returning to Miletus in advanced middle age. According to one account he had learned all he knew of geometry from these very priests; according to another he was entrusted by the Pharaoh with the task of determining the height of the Great Pyramid, a problem which the priests had vainly tried to solve.
The accounts on his views, social, political or philosophical, are just as confusing. Some tell us that he was a confirmed bachelor, that he had found an outlet for his paternal instincts in adopting his sister's family, that once when asked why he did not marry and have children of his own since he loved children so much, he replied: "Just because I love children so much." Other biographers, however, assure us that Thales had married and lived to be a patriarch, surrounded by children and grandchildren. Plutarch holds that Thales had democratic leanings, in support of which he quotes his letter to Solon. There Thales invites Solon to make his home in Miletus, apologizing, at the same time, that his native city is under the rule of a tyrant. Again, once when asked what was the strangest sight his eyes had perceived, he allegedly replied: "A tyrant ripe in years." But other sources have it that upon his return from Egypt Thales made his home with the Milesian tyrant, that for many years he acted as the latter's counsellor at large, and that it was, indeed, on the advice of Thales that the dictator had wisely declined a tempting alliance with Croesus.
The accounts on Pythagoras may be less at variance, but they are bewildering enough in other respects, for, in addition to the confusing versions of the chroniclers, we have to contend here with disciples who would put into the mouth of their dead Master anything that fitted an occasion or proved a point. Indeed, Pythagoras became the centre of a cult which persisted for many centuries and exerted a tremendous influence on scientific and religious thinking.
It is claimed that not only had he visited Egypt, but that his travels had taken him much farther East; that, in fact, much of the knowledge which he later conveyed to the Hellenes had been imparted to him by Persian Magi and the priests of Chaldea. One is almost willing to believe this after examining the medley of views and taboos ascribed to Pythagoras. Yes, taboos —since many of the rites of the sect later rationalized by his followers into principles have all the earmarks of taboos.
A case in point is the alleged Pythagorean aversion to animal flesh. I say alleged, for, on this score, too, there is no unanimity, some biographers asserting that Pythagoras celebrated his mathematical discoveries by sacrificing oxen to the gods, while others go so far as to claim that he was the first to introduce meat into the diet of Greek athletes who hitherto had been training on figs and butter.
Some followers of Pythagoras traced the interdiction to his doctrine of metempsychosis. According to them, the Master taught that of the three attributes of the soul only reason was exclusively human, while emotion and intelligence belonged to animals as well; that upon man's death his soul migrated from animal to animal; and that, consequently, by killing an animal one might mutilate a soul. All this makes beautiful reading but fails to explain why Pythagoras extended his dietetical prohibitions to beans. Indeed, according to Diogenes Laertius, this bean cult was the indirect cause of his death. Here is the story for what it is worth.
When Pythagoras returned from his Oriental travels, he found his native Samos under the rule of a tyrant. He then proceeded west and settled in Crotona, a prosperous city on the heel of the Italian Boot. There he eventually established a school and, incidentally, acquired great political power. Now, those were the days when totalitarianism was making serious inroads into Greek democracy, and so, as time went on, an opposition party arose which accused Pythagoras of dictatorial designs. A frenzied mob set fire to his mansion. The Master managed to escape, but having reached in the course of the ensuing pursuit a field of beans, he chose to die at the hands of his enemies rather than to trample down the sacred plants.
The reader will have realized by now what a formidable task it would be to pick the few sound kernels from this biographical chaff, let alone use the material to analyse the achievements of the Founders. And yet, behind the hazy mist of these fanciful tales and legends the portraits of the two men emerge, mere silhouettes perhaps, but silhouettes that become much less elusive and confusing when viewed as parts of the larger panorama of classical mathematics.
Excerpted from Mathematics in Ancient Greece by Tobias Dantzig. Copyright © 1983 George Dantzig. Excerpted by permission of Dover Publications, Inc..
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