CHAPTER 8

Plato’s Search for Pythagoras

Fourth Century B.C.

In about the year 389 B.C., Plato left his home in Athens and boarded a ship setting sail westward into the Ionian Sea. His destination was Tarentum, one of the old colonial cities of southern Italy, in the coastland known to him as Megale Hellas. He was going in search of Pythagoras.1

In the 110 years since his death, Pythagoras had become the stuff of legend. Some believed he had been the wisest man who ever lived, almost a god. There were stories that a wealth of precious knowledge had perished with him and his followers in upheavals that had destroyed their communities in 500 and 454 B.C. Though no one alive was old enough to have known Pythagoras, Plato had heard that in Megale Hellas there were still men calling themselves Pythagoreans. So, in his thirty-eighth year, he sailed to the shores where Pythagoras at about that same age had preceded him and walked and taught and died. The stones of the promontories, the pleasant coastlines, the very dust of the roads, ought to remember him.

Plato’s investigation began in Tarentum, on a small peninsula at the western extreme of the instep of the Italian boot, the first port of call for ships crossing from Greece.[1] The only story connecting Pythagoras with that city was that he had convinced a bull there not to eat beans, but Tarentum had been far enough from Croton for refugees from the fifth-century attacks to have settled, felt reasonably safe, and started their own exile Pythagorean community. It had survived, and Plato knew that its most prominent member now was Archytas of Tarentum – ‘Archytas the Pythagorean’.

In Archytas, Plato found a man who embodied Pythagorean ideals both in his lifestyle and his studies. Archytas was an outstanding scholar and mathematician working in the Pythagorean mathematici tradition, and also an able civic leader. Meeting him must have confirmed for Plato that the years of Pythagorean rule in Megale Hellas had been an era of peace and stability, strengthening his conviction that men who knew philosophy and mathematics made splendid rulers. Plato and Archytas were within a year of each other in age. The visit in 389 was the first of several during which Plato conversed with him and his Pythagorean friends, absorbing knowledge and information that only a handful of men in the world could have given him. Megale Hellas would continue to draw Plato, not only because of Archytas.

At the time of Plato’s first visit, the southern Italian cities were living under the encroaching shadow of a formidable enemy – Dionysius, tyrant of Syracuse, close across the water in Sicily. ‘Tyrant’ did not necessarily have negative connotations then. The term meant a ruler whose claim to power was not hereditary, and, indeed, Dionysius had begun in the lowly position of clerk in a city office. However, he also fit the later, ugly definition. Tactics that made him hugely successful shocked even his contemporaries. Dionysius reigned for nearly forty years, preserving Syracuse’s independence during repeated invasions while most of the rest of Sicily fell to the Carthaginians from North Africa. Syracuse became one of the most powerful cities in the world, her fleet for a time the strongest in the Mediterranean. It was certain that if Dionysius chose to move against his Italian neighbours, no one could stop him. Plato had come to an unstable, dangerous region, but instead of heading directly back to safer Athens, he decided to experience at first hand the court of a powerful, gifted ruler. Here was no theoretical governance. It was the real thing.

Dionysius’ capital was, or was in the process of becoming, a splendid, well-fortified city, built strategically on an island separated from the mainland of Sicily by a narrow swath of water. There was a Pythagorean community in Syracuse, begun like the one in Tarentum by fifth-century Pythagoreans who in this case had fled west across the Gulf of Messina, but Plato was more interested in the court of Dionysius. He was becoming increasingly intrigued with public affairs, and he seems to have enjoyed – perhaps too well for his own good – rubbing shoulders with powerful courtiers among whom he felt more than able to hold his own. On this first visit, Plato met one of the most influential men in Syracuse, the tyrant’s brother-in-law Dion. Plato was impressed with Dion . . . and Dion with Plato.

Not long after Plato’s visit, Dionysius’ invading forces wreaked devastation on the south Italian cities, and the entire region fell to Syracuse. In terms of the map, the football had kicked the boot. Meanwhile, back in Athens, Plato went on to establish his Academy, adopting a ‘Pythagorean curriculum’ that he had learned from Archytas: a ‘quadrivium’ of arithmetic, geometry, astronomy, and music. The inclusion of music was an exceptionally Pythagorean touch.

The ruthless Dionysius died in 367, survived by his son, Dionysius the Younger. Unfortunately for Syracuse – though perhaps to the relief of many in the region – the son was a much less able leader than the father. Plato’s acquaintance Dion, the new ruler’s uncle, was dubious about his nephew’s ability to keep Syracuse as dominant as the old tyrant had left it. For whatever well-meaning or devious reasons (history records the events but not the motivation) Dion decided to improve his nephew by seeing to his belated education. The father had been an innately brilliant leader with literary pretensions (though his writing was widely judged to be embarrassingly bad), but the son needed assistance if he was to rule effectively and continue to frustrate the Carthaginians’ desire to complete their takeover of Sicily. Dion recalled his conversations with Plato twenty years earlier and some of Plato’s dialogues that he had read since then, in which Plato had been developing the idea that men like Pythagoras and Archytas – philosophers for whom the ‘quadrivium’ was bread and butter – should be the political rulers. To fill such shoes and be a ‘philosopher king’, as Plato coined the term, Dionysius the Younger needed training only Plato could provide. Dion decided to try to convince Plato, by then sixty-one and famous in Athens and far beyond, to return to Syracuse and tutor him.

In spite of what must have been a yearning to foster a philosopher king in a world power like Syracuse, Plato was not initially keen about Dion’s proposal, thinking it would be a risky undertaking and unlikely to succeed. Archytas convinced Plato to change his mind. Partly tempted by the opportunity for more conversations with Archytas, Plato sailed for Syracuse. For a while, he was on sufficiently good terms with Dionysius the Younger to do some networking on Archytas’ behalf. A friendly relationship between Dionysius and Archytas was advantageous for the city of Tarentum. However, Dionysius did not study with Plato long. Before the year 366 ended, he banished Dion; Plato, suspecting that his own best interests did not lie in this court, prudently took his leave.

Yet five or six years later, in 361–360 B.C., Plato was back, invited by the tyrant himself. Dionysius sent an emissary named Archedemus, a friend of Archytas, on a special ship to summon Plato. The banished Dion also had a clandestine hand in his return. He asked Plato to engineer a reconciliation between him and Dionysius.

Plato arrived and Dionysius’ lessons resumed, but any hope of transforming Dionysius into a philosopher king was, again, short-lived. It cannot have helped that Plato was at court partly at the behest of the banished Dion. Plato was soon not only out of favour but in danger for his life. He got word to Archytas, and that resourceful man, using the influence he retained with Dionysius, sent an ambassador with a ship from Tarentum and persuaded the tyrant to release Plato. Afterwards Archytas was not only known as ‘Archytas of Tarentum’ or ‘the Pythagorean’ but also as ‘Archytas who saved Plato’s life’.

Dion captured Syracuse three years later and was assassinated three years after that at the behest of another Syracusan acquaintance of Plato. Dionysius regained control for a short period, but he seems never to have had much talent or inclination for ruling, and it may have come as a relief to him in 344 when the Corinthian general Timoleon compelled him to surrender and retire to Corinth. There he became a language teacher. Perhaps Plato’s efforts had not been entirely wasted and a former tyrant was well qualified to teach.

In Corinth, Dionysius met Aristotle’s pupil Aristoxenus, who was collecting information about Pythagoras and the Pythagoreans. Aristoxenus would turn out to be one of the earliest and most valuable sources, for Tarentum was his birthplace and he said his father knew Archytas. From Dionysius, who had been rather useless at nearly everything else, Aristoxenus was able to glean firsthand knowledge about Pythagoreans in the fourth century in Syracuse, not far from the area where the society had originated.

As Dionysius told the story to Aristoxenus, some of his courtiers in Syracuse had spoken disparagingly of the local Pythagoreans as arrogant, pious fakes whose rumoured moral strength and superiority would evaporate in a crisis. Other courtiers disagreed, and the two sides contrived a way to settle the dispute. Would one Pythagorean be willing to stake his life on the dependability and faithfulness of another? Would the other’s faithfulness and dependability – to the death – prove deserving of such trust?

The courtiers accused a man named Phintias, a member of the local Pythagorean community, of plotting against Dionysius. When Dionysius sentenced him to death, Phintias asked for a stay of execution for the remainder of the day, long enough to settle his affairs. It was a Pythagorean custom, established by Pythagoras himself, to keep no private property but own all things in common. Phintias was the oldest among the local brotherhood and chiefly in charge of the management of finances. Dionysius and his court, following their plan, allowed him to send for another Pythagorean, Damon, to remain as hostage until his return. To the astonishment of the court, Damon willingly came to stand as personal surety for Phintias. Phintias departed, and the courtiers – sure they had seen the last of him – sneered at Damon for being such a trusting fool. But the faithful Phintias returned at sunset to face his death rather than leave his friend to be executed in his stead. ‘All present were astonished and subdued’, reported Dionysius, who was so impressed that he embraced the two men and asked to be allowed to join their bond of friendship. Not surprisingly, ‘they would by no means consent to anything of the kind’. What happened to them then is not known. Plato, so often at court in Syracuse, also likely heard about this incident, but he never wrote about it.

Plato’s activities in Megale Hellas went beyond learning about Pythagoras and Pythagorean teachings, experiencing day-to-day reality in a tyrant’s court, and abortive attempts to tutor Dionysius. He helped Archytas strengthen his position in Tarentum as a minor philosopher king. Archytas went on to play a prominent role in political affairs among the cities of Megale Hellas and Sicily, in accordance with the Pythagorean tradition of wise and able involvement in public service.

In a search for the real Pythagoras and the Pythagoreans and what they believed and taught, the information about Archytas, Plato, and Dionysius the Younger provides valuable clues. Most significantly, it reveals a link between Plato and a Pythagorean community that still existed in the fourth century B.C. in the region where Pythagoras and his followers had had their golden age in the late sixth century. Plato knew, and knew of, other fourth-century Pythagoreans, but after his visits to Tarentum, when he thought ‘Pythagorean’ he was probably mostly thinking of Archytas and his associates. When he thought ‘Pythagorean mathematics and learning’ he was thinking of the mathematics and learning of Archytas.

What was he like, this man who was, for Plato, the best available evidence of what it meant to be Pythagorean and what ‘Pythagorean knowledge’ was? What could Plato have learned from him about Pythagoras and what Pythagorean teaching had been more than a century earlier?

Archytas was known to be a mild-mannered man who ruled in Tarentum through a democratic set of laws – information deduced from the news that these were not always obeyed: Though the ‘laws’ said a man should serve no more than one year, the city ‘elected’ Archytas seven times to the office of strategos, or ruling general.2 Aristoxenus wrote that Archytas was never defeated as a general except once, when his political opponents forced him to resign and the enemy immediately captured his men. Archytas, said Aristoxenus – whose father, he claimed, had known the man in person – was ‘admired for excellence of every sort’.

There can be no doubt that as a scholar Archytas lived by the great insight that set the Pythagoreans apart from other ancient thinkers: that numbers and number relationships were the key to vast knowledge about the universe. Archytas was a rigorous mathematician who solved an infamous problem in Greek mathematics known as the Delian problem, or doubling a cube, that is, creating a new cube twice the volume of the first. Archytas’ solution was sophisticated, requiring new geometry using three dimensions – ‘solid’ geometry – and involving the idea of movement.3 [2]

Diagram showing how Archytas solved the Delian problem, evidence of how advanced Pythagorean mathematics and geometry had become in little more than one century.[3]

Viewing the world through the eyes of his Pythagorean forebears, Archytas could not avoid pondering the possible hidden, underlying numbers and geometry. ‘Why are the parts of plants and animals (except for the organs) all round?’ he asked, ‘of plants, the stems and branches; of animals, the legs, thighs, arms, thorax? Neither the whole animal nor any part is triangular or polygonal.’ He suspected there was a ‘proportion of equality in natural motion, since all things move proportionately, and this is the only motion that returns back to itself, so that when it occurs it produces circles and rounded curves.’

Later scholars, among them Euclid and Ptolemy, agreed that Archytas’ precise work in the mathematics of music was fundamentally linked with the earliest Pythagorean mathematics and music theory.4 Archytas extended the study of numerical ratios between notes of the scale and showed that if you defined a whole tone as the interval separating the fourth and fifth notes of the scale (such as F and G in a scale beginning with C), as Greek music theorists were doing, then a whole tone could not be divided into two equal halves.[4] This had dramatic implications, for it was an example of something obviously present in the real world that could not be measured precisely. A different example, discovered in the right triangle, had famously caused the first Pythagoreans to have a devastating crisis of faith in the rationality of the universe, but incommensurability seemed no longer to disturb Pythagoreans like Archytas in the fourth century B.C.

In astronomy, Archytas puzzled over the question of whether the cosmos is infinitely large, and was notorious for asking: ‘If I come to the limit of the heavens, can I extend my arm or my staff outside, or not?’ He replied that whatever the answer – yea or nay – if he were out there performing this experiment, he could not actually be at the limit of the heavens. If he could not extend his arm or staff farther, something beyond the supposed limit had to be stopping him.5

In archaic-sounding litanies, the first Pythagoreans had asked ‘What are the isles of the blessed?’ and answered ‘The Sun and the Moon.’ Archytas brought this up-to-date in a more sophisticated catechism, asking ‘What is calm?’ and answering as a parent might answer a child, with an example: ‘What is a man?’ ‘Daddy is a man.’ Similarly, Archytas’ reply to ‘What is calm?’ was ‘Smoothness of the sea.’ His catechism, however, implied more than ‘example answers’, for he liked to connect the specific with the general, reflecting the Pythagorean doctrine of the unity of all being, and he enjoyed thinking about the relationship between the whole and the parts or particulars. His questions and answers about the weather and the sea were particular cases of deeper questions about smoothness and motion. The problem of dividing a whole tone into equal halves was a particular case of a mathematical discovery about ratios that could not be equally divided. His observations about the roundness in trees, plants, and animals were particular manifestations of a ‘proportion of equality in natural motion’. Archytas was convinced of a tight connection between understanding the universe, or anything else, as a whole and understanding the details. Plato regarded such ideas as the following, from Archytas, as the teaching of the Pythagoreans:

Those concerned with the sciences seem to me to make distinctions well, and it is not at all surprising that they have correct understanding of individual things as they are. For having made good distinctions concerning the nature of the whole, they were likely also to see well how things are in their parts. Indeed, concerning the speed of the stars and their risings and settings, as well as concerning geometry and numbers, and not least concerning music, they have handed down to us a clear set of distinctions. For these sciences seem to be akin.6

When Archytas wrote about such matters as smoothness and nonsmoothness of the sea he was reflecting another Pythagorean traditional favourite – opposites (smoothness/lack of smoothness; motion/lack of motion) – and for him that line of thought inevitably led back to thinking about infinity. Can something be infinitely calm? Or infinitely uncalm? Or infinitely smooth; infinitely rough?

As a politician and general, Archytas was convinced of what he was sure his Pythagorean forebears had demonstrated: The unity of all things had to include ethics and politics. The value of mathematics extended to the political arena. In the following fragment, ‘reason’ could also be translated as ‘calculation’. To a Pythagorean like Archytas, the two meanings were probably synonymous.

When reason/calculation is discovered, it puts an end to civil strife and reinforces concord. Where this is present, greed disappears and is replaced by fairness. It is by reason/calculation that we are able to come to terms in dealings with one another. By this means do the poor receive from the affluent and the rich give to the needy, both parties convinced that by this they have what is fair.

Plato, of course, could not have agreed more. The ability to use ‘reason’ or ‘calculation’ would make a philosopher king a superbly able ruler.

For Archytas, the concept of unity meant he should also apply a Pythagorean search for deeper levels of mathematical understanding to optics, physical acoustics, and mechanics. His is the earliest surviving explanation of sound by ‘impact’, with stronger impacts giving higher pitches, but he nodded to his Pythagorean forebears by insisting this was a theory that had been handed down to him. By ‘impact’, Archytas meant impact on the air – whipping a stick through the air, playing a high note on a pipe by making the pipe as short as possible (making a stronger pressure on the air, he thought), and the sound of the wind whistling higher pitches as its speed increased, or a ‘bull-roarer’. That last was an instrument used in the mystery religions, a flat piece of wood on the end of a rope. Whirling it around in the air like a giant slingshot produced a fearsome howling sound; the faster the whirling the higher the pitch.7

One of the most widely known, influential, and enduring Pythagorean ideas passed down through Archytas to Plato was the concept of the ‘music of the spheres’, the music Archytas and his Pythagorean forebears thought the planets made as they rushed through the heavens. Here is Archytas’ explanation for why humans never hear it:

Many sounds cannot be recognised by our nature, some because of the weakness of the blow (impact), some because of the great distance from us, and some because their magnitude exceeds what can fit into our hearing, as when one pours too much into narrow mouthed vessels and nothing goes in.

According to Pythagorean tradition, only Pythagoras could hear this music.

Archytas was a generous man, kind to slaves and children. He invented toys and gadgets, including a wooden bird (a duck or a dove) that could fly. Aristotle was impressed by ‘Archytas’ rattle’, ‘which they give to children so that by using it they may refrain from breaking things about the house; for young things cannot keep still.’8

This, then, was the science, mathematics, music theory, and political philosophy that Plato, from Archytas, learned to think of as Pythagorean. Through Plato, much of the image of Pythagoras and Pythagorean thought in Western civilisation is traceable to Archytas’ window on Pythagoras.

How unclouded was this window? Archytas regarded himself as an authentic Pythagorean, true to the earliest traditions and teachings. In his era, oral accounts could still be accurate, especially in a continuing community that considered it vitally important to keep an ancient memory alive and clear. In many ways, Archytas was probably a good reflection of what it had meant to be Pythagorean when Pythagoras himself walked the paths of Megale Hellas. However, he was one of the mathematici, the school of Pythagoreanism that believed following in Pythagoras’ footsteps meant diligently seeking and increasing knowledge. The Pythagorean ideals that underlay Archytas’ thought and work led him to newer discoveries. He was among the great scholars and mathematicians of his era, by reputation the teacher of the mathematician Eudoxus. If Archytas had focused only on the knowledge of the first Pythagoreans, this would have been impossible.

Plato himself provided a window through which we view Archytas. No matter how accurately Archytas reflected Pythagoras and Pythagorean thinking, we see him through Plato’s eyes and with Plato’s mind, the eyes and mind of one of the most creative thinkers in all history. It is in the nature of such a man, if he is impressed with an idea, to take the ball and run with it – to say, ‘This is, of course, what you mean’, and restate someone else’s good idea with a spin that makes it absolutely brilliant – and his own, not the original. Assuming Archytas was an exemplary Pythagorean, when Plato got the ball to the other end of the field, was it anything like the same ball he had caught in the pass from Archytas? That is one of the most debated questions in all the long history of those who have yearned to know what Pythagoras himself, and Pythagoreans before Plato, really discovered and thought.

On one significant issue, Plato disagreed with Archytas, and that disagreement is a welcome clue, a clear indication of something in pre-Platonic Pythagorean thinking, undiluted by Plato, that differed from Plato. Archytas, Plato complained, was too concerned with what one could see and hear and touch, and with searching for mathematics and numbers to explain it. For Plato, the goal of studying mathematics was to turn away from experience that humans have through their five senses to a search for abstract ‘form’, out of reach of sensory perception. Numbers and mathematical understanding were a venture into abstract form, but not the same, he thought, as his own concept of the ultimate understanding of ‘the beautiful and the good’. This difference, in Plato’s view, made Archytas an inadequate philosopher and himself a better one.

Plato’s knowledge about Pythagoras and Pythagoreans was not confined to what he learned through Archytas. There is evidence in his dialogues that he heard about them from Socrates; also, Plato and Archytas both knew of Philolaus. If the characters in Plato’s dialogue Phaedo are not entirely fictional, he was acquainted with contemporaries who were ‘disciples of Philolaus and Eurytus’ in Phlius, a community west of Corinth, as well as with Echecrates, who speaks for them in the dialogue, and Simmias and Cebes. Plato also knew about Lysis of Tarentum who, like Philolaus, had emigrated to Thebes. The Pythagorean community there was apparently still in existence in Plato’s time.

Plato could not have avoided also knowing about acusmatici Pythagoreans who did not agree that scholars like Archytas were Pythagoreans.9 The Greek public in the fourth century B.C. generally failed to recognise a distinction between mathematici and acusmatici and lumped all ‘Pythagorists’ together as an eccentric lot. Athenian comic dramatists lampooned them as unwashed, secretive, arrogant characters who abstained from meat and wine and went about ragged and barefoot. Plato’s pupils, educated at his Academy in the ‘Pythagorean sister sciences’ – the quadrivium of arithmetic, geometry, astronomy, and music – spoke of their philosophy and that of Pythagoras as one and the same and featured him in their books. They were certainly more in the mathematici tradition than in the acusmatici, but they nevertheless were the targets of the same jibes.

The unshakable conviction of the men who inspired the caricature – that they were following in the authentic footsteps of Pythagoras and preserving a precious tradition – caused some of their contemporaries to feel, a bit uncomfortably, that even the most eccentric were favoured by the gods and privy to mystical secrets. Antiphanes, in his play Tarentini (the title connects it with Tarentum) spoke of Pythagoras himself as ‘thrice blessed’, and Aristophon had one of his characters report:

He said that he had gone down to visit those below in their daily life, and he had seen all of them and that the Pythagoreans had far the best lot among the dead. For Pluto dined with them alone, because of their piety.

Lest anyone conclude that Aristophon approved of Pythagoreans, another character commented that Pluto had to be a very easygoing god, to dine with such filthy riffraff.10

Diodorus of Aspendus, who was not fictional, was described as a vegetarian with long hair, a beard, and a ‘crazy garment of skins’ who with ‘arrogant presumption’ drew followers about him, although ‘Pythagoreans before him wore shining bright clothes, bathed and anointed themselves, and had their hair cut according to the fashion.’11

Aristoxenus – who interviewed the tyrant Dionysius in his Corinthian ‘retirement’ – would have none of this. He was effectively a propagandist for the mathematici, taking pleasure in contradicting the acusmatici by insisting that Pythagoras ate meat and that the aphorisms were ridiculous, and he tried to disassociate ‘true Pythagoreans’ from what he saw as this unsavoury, superstitious group who were giving the movement a bad name. He listed the pupils of Philolaus and Eurytus and called them ‘the last of the Pythagoreans’ who ‘held to their original way of life, and their science, until, not ignobly, they died out’. Because these men died a few decades before the comic allusions in the plays, dubbing them ‘the last of the Pythagoreans’ was making the point that the butts of the jokes were only pretending to be Pythagoreans.

Aristoxenus’ public relations efforts did not succeed well. Through the fourth century B.C., the popular image of Pythagoreans continued to resemble the acusmatici more than the mathematici. But after the fourth century, the acusmatici, with a few exceptions, dwindled and vanished from notice. Had there been no other Pythagorean tradition than theirs, and if they did indeed represent the truer image of Pythagoras and his earliest followers, it would be almost impossible to explain how such an odd cult figure, not far different from others in antiquity, became so dramatically and rapidly transformed in the minds of intelligent men and women as to inspire deep and effective scientific thinking and seize the imagination of centuries of people to come.

Was it all due to Plato? Did he get so excited about something that was mainly legend that he elaborated on it himself until he had made it hugely significant? At a minimum there had to have been the discovery in music of pattern and rationality underlying nature, and the accessibility of that rationality through numbers – and that was of no small significance. It seems much more reasonable to conclude that Pythagoras, responding to different types of interest and intelligence among his followers, encouraged both kinds of thinking – acusmatici for those who needed something naive and more regimented and conservative, and mathematici for those with minds eager to grasp difficult, nuanced concepts and explore their implications. He was personally, perhaps, not entirely unlike either group.

However that may be, from the time of Plato, what survived as ‘Pythagorean’ and ‘Pythagoras’ was largely mathematici, and that included the conviction that right from the time of Pythagoras himself, and attributable to him, there had been a truly remarkable new approach to numbers, mathematics, philosophy, and nature.

[1]Tarentum was the only colony established by Sparta, and Plato greatly admired the Spartan system of government. However, the people who had colonised Tarentum in 706 B.C. had come there under unusual circumstances and might not have shared Plato’s enthusiasm for Sparta. They were sons of officially arranged marriages uniting Spartan women with men who were not previously citizens. The purpose was to increase the number of male citizens who could fight in the Messenian wars. When the husbands were no longer needed as warriors, the marriages were nullified and the offspring forced to leave Sparta.

[2]For an example of the use of movement in geometry, take a straight line, fasten down one end of it, and swing the other end about. The result is an arc. Take a right triangle and stand it upright with one of the sides serving as its base; swivel it around the upright leg and the result is a cone. (The ancient scholar Eudemus used this explanation in his description of Archytas’ solution.)

[3]A lengthy text is needed to understand it and is available in S. Cuomo, Ancient Mathematics, Routledge, 2001, pp. 58 and 59, and on the Internet at http://mathforum.org/dr.math/faq/davies/cu/bedbl.htm

[4]More generally, ratios such as 5:4, or 9:8, in which the larger number is one unit larger than the smaller (mathematicians call these superparticular or epimeric ratios), cannot be divided into two equal parts.