[2]There were more than one Apollodorus, but this one was probably Apollodorus of Cyzicus, who lived in the fourth century B.C.
[3]The claim has never been that Pythagoras discovered the right angle or right triangle, but that he discovered the relationship between the three sides of a right triangle – what we call the Pythagorean theorem.
[4]You can think of 3–4–5 as 3 inches, 4 inches, and 5 inches, though it could just as well be centimetres, miles, parsecs or any other unit of measurement.
[5]Unfortunately, most of Babylon of the early second millennium B.C. cannot now be excavated because it is well below the water table.
[6]The tablet is in the Iraq Museum in Baghdad, listed in the register as 55357.
[7]This mechanism, used probably in preparing calendars for planting, harvesting, and religious observances, was discovered in the wreck of a Roman ship that sank off the island of Antikythera in about 65 B.C. It was more technically complex than any known instrument for at least a millennium afterwards.
[8]Political and social upheaval may have created disruptions. Or the fault may lie with modern scholarship, for few sites have been dug from these periods. They do not attract many scholars, partly because the documents are terribly difficult to decipher. Furthermore, as the very complicated cuneiform script gave way to alphabetic Aramaic, documents tended to be written on perishable and recyclable materials. The old Sumerian, Akkadian, and the cuneiform script were used for fewer purposes, mathematics apparently not being one of them, and even where cuneiform was used, it was often on wax-covered ivory or wooden writing boards that were erased for reuse or have not survived.
[9]A rational number is a whole number or a fraction that is made by dividing any whole number by another whole number: 1⁄2, 4/5, 2/7, etc. An irrational number is a number that cannot be expressed as a fraction, that is, as a ratio of two whole numbers. The square root of 2 was probably found by Pythagoreans, working from their theory of odd and even numbers, possibly as early as about 450 B.C., and surely by 420, fifty to eighty years after Pythagoras’ death. Plato knew of the square roots of numbers up to 17.
[10]Though Heraclitus seems forthright and outspoken in the fragments about Pythagoras, he was known to be no easy read. His contemporaries dubbed him Heraclitus the Obscure and Heraclitus the Riddler. A story circulated in the time of Diogenes Laertius that when Socrates received a copy of a book by Heraclitus, he commented: ‘What I understand is splendid; and so too, I’m sure, is what I don’t understand – but it would take a Delian diver to get to the bottom of it.’
[11]See Chapter 9.
PART II
Fifth Century B.C.– Seventh Century A.D.
CHAPTER 7
A Book by Philolaus the Pythagorean
Fifth Century B.C.
After Pythagoras’ death, the demise of the Pythagorean brotherhood in southern Italy did not take place overnight or in a few short years. Many Pythagoreans survived the violence that ushered in the fifth century B.C., and the Pythagorean drama, minus its lead character, continued in the colonial cities. Only one Pythagorean is known for certain by name from that period: Hippasus of Metapontum. He was a scholar and perhaps a brilliant one, apparently part of the Pythagorean inner circle, who worked in music theory, mathematics, and natural philosophy and considered fire a first principle. One report credits Hippasus with constructing the dodecahedron, the twelve-sided solid – he ‘first drew the sphere constructed out of twelve pentagons’. He may have taught the cantankerous Heraclitus. However, after Pythagoras’ death, Hippasus fell from grace, and he is chiefly remembered as an ill-fated, perhaps ill-intentioned figure.
When the Pythagoreans discovered that mathematical relationships underlie nature, they did not announce this to the world. Secrecy was their custom. Hippasus, however, had to have been privy to the discovery, because he performed the successful experiment with bronze disks. Some stories connected him with the discovery of incommensurability, or even made him the unlucky discoverer. Accounts differed about how he erred, but somehow, while all good things were attributed to Pythagoras, all bad things seemed to get hung on Hippasus. His transgression was revealing a secret of geometry, or discovering incommensurability, or effrontery to the gods by making a discovery in geometry (that could have been the dodecahedron), or taking credit for a discovery instead of attributing it to Pythagoras.
The most nuanced and authentic-sounding material about Hippasus comes from Aristotle and Aristoxenus, and it links Hippasus with a fault line that developed in the Pythagorean brotherhood between two factions calling themselves the acusmatici and the mathematici.1 The antagonism, which may have had its roots in a two-level hierarchy initiated by Pythagoras, the better to organise his brotherhood according to interests and abilities, split the community into opposing camps.
The acusmatici were devoted to rote learning. Their philosophy (quoting Iamblichus) ‘consisted of unproven and unargued aphorisms, and they attempted to preserve the things Pythagoras said as though they were divine doctrines.’ Scholars surmise that these aphorisms were relics of the most elementary, easily remembered part of Pythagoras’ teaching. Some were folk maxims with added interpretations, with knowledge of the interpretations possibly serving as passwords or signifying rank in the community. There were three kinds of aphorisms (this according to Iamblichus): Some asked what something was: ‘What are the Isles of the Blessed? The sun and the moon.’ A second kind indicated superlatives: ‘What is most wise? Number.’ ‘What is most truly said? That men are wretched.’ A third concerned minutiae about ‘what one must do or not do.’ Many sounded pointless to anyone unaware of the secret interpretations. ‘Do not turn aside into a temple’ meant ‘Do not treat God as a digression.’ ‘Do not help anyone put down a burden; rather, help him take it up’ meant ‘Do not encourage idleness.’ ‘Do not break a loaf of bread’ because ‘it is disadvantageous with regard to the judgement in Hades.’ Iamblichus threw up his hands at that and called it ‘far-fetched’. Acusmatici evidently understood the connection. They claimed the title ‘Pythagorean’ for themselves exclusively.
The mathematici, on the other hand, were willing to admit the acusmatici under the banner of the brotherhood, but they preserved and extended a different kind of Pythagorean knowledge. Though not always agreeing among themselves, they shared a conviction that the acusmatici’s refusal to allow knowledge to develop further was contrary to the spirit in which Pythagoreanism had been practised when Pythagoras was alive.
According to Aristotle and Aristoxenus, Hippasus was one of the mathematici, or one of those who would be labelled mathematici when the groups became fully polarised. The opposing camp – those who would be known as acusmatici – frowned on his work as new and subversive. The mathematici might have been expected to defend Hippasus, but they were engaged in delicate manoeuvres, insisting they were not introducing new doctrines, merely working on explication of the doctrines of Pythagoras. They disassociated themselves from Hippasus, to no avail since the acusmatici continued to accuse them of following him rather than Pythagoras. Hippasus was caught in the crossfire. His punishment, from the gods or the Pythagoreans, depending on which story to believe, was drowning at sea, expulsion from the community, and/or the construction of a tomb to him as though he were dead.
Hippasus’ story provides a clue for dating some of the Pythagorean discoveries. Modern historians are sceptical about the claim that Hippasus taught Heraclitus, but they think the fact that many believed he did dates Hippasus reliably. He was supposed to have taught Heraclitus, not the other way around, and since Heraclitus’ lifetime overlapped Pythagoras’, Hippasus must have been an even earlier contemporary of Pythagoras. Furthermore, Hippasus’ disk experiment had to have happened after the discovery of the ratios and before Hippasus’ disgrace, which occurred (dated by t
he split in the brotherhood) shortly after Pythagoras’ death. This chronology makes it impossible for the discovery of the musical ratios to have been made later, in the next generation. They were an authentically early Pythagorean discovery.
Hippasus’ disgrace and the mathematici/acusmatici conflict are not the only evidence that some Pythagoreans who survived the turn-of-the-century upheavals stayed in Magna Graecia. Remnants of the brotherhood persisted throughout the region. Iamblichus had information that Pythagoras’ ‘successor’ was Aristaeus, who married his widow Theano, ‘carried on the school’, and educated Pythagoras’ children. A son of Pythagoras named Mnesarchus reputedly took over the school when Aristaeus became too old. If folk memory in Metapontum had it right, Pythagoras survived for a while in exile and established a school there.
Some Pythagoreans continued to hold, or rapidly regained, positions of political importance and possibly extended their influence over an even wider area than before, but as these leaders became influential again in the government of the cities, they courted disaster by ruling more and more autocratically. A revolution unseated them in mid-century, about 454 B.C. The second century B.C. historian Polybius repeated a description he found in earlier accounts: ‘The Pythagorean meeting places were burned down and general constitutional unrest ensued – a not unlikely event, given that the leading men in each state had been thus unexpectedly killed. The Greek cities in these regions were filled with bloodshed and revolution and turmoil of every kind.’ The result this time was a Pythagorean diaspora – to Thebes, to Phlius (near Corinth), to Syracuse, and elsewhere. The curtain fell for the second and final time on the Pythagorean golden age in Magna Graecia. The original community that Pythagoras had taught no longer existed.
In a larger context, the story had only begun. From about this time, there were two discernible contrasting strands of thought in the ancient Mediterranean world: ‘Ionian’, from mainland Greece and that area of the Mediterranean; and ‘Pythagorean’ or ‘Italian’, stemming from southern Italy. Through members of Pythagorean refugee communities and their intellectual descendants – and men like Plato who were drawn to their ideas – the remnants of the thinking of an obscure ancient group became a powerful worldview. By late antiquity, no one could claim to be a serious thinker and ignore the ‘Pythagorean’ or ‘Italian’ school.
Meanwhile, the acusmatici/mathematici split nevertheless continued to infect the scattered brotherhood, and the disagreement about who reflected the spirit and work of the first Pythagoreans still causes difficulty for anyone trying to discern the truth about that earliest era. Most educated people through the centuries would insist that the mathematici were the true Pythagoreans, preserving and extending the great Pythagorean mathematical legacy. The reason for this certainty is that it was the mathematici tradition that Plato handed down to the future. He made the choice for Western civilisation.
Aristotle, a generation later than Plato, was well acquainted with both Pythagorean varieties and described an acusmatici legacy that in addition to the aphorisms included the miraculous legends, the doctrine of reincarnation and Pythagoras’ memory of his past lives. The mathematici legacy accepted most of that, too, but emphasised the different approach to the world and the soul through numbers, mathematics, and music. The mathematici had preserved historical information: that Pythagoras came to Croton during the reign of Polycrates on Samos and had a powerful influence on the leaders of his new home city. Aristotle never traced a heritage of knowledge and mathematics to Pythagoras himself by naming names in succeeding generations, but he had no quarrel with the mathematici’s claim that this unbroken heritage existed.2 Plato attributed a sophisticated version of the Pythagorean mathematici number theory not just to Pythagoreans but to Pythagoras himself.
The second half of the fifth century B.C. (450 to 400) is still much alive in the cultural memory of the modern world. Greek tragedy had blossomed with Aeschylus and was continuing with the plays of Sophocles and Euripides, raising issues that need no modern context to make them relevant today. Aristophanes was scandalising his delighted audiences, satirising public affairs and leaders in brilliant, flagrantly indecent comedies. Though these would soon be dubbed ‘Old Comedy’ as newer forms and subjects become fashionable, in the twenty-first century his The Frogs became a Broadway musical. The physician Hippocrates was working and writing, and medical school graduates more than two millennia later repeat the oath attributed to him. Athens, in mainland Greece, was picking up the pieces after a long conflict with the Persians and enjoying an interval of peace, growing rich from silver mines and tribute from other members of the Delian League, former allies in the Persian Wars. Unaware how short this respite would be before they became involved in the Peloponnesian Wars, Athenians restored their city, which the Persians had burned, and erected the Parthenon. In this half-century, Plato was born and grew to manhood, and Philolaus the Pythagorean, nearly fifty years Plato’s senior, wrote the first Pythagorean book – or at least the first that was destined to survive.
Philolaus was one of the refugees who left Croton or Tarentum at mid-century. He settled in about 454 in Thebes, a powerful old city northwest of Athens whose ancient origins made her a favourite setting for Greek dramas. She had once been the seat of the real King Oedipus. Politically, Thebes’ only consistent policy was hatred of Athens. She had sided against Athens in the Persian Wars and then collaborated with Sparta against her, an alliance that would last until nearly the end of the Peloponnesian Wars at the close of the century. Thebes and Sparta would finally part ways when Thebes suggested the defeated Athenians be totally annihilated and Sparta disagreed.
It would appear that Thebes was not a particularly serene location for a fledgling brotherhood to pursue peaceful studies, but Philolaus founded a new exile Pythagorean community there. He had either died or moved elsewhere by the end of the century – information that comes indirectly through Plato, who in his dialogue Phaedo had a character named Cebes comment, ‘I heard Philolaus say, when he was living in our city. . .’ Cebes’ city was Thebes, and this conversation was supposed to be taking place the day Socrates died in 399 B.C. If Philolaus was still alive then somewhere else, he was seventy-five, but that reference to him was the last that has survived.
Some time between 450 and 399 B.C., probably in Thebes, Philolaus set down an extensive written record of Pythagorean thought, something no Pythagorean had done before as far as anyone has been able to discover. The only traces of it today are fragments, mostly references in the writing of scholars during the first century A.D., long removed from his time.[1] In the nineteenth century there was controversy about whether Philolaus wrote a book and whether the fragments are genuine, but in 1893 a papyrus came to light with excerpts from a medical history by Menon, a pupil of Aristotle in the fourth century B.C., referring to a book by Philolaus that already existed then. Since that discovery, scholars have analysed the Philolaus fragments in the context of the fifth century B.C., Philolaus’ century, and they largely agree about which are authentic.3
An anachronistic, late fifteenth-century A.D. drawing, from a music theory book by Gaffurio, reveals how scholars of that era envisioned Pythagoras (and Philolaus, who was not actually Pythagoras’s contemporary) studying the ratios of musical harmony.
Though it hardly seems fair to Philolaus, anyone looking for specifics about Pythagoras and what he taught is frustrated by the fact that Philolaus was a splendid thinker in his own right. He was writing his own book, not recording the discoveries or words of another man, and included his own thinking as well as what had evolved in the Pythagorean mathematici communities since Pythagoras’ death. Nevertheless, Philolaus definitely considered himself a Pythagorean, and, given the time frame, much of the science and doctrine in his book must have been a direct reflection of Pythagoras and his earliest followers. Philolaus was almost a direct link, for Pythagoras had died or disappeared from public view in 500 B.C., only twenty-
five years before Philolaus’ birth. Philolaus’ teachers and acquaintances as he grew up in Croton or Tarentum must have been almost exclusively Pythagoreans, and some of the older of them would have known Pythagoras.
Unfortunately, Philolaus treated all of his material as a unified body of knowledge, making no distinctions between earlier and later, between the time Pythagoras was alive and the time of Philolaus’ writing, or between himself and others. He was not being careless. For a Pythagorean there was unity to truth, and unity to the search for it. The path to knowledge about the universe and the path to reunion with the divine were one and the same path. Truth about nature, and divine truth, were one and the same truth. In such a context, even if Pythagoras himself had not made a particular discovery, one could assume it had been implicit in his teachings. Furthermore, there was a form of ancient one-upmanship that Pythagoreans like Philolaus shared with their contemporaries. It was demeaning to an idea or discovery to call it new or original. Knowledge became more credible the older it was and the more it could be attributed to a great figure. Philolaus would have been loath to identify any source other than Pythagoras, even if it was himself.
Nevertheless, Philolaus was not without an agenda of his own. One of the clues that place his writing in the late fifth century B.C. was that he was trying to present Pythagorean ideas in a way that responded to a stalemate arising from ‘Eleatic’ teaching.
The philosopher Parmenides was from Elea (hence ‘Eleatic’), a Greek colony north of Croton on Italy’s west coast. According to Plato he was born in 515 B.C., but Greek chronicles say about 540. In either case, he was a younger contemporary of Pythagoras, but remarkably, in spite of the overlap of their lifetimes, the close proximity of Elea to Croton, and a passage in Plutarch that says Parmenides ‘organised his own country by the best laws’, only one early source gave Parmenides even the remotest link with Pythagoras or the Pythagoreans. The link was indirect, in Diogenes Laertius’ third-century-A.D. biography of Pythagoras:
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