by Mark Adams
I’d even found the obscure genius who could help me make the final leap. I hopped a train to Washington, DC, and headed to the apartment of ninety-five-year-old Ernest McClain, a retired music professor who had written a book titled The Pythagorean Plato. In it, McClain described the Atlantis tale as “a sophisticated entertainment for Pythagoreans only” and lamented that “for the musically innocent, it is and must remain merely a Platonic fairy tale, incomprehensibly loaded with absolutely meaningless numerical detail.”
The book was tough sledding; a leading American musicologist who specializes in Pythagorean harmonics later described it to me as “completely impenetrable.” I spent a pleasant four hours nodding my head and listening to McClain weave a word tapestry about the relationship between ancient Sumerian octaves, the Rig Veda, the Babylonian base-sixty number system, and the music of the spheres. I sat at his table and briefly held hands in silent prayer with him and his home health care aide before eating a bowl of chicken soup and a sandwich, and returned to Union Station as musically innocent as when I’d arrived. About the only thing I’d understood was McClain’s repeated exhortation: “I know you’ve come for a shortcut, but some things you can’t understand until you figure them out for yourself!”
• • •
A few weeks later, I made my way to the Green Mountains of Vermont, where John Bremer had offered to try to untangle all this stuff for me. Bremer was a retired educator; like Plato he had traveled widely and founded an institution of higher learning, Cambridge College, just outside of Boston. He had spent decades thinking about Plato. He knew Ernest McClain personally and considered him brilliant. A half century before Jay Kennedy began his computer stichometric analysis, Bremer was counting syllables of ancient Greek by hand. Like Kennedy, he noticed patterns that seemed too striking to be coincidental.
Still dashing at age eighty-six, with longish hair, a posh British accent, and a peach-colored shirt unbuttoned to his sternum, Bremer continued to wrestle with Plato daily. We met on an unseasonably hot day, so we retreated to his basement, where his large desk sat amid a forest of bookshelves, one devoted entirely to Plato. He called himself a Socratic philosopher, which I supposed meant he asked a lot of questions in order to reach the truth. We assumed the familiar positions of teacher and student, he behind the desk and me across, frantically scribbling lecture notes.
“Have you met Plato’s Divided Line?” he asked, as if making introductions at a party.
We were acquainted, the Divided Line and I, though I wouldn’t go so far as to say we were friends. Robert Brumbaugh had called Plato’s Divided Line “probably the most famous and most often drawn diagram in the whole history of philosophy.” It is a deceptively simple geometric image that helps explain one of Plato’s most important principles: that knowledge can be classified into four types. The description is spoken by Socrates: “Take a line which has been cut into two unequal parts, and divide each of them again in the same proportion.” Each of these sections represents one of Plato’s four ascending levels of knowledge. If the line is drawn vertically, it looks something like this:
The top two sections of the line represent the intelligible world; the bottom two the visible world. The lowest of the four is Eikasia, or conjecture, which Brumbaugh equates with guessing, as in “I guess so”; it is opinion based on hearsay or stories. The next highest level is Pistis, or belief “based on first-hand experience,” Brumbaugh writes. Above that is Dianoia, or understanding, using reason to reach conclusions. Mathematics falls into this category.
The highest level, Noesis, moves beyond following rules to reach correct answers to comprehending the true essence of the eternal forms, those perfect examples that exist beyond time and space. Plato says this fourth state is attainable only “by the power of dialectic,” or rigorous philosophical discourse. The highest example of this level of knowledge is the form of the good, a concept so abstract that only philosopher-kings can fully understand it. The closest Plato comes to explaining it is by comparison to the role of the sun in the visible world. Instead of illuminating the observable world, the form of the good illuminates truth.
Got all that? I didn’t either the first time Bremer explained it, and he was a very skillful explainer. It was only much later that I realized the Divided Line was where the worlds of Plato, Pythagoras, and the Da Vinci Code truly did collide.
Bremer pulled out a large sheet of paper filled with columns of neat, tiny handwritten numbers. It looked like a page from the ledger of a particularly prosperous Victorian merchant. On closer inspection I saw it was his hand-counted tally of syllables in the Republic. Sometime during the second Eisenhower administration, Bremer had sat down and determined that the Republic had taken twelve hours to read aloud during Plato’s lifetime, and then broke the dialogue down into 240 units of three minutes each. When he examined the content of each of these sections, he found that Socrates’s explanation of the Divided Line fell between the sixty-first percentile and the sixty-third percentile of the Republic. Jay Kennedy’s computer analysis fifty years later confirmed Bremer’s work. Both men’s calculations placed Plato’s discourse on the Divided Line almost exactly at what is known as the golden section (or golden ratio), a mathematical ratio usually represented by the Greek letter phi. The ratio is approximately 1:1.618, or the equivalent (for our purposes) of 61.8 percent. For a geometer like Plato, obsessed with uncovering the eternal mathematical laws guiding the universe, the golden section must have been like a glimpse into the mind of the Divine Craftsman.
The golden section can be found by dividing a line into two parts, so that the length of the longer section divided by the length of the shorter one is equal to the entire length divided by the length of the longer section. (The ratio should be approximately 0.618:0.382.) If one takes a golden rectangle, a parallelogram whose sides are in proportion to the golden section, and subtracts the area of a square whose sides include one of the shorter sides of the rectangle, the remaining parallelogram will also be a golden rectangle. This process can be repeated forever. (See the diagram on the facing page.)
Another method of reaching the golden ratio is via the Fibonacci sequence. This is a series of numbers in which the sum of any two consecutive numbers adds up to the next number in the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on to infinity. As the sequence progresses and the numbers get larger, the result of any number divided by its predecessor edges ever closer to the irrational number 1.618. . . . (Thus: 8 ÷ 5 = 1.6; 13 ÷8 = 1.625; 21 ÷ 13 = 1.615; and so on.)
What Bremer determined is that the allegory of the Divided Line, which appears roughly 61.8 percent of the way through the Republic, separates the dialogue into two distinct sections. Everything up to that point deals with the sensible world. Everything after it deals with knowledge that exists only in the intelligible realm.
Though Plato doesn’t name the golden section explicitly in his works, he quite clearly refers to its perfect proportions in the Timaeus when introducing the elemental particles of air, fire, water, and earth. The ratio 1:1.618, like the Pythagorean harmonies, is one of those cosmic intersections where the natural world and mathematics converge. It appears frequently in nature, most famously in the spiral of a nautilus shell. Its aesthetically pleasing proportions can be found throughout the human body, in everything from the dimensions of a beautiful face to the comparative lengths of your finger bones. The Parthenon and its gigantic statue of Athena were likely designed to adhere to its laws of symmetry.
When Plato lists his five elemental polyhedrons in the Timaeus, four of them are matched to the basic elements of fire, air, earth, and water. He singles out the fifth, the twelve-sided dodecahedron, as that “which the god used for embroidering the constellations on the whole heaven.”15
The faces of a dodecahedron are pentagons; the ratio of any diagonal drawn between two of a pentagon’s five interior angles to the length of any of its five sides is the gol
den section.
Mephistopheles himself would get bored counting the various golden sections hidden within a Pythagorean pentagram.
In a Platonic dialogue, this might be the point in the conversation at which Socrates raised an important question: So what? Bremer had already provided an answer of sorts in an essay titled “Plato, Pythagoras, and Stichometry.” The mathematical patterns he’d uncovered in Plato’s works were not “simply a kind of ornamentation, a pleasurable addition to the content of the dialogue, a literary device,” he wrote. Instead, they were “an essential, perhaps the essential, part of the dialogue.”
Which brings us back to Bremer’s basement, and to Atlantis.
“I wanted to mention something to you in the Critias,” Bremer said, flipping through the pages of a translation. “Take a look at the paragraph that states ‘Now first of all we must recall the fact . . .’”
I had practically memorized the words that follow, since they are some of the most important in Atlantology: “. . . that nine thousand is the sum of years since the war occurred.”
“The number of syllables from there to the end of the Critias is nine thousand. Nobody in the world—except me, and now you—knows that.”
“There’s no chance that it’s a coincidence?”
“Oh, no, of course not! Plato is far too careful a writer. You must understand that in this stichometric game we’re dealing with a text that was sort of established in the sixteenth century, which in turn was based on manuscripts which were written in the ninth and tenth centuries, fifteen hundred years after the death of Plato. So I don’t feel very upset if it turns out there are 9017 syllables. With all of my numbers, if it’s within 1 percent, it’s probably intentional.”
What the numbers were not, Bremer said, was a secret code that would lead me to the Temple of Poseidon. Mathematics was simply one method Plato used to convey important information to those prepared to receive it.
“In a way, the Timaeus is an invitation to those with a disposition of a certain kind to enter into the inquiry into the nature of the cosmos. What the Critias tells you is how far you can go. It ends abruptly. Zeus assembles the gods and begins to address them, and then there’s a period. That’s all there is! Don’t try to go beyond this. It’s not the dialogue that’s incomplete—it’s human knowledge!”
“So don’t look for what’s not there?”
“Yes. These are the limits of that kind of knowledge. You can’t go any further.”
In other words, I’d bumped up against my own Divided Line. Nec plus ultra. Plato even had a name for this unsatisfactory moment in a philosophical inquiry—aporia, or impasse. In the Meno Socrates applauds the purgative effect of aporia, for only once someone confronts the dead end of his ignorance can he begin to move forward.
“What would you say if someone came in here and told you they’d found Plato’s Atlantis by following the numbers from the Critias?” I asked.
“What you’re describing is what I would regard as a teaching/learning situation,” Bremer said, leaning forward and folding his hands. “On the whole there’s no point in saying, ‘You’re loony.’ But I think they would be profoundly mistaken.”
Bremer’s landscaper came to mow the lawn and the buzzing drowned out our conversation. We walked upstairs into the humid late afternoon air and I prepared to drive back home. Bremer walked me to my car, gave directions back to the highway, and, Socratic that he was, left me with a question of his own.
“What fascinates me about what you’re doing is why are all these folks devoting untold energy trying to figure out whether this was a historic place or not? What do they think they’re up to? What is it that makes them search for this thing, when rationally they must know there are a very large number of people who consider that they are wasting their time?”
Or are they?
CHAPTER TWENTY-NINE
True or False
Atlantis
Tony O’Connell was on the line from Ireland, describing the new house he and Paul had just moved into right outside their village in County Leitrim. “I suppose you could say we’re now stumbling distance from both pubs,” he said. “Well situated for your next visit.”
My Atlantis odyssey finally having ended, I’d called the wise man of Atlantology seeking some guidance on my preliminary findings. Tony was a bit like Socrates—and unlike virtually every other person I’d spoken with about Atlantis over the past two years—in that he was primarily interested in the things he didn’t know about Plato’s lost city. He’d written more than a thousand detailed entries for the Atlantipedia, and while he was as certain as ever that Plato’s story was “generally reliable,” he still hadn’t settled on a single location hypothesis.
We briefly debated the various possibilities for the Pillars of Heracles, but after a few minutes Tony stopped short and said, “Mark, you should be perfectly happy to come up with your own conclusions whether they agree with mine or not. There’s no one dealing with this subject who isn’t speculating.”
Speculation doesn’t have to devolve into theories based on alien visitors or secret rooms beneath the Sphinx’s paw; after all, the Timaeus is largely a speculative work. Tony suggested a jurisprudential image to illustrate the burden of proof Atlantologists needed to meet. Because the evidence for Atlantis is “at best circumstantial,” he said, its existence cannot be proved beyond a reasonable doubt, as it would need to be in a criminal case. In a civil court, however, the legal standard is whether something is more likely to be true than not true. By that measure it should be possible, Tony said, to build a convincing case “for the time and location of Atlantis” from largely circumstantial evidence.
Following Tony’s lead, I decided that to reach a verdict about Atlantis, five general points needed to be addressed:
Plato’s numbers, especially the nine thousand years
The island’s physical characteristics, including its concentric rings, mountains, large plain, and canals
The conflict between Atlantis and Athens
The Pillars of Heracles and the impassable shoals of mud beyond
The cataclysmic event that destroyed Atlantis
Plato’s numbers were exhibit A, since they are essential to almost every location hypothesis. If some future Heinrich Schliemann ever uncovers a three-ringed coastal city with a central island five stades across and a temple of Poseidon that matches Plato’s precise 2:1 dimensions, the Atlantis case will be closed.
I think that’s extremely unlikely to happen.
The Timaeus and Critias are dripping with Pythagorean influence. The dialogues begin with Socrates making a reference to the sacred tetractys: 1, 2, 3, 4. The speaker Timaeus is a Pythagorean who explains to his friends how all matter can be broken down into minuscule right triangles. In the cosmology he lays out, the heavenly bodies move according to the same mathematical harmonies that Pythagoras supposedly discovered in a blacksmith’s shop. About the only thing that could make the Timaeus and Critias more Pythagorean would be for Poseidon to carve the words DON’T EAT BEANS into the Atlantean plain with his trident.
The philosophical meaning of Plato’s Atlantis numbers has been lost to nongeometers, possibly forever, unless a copy of his Critias lecture notes turns up beneath a swing set in the Athens park where the Academy once stood. What hasn’t been lost is Aristotle’s reminder that to the Pythagoreans, numbers were not just amounts but things. Robert Brumbaugh’s conjecture that Plato’s opposition of evens and odds symbolized Atlantis’s degeneration, like a black hat on a villain, makes much more sense to me than the idea that such specific numbers had been passed down through the millennia like a land surveyor’s report.
The three-ringed city was almost certainly intended as a geometric metaphor. Plato loved circles, which exemplified the otherworldly perfection of his eternal forms. In the Timaeus, the world is a sphere because that is th
e ideal shape. Both the individual human soul and the soul of the living cosmos—the circles that the Divine Craftsman scissors out of the World-Soul like a chain of paper rings—are said to move in a circular motion. That doesn’t necessarily mean Plato didn’t have a real-world model in mind. Santorini’s rough bull’s-eye shape could have inspired the concentric circles of Atlantis, and Michael Hübner’s giant stone donut in Morocco was certainly intriguing. Better matches could have been found in the ancient Mediterranean, though. Carthage was famous for its annular naval harbor, constructed around a circular center island, with a single entrance like that of Atlantis. Plato would have been quite familiar with Carthage, since his tyrannical host in Syracuse, Dionysius, was at war with the Carthaginians. But if you want my honest opinion—and if you’ve read this far, you presumably do—I think Plato just had a thing for circles.
Viewing Plato’s numbers and geometric shapes as symbols instead of raw data allowed me to avoid the explanatory gymnastics required to squeeze the key figure of nine thousand years into any hypothesis. With a stroke, some of Atlantology’s biggest problems vanished: the lack of evidence for a Paleolithic Athens; the reliance on Egyptian lunar calendars;16 the parsing of Plato’s words to mean seasons instead of years. The complicated interpretations required to reconcile nine thousand years with dates like 2200 BC in Malta and 1500 BC for the Thera blast simply disappeared. Poof. When did Atlantis sink? We don’t know. Or rather: We don’t know yet.