Visually mapping Cusanus and Bessarion…

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As I mentioned here recently, I’ve been trying to grasp the structure of the humanist community of astronomers / mathematicians orbiting around Nicholas of Cusa and Cardinal Bessarion in Rome… but so far haven’t found any definitively useful books on the subject. Thony Christie has a nice article here, and there’s a book on 15th century Viennese astronomy here (for Regiomontanus and Peurbach), but sadly not a great deal else that rises far above Wikipediaesque factoids.

All the same, here’s the connection map I’ve put together: it’s far from complete, but it’s probably a decent enough starting point. Doubtless you’ll note plenty of familiar names!

Map of the community around Nicholas of Cusa and Bessarion

Also, I found a nice blog post containing pictures of Bessarion: mirroring his life-long interest in astronomy, the Greek epitaph on his tomb (below) says “I, Bessarion, raised this tomb to hide my bones; my soul will seek the stars whence once it came.” Not particularly religious for a Cardinal, perhaps, but I like it all the same!

The greek epitaph on Cardinal Bessarion's tomb

David A. King’s Regiomontanus acrostic theory…

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Though Professor David A. King is best known, academically speaking, for his detailed study of astrolabes, I first ran across him via his epic (2001) tome “The Ciphers of the Monks” (summarised here): there, what happened was that one particular 14th century astrolabe from Picardy had some markings in an unusual number system first devised by Cistercian monks, and – like the proper devotee of historical arcana he assuredly is – King began collecting all the occurrences of that system he could find, which culminated in his book on the subject. There’s also a nice paper here on how the same number system was also used for tallying / gauging wines in the late Middle Ages.

All of which provides a suitable introduction to King’s most recent excursus beyond mainstream astrolabe history, because for this too the herald for his ‘call to adventure’ was an astrolabe with unusual markings on it (this time, an angel and an apparently acrostic dedication). But this second astrolabe also had a remarkable provenance – that Renaissance king of astronomers Regiomontanus had constructed it and presented it to his patron Cardinal Bessarion. Once again, David King set out to try to uncover the meaning of an astrolabe’s curious engravings – but his research journey carried him onwards to the artist Piero Della Francesca, right at the heart of the Renaissance project…

The angel part of the astrolabe engraving looks straightforward enough (note the parallel hatching on the top wing-edges, the trendy crosshatching in the background, the mid-Quattrocento “^” for “7” in the 60/70/80/90 sequences, and the early-Quattrocento ‘4’ shape at the bottom):-

Similarly, the dedication looks straightforward enough too at first glance (note the looped early-Quattrocento ‘4’ in 1462):-

 SVB DIVI BESSARIONIS DE
CARDINE DICTI PRAESI
DIO ROMAE SVRGO IO
ANNIS OPVS :~ 1462

King translates this as: “Under the protection of the divine Bessarion, said to come from the cardo, I arise as the work of Ioannes in Rome in 1462“. But the clever part, King believes, is that Regiomontanus’s slightly clunky Latin manages to cleverly conceal a number of additional messages to his new patron Bessarion:-

Here eight hidden vertical axes of an acrostic contain all sorts of hidden messages that would have especially pleased the Cardinal once he had figured them out: references to himself and his rank, to Regiomontanus, and to an old Byzantine astrolabe that he had shown to the young German. The angel is Bessarion, but not the Cardinal. There are several plays on the Latin word cardo, meaning “hinge, axis or pole”. In brief, two astrolabes come together in one, two poems, two languages, two Bessarions, two men who used the name Ioannes, two places, Rome and Constantinople, all come together in one.

Ummm… eh? It’s just a dedication, isn’t it? When I first looked at it, all I really noticed was the word “DEI” vertically hidden at the end: but Professor King thinks that the wobbly spacing and stretched letters indicates that there is much, much more going on here. However, it has to be said that after it was auctioned by Christies in 1989, precisely the same evidence was used to argue that it was “suspicious”, and that it even might be a “19th-century fake”: so be aware that we are now entering the kind of is-it-a-cipher/is-it-a-hoax territory that should be eerily familiar to Cipher Mysteries readers…

The other thing you need to know is that Bessarion also owned a spectacular Byzantine astrolabe dated 1062, which also had “:~” on one of its engraved bands of text: King agrees with Berthold Holzschuh that the two astrolabes are connected in some way, and that perhaps part of the reason for Regiomontanus’ presenting it to Bessarion in 1462 was to mark the 400th anniversary of the making of the magnificent Byzantine astrolabe.

So, let’s take a deep breath and dive deep into King and Holzschuh’s acrostic world, to see if his theories hold water (or if they sink like a stone)…

Firstly, Holzschuh suspects that the primary secret message held here emerges if you reorder the words to mirror the start of the Greek text on the Byzantine astrolabe:.

SVB BESSARIONIS PRAESIDIO
SVRGO
IOANNIS OPVS DICTI DE CARDINE DIVI
ROMAE 1462

…which (because the Latin word ‘cardo’ means hinge or axis) he translates as “Under the protection of Bessarion, I arise in Rome in 1462 as a work of Ioannes explaining the rotation of the universe“. He also notes that the angel’s fingers “point to 4 and 8 hours on the horizontal scale of the markings below it, suggesting we should look for eight items in the four lines of the epigram“. The eight vertically hidden messages he highlights look like this:-

(Note that this is adapted from p.12 of David King’s Regiomontanus theory webpage, but that the DEI ESIO (‘ESIO TROT‘, surely?) acrostic ringed in red was mislabelled VIII).

Now, I have to say that I really am particularly impressed with the acrostics marked I (SVB CD ANNIS) and VIII (1062 / 1462), in that these not only tie in neatly with the “:~” on Bessarion’s 400-yearold Byzantine astrolabe, but also nicely explain (a) the curious starting position of “SVB” on the top line and (b) why IOANNIS is split over two lines. However, sorry to be a dreadful cipher party pooper but I don’t actually buy into a single one of the other acrostics suggested, nor into any of the hundreds of tenuously complex patron saint / birthday / symbolism / IO / 1407 / golden section etc arguments that are used to support them. No, not even the angels’ fingers.

To my eyes, the astrolabe’s acrostic angle does tell a hidden story: that Regiomontanus presented this to his patron in 1462 with a silent nod to the 400th anniversary of Bessarion’s Byzantine astrolabe. Perhaps there’s even “DEI” hidden on the right (though this seems way too prosaic and straightforward for the needs of King’s complicated exegesis). However, I honestly don’t see any evidence of anything else hidden in the message that is beyond pure chance, i.e. that you could not also extract from just about any other Latin inscription of comparable size and date.

Hence, I just can’t make the giant leap over to the second plank of King’s narrative, which connects Bessarion’s patronage of Regiomontanus (as expressed in the 1462 astrolabe) to Piero Della Francesca’s epic painting “The Flagellation of Christ“, which Martin Kemp (1997) rightly described as “sumptuously planned”. This single work has caused more art history ink to be spilled in vain than perhaps any other painting (yes, even more than the Mona Lisa): King tabulates (pp.23-24) over forty subtly nuanced theories about who the various characters represent, before adding Berthold Holzschuh’s (2005) theory to the list – that the bearded man at the back being whipped is Cardinal Bessarion, and that the man in red at the left of the foreground group is Regiomontanus.

Nope, sorry – this theory doesn’t work for me either. There’s maths and geometry aplenty in Piero’s work, sure, but I completely fail to see how it links to Bessarion and Regiomontanus on any level. Perhaps my idea of what constitutes evidence is just too limited, or maybe I’m just too stupid to grasp how these two objects do really form part of a vast Renaissance patronage fugue. 🙁

If, however, you’re still intrigued by all this, there’s a nice set of slides on King and Holzschuh’s theory here: and a 2007 book by David King on the subject, with the snappy title Astrolabes and Angels, Epigrams and Enigmas – From Regiomontanus’ Acrostic for Cardinal Bessarion to Piero della Francesca’s “Flagellation of Christ” (Stuttgart: Franz Steiner) that you can buy on Amazon, though be warned that even a second-hand copy is a hefty £120 (yes, really!) Enjoy!

The Chaocipher got Slashdotted!

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The Internet is a strange thing, a virtual photographer’s jacket crammed with countless pockets of enthusiasts. For example, you beautiful cipher mysteries fans circulate within one bijou (but nicely-appointed) pocket, while the massed legions of Slashdot fans have a Tardis-style hyperzoom lens pocket all of their own. But… what would happen if these two worlds collided?

A chance to find out came in December 2009, when Edith Sherwood’s The-Voynich-Manuscript-was-made-by-Leonardo-da-Vinci-so-it-was website got picked up by Slashdot. From the 4900 overspill visits Cipher Mysteries got at the time, I estimated that she must have had “(say) 30000 or more” visits. This was probably about right, because in the few days since the same thing happened to Cipher Mysteries last weekend, its visit counter has lurched up by 38,000+. The onslaught started on Saturday night, when at its peak the Cipher Mysteries server was getting a new visitor roughly every second. By late Sunday, however, the story had finally slid off the bottom of the Slashdot front page (which only ever lists the ten most recent news items), at which point the tsunami turned into merely a large river. 🙂

According to the server logs, my Slashdotted Chaocipher page was read in 132 countries (USA 52%, Canada 8%, UK 7.5%, Australia 5.4%, etc), while US Slashdotters were mainly from California, Texas, New York, Washington, followed by another long tail. And OK, I know it’s a biased sample, but it was nice to see Internet Explorer in less than 8% of the browsers. One long-standing stereotype did fall by the wayside, though: there was a relative absence of trolls leaving snarky comments. Might Slashdot be *gasp* growing up? 😉

Actually, the nicest thing about the whole episode for me was that Moshe Rubin’s brother in Florida was unbelievably impressed when he saw Moshe’s name pop up on Slashdot. I know it’s only a small thing, but I’m really pleased for the guy, he deserves credit for his hard work and persistence bringing the Chaocipher out into the light.

* * * * * * *

Some quick follow-up thoughts on the Chaocipher…

It strikes me that Byrne’s neologism “Chaocipher” was remarkably prescient for 1918, because the whole idea of “chaos theory” – as per Wikipedia, “the behavior of dynamical systems that are highly sensitive to initial conditions“, AKA ‘the butterfly effect’ – had not long before that been started by Henri Poincaré. The French mathematician had shown that the classical three-body problem sometimes yielded tricksy outcomes that never converged (i.e. to a collision) nor diverged (i.e. to increasing distance from each other), but where the three bodies were somehow trapped in a dynamically constrained yet utterly mad-looking (OK, he actually said ‘nonperiodic’) manner. Yet after this promising beginning in the 1880s, the ‘chaos’ concept’s journey onwards was a particularly arduous (and non-obvious) one: even though people noticed the signatures of this odd behaviour in many different contexts, they had no comfortable vocabulary to describe it until well after Benoit Mandelbrot and Edward Lorenz in the 1960s.

And so I find it neatly uncanny that the Chaocipher appropriates the “chaos” word 50 years earlier than it should, while at the same time exactly demonstrating the properties that contemporary mathematicians now ascribe to it (i.e. “deterministic chaos”). As the cipher’s twizzling steps subtly mangle the order of the letters on the two rotors, both the error propagation and the cipher system complexity sharply ramp up over time, in a (quite literally) chaotic way: to my eyes, Byrne’s Chaocipher is no less artful and pleasing than any Mandelbrot set I’ve ever seen. However, because its mechanism was not disclosed until this year (2010), it is perhaps best thought of part of the secret history of applied chaos: by way of comparison, the earliest paper on “chaotic cryptography” I’ve found was Baptista’s “Cryptography with chaos” in Physics Letters A (1998) [mentioned online here].

So, it might be that as the full story behind the Chaocipher emerges from Byrne’s papers, we’ll discover that he cleverly applied Poincaré’s and Hadamard’s ideas to cryptography: but – between you and me –  I somehow doubt that this is what really happened. In my mind, there’s something both ham-fistedly mathematical and deviously mechanical about the Chaocipher, that makes its mongrelly combination of Alberti’s cipher wheel and movable circular type something that could (in principle, at least) have been devised any time since about 1465. All the same, I think that the single aspect of the Chaocipher that most makes it resemble an out-of-place artifact is that it is a pure algorithm made solid – a bit like a programming hack devised by someone who had never seen a computer. Perhaps programming is closer to carpentry than we think!

Without doubt, the Chaocipher lies just outside the rigid mathematical confines of the cipher development path laid down by the sequence of crytographers since Alberti: and so for me, the most inspiring lesson to be learned from it is that genius need take only a single step sideways to become utterly unrecognizable to the mainstream. Thinking again about the Voynich Manuscript’s cipher, might that too merely stand a single conceptual step beyond our tightly-blinkered mental range? Furthermore, might that also ultimately turn out to be part of the same secret history of applied chaos? It’s certainly an interesting thought…

The Chaocipher revealed!

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“The Chaocipher” is a devious cipher system invented in 1918 by John F. Byrne: allegedly, it was so complex that nobody could crack his challenge ciphertexts (even with the plaintext to refer to!), yet was so simple that its mechanism was claimed to comprise only two rotating disks small enough to fit in a cigar box, and could be operated by a ten-year-old (admittedly a diligent, determined and well-practised one) to encipher and decipher texts.

Hence, the Chaocipher’s long-standing mystery revolved around three questions:

  1. Was the Chaocipher for real? (i.e. could something so simple really produce such tricksy ciphertext)?
  2. Was it more secure than, say, the Enigma machine?
  3. More to the point, is the Chaocipher actually an unbreakable cipher?

As of a few years ago, only three people knew the Chaocipher’s secrets – John Byrne Jr (the inventor’s son), and two Cryptologia editors (who saw it in 1990 but were sworn to silence). Yet as Chaucer noted, time and tide wait for no man (not even Cryptologia editors) – so there was a very real (and growing) possibility that the secrets of the Chaocipher might somehow get lost forever.

Hence last August, Moshe Rubin – who CM readers may well recall as the zesty Israeli software / crypto guy who not long before had set up the Chaocipher Clearing House website – decided to try to contact John Jr before it was too late, and so cold-called his way through the list of Byrnes living in Vermont. Before long, Moshe found himself in contact with Patricia Byrne (John Jr’s wife) from whom he discovered the sad news that her husband had passed away a year or two previously.

However, because Pat Byrne was already looking for a buyer for her husband’s cryptological material, Moshe put her in contact (via David Kahn) with David D’Auria, the chairman of the National Cryptological Museum’s Acquisitions Commitee. Somewhat surprisingly, after a couple of months Pat Byrne very generously decided to donate the whole set to the NCM, a terrific gesture which I (for one) highly appreciate (and I hope that you do too!)

And so it came to be that Moshe Rubin found himself allowed what he describes as “preview access to some of the material“.  Although he found that the precise setup John Byrne Sr had employed was not immediately obvious from the material to hand, Moshe burnt a load of midnight oil (is elbow grease more or less inflammable?) before finally managing to reconstruct the original algorithm in all its subtly obfuscatory glory.

Just as Byrne had described, his Chaocipher used two rotors (with the plaintext alphabet on the right rotor and the ciphertext alphabet on the left rotor) BUT with both alphabets altered slightly (let’s call this process ‘twizzling’, for want of a better word) after processing each letter. I’ve hacked together a 30-second Chaocipher animation on YouTube to try to demonstrate Byrne’s twizzlification…

Rather than go through the fine details here, I’m happy to refer you to Moshe’s detailed (and very readable) description of the process here: the only significant difference between my video and his text is that because the rotors mesh (and hence physically rotate in opposite directions to each other), the numbering sequence on each rotor is reversed relative to the other – i.e. even though #1 is at the top of each rotor, #2 and #3 proceed clockwise on the right (plaintext) rotor but anticlockwise on the left (ciphertext) rotor. Whereas in his text, both numbering systems run in parallel to each other (which might confuse you, it certainly confused me a little).

Of course, the obvious practical weakness of the Chaocipher is that any errors in enciphering, transmission, and deciphering get near-irreversibly propagated through the rest of the message: which probably makes the whole system too fragile to use in wartime, however cryptographically secure it may be (and, answering the second question above, I suspect that it may well prove to be more complex than Enigma, for it really is quite a fiendish system).

But is it (practically) unbreakable? Well, the obvious answer would be that if it has now been released into the wild, you’d have thought someone in a three-letter-agency (or GCHQ, naturally) would have worked out a clever way in. However, I’m not 100% sure that has happened yet… so, interesting times.

All credit to Moshe Rubin, then, for his persistence and hard work bringing this cipher mystery into the light: he has a Cryptologia paper coming up, and plenty more work to do over coming months (or years?) fleshing out the behind-the-scenes story from the stack of Byrne’s papers now in the NCM. It’s a fascinating slice of cipher history, and I wish him the very best of luck with the inevitable book and selling the movie rights! 😉

* * * * * * *

Update: I’ve added a follow-on Chaocipher post here, discussing the intriguing parallels between the Chaocipher and chaos theory…

Astrolabes, nocturnals and Voynich Manuscript page f57v…

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For a decade, I’ve wondered whether any of the Voynich Manuscript’s circular drawings depict astronomical instruments – for before satnav there was celnav (“celestial navigation”). Here’s a brief guide to three key instrument types from the VMs’ timeframe, and my current thoughts on the enigmatic circular diagram on f57v…

* * * * * * *

A key navigational problem of the 15th century was determining your latitude. Though many different instruments (such as the quadrant, the cross staff, and the back staff) came to be used to do this around this time, I’m restricting my observations here to the three purely circular ones – the astrolabe, the mariner’s astrolabe, and the nocturnal.

(1) Though astrolabes were originally used for determining the positions of planets and stars, people realised that they could also be used for telling the time (if you knew your latitude), or for working out your latitude (if you knew what time of day it was). Astrolabes were constructed from a complex (but well-known and well-documented) set of multilayered rotating components:-

  • A backplate (the mater) whose edge (the limb) is marked round with 24 hours or 360 degrees
  • A large circular central recess (the matrix, or womb) in the mater, into which you insert…
  • A disk (the tympan) containing a stereographically projected map of the sky for a particular latitude
  • On top of the tympan goes a rotating spidery net-like thing (the rete) containing easily recognizable stars;
  • On top of the rete goes a long rotating rule (the rule)
  • On the back goes a second rotating rule-like thing with two sighting holes / marks (the alidade)

If you haven’t seen an astrolabe dissected, there’s a nice annotated diagram on the Whipple Museum website.

My understanding is that most medieval European astrolabes were inaccurate because they were made of wood, though this improved when they started to be made of metal (an innovation which I understand mainly began in the 15th century). Yet even with well made astrolabes to hand, using them can be a bit tricky, particularly when you are at sea: and they’re not very convenient to use at night either.

(2) So, step forward the mariner’s astrolabe (or sea astrolabe or ring). Though this was little more than a cut-down version of the astrolabe, its key design feature was that it was built to be particularly heavy (and so was much more stable at sea). In contrast to the thousands of astrolabes out there, only 21 mariner’s astrolabes are known: the earliest description of one is from 1551, while historians suspect they came into use in the late 15th century.

Really, this was little more than a superheavy astrolabe limb hanging from a ring and with an alidade on the front: but it did the job, so all credit to its inventor… whoever that may be. The Wikipedia mariner’s astrolabe page notes that it might possibly have been Martin Behaim (1459-1507), but because it seems he was adept at relabeling other people’s discoveries and inventions as his own, probably the most we can pragmatically say is that the idea for the mariner’s astrolabe was ‘in the air’ in the mid-to-late 15th century.

(3) Solving the astrolabe’s other major shortcoming, the nocturnal (or nocturlabe, nocturlabium, or horologium noctis) was specifically designed to be used at night. A 2003 paper notes that the first evidence of nocturlabes was not a textual mention in 1524 (as was long thought), but rather a series of actual devices made by Falcono of Bergamo and dating from 1504 to 1507 (who also made astrolabes, such as this one from the British Museum). For a nice picture, the National Maritime Museum has a 17th century nocturnal here (D9091).

As far as construction goes, a nocturnal consisted of: a rotating outer ring marked both with the months of the year and with the 24-hour time; a hole in the middle of the central pivot that you could see through; and a second rotating ring with one, two, or three pointers. Once you had rotated the outer ring to closely match that day’s date, you would hold your nocturnal at arm’s length, line Polaris up through the central hole, and then align the second rotating ring so that its pointers pointed at some well-known stars (normally Shedar [α Cassiopeia], Dubhe [α Ursa Major], and Kochab [ß Ursa Minor]): there’s some nice discussion here on why these were chosen.) Once you had done all that, you would find (as if by high-tech magic) that the major pointer on the second ring would be pointing to the current time of day marked on the first ring. (Well… pretty much, anyway.)

Here’s a simplified look at the night sky, highlighting the four key stars referred to on a typical nocturnal:-

Incidentally, an open history of science question is whether Columbus had a nocturnal on his well-equipped voyages of discovery. This well-informed page seems to imply that he did, and that it was used to determine midnight – the ship’s boy would then turn over an “ampoleta” (a little sand-glass that would take half-an-hour to empty) to start counting out the daily cycle of shifts. Unfortunately, it turns out that Columbus didn’t properly understand how to use his various astronomical instruments, and that he faked a number of his latitude records. Oh well!

To summarize: though the astrolabe had been used and developed since antiquity, there was little about it that was secret circa 1450. However, this was the moment in history when people were starting to apply their formidably Burckhardtian Renaissance ingenuity to get around the limitations of the traditional astrolabe, by adapting the basic design for use at sea and at night. Yet for both the mariner’s astrolabe and the nocturnal, the documentary evidence is silent on who made them first.

* * * * * * *

What, then, of the Voynich Manuscript?

I have been trying to get under the skin of the ringed diagram on f57v for many years: even by the VMs’ consistently high level of (well) anomalousness, this page has numerous anomalies on display that seem to promise a way in for the determined Voynich researcher:-

  • Its drawings most closely matches the circular astronomical drawings in Q9 (‘Quire #9‘), yet its bifolio is bound in the middle of the herbal Q8
  • It has a curious piece of marginalia at the bottom right
  • There’s a spare ‘overflow’ word at the top left [marked green below]
  • The second ring comprises essentially the same 17-character sequence repeated four times
  • Each 17-character sequence contains an over-ornate anomalous “gallows” character [marked red below]
  • The 17-character sequence contains a number of low-instance-count letter-shapes
  • The fourth ring contains another long sequence of single characters [marked blue below]
  • It has four strange ‘personifications’ drawn around its centre (seasons? winds? directional spirits?)
  • It is far from clear what the four personifications are depicting, let alone representing
  • Finally, it has a ‘sol’-like dotted sun at the centre

I therefore think that any proper account of f57v should therefore not only offer a high-level explanation of its intent and content, but also a low-level explanation of these anomalous features. The problem is that any reasoning chain to cover this much ground will almost inevitably require a mix of codicology, palaeography, history, astronomy, and historical cryptography… so bear with me while I build this up one step at a time.

First up is codicology: Glen Claston and I agree that f57v was probably the very first page of the astronomical section Q9 – by this, we mean that the two bifolios currently forming Q8 have ended up bound upside-down. So, even though the current folio order is f57-f58-(missing pages)-f65-f66, the original folio order ran f65-f66-(missing)-f57-f58. The page immediately preceding f57v (i.e. f57r) has a herbal picture on it, which is why Glen and I are pretty sure that f57v formed the first page of the astronomical section: while both sides of f58 have starred paragraphs (and no herbal drawings), which also makes it seem misplaced in the herbal section.

A second clue that this is the case is the marginalia mark at the bottom: I think this is a scrawly “ij” with a bar above it (i.e. secundum), indicating the start of Book II (i.e. where Book I would have been the herbal) – this probably isn’t a quire mark because it doesn’t appear on the end folio of a quire. And a third clue is that the page we believe originally facing f57v (i.e. f58r) has an inserted blank block at the start of the first paragraph, which I suspect is a lacuna [highlighted blue below] deliberately left empty to remind the encipherer that the unenciphered version of this page began with an ornamented capital.

As for the odd word at the top left, the odds are that this is no more than an overflow from the outermost text ring: a similar overflow word appears in one of the necromantic magic circles famously described by Richard Kieckhefer as I described in “The Curse” (though of course this doesn’t prove that this page depicts a magic circle).

I think codicology can also help us to understand the mysterious 17-glyph repeating sequence, a pattern that has inspired many a high-concept numerological riff over the years: for if you look carefully at the four over-ornate gallows, you might notice something a bit unexpected…

Even though I’d prefer to be making this judgment on the basis of better scans (which seem unlikely to be arriving any time soon, unfortunately), I’m pretty sure that what we’re seeing here is a pair of characters which have been joined together to resemble a non-existent gallows. I’d even go so far as to say that I think that the decision to make this change was probably made while the author was still writing the page: from which I infer that 18 x 4 would have been too obvious, but 17 x 4 was obscure.

If you accept that this is right, then this changes the number patterns completely, because whereas 4 x 17 = 68 doesn’t really have much numerical (as opposed to numerological) significance, 4 x 18 = 72 does – for you see, 72 x 5° = 360°. And if we are looking at some kind of 360° division of the circle, then all of a sudden this page becomes a strong candidate for being some kind of enciphered or steganographically concealed astronomical instrument, because division into 360° has been a conceptual cornerstone of Western astronomical computing for millennia.

For several years, I therefore wondered if f57v might be depicting an astrolabe: but I have to say that the comparison never really gained any traction, however hard I tried. However… the question now comes round as to whether f57v’s circular drawing might instead depict a mariner’s astrolabe or a nocturnal.

That this might be a mariner’s astrolabe is perfectly plausible. The ‘overflow word’ might denote a ring, the second 360° ring could be the scale round the edge, and the four people in the middle could simply be decorative “fillers” for the four holes normally placed in the middle.

Comparing f57v with a nocturnal, however, is particularly interesting. The obvious thing to hide in the central design would be depictions or denotations of the constellations and the sighting stars so crucial to the operations. Given that there are plenty of different strength lines and curious shapes in the four characters to be found there, let’s take a closer look…

Now, the four elements we’d expect to see in a description of a nocturnal are Cassiopeia, Ursa Major, Ursa, Minor and Polaris: and I suspect that this is what we have here. Look again at the woman’s face on the left, and I wonder whether her name has been quite literally written across her face:-

As for the top and bottom characters here on this page, they have long puzzled Voynich researchers – why are they so wildly hairy and apparently facing away? What kind of a person is being shown here? Perhaps the answer is simply that these represent not people but bears, specifically the Great Bear (Ursa Major) at the top and the Smaller Bear (Ursa Minor) at the bottom.

The final character of the four would represent Polaris (short for stella polaris), which in the 16th Century (?) came to be called ‘Cynosura’ (the Greek mountain nymph who nursed Zeus in Crete). I have to say that I don’t really know what is going on here – perhaps other people better versed in astronomical history or mythology might be able to tell me why this person should be carrying a ring or an egg (?), and what the character’s curious strong lines (nose and top of upper arm) might be denoting.

Yet perhaps the biggest clincher of all, though, is the ‘sol’-like shape right at the centre of f57v. We might be able to discount the possibility that this represents the astrologers’ glyph for the sun, because this only came into use around 1480 (as I recall). For in the context of a drawing of a circular astronomical instrument, is this not – almost unmistakeably – a depiction of Polaris (the dot) as viewed through a hole in the pivot (the circle)?

As always, the evidence is far from complete so you’ll have to make up your own mind on this. But it’s an interesting chain of reasoning, hmmm?

Spookily, the kind of analogue computing embedded in nocturnals has a thoroughly modern equivalent. Polaris does not sit precisely on the Earth’s pole but rather rotates around it very slightly, and so requires a correction in order to be used as a reference for true North (on a ship, say). Hence a spreadsheet can be constructed to make this fine adjustment – essentially, this is a nocturnal simplified and adapted to yield the north correction required. Some good ideas can remain useful for hundreds of years!

Some anonymous Voynich quotes for you…

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Today’s first Voynich quote was overhead yesterday by Bill Tozier in an Ann Arbor restaurant (I presume?):-

I’m gonna find some fascist architecture!

Hmmm… might this have been that rarest of things, a Cipher Mysteries reader caught in the wild? Better still, might it have been a CM reader happy to ‘fess up? The comments section is ready and waiting for you here. 🙂

Of course, there has already been a theory linking the Voynich Manuscript with Michigan (for the simple reason that there is a theory linking everything with the VMs, if you’re bothered to look long enough). Specifically, Jan Hurych emailed me back in May about the f116v michiton oladabas page, saying:-

I got it – “mich::” stands for Michigan, apparently where the first sunflower came from 🙂

Today’s last Voynich quote (again from an unknown author) popped up on Google just over a month ago, courtesy of Wisam Mohammed:

Discoveries may excite our blood but mysteries sustain our soul. When we’re strong and arrogant, mysteries remind us how little we know of God’s world. And when we are weak and desperate, they only encourage us to believe that anything is possible.

So… who wrote that, then? 🙂

Circa 2010, what are the frontiers of Voynich knowledge?

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I recently blogged here about the difference between skepticism (which has at its heart both a guarded optimism and a realistic take on the practical difficulties involved in gaining knowledge) and cynicism (which by way of contrast is a denialist position, that says it is safer to believe nothing rather than get hurt by believing something that will turn out to be incorrect): but what I didn’t really go on to say was that I think there’s currently rather more cynicism at play in the Voynich research world than is properly healthy – and that perhaps the Wikipedia article simply reflects this critical imbalance.

So here’s my small wish for the day: that Voynich experts should try to use their insightful brains and creative historical imaginations not to construct yet more reasons why existing theories are wrong (which is, lets face it, about as hard as machine-gunning fish in a barrel), but instead try to construct questions they would really like to see answered. By doing this, we can start to map out the edges of our collective knowledge, and get some kind of frontier research mentality going again – perhaps it is simply this which is currently most conspicuous by its absence of late.

In this spirit but putting the codicological and palaeographical frontiers to one side (because the Beinecke doesn’t seem to be at all interested, and I suspect it will start to become clear over the next few months why this is so), here’s my proposal for an entirely new research front to open right up: Rome 1465-1467.

* * * * * *

The central cryptographic paradox of the Voynich Manuscript is that it manages to combine the simplicity of 14th century monoalphabetic ciphers (language-like and with a restricted alphabet size) with the mathematical inscrutability of 16th century polyalphabetic ciphers, yet has a (claimed) radiocarbon dating that sits between the two. Similarly, it contain a cipher letter pair (‘4o’) which was in use around Milan between 1440 and 1460, yet its cipher system is tangibly more sophisticated than anything found in the cipher ledgers of the day.

I’m going to put the radiocarbon dating on one side for the moment, and run with 1465-1467 – this was specifically when Leon Battista Alberti started researching in Rome not only how to break ciphers, but also how to make unbreakable ciphers. In fact, this precise time and place marked the birth of polyalphabetic ciphers, and arguably of modern cryptographic (and cryptologic) practice.

So far so well documented. But there’s a crucial element missing from this – the company Alberti kept in Rome while he was doing this. One of the only things I learnt from Gavin Menzies’ dismal “1434” (which I can’t even bring myself to review) was that while Regiomontanus was in Rome between 1461 and 1465, he often met up with Alberti and Paolo Toscanelli at Nicholas of Cusa’s house, though the mystery is what he was doing between 1465 and 1467  when “he seems to have disappeared”. [p.143] Of course, Nicholas of Cusa died in 1464, and though Toscanelli was a good friend of Nicholas, he only rarely ventured out of Florence, so this is already something of a simplification.

Yet here we have a critical moment when four polymathic giants of the Renaissance did somewhat more than cross paths (and one might throw others such as Filelfo, George of Trebizond, and [dare I say it] Filarete into this same mix): one might even speculate whether combining Nicholas of Cusa’s interest in concave lenses (De Beryllo, 1441) with Regiomontanus’ astronomy and with Toscanelli’s cosmography did indeed provide the conceptual spark that was to grow into the telescope (and then the microscope) during the course of the following century (even if the raw technology to make such an object was not yet there).

Might this intellectually rich time and place in some way be the loamy bed in which the seed of the Voynich Manuscript grew to its full fruition? To my eyes, there’s something innately multidisciplinary about the VMs, that speaks of subtle collaboration – people contributing to make something more than merely the sum of its parts.

Hence the new research frontier I propose is based on a single question: what are the archival resources that historians have used to reconstruct these meetings (and this community) in Rome in 1465-1467? Perhaps if we now revisit these same resources, we might notice a fleeting mention of the VMs in conception, in construction, in motion, or in retrospect, who knows?

* * * * * *

So, what question would you like answered? What research frontier would you like opened up in 2010?

Jim Gillogly’s Beale sequence revisited…

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Some more thoughts on the curious “key” sequence in the Beale Papers…

Back in 1980, Jim Gillogly applied the Declaration of Independence codebook for the second Beale Paper (“B2”) to the first Beale Paper (“B1”), and discovered a very unlikely sequence in the resulting text: ABFDEFGHIIJKLMMNOHPP. The chance of the middle section alone (“DEFGHIIJKLMMNO”) occurring at random is about one in a million million, and what is even spookier is that the two aberrant letters in the longer sequence (“F” near the beginning, and “H” near the end) are one entry off from correct letters in the codebook (195 = “F” while 194 = “C”, and 301 = “H” while 302 = “O”).

Gillogly attributed these to encoding slips: but given that I’m wondering whether this string is perhaps a code-sequence of some sort, could it be that the encoder used a slightly different transcription of the Declaration of Independence from the one he/she used for B2? This would yield systematic single-number shifts: so let’s look again at the key-sequence and the adjacent letters in the B2 codebook:

112 T R G A I
18  P B W H C
147 T A O T A
436 L B A P U
195 C F L A T  <-- Gillogly's first apparently offset code
320 I D O T E
37  A E S T W
122 P F S T W
113 R G A I A
6   O H E I B
140 I I T R O  <-- this code might possibly be offset too?!
8   E I B N F
120 T J P F T
305 P K O G B
42  T L O N A
58  O M R R T
461 H M H H D
44  O N A O N
106 P O H T T
301 T H O T P  <-- Gillogly's second apparently offset code
13  O P T D T
408 O P U T P
680 C A U B O
93  C W C U R

Today’s observation, then, is that if the errors in the Gillogly key sequence arose from having used a slightly different codebook transcription of the Declaration of Independence and that the key string should have been ABCDEFGHIIJKLMMNOOPP (as seems to have been intended), then we have two definite (but possibly even three) places where the B1 codebook transcription may have slipped out of registration with the B2 codebook transcription: the code used for the first “I” (141) could equally well have been 140, because that also codes for “I”.

Yet because the sequence is long enough to contain codes that seem correct either side of these errors, we have the possibility of determining the bounds of those stretches in the B1 transcription where the variations (in this scenario) would have occurred. Specifically:-

122 P F S T W
 ?? -1
140 I I T R O
 ?? +1
147 T A O T A
147 T A O T A
 ?? -1
195 C F L A T
 ?? +1
 ?? +1
301 T H O T P
 ?? -1
305 P K O G B

So, if this scenario is correct, it would imply that (relative to the B2 codebook) the B1 codebook transcription dropped a character somewhere between #147 and #195, gained two somewhere between #195 and #301, and then lost another one between #301 and #305. There’s also the possibility that a character was dropped between #122 and #140 and then regained between #140 and #147… not very likely, but worth keeping in mind.

Between #147 and #195, the B1 code usage table looks like this (20 instances):-

148
150 150 150 150 – 154
160 – 162
170 – 172 – 176 176
181 – 184 – 189
191 – 193 – 194 194 194

Between #195 and #301, the B1 code usage table looks like this (64 instances):-

200 200 – 201 201 – 202 – 203 – 206 – 207 – 208 208
210 – 211 211 212 212 – 213 213 – 214 – 216 216 216 216 216 216 216 – 218 218 – 219 219 219 219
221 221 –  224 – 225 – 227
230 230 – 231 – 232 232 – 233 – 234 234 234 – 236
242 – 246 – 247
251
261 – 263 – 264
275 275
280 280 – 283 283 – 284 284 – 286
290 – 294

So, this proposed mechanism would offset up to 84 codes from B1, which may be sufficiently disruptive to have caused B1 to appear undecodable to cryptological luminaries such as Jim Gillogly. It is also entirely possible that (just as with the B2 codebook) there are other paired insertions and deletions to contend with here.

There’s an interesting observation here that many of the transcription errors in the B2 codebook fell close to 10-character (line) boundaries: if this is also the case for some of these (putative) B1 codebook transcription errors, then we should be able to reduce the number of possible variations to check.

The Beale Papers Paradox…

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It seems as though penetrating public cryptographic analysis of the three Beale Papers (B1, B2, and B3) halted abruptly in 1980 when Jim Gillogly pointed out a problem with B1. If, as he pointed out, you apply to B1 the same dictionary code used for B2 (famously derived from the Declaration of Independence), you get a ciphertext with some distinctive properties:- 

SCS?E TFA?G CDOTT UCWOT WTAAI WDBII DTT?W TTAAB BPLAA ABWCT
LTFIF LKILP EAABP WCHOT OAPPP MORAL ANHAA BBCCA CDDEA OSDSF
HNTFT ATPOC ACBCD DLBER IFEBT HIFOE HUUBT TTTTI HPAOA ASATA
ATTOM TAPOA AAROM PJDRA ??TSB COBDA AACPN RBABF DEFGH IIJKL
MMNOH PPAWT ACMOB LSOES SOAVI SPFTA OTBTF THFOA OGHWT ENALC
AASAA TTARD SLTAW GFESA UWAOL TTAHH TTASO TTEAF AASCS TAIFR
CABTO TLHHD TNHWT STEAI EOAAS TWTTS OITSS TAAOP IWCPC WSOTT
IOIES ITTDA TTPIU FSFRF ABPTC COAIT NATTO STSTF ??ATD ATWTA
TTOCW TOMPA TSOTE CATTO TBSOG CWCDR OLITI BHPWA AE?BT STAFA
EWCI? CBOWL TPOAC TEWTA FOAIT HTTTT OSHRI STEOO ECUSC ?RAIH
RLWST RASNI TPCBF AEFTB

Here you can see not only tripled letters (AAA, PPP), quadrupled letters (TTTT) and even quintupled letters (TTTTT), but also (and this is the part that ignited Gillogly’s cryptographic curiosity) the sequence ABFDEFGHIIJKLMMNOHPP. Even if you restrict your view to the DEFGH IIJKL MMNO monotonically increasing sub-sequence in the middle, the chances of that appearing at random would be (he calculates) about one in a million million. Making it even more improbable is the fact that the aberrant “F” near the start has code 195 where code 194 is “C”, and the aberrant “H” near the end has code 301 where code 302 is “O”, which makes it look a great deal as though these were simply encoding slips. And if these were intended to be C and O respectively, the unlikeliness of the sequence vastly increases again. 

Yet as far as the multiple letter groups go, we can do some simple probability calculations based on the 1321 characters Gillogly lists for the B2 codebook. From frequency analysis – T 255, A 167, O 145, H 80, I 69, S 62, F 62, P 59, W 59, C 53, B 48, R 41, D 37, E 36, L 35, M 30, U 28, G 19, N19, J 10, K 4, V 2, Y 1, X 1, Q 1, Z 0 – you can see that T, A, and P occur 19.3%, 13.5%, and 4.46% (respectively) of the time in the codebook. So, if the text letters were picked at random (as would pretty much be the case if B2’s codebook was completely the wrong codebook for B1), the chances of these patterns occurring randomly at least once in a 520-character sample would be something like this:- 

  • prob(TTTTT) = 1 – (1 – 0.193^5)^(520-(5-1)) = 12.9%
  • prob(TTTT) = 1 – (1 – 0.193^4)^(520-(4-1)) = 51.2%
  • prob(AAA) = 1 – (1 – 0.135^3)^(520-(3-1)) = 72.1%
  • prob(PPP) = 1 – (1 – 0.0446^3)^(520-(3-1)) = 4.5%

You would also expect to see a copious amount of TT and AA pairs scattered through the text, which is in fact exactly what we see (13 x TT and 10 x AA, quite apart from the TTTTT, TTTT and AAA listed above). 

And therein lies the basic Beale Papers paradox: though the distribution and clustering seem to imply that B2’s codebook was not B1’s codebook, the ‘Gillogly sequence’ seems to imply that the two are linked in some way. So, what’s it to be? Damoclean swords aside, how can we unpick this cryptologic knot? 

My observation here is that if there is also some kind of monoalphabetic substitution going on (i.e. in addition to the Declaration of Independence codebook), then it’s quite possible that the Gillogly sequence represents the keyword or keystring used to generate that substitution alphabet. This might well explain the doubled letters within the keystring (i.e. the II MM and PP): if so, we would be looking for a keystring with four doubled letters but where none of the vowels repeat. 

ABCDEFGHIIJKLMMNOOPP 

Hmmm… there can’t be many English words ending with two adjacent doubled letters: in fact, the only two I can think of are coffee and toffee (please let me know if you can think of any others!) ‘Toffee’ doesn’t sound very promising, so could it be ‘coffee’? The previous word would then need to end with “C” to make a doubled letter… not hugely promising, but perhaps it’s a start!- 

ABCDEFGHIIJKLM MNOOPP
xxxxxxxxxxxxxC COFFEE
xxxxxxxxxxxxxT TOFFEE

Alternatively, it might be a three letter word, like “TOO” or “OFF”. Had Eric Sams considered this, doubtless he would have happily constructed all kinds of valid key phrases that fit these constraints, such as:-

ABCDEF GHIIJ KLMMNO OPP
CLUNKY SPEED RABBIT TOO

OK, it’s true that the key phrase to the Beale Papers is not going to be “CLUNKY SPEED RABBIT TOO”, but maybe (just maybe) it’s a step in the right direction. 🙂 

Incidentally… the Wikipedia Beale Papers page notes that “In 1940, the famous cryptanalyst, Dr. White of Yale University, came close to solving the Beale ciphers after tracking down the suspected key hidden by Beale in St. Louis—he never spoke of his findings.” Though I did a bit of Internet sleuthing to try to work out who this Dr White was, I didn’t really get anywhere – I don’t think he was the Maurice Seal White (b.1888) who wrote the 1938 book “Secret writing : how to write and solve messages in cipher and code” (which I found listed in Lou Kruh’s bibliography and Worldcat) and who was a Columbia alumnus in 1920 (see p.212 here), but it’s hard to tell. Please let me know if you find out!