O’Donovan notes #2b: the ‘4o’ Revised and updated edition.

math numerals 40 in 1375 Majorca
Header image. Numeral ’40’ –  c.1375 AD.    Majorca.   Jewish work. Made for the court of France, attributed to Abraham Cresques. The present writer has already, in work previously published, explained in detail the points of connection between the Voynich map and Cresques’ masterly work. This was an original contribution to the study, and its source in the present author’s work, should be credited in the normal way.

Original post published November 19th., 2021. Updated and revised version – November 25th., 2021.

READERS PLEASE NOTE – this post contains original work and references to an ongoing research ‘conversation’ between three researchers. Be good enough not to pretend the findings are just an ‘idea’ but if you wish to repeat the information, attribute it correctly and give the source accurately. To do otherwise is dishonest.

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Unresolved problem – the ‘4o’ glyph.

Some very recent comments made in a conversation between Mark Knowles and Nick Pelling, together with a little cross-checking of my own on other points, together leads me now to issue a revised and corrected version of this post’s second part.  

This post considers comments made about the still-unresolved question of the ‘4o’ glyph or string.

A paper up at academia.edu dismisses the question in a single short paragraph.

Hannig is saying that it ‘seemed obvious’ that the sign ‘4o’ was Latin and represented the sound ‘qu’ [sic] but that if you regard it as deriving from Hebrew ….

A reader would not gain from this any sense that the ‘4o’ had ever been studied in depth. Though Rainer Hannig is described as a Faculty member at Philipps University, Marburg, he has provided nothing by way of footnotes or references to help his readers get a clear idea of what has been said before. The impression given is that this posited ‘Latin’ rendering is owed to no-one and is not supported by any research. ‘It seemed commonsense’ is not a reasonable argument but another of those annoying ‘believe it or else’ statements.

In fact, the ‘4o’ is mentioned in d’Imperio’s book (1978), in conversations at Reeds’ mailing list etc. (for these, see ‘Constant References’ in the Table of Contents page).

The most informative comments I’ve seen about it, in Voynich writings from the first hundred years of this manuscript’s study were those in Pelling’s book of 2006. If any reader thinks I should mention another researcher’s discussion please leave a comment.

Written before the radiocarbon dating, Pelling’s work identifies an early example of non-numerical use for a ‘4o’ glyph or string in Urbino, in a ledger dated 1440. In that example, the reading is given: “quo”.

So the Latin reading wasn’t a product of ‘commonsense’ but of historical investigation, with a specific historical document explaining why the reading ‘quo’ was offered in Italy in 1440. Pelling wrote:

“Though you might think it fortuitous to find it [the ‘4o’] even once, it actually turns up in two [cipher-] ledgers, and in at least four separate [northern Italian] ciphers. The earliest mention is in the Urbino cipher ledger (in a cipher dated 1440), as well as in the main Milanese cipher ledger in ciphers dated 1450, 1455 … and 1456.   Later on, the same shape crops up in numerous other number-based ciphers, but by then it was used simply as a number. By contrast, the four earlier ciphers are all linked, because although they contain both ‘4’ and ‘4o’ symbols, relatively few other numbers appear. (p.177).

Hannig’s failing to direct readers to the sources he consulted was unfortunate, but worse is that he cites not a single source, medieval or later, to support his theory that the ‘4o’ is an abbreviation from Hebrew. A scholarly paper provides readers with the means to check whether the writer has done his ‘homework’ and explains how an opinion was first arrived at by giving details of the author’s background reading, and documents in proof.

As Pelling’s commentary did but Hannig’s. did not.

NOTE. (added 22 Nov. 2021). My point in the paragraph above was that scholars should not lower their standards when commenting on Beinecke MS 408.  I gather some individuals have created or adopted a theory that my paragraph was designed to cast doubt on Hannig’s position in Marburg.  It would have been better, from my point of view, to have that bit of fantasy put here as a comment, and I given the chance to set the memers straight.  That meme has merely produced a theoretical interpretation about a comment made about a theoretical comment,  in one paragraph of one paper, and concerning a single Voynich string/glyph.  *sigh*.

Now, this doesn’t necessarily mean that either Pelling or Hannig is right, and the other wrong, Their opinions are not necessarily exclusive of one another.

The example from Urbino (1440 AD) may be the earliest example of non-numerical use known so far for that closed ‘4’ like form, but there may be earlier examples to be found and – these may occur in diplomatic or commercial documents involving merchants or diplomats from anywhere within a very wide geographic range.

Interested in whether the non-numerical ‘4’ form came from the example of printing, a check against Smith’s History of Mathematics and pers.comm. with a master printer said the same – not before 1440.

However, I find reason to think that use of the closed ‘4’ occurs in a commercial context during the fourteenth century. More on that below.

As regards the Jewish population of Urbino during the time of interest to us, the entry at JVL says in part:

URBINO, town in central Italy, formerly capital of an independent duchy. The earliest record of Jews dates from the beginning of the 14th century, when Daniel of Viterbo was authorized to trade and open a loan bank. His family long continued to head the community. Other loan bankers, ultimately eight in number, received authorization to operate later. .. In the 15th century the dukes of the house of Montefeltro favored Jewish scholars and were interested in Jewish scholarship; Federico II [Montefeltro] collected Hebrew manuscripts.

Urbino had unusually strong and constant connection to the Iberian peninsula, and directly linked by an old road to Florence and the Mediterranean.

As for MILAN, (this comes from JVL)

In 1387, Duke Gian Galeazzo Visconti granted privileges to the Jews in the whole of Lombardy; these were confirmed by Francesco Sforza and his successors. An important court Jew was Elia di Sabato da Fermo, who, in 1435, became the personal physician of the duke Filippo Maria Visconti. When in 1452 Pope Nicholas V approved the Jewish right of residence in the duchy, he specifically authorized the construction of a synagogue in Milan.

Which shows that about the time the content was gathered that is now in Beinecke MS 408, conditions were especially favourable (and in the case of Milan, suddenly so) for commercial and diplomatic or political links across the religious barriers.

Pelling kindly clarified details of the Milanese cipher ledgers recently, because I was unclear whether they were only a record of ciphers used, or were enciphered commercial ledgers or ledgers recording items of diplomatic correspondence. He replied that “one is an ambassadorial ledger and the other is more of a local cipher ledger. So the story is a mix of both”.

So it is quite possible that there are, or were, earlier ledgers of some kind in which non-numerical ‘4o’ forms might be found, though not necessarily only the Italian ledgers. The practice might have been adopted by Urbino from some earlier, and other, example and from that example, later, by Milan.

It is no offence against history or logic to suggest, as a possibility to be investigated, that the instance in Urbino might have derived from commercial or diplomatic writings of Jews, but while that possibility might inspire some researcher to investigate in more depth, Hannig’s omitting to cite a single item of historical evidence means that for the meantime his idea can bear no weight, whatever its real merits might be.

We can be sure that within commercial and trade schools, in some Italian states at least, ‘4’ like numerals were in use by the fourteenth century. A merchant’s handbook now in the Beinecke Library (as MS 327) is one to which I’ve been referring Voynich researchers for a decade or more, for one reason after another – and here it is again helpful:

One thing of which we can also be sure, is that the ‘4’ does not derive from works produced by European printers. On this point, Smith’s History of Mathematics is clear and because that was first published almost a century ago, I’ve checked with a master printer who confirms it. The printed ‘4’ form comes much later.

On the political front, Milan’s relations with the two trading cities of Venice and Genoa are of particular interest. As a basic outline, and one that can be checked against any solid history of the period, I’ll quote briefly from a non-Voynich wiki article:

The tide of the war [between Genoa and Venice] reversed when in 1353 the Genoese navy suffered a defeat .. [loss of a fleet then] sparked civil unrest in Genoa, …. To combat this discord, the republic was temporarily dissolved and Genoa came under the rule of the Duke of Milan. …

A peace treaty was signed between Venice and Milan in 1355 [but hostilities between Genoa and Venice continued] ..Genoa broke free from Milanese control following the conclusion of the war, [specifically in December 1435] and the republic was reestablished. – from wiki article, ‘Genoese navy

Historians today refer often to ‘entanglements’ and place less emphasis on quasi-national boundaries than was the habit in nineteenth-century histories. The connections of Urbino and f Milan, of Milan and Venice, and those to Genoa, are as mutable as diplomatic chess between nations and more enlightening than focus on supposing ‘the Latin’ necessarily opposed to ‘the Jewish’ interpretations of ‘4o’.

In this case, we see that there is no reason to deny a possibility that the ‘4o’ form’s non-numerical use, might have originated among persons literate in Hebrew and so been adopted by patrons or commercial partners as a Latin ‘cipher’.

But that’s all it is – a possibility. Pelling’s opinion is derived from specific historical example and so must be granted greater weight than the self-referential and theoretical explanation by Henning. For all that, the two opinions are not necessarily opposed, but might prove complementary.

Only research can clarify the still-unresolved problem.

Should any reader feels attracted by the idea of investigating it, I’d suggest thinking about possible reasons why the ‘4o’ should have been employed as a non-numerical string or glyph. Among the possibilities are:

  • habit – the scribe was accustomed to writing the ‘4’ style numeral, and used it for the number and for ‘q’. This is a palaeographic question.
  • the Voynich ms ‘4o’ is unrelated to the numerical ’40’ but might have been deliberately adopted as form for ‘q’ for reasons unknown.
  • It may have been adopted – as a cipher – for the sound associated with that form, or pair. A correspondent suggests that it may have been used to remind readers that the ‘q’ was to be read as a glottal stop, not like the softer ‘q’ of quo, or quatre etc.
  • In the Vms, use of the ‘4’ shape might indicate a cipher in which forms from foreign or rare (or even imaginary) alphabets* were employed. Mixing alphabets was a well-known custom, both in aiding memorisation of texts and as a form of encipherment – the method is fourth among those described by Roger Bacon in the fourteenth century (See earlier post ‘here).
  • It may have been adopted by some for the number ’40’s significance. Personally, I don’t think it likely, in the Vms, but to illustrate what I mean by significance, I add an example below. I’ll give an example further below.
  • In the Voynich manuscript, it may not indicate cipher, but simply a ‘q’. Statistical analyses of the Voynich text, however, present objection to that idea.
  • I recommend those interested in recent discussion about the Milanese cipher ledgers see M.R. Knowles’ recent comment to Nick Pelling’s post ‘New Paper on Fifteenth century Cryptography‘ ciphermysteries, (July 8th., 2017). Knowles’ comment is date-stamped November 22, 2021 at 8:21 pm, the conversation between himself and Pelling continuing to the time of writing.

On rare and imaginary alphabets used in medieval Europe see ‘alphabets’ in Mary Carruthers’ The Book of Memory. 

One earlier alphabet (at least) contained a ‘4’ shape, the alphabet known as ‘Old Hungarian’ or as  Székely-Hungarian Rovás, derived from a Turkic script of inner Asia. Another ‘Rovas’ alphabet was the Khazarian, of which very little is known, though according to Omniglot it was ‘possibly used as late as the thirteenth century’.  Both links given above are to the Omniglot entries.  Personally, I like the idea of the numeral ‘4’ form having origins distinct from use in cipher, and the use in cipher deriving from a rare alphabet – seems to connect pretty well with other pointers to the Black Sea region and various Turkic languages… but questions about the Voynich manuscript’s written text are for others to explore. Not my field.  

This is the point at which it’s a good idea to check your skill-set against those possibilities – and others which may occur to you.

‘Commercial …’.

For someone unfazed by technicalities of commerce and accounting, a broader context for fifteenth-century Urbino and Milan can be found in e.g.

  • Raymond de Roover, Banking and Credit in Mediaeval Bruges (1948).
  • Quentin van Doosselaere, Commercial Agreements and Social Dynamics in Medieval Genoa. New York, New York: Cambridge University Press, 2011.

Don’t be offput by the publication date for the first reference. Herbert Heaton’s review, published in The Annals of the American Academy of Political and Social Science, Vol. 263, (May, 1949), p. 230, says,

[de Roover] found about 2,400 folios of ledgers or journals belonging to two Bruges money-changers, dated between 1367 and 1370. To decipher, interpret, and convert into living story this dry as dust collection, it was necessary not merely to scour the Belgian archives for further data but also to search in those Italian cities-Florence, Genoa, Lucca, and the rest-from which Italian traders came to trade in Bruges.

with regard to numerals in commercial bookkeeping of late medieval Europe, I notice that john Durham concludes than in account-books from medieval Europe, Arabic numerals are found ‘at least’ by the early fourteenth century, when he sees it as an innovation in European book-keeping and ‘probably a Pisan innovation’.  He’s not speaking specifically of the ‘4’ shape.  

  • John W. Durham, ‘The Introduction of “Arabic” numerals in European Accounting’, The Accounting Historians Journal, Vol. 19, No. 2 (December 1992), pp. 25-55.[JSTOR}.

Byzantine accounting.

  • Edward Peragallo, ‘The Ledger of Jachomo Badoer: Constantinople September 2, 1436 to February 26, 1440’, The Accounting Review, Oct., 1977, Vol. 52, No. 4 (Oct., 1977), pp. 881-892. [JSTOR]

Location of later diplomatic documents:

  • Vincent Ilardi, ‘Fifteenth-Century Diplomatic Documents in Western European Archives and Libraries (1450-1494)’, Studies in the Renaissance, Vol. 9 (1962), pp. 64-112 (49 pages) [JSTOR]

Sound’ and Linguistics.

What sound (apart from a Latin ‘quo’) might have been linked with the ‘4o’ by those who used it? If anyone has investigated this question, their findings haven’t yet hit the headlines, so if your skills and inclinations suit this angle of approach, you might find helpful E.M. Smith’s posts:

Associations for the number ’40’.

As I say, I don’t think the form likely to have been adopted for its significance but with cryptology you never know, so I’ll Hopper as illustration of cultural significance for number. He is not the first or last source that might be consulted.

“The forty days of Christ’s temptation harks back to the 40 days of Elijah’s solitude, or the forty days of trial by flood” (p.71) see also p.13, 15, 25, 26, 127.

  • Vincent Foster Hopper, Medieval Number Symbolism. (various editions)

If any researcher does find examples of non-numerical uses for the ‘4o’ before the example from Urbino, we may be very close to understanding the context from which we have matter now in the Voynich manuscript’s written text.

Remember..

Any Voynich writer who presumes that you will accept some proposition with no shred of historical evidence offered is a person who may have a transmissible case of ignorance. Treat with caution.

Item from the present author’s research. Fifteenth-century copy of a thirteenth-century text. England.

Postscript – ,my apologies to Rainer Hannig. The mis-spellings of his surname in the earlier version of this post was due entirely to my own dreadful handwriting, not the typist’s occasional errors in reading it.

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