Consider this.. (cont.) Doing the math.

This post/essay is more than 3,600 words.

THIS SERIES of essay-length posts is prompted by questions about the form of one sharp, angular glyph resembling the modern short-stemmed ‘4’. Our paradigmatic example being:

This post outlines the communities and inter-connections between them over time which would finally see emerge the same ‘4’ shape numeral and other matter whose reflection is found in our present, fifteenth-century manuscript with its many unusual features.

In the instance seen above, the long bar above it makes it easy to interpret the glyph as alphabetic, and so take this pair as abbreviating some such word as q[u]o – yet the glyph’s form is not written as a Latin ‘q’ of the early fifteenth-century and allows us to suggest that even if, here, the ‘4’ glyph wasn’t intended for the numeral ‘4’, it has been written by a hand accustomed to writing the numeral in that way.

Before 1440 ‘four’ represented by this shape was still uncommon – unattested (to date) in Germany before the Voynich manuscript was made (1400-c.1440), and rare;y in England. Thus, so far, we must attribute it to the south-western Mediterranean and to the communities having attested ties to Majorca at the time our earliest clear example of the ‘4’ numeral occurs there (1375 AD).

The following passage, appended as a comment to the previous post, deserves greater prominence.

“James I appears to have chosen Majorca as his first target because of the island’s geographical importance and its closeness to the Spanish coast. Almost equidistant from Catalonia, the north of Africa, and Sardinia, the island’s ports dominated the trade routes of the western Mediterranean. James’s army included … large numbers of townspeople from the main trading cities of Catalonia and southern France, especially Barcelona, Marseilles, and Montpellier. Unlike the barons …many townspeople actually settled in Majorca and contributed to its prosperity. Some of the settlers came from further afield. While Catalans were the most numerous, there were also Aragonese, Navarrese, men from southern France, Italians (from Naples, Sicily, Genoa, and Pisa), Castilians, and Portuguese. In addition to the conquered Muslims, there was also an important Jewish community in Majorca from very shortly after the Christian conquest of 1229. This community had ties not only to Catalonia and southern France, from which many of its members had come, but also to north Africa, and Italy.” (p.335)

  • passage from J.N. Hilgarth, ‘Sources for the History of the Jews of Majorca’, Traditio, Vol. 50 (1995) pp.334-341, though other recent sources will include the same information.

To do a reality-check here – to ensure we’re not straying too far from evidence and veering from historical research into merely hunting support for a theory – we now test our present emphasis on the south-western Mediterranean against earlier informed opinion about Beinecke MS 408.

The set of connections exemplified by the Majorcan population accords with Erwin Panofsky’s attributing the manuscript -or perhaps something about its vellum and style of drawing – to ‘Spain or somewhere southern, with Arab and Jewish influence’ and shows there need be no opposition supposed with the consensus opinion of specialists in manuscript studies who were known to H.P. Kraus and his assistant Hellmut Lehmann-Haupt in the 1960s – their focus having been clearly on Italians.* Again, the month-names’ orthography has been variously described as Occitan (also spoken in Genoa), and as Judeo-Catalan, and so forth. (These things have been treated as separate issues in earlier posts. For a list, with links, see Table of Contents page in the top bar.)

*The views relayed to John Tiltman by Lehmann-Haupt, research assistant to the bookseller H.P. Kraus, are recorded by Mary d’Imperio, Elegant Enigma p.7 and 8).

Checking, again, if the class of text early using ‘4’ numerals is appropriately described as practical, navigational and/or commercial, those parameters easily present us with another instance prior to 1440.

That detail comes from manuscript known as the ‘Handbook of Michael of Rhodes’. It is in private hands but has been studied and summaries of the study are online.

  • Michael of Rhodes’ book website (here).

Michael’s education was gained as a mariner, his education by apprenticeship and in keeping with ‘tradesmans’ mathematics taught by schools of the kind known to the Italians as ‘abaco’ or ‘calculation’ schools. He began writing an account of his life and nautical-commercial calculations in 1434, his year of death being 1445. (see pages at the Galileo Institute site). As one might predict, he served one of the Italian maritime city-states – Venice.

Appropriately enough, his example for that calculation (partly illustrated above), is about the purchase of pepper – obtained by Venetians of his time from Alexandria or Tunis, but which had been traded since Roman times (at least) into the Mediterranean via Egypt from commercial pepper plantations in southern India. Alexandria remained a principal centre of that trade in Michael’s time, with Tunis, though in the earlier medieval period, the carriage of eastern products to Christian Europe had been principally in the hands of Jews and others classed and taxed as Jews in areas under Muslim governance.

Such links with Egypt and exotic goods naturally again reminds us that Georg Baresch believed the manuscript’s content had been gained ‘in the east’ and contained matter that was – in some sense unspecified – both Egyptian and ‘ancient’. He also said that the plant-pictures referred to ‘exotics’ whose forms were still unknown to German botanists in his time, when Germans led all Europe in that field.

My study of Beinecke MS 408 also found much to support Baresch’s opinion about the plant-pictures.. but presently we are not concerned with meaning so much as with forms – the form of the written text, of the pictorial text and the manuscript’s presentation.

On that basis, we may lay aside (pending possibly better information) such Voynich theories as the ‘Norwegian’ or the ‘central European’ or the ‘New World’ theories, which offer no comparison for the ‘4’ shaped glyph, or for the apparently anomalous ‘gallows glyphs’ with their elongated ascenders (if that’s what they are), nor comparable styles of script, drawing, page-layout or -disposition, nor the presence in any such manuscript noted so far of quires both quinion and septenion as we do see in the Voynich manuscript and have also found in Italy and in Hebrew manuscripts from the south-western Mediterranean – on paper, on membrane and in a combination of both (see earlier posts).

Even within Italy, it seems at present that perhaps we should discount the higher levels of education and of society, since the only instance of a ‘4’ shape which might be associated with nobility or bureaucracy known so far, is in one cipher-ledger from Urbino dated to 1440, brought to notice by Nick Pelling in 2006. But 1440 is sixty-five years (nearly three generations) after our earliest clear instance of that ‘4’ in Abraham Cresques’ Majorcan ‘Atlas’ of 1375 and almost a century and a half after one brief appearance in Florence, in a copy of the Liber abaci.*

*The bankers of Florence were strongly opposed to use of the Hindu-Arabic numerals, attempting and succeeding for a time in blocking their use.  I don’t have an English-language source for this, but see  Anna Maria Busse Berger,Lada Hordynsky-Caillat and Odile Redon, ‘Notation Mensuraliste et Autres Systèmes de Mesure au XIVe Siècle, Médiévales, No. 32 (Spring, 1977), pp. 31-46 and particularly p. 34. [JSTOR]

On the other hand, while the written text’s inclusion of that ‘4’ glyph in Beinecke MS 408 directs our attention to the commercial and maritime interests of communities whose people are found settled in thirteenth- and fourteenth-century Majorca, including those maintaining ties with Italian cities, it is Panofsky’s “Spain or somewhere southern” which is given clearest support by content in our fifteenth-century document.

When I cited the Codex Vigilanus among examples explaining the Voynich pages’ page layout and dispositions, I found no previous allusion to it in any ‘Voynich’ writing though I should not be surprised to find it mentioned elsewhere today.

It crops up again now because the same manuscript is referenced in Hill’s Tables and in the review of Hill’s work by Louis C. Karpinski, who was at that time (1915) the foremost scholar interested in the history of European forms for the numerals.

As introduction and context for quoting from Karpinski’s review, I’ll reproduce a paragraph from one earlier post from voynichimagery. In it, I was making the point that the Voynich page design, especially but not only in the ‘bathy-‘ section, differs markedly from the consciously ‘Greek and antique’ simplicity of Italian ‘humanist’ manuscripts, yet it finds echoes in other times and places, including tenth-century Spain.


excerpt from: D.N. O’Donovan, ‘Fold-outs in Europe – Afterword’, voynichimagery 20th June 2015.

(detail) Codex Vigilano [~Vigilanus] Albeldense fol.59. Spain. Mozarabic. Compilation 9th and 10thC

.. another example. This from Spain, in a volume containing material of the the 9th-10th centuries. Notice how these illuminations fill the sides of the page, and how the text seems to be fitted around the central figure, a little irregularly, as if the imagery had been set first, and the text written later – the very opposite method from that used in manuscripts from most of Latin Europe, but this was in Spain, under Muslim rule – though the degree of influence from Baghdad as against that of the Berbers from North Africa is debated along sectarian lines. However, that non-Latin character in contemporary Spain may explain the way these pages are planned, uncharacteristic of Latin texts per se, despite the language in which it is written. … these pages’ design offers points of comparison with MS Beinecke 408. Most particularly, in my opinion, with the ‘bathy-‘ section, which [because of anthropoform ‘ladies’] implies again connection with the [month diagram] foldouts … Note here, once again, that same convention [seen in Yale, Beinecke MS 408] of using roughly-parallel curved lines to denote curve and volume. … [and the makers’ familiarity with the ‘false-bearded’ face and the concept of a bicorporate form, all of which occur in Beinecke MS 408 –  D.]

Unitalicised text in the passage above  added  8th/9th December 2021.

excerpt from: D.N. O’Donovan, ‘Fold-outs in Europe – Afterword’, voynichimagery 20th June 2015.


Having previously cited that codex, it was pleasant to find it referenced by Hill and commented on by Karpinski, who said,

The earliest European forms are doubtless found in the Codex Vigilanus, written in 976 AD in the monastery of Albelda near Logrono in Spain. A second Spanish manuscript of about the same date, not described by Mr. Hill, also contains similar forms, and facsimiles. Both are to appear in the next issue of Professor John M. Burnam’s Palaeographia iberica.

from: ‘The Development of the Arabic Numerals in Europe Exhibited in Sixty-four Tables. by G. F. Hill. Reviewed by  Louis C. Karpinski’ for The American Mathematical Monthly,  Vol. 22, No. 10 (Dec., 1915), pp. 336-337.

Hill’s reference to the Codex Vigilanus was a note to his Table 1: 

1. 976. Escorial d I 2. Codex Vigilanus, written in the year 976 in the monastery of Albelda near Logrotio. See P. Ewald, Neues Arcbiv der Gesellsch. /. alt. deutsche Geschichtskunde, viii (1883), p. 357. Cp. Smith and Karpinski, p. 138. The forms are described as the Indian figures, quibus designant unumquemque gradum cuiuslibet gradus. Quarum hec sunt t”orm(e): 987654331. Ewald connects the form for 5 with the Roman V. Since he does not say that the year 976 is that of the Spanish era, we must assume that it is of the usual Christian era.

I have not sighted Burnam’s Palaeographica iberica.

Already, by the tenth century, mathematical studies were advancing within Spain as in North Africa. While few scholars consider any matter in terms of Christian, Jewish and Muslim scholarship altogether, the separate studies of western numerals and mathematics have, independently, commented on the situation in tenth-century Spain. It was from there that – for example – Gebert d’Aurillac was said to have learned his calculating ‘arts’* though I suspect that his original ‘abacus’ with its significant factors – 9 and 27**– is less related to that form later given his name than to something he first encountered during the period when Barbary pirates had him.

*D.E. Smith. A History of Mathematics (Vol.2, p.75) says ‘there is good reason for thinking Gebert’s knowledge of the numerals was gained in Ripol, at the convent of Santa Maria de Ripol.

**the ‘9 and 27’ are rarely mentioned in secondary accounts today. I have no English-language reference for it to hand, but see the review of O. Chasles, ‘Histoire de l’arithmétique. Explication des traités de l’Abacus, et particulièrement du traité de Gerbert; Extrait des comptes rendus des séances de l’Académie des sciences’, Reviewed by H.G. in Bibliothèque de l’École des chartes, Vol. 4 (1842-1843), pp. 382-386.

But that’s by the way.

With regard to the Maghrib, I quote from Ahmad Djebbar’s studies, which do show that we do better to focus on lines of connection over time and distance, than defining matters in terms of a static parochial ‘nationality’.

Abū ‘l-Qāsim al-Qurashī … was a native of Seville, in Spain, spent a period of his life in Bougie (Béjaïa) where he died in 1184. The biographers who evoked him consider him a specialist in Algebra … [in which subject] al-Qurashī is known for his commentary on the book of the great Egyptian mathematician Abū Kāmil (d. 930). This commentary has not yet been recovered but its importance is confirmed by the historian Ibn Khaldūn (d. 1406) who considered it one of the best treati[s]es written on the book of Abū Kāmil.

Djebbar does not consider the works of Christian or of Jewish scholars relevant to his study, yet traces the evolution of mathematical studies in the Maghrib to Kairouan, which had been a community of unusually ascetic Jews until mention of them and of Kairouan in a narrative known as the ‘Night Journey’ linked Kairouan directly to the Prophet of Islam, reverence for whom saw the town declared a holy place and the original inhabitants expelled.*

*In this connection, I’d mention that D.E. Smith (op.cit., loc.cit.) says the names given the Ghobar numerals in the earliest Latin texts are: igin, andras, ormis, arbas, quimas, calctis, zenis, temenias, celentis and sipos, which Smith says appear to be Semitic. 

By the tenth century there were remarkable Jewish scholars working from the region presently of interest to us, but before considering one Jewish mathematician of the fourteenth century – that is, one who lived at the time we see the early emergence of that ‘4’ shape, it will be as well to pause again to check our bearings.

So far, it appears that what we have as the content in Beinecke MS 408 may be – again to quote Panofsky – “considerably earlier” matter within the material which was copied to provide the quires of our present fifteenth-century manuscript, and if the copies were not inscribed within Italy itself (as is possible), I think that by considering other matter in the manuscript we may posit with some confidence that the material as we now have it was copied for an Italian sponsor – whether Christian or Jew – during the period 1400-c.1440.

It is also possible that the manuscript’s written part, being added to the page after the pictorial text in a custom contrary to that of mainstream Latins’ work, may have taken its form as ‘Voynichese’ not much earlier than our present copy and thus to require study within parameters different from most of the imagery in which so few details express the Latins’ worldview iand so much speaks to earlier and other customs.

A relatively late creation of the ‘Voynichese’ script offers us one reasonable explanation for the apparent discrepancy between (i) disposition of image and text, and primacy given image over text, against (ii) the scribes’ evident familiarity with a straight and short-stemmed ‘4’ form characteristic of western works from the fourteenth century and later, whether that form is used here as alphabetic or numeric. Of course, that is not the only possible explanation we could call reasonable. We have yet to consider scripts from other parts of the greater Mediterranean (let alone the world) in which a ‘4’ form occurs.

Yet Spain and the example of the Codex Vigilanus allows us at least to suggest that the earlier models might date from as early as the time when ‘elongated ascenders’ still appear in such documents as the Papal charter establishing the convent of Ripol.

The fact is that we can’t be certain, at this stage of investigation, that the Voynich ‘gallows glyphs’ do have their form such ‘elongated ascenders’.

They might – for example – imitate scribal conventions from some other language altogether. I expect that there were some readers who sat up on seeing one not-quite-match between the form of a Voynich ‘gallows glyph’ and a Greek form in that detail from Codex Vatopedinus 655 which is in the previous post.


A letter whose chief theme was the lamentable decline of mathematical studies among the Jesuit scholars of Prague was sent to Athanasius Kircher in 1667 from Aloysius Kinner, about six months after the Voynich manuscript had been sent from Prague to Kircher in Rome.

Kinner refers to the manuscript and to Marcus Marci, on whose behalf the manuscript had been sent.

Marci had included with it a covering letter whose final paragraph reported, but declined to endorse, a rumour that – as Marci recalled it – was told to him several decades before, and – as he remembered it – by Rafel Mnishovsky. Evidently sent in 1666 (though dated August 1665) that paragraph in Marci’s letter remains the sole basis for any alleged connection between the manuscript and Rudolf II.

In January of 1667, then, Kinner writes in connection with mathematics:

Our own Marcus, so widely known for his writings in mathematics and other studies has now fallen into the second infancy of old age. He barely understands everyday necessities, as I note with much sadness and distress whenever I happen to visit him…. Now these men are gone scarcely any are left who could be called mathematicians and those few are totally occupied with other studies and are obliged to sneak their glances at mathematics….There is a deep silence, not to say ban, on Euclid and Appollonius in this university so that we are now not even supposed to know the names let alone the thing….And now for other matters. Dominus Marcus has lost his memory of nearly everything but still remembers you. He very officially bids me salute you in his name and he wishes to know through me whether you have yet proved an Oedipus in solving that book which he sent via the Father Provincial last year and what mysteries you think it may contain. It will be a great solace to him if you are able to satisfy his curiosity on this point….I do not know whether you are interested in having your Organum Mathematicum which you once prepared for our Archduke Carolus…

It only remains, now, to compensate a little for the habit of historians of ‘parochializing’ specific studies. I’ll mention just one medieval Jewish mathematician – Immanuel ben Jacob Bonfils (c. 1300 – 1377).

In 1936 an optimistic George Sarton wrote,

It is extremely interesting that the streams of thought which led eventually to decimal calculations on the one hand and to exponential calculations and logarithms on the other, had apparently two main sources, a Christian one and a Jewish one – both being ultimately derived from the same Greco-Arabic fountain head.. Gandz and I have now placed him [Bonfils] – and forever- among the great mathematicians of the fourteenth century, in the company of Oresme and John of Meurs. Henceforth the city of Tarascon should not remind us only of the famous Tartarin but also of one of the great mathematicians of the Middle Ages, the Provencal Jew, Immanuel ben Jacob Bonfils.

The remarks speak of Sarton’s acquiring a photostat copy of BNF Hebrew Ms IO54.6 and Gandz’ translation of the text (‘Derek (i) hilluq’). Gandz’ prefatory remarks, outlining earlier errors of the text’s description and interpretation incidentally offer another good example of that complex mix of forward and backward movement by which an historical study so often proceeds.

On the mathematical text, Gandz writes:

The invention of Bonfils introduces two new elements: the decimal fractions and the exponential calculus. In the latter case he substitutes the addition and subtraction of the exponents, or of the denominators of the degrees, as Bonfils calls them, for the multiplication and the division of the decimal powers. Our impression is that Bonfils is primarily interested in the demonstration of this method of the exponential calculus.

As you’ll see by consulting wiki articles about Algebra or Calculus, Sarton’s confidence was misplaced. We are yet to see Bonfil’s role properly acknowledged in mainstream narratives.

Quotations from Sarton and from Gandz from

  • George Sarton and Solomon Gandz, ‘The Invention of the Decimal Fractions and the Application of the Exponential Calculus by Immanuel Bonfils of Tarascon (c. 1350)’, Isis , Vol. 25, No. 1 (May, 1936), pp. 16-45.

I haven’t yet spoken about that illuminating recent article (mentioned in last week’s post) but it will have to wait.

More recommended readings.

  • Yakir Paz and Tzahi Weiss, ‘From Encoding to Decoding: The AṬBḤ of R. Hiyya in Light of a Syriac, Greek and Coptic Cipher’, Journal of Near Eastern Studies , Vol. 74, No. 1 (April 2015), pp. 45-65. A recent study of medieval Jewish atbash [JSTOR]
  • Tony Lévy and Charles Burnett, ‘”Sefer ha-Middot”: A Mid-Twelfth-Century Text on Arithmetic and Geometry Attributed to Abraham Ibn Ezra’, Aleph, 2006, No. 6 (2006), pp. 57-238. with regard to the practical mathematics involved in cartography. [JSTOR]
  • Ptolemy’s Table of Chords‘ – wiki article.
  • Pamela O. Long, David McGee and Alan M. Stahl (eds.) of The Book of Michael of Rhodes: A Fifteenth-Century Maritime Manuscript. (2009).
  • Frederick M. Hocker & John M. McManamon, ‘Mediaeval Shipbuilding in the Mediterranean and Written Culture at Venice’, Mediterranean Historical Review, Volume 21, 2006 – published online in Issue 1, 24 Jan 2007.

2 thoughts on “Consider this.. (cont.) Doing the math.

  1. There have been so many studies of the ‘silk and spice’ routes that it may not be necessary to footnote my description of the pepper route, as it was over the period between about the tenth to the mid-twelfth centuries, but here’s a passage from a paper by David Jacoby,

    Beginning in the late tenth century the main westward flow of spices, aromatics and colorants from the Indian Ocean was increasingly diverted from the Persian Gulf, plagued by political instability, to the Red Sea. The Fatimids, who ruled Egypt from 969[A.D.}, encouraged, protected and controlled this lucrative trade in precious commodities, which in addition to frankincense and myrrh produced in the south of the Arabian peninsula flowed through their territory. As a result Egypt established itself in the eleventh century as the main supplier of these costly Oriental goods to the Mediterranean region, including other Levantine countries and Byzantium. Concomitantly, there was a sharp decline in the role of cities in Islamic Syria and of Byzantine Trebizond in that context. A surprisingly large quantity of pepper was found in May 1101 by the Christian forces that captured Caesarea, which was not a commercial center.
    from: David Jacoby, ‘The Economic Function of the Crusader States of the Levant: a New Approach’, in
    Fondazione Istituto Internazionale di Storia Economica “F. Datini” Prato, Serie II – Atti delle “Settimane di Studi” e altri Convegni [no] 38 pp.159-191.
    The paper is accessible through Persee.

    The situation with regard to Trebizond vs Alexandria would reverse for a time during the period of Mongol control in the Black Sea, and we see the focus shift among the east-west traders towards Tabriz, Trebizond and the ancient ‘crossing-over’ point of Chersonesus, one of whose centres was the Genoese ‘colony’ of Caffa. A large part of this shift was due to Mamluk Egypt’s having driven the last of the Latin occupiers out of the Levant in 1291, though it is clear that those unable or unwilling to take part in the longer-distance maritime trade continued (despite intermittent papal bans and Mamluk bans) to trade through Alexandria and Tunis, Venice having for a time, as special a status in Alexandria as Genoa did – for a time – in the Black Sea.


Leave a Reply

Please log in using one of these methods to post your comment: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.