Tag: Abraham Cresques

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.

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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.

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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.

Prague

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.

Consider this.. (cont.). Numerals, networks, Spain and something of Kabbalah.

This post is almost 2800 words.

The earliest close examples of an upright ‘4’ numeral noted so far come from Florence at the beginning of the fourteenth century and then in Cresques’ great pictorial compendium of 1375, which includes various diagrams and a great worldmap, gridded by the ‘rose’ and containing what is still the first known inset ‘compass rose’ (see further below).

Contrary to what you might find said elsewhere, Cresques’ map is not a ‘mappamundi’ and its connection to the rutters or ‘portolans’ is certainly debateable, given that the same notion was rejected on technical grounds as early as the mid-twentieth century.

The recent, astounding assertion by one writer, on a nicely presented and official looking website was that Cresques had all his information from a couple of northern European Christian missionaries and that is surely pure invention. The sources of Cresques’ inscriptions for that map are already known, and include sources such as Ibn Jubayr’s journeys and the Alf Layla wa Laya. Allusions are also made to Jewish legends, such as that Noah settled north Africa after the flood and began viticulture again from there.

One cannot now discover how that modern author came to entertain the ‘Christian missionaries’ notion, for he died during the pandemic and I know only that he claimed some connection to the Central European university of Prague. With both authors of that project now lost, the translation of all the labels into English may be stopped or at least halted for the foreseeable future, but we do not have to rely on that material to consider the problem of the ‘4’.

IN the same way that Genoa was under Milanese control by the time the Voynich manuscript was made, so it was with two more of the four once-independent Italian maritime states.

Amalfi had earlier been taken by Pisa (August 6th., 1136) and in 1406 Pisa itself was taken, by stealth, by the Florentines. Amalfi had been absorbed into the Kingdom of Naples during the late fourteenth century.

Oddly enough, Florence did not develop Pisa as its maritime arm, but on the contrary suppressed the naval activity of both Amalfi and Pisa.

The significance of Florentine control of Pisa, Neapolitan control of Amalfi and Milanese rule in Genoa, is that direct political power meant access to all technical secrets, skills and any system of ciphers earlier held exclusively by the neighbour. Trade secrets were treasure then, just as now.

When we find the ‘4’ form appear briefly in Florence, early in the fourteenth century, within a copy of the Pisan ‘Liber abaci’ we know the exemplar might have been a local copy, or an earlier one acquired directly from Pisa or indeed from Amalfi, noted for its schools of mathematics. The best copies were known to be ones closest to the date of composition.

It should be noted here too that (to quote an online tourist site) “by about the 1230s Amalfi became one of the first locations in Europe to produce paper…. [which] was soon sold all over the Mediterranean. Paper making continued as an important local trade throughout the Middle Ages and Renaissance.”

My own view of the ‘4’ numeral’s history, at present, is that we shall probably never know absolutely who first wrote the numeral as ‘4’ in Europe and that if there were a single key to the problem it may well have been lost in 1343, when a tidal wave obliterated Amalfi’s harbour and lower town, ushering in a period of decline from which the town never recovered. It s relevant, in my opinion, that all four – Genoa, Amalfi, Pisa and Venice had allotted quarters in medieval Constantinople. (see interactive map by Saffran and Nicolescu)

However, we may still regard Amalfi or Genoa as likely to have brought that ‘4’ form to Italy, because of a demonstrable connection between those maritime states and Cresques’ great work.

The semi-legendary Amalfitan surnamed ‘Gioia’ is popularly credited with having first placed a magnetised needle over a diagram of the western wind-rose and enclosed all in a glass-covered box. Properly, that was not the ‘compass’ but the bussola (buxula), and the true navigational compass (as qumbas) the eastern navigator’s ‘rose’ whose points were named for stars. In my opinion it was in that sense Cresques describes himself as ‘master of bussola and compass’. The islands of Majorca and Minorca were remarkably cosmopolitan at that time and Arabic had been the island’s official language until just 70 years before. What is more, the original ‘Barbary’ pirates of the north African coast around Sicily, who were Berber and Arabs, are said by Ibn Majid to share the same skills and know-how as he – an Arabs master of the eastern seas.

Seen in daylight, Cresques’ great worldmap shows the world below, but at night with no illumination save a few candles what one sees is as if a veil scattered with golden dots were laid over the darkened world. Makers of terrestrial maps and marine charts also made maps of the heavens. Bussola and qumbas.

You may think such technicalities aren’t needed here but, as I first discussed some years ago in the course of providing a detailed analysis and commentary for the Voynich map, there is a precedent for Cresques’ inclusion of his ‘angel of the rose’ and for his map’s mirroring along its centre line. I won’t elaborate on the last point here, but refer again to the fourteenth century Genoese cartographer, Pietro Vesconte (sometimes found as ‘Vesconti’).

In one of his ‘rose-gridded’ charts, composed in 1311 1318 – that is, about or shortly after the time of that Florentine ‘4’ – there is another faint allusion to the same motif found in Cresques’ work and in the Voynich map and in all three cases – that is, the Vesconte carte marine, Cresques’, and the Voynich map, the motif of its ‘surveying angel’ is even placed within the same – north-west – quarter.

Note (added 5th. Dec. 2021] Pietro Vesconte’s date of birth is uncertain, but he is described as ‘flourishing’ c.1310-1330. Genoese by birth and education, his earlier charts and maps were produced there, but most of his extant work was produced in Venice.

This is less obvious in the Voynich map because it presents with its east and west reversed from the European norm. Western custom permits such east-west reversal with a constant North only in literal representations of the night sky.

I apologise to long term readers for again repeating points first made by me and in research published before 2020, but some of that research was treated as if its conclusions were just ‘an idea’ to be imitated, and its illustrations to be re-used without their context – so what was ‘lifted’ by the imitators was invariably – perhaps inevitably – badly mangled, and has never been well used by Voynich theorists and so must be repeated here. I regret having to deface the illustration for the same reason.

(left) detail from the Voyich map, its North-west roundel (upper right) detail from Abraham Cresques rose-gridded world-map, its north-western quadrant; (lower right) detail from a chart by Pietro Vesconte (sometimes found as ‘Vesctonti’, its upper-west corner). From the 1318 Vesconte atlas (Museo Correr, Venice)

and so, again..:

Since I have already said that the final recension of the Voynich map should be dated to c.1350, with our present copy dated to the early fifteenth, I think here again we may narrow the environment for the ‘4’ shape as numeral and, just possibly its use in the Voynich manuscript, to the specific environment of calculation and geometry gained in service to navigation and trade.

In other words to apprenticeships and the ‘abaco’ school rather than in schools offering a more literary, theoretical or philosophical education. More evidence may demand that opinion be altered, but that’s where I stand so far.

It might even be that the Voynich ‘4’ shape is meant in the manuscript as a numeral, even if also used, or originating, as an alphabetic sign, though I should be wary of assuming that the Voynich script’s other glyph of similar form – that with a more curved ‘eye’ – is necessarily to be read as it is.*

*a question I’ve not ever looked into, but which arises from time to time, is a possible origin for the ‘Cistercian’ numerals in a version of Syriac script. See later note on a mixed alphanumeric system.

For those who enjoy the slog of using pictorial archives of kind typified by the Index of Christian Art (as was), it might be fun to see what else turns up for ‘4’ in European sources around 1300.

In any case, the story which puts Leonardo of Pisa and his ‘Liber abaci’ centre stage is an over-simplified one. That story’s short version runs something like ‘Arabs brought the Hindu numerals westwards. Leonardo (‘Fibonacci) saw them, and brought them to Europe’.

But Leonardo didn’t use that ‘4’ shape. His relevance to our present problem is rather the pattern of his travels, which illustrate nicely contemporary networks of trade and travel.

The Pisan Leonardo first learned Arabic numerals in a major Berber-speaking city of North Africa, during the last decade of the twelfth century. His sobriquet ‘the traveller’ was well earned.

Fibonacci states that his father wanted him to stay and be taught “for some days” in a “calculation school” in Bejaïa, where he was introduced to the “art [of calculation] by the nine figures of the Indians”. The knowledge of this art pleased him so much that he learned all he could about how it was studied in Egypt, Syria, Greece, Sicily and Provence when going there for the sake of trade.

So there it is. Those ‘Indian’ numerals were already known in Greece, in Sicily and in Provence before the end of the twelfth century. I’ve used the quote only so I can reference:

  • Charles Burnett, Numerals and Arithmetic in the Middle Ages (Variorum Collected Studies Series CS967) 2010.

There were especially close ties between Pisa and Béjaïa during the twelfth century. In c.1350, it was rather with Tunis and Cairo that the Venetian traded who wrote the zibaldone now Beinecke MS 327.

Béjaïa, formerly Bougie and Bugia was by Roman times known as  Saldae.    Béjaïa is still among the centres where the Berber language Kabyle is in daily use.  

Poor Ramon Lull would arrive in Béjaïa during the early fourteenth century (in 1314) as an 81 year old Dominican friar hoping to make converts to the Latin Christian church. He was dead within a twelvemonth, though accounts of his death differ, some saying he was executed for trying to persuade Muslims to become apostates to their faith – something prohibited in every region under Muslim governance as indeed it would have been in regions under Latin Christian governance had the reciprocal occurred.

Other accounts have Llull dying on the ship returning him to Majorca.

——-

Correction. (December 15th., 2021).

I see that my sources are out of date, superseded by an updated (Feb.2021) entry in Stamford University’s Encyclopaedia of Philosophy, where it says that Llull did not enter the Dominican order, nor the Franciscans’ and gives the following account of his fruitless trip to Béjaïa.

‘De vita’ narrates this trip in detail. Llull spent most of the time in prison. Instead of seeking to meet intellectuals, as he did on his first trip to Tunisia, right after disembarkation, he went to the main square and harangued passersby and anyone present at the time. The crowd was infuriated, and Llull was placed under arrest. The authorities questioned and imprisoned him. He would stay there for six months, receiving visits from sages who sought to convert him to Islam. He was later expelled from the city, but his hardships would not end there. His ship sank on the trip back to Genoa, but Llull and another passenger managed to survive by reaching the coast. He would then remain in Pisa, where he would finish texts he had previously began writing, such as ‘Ars generalis ultima’.

——-

Llull has his place in western Europe’s history, but unless one of his works contains examples of Majorcan-Florentine ‘4’ he is less relevant to our present question than the more congenial, secular, interactions between Berbers, Jews, Arabs and Italians before 1300, including within the naval, commercial and cartographic schools.

Voynich writers interested in the possibility that the Voynich ‘alphabet’ may be composed of elements taken from a number of other systems may be interested in an account of the invention, during second quarter of the twelfth century, of a new mixed system of mathematical notation.

Burnett writes:

*Charles Burnett, ‘The Semantic of Indian Numerals in Arabic, Greek and Latin’, Journal of Indian Philosophy, Vol. 34, No. 1/2 (April 2006) pp.15-30. [JSTOR]

 For those who’d like to see what Greek script of the fourteenth century looked like, here’s a detail from a Greek fourteenth-century map in Codex Vatopedinus 655.

“Europe gained its numerals from the Jews”

“The Jewish community… reconstituted in 1306” from ‘Amalfi’ in the Encyclopaedia Judaica JVL online. 2005.

As early as 1891, when European scholars were just beginning to enquire into the history of the forms for their numerals, it was already being accepted as fact that they had come from Spain. (See for example the off-hand “or rather, from Spain” in a review published in the Scottish Antiquary (Vol. 6, No. 22, (1891) p.54).

But even more unexpectedly, an association was made with Kabbalah as early as 1839!

I’ve just learned the last fact thanks to Phineas Mordell’s meticulous documentation of his sources and precedents in a very brief note of 1925. For its historical value, I’ve reproduced this note in full.

  • Phineas Mordell, ‘Note on the Theory of the Kabbalistic Origin of “Arabic” Numerals’, The Jewish Quarterly Review, Vol. 16, No. 2 (Oct., 1925), p. 207.

Of course it is possible that by 1932, Erwin Panofsky had read one or more of those sources listed above, or even an article published late in 1931; in addressing Friedman’s list of Questions more than twenty years later, Panofsky mis-remembered the year in which he’d seen the manuscript – writing ‘1931’ when it can only have been in 1932, as explained in an earlier post.

  • Solomon Gandz, ‘The Origin of the Ghubār Numerals, or the Arabian Abacus and the Articuli’, Isis, Nov., 1931, Vol. 16, No. 2 (Nov., 1931), pp. 393-424.

Panofsky was never so vapid as to mistake for an ‘idea’ the conclusions of genuine research, nor was he obliged to parrot others’ assertions for want of ability to form balanced and well-informed opinions of his own – but he may well have read one or more of those articles for the same reason that G.F. Hill wrote his monograph (see previous post) that is – to assist in accurately provenancing artefacts and quickly identifying fakes.

I think Panofsky could have known those precedents- not that he would say anything of the sort unless it were suggested to him by aspects of the materials, pigments, imagery and vellum which he observed during the two hours he spent studying the manuscript itself. But now to that list of things observed we may add (with a query) the form of one or more of the Voynich glyphs – perhaps even the ‘4’. We don’t know. All we do know is what some long-term readers of my blogs probably know by heart now, but for newcomers..

Panofsky’ freely-given opinion was given to Mrs. Voynich and Anne Nill, the latter soon reporting it in a letter to her friend, Herbert Garland. She wrote*

“he is convinced that the MS. is Spanish (or something southern near Spain) and shows strong Arabic and Jewish influences. He thinks there is some influence of the Kabbala in it.!!!!!

**for details of Nill’s letter, see the transcription made by Rich Santacoloma which I believe was the first published transcription. See his post. ‘Anne Nill speaks‘.  For my earlier comments, in this blog, about the letter and about Rich’s thoughts see here. 

To the question,  ‘What exactly had Panofsky seen in the manuscript which led him to mention Kabbalah’? I never did find a clear answer, and ran into unexpected difficulties, such as the lack of modern scholarly articles about the medieval commentaries or even modern translations of those commentaries that I could quote in an English-language blog.

As with many other research questions, one sometimes has to leave a problem aside for a time, until new information or pure serendipity offers a way forward.  Very recently, a single article in n online journal has explained the apparent barriers and, quite incidentally, offered a line to another and quite different question that I’d laid aside pending better information. I’ll try to get to that journal article in the next post.

For a short comment and an initial bibliography for the question of any influence from Kabbalah in Beinecke MS 408 see  (Post #15). 

With this question, as with the history of European forms for its numerals and most other fields of historical research, the public’s idea of a positivistic ‘forward march’ is deceptive.

Very often a study moves over time more like a pretty complicated sort of quickstep, involving  not a few trodden toes, losses of direction and ‘excuse-me’ interruptions, backwards moving which takes one forwards and some few straight-forward passages.  In the history of European numerals, for example, there was a period in the 1950s and in America, where the story was badly misdirected by an ideological fixation on the Babylonians and a transmission-theory gone berzerk.  As example, here’s one such paper, though if you don’t feel like reading it all, here’s a taste of that author’s ‘commonsense amateur theory’ approach.

… a casual inspection of the Arabic numerals suggested that these symbols might have evolved from forms such as are shown in Fig. 10, hereafter termed Ancestral Arabic numerals. It is evident that they are a variation of the Prototype numerals which the writer later derived from hand-signs, and still later discovered had been widely employed..

from: W. Clyde Richey, ‘On the Origin and Development of the Arabic Numerals’,  Proceedings of the Pennsylvania Academy of Science , Vol. 26 (1952), pp. 134-146. [quote shortened .. 5th Dec. 2021]

Not ‘handsigns’ but methods of finger-calculation may, in fact, prove relevant to our efforts to read Voynichese.

But I digress.

When quite early insights are overlooked or shrugged away in favour of worse ones, as happens more often than laymen suppose, it is also true that they may be recovered or re- discovered again later thanks to properly documented work in the meantime, or because the sum of historical evidence requires it.  

As example, here’s Charles Burnett, writing in 2006, and after years of close study of the question…  and evidently arriving at a view held by at least one person in 1891, in Scotland. 

One can observe, too, that, during the course of the twelfth century, alternative forms of the Indian numerals dropped out of use, especially the ‘eastern forms’ which were briefly shared by Arabic scholars in the Eastern Mediterranean, Greek scholars, and Latin scholars in Italy. The forms which drove out their competitors (in my view) were developed by scholars in Toledo, and quickly spread to northern Italy, where they were used by Fibonacci. (p.21)

  • Charles Burnett, ‘The Semantic …’ op.cit. 

… except for that form of ‘4’, which Fibonacci did not use….

(detail and enlargement) Brit.Lib. Burney MS 275 f. 334

MS Burney 275 is described as

Scholastic miscellany, Central France (Paris), 1309-1316. Priscian, Cicero, and Pseudo-Cicero, Boethius, Aristotle, Euclid, Adelard of Bath, Ptolemy, translated [nominally – D.] by Gerard of Cremona.

Note – after some thought, I’ve altered the spelling of the Genoese cartographer’s name from ‘Vesconti’ to ‘Vesconte‘ as less likely to create confusion with the Milanese Visconti family, though researchers hunting secondary sources should search both versions of Pietro’s name.