Consider… Maths & memory Pt 1.

Four posts in one. Take your time. Hope to see you in a month’s time.

One thing to emerge so far, while tracking use of the simple ‘4’ shape as a numeral – and we haven’t yet begun to track its use as an alphabetic form – is that, before the Voynich manuscript’s date-range of 1404-1438, it has been found only among the commercial and working classes of the south-western Mediterranean, and chiefly in the Majorcan kingdom with its Jewish cartographers and residents of certain maritime and trading cities of Italy – Venice not being among the earliest to show it.

Since that particular short-stemmed and angular ‘4’ shape, as a numeral, appears earliest in that region and it was also in the south-western Mediterranean that Kabbalism flourished among sectors of the Jewish population, in the thirteenth and fourteenth centuries, it is not surprising that there might be in matter now in the Voynich manuscript, as Erwin Panofsky thought, ‘something of Kabbalah’.

NOTE – throughout these posts I mean by ‘the Mediterranean’ the greater Mediterranean, containing all the waters from the Black Sea to Gibraltar. The ‘south-western’ region is defined as west of Sicily, between the coasts of Italy and of north-west Africa.

Nevertheless, aspects of the manuscript’s drawings and codicology make clear that wherever and by whomever the current content was put together to make Beinecke MS 408, much of the copied material originated outside the Latin domains.

I would hope that, in the third decade of the twentieth century, Voynich researchers will have no difficulty accepting a possibility which earlier Voynich writers found inconceivable – that is, that the manuscript may have no direct connection to those texts which for Wilfrid, Newbold, the Friedmans, d’Imperio and others moulded by nineteenth-century attitudes, defined the scheme of Euope’s intellectual history.

Fixation on ‘high culture’ as on ‘high society’ was for many decades a mental barrier to the manuscript’s proper study – and its effects linger. This is why (for example) no other form of art save manuscript art, nor any type of manuscript save in the official herbals was ever considered when attempting to read the Voynich plant-pictures, despite the fact that even within Latin Europe vegetable images appear in a variety of forms, from attempted naturalism to the fantastic and in media as diverse as stone, wood, embroidery, gem-engraving, and frescos.

Nor should we now, in the third decade of the twenty-first century, maintain another idea long outdated in historical studies – namely the idea that nothing foreign could enter Europe’s mental horizon unless some Latin went elsewhere, selected and ‘fetched’ it, or at least acted as a sort of customs agent at the gate of a non-existent ‘white-walled Europe’.

It is now well-known, if not widely admitted in works for the general public of Europe and America, that medieval Europeans were not rarely passive beneficiaries of information, ideas and goods conferred upon the west by ‘foreigners’.

Nominating some single Latin figure in the role of sole agent and gate-monitor has a long history in Europe. Nestorian Christian works, for example, were often attributed to one John of Damascus; Gerard of Cremona was (and still is) credited as if author of translations from Arabic, Hebrew and other languages though the translations are known to have been made by multilingual Jews and Muslims, and the same works to have been translated previously or subsequently without any such ‘monitoring eye’.

In this way, too, the English nominated Roger Bacon, and the Germans a semi-mythical ‘Meister von Kriechenland Niger Berchtoldus’ to substitute for the Chinese as responsible for Europe’s acquiring knowledge of how to make gunpowder.

The habit has been as consistent as it has proved persistent. It is solely to serve as such a ‘gatekeeper’ between Jewish Kabbalism in north Africa and the Iberian peninsula on the one hand, and the Latins of mainland Europe on the other, that Ramon Llull has been imagined as knowing anything of Kabbalah, and why – despite the testimony of Leonardo of Pisa that knowledge of Arabic numerals and their calculation-methods was already known in ports of North Africa from Bejaïa to Egypt, and “Syria, Greece, Sicily and Provence” in all of which (as he says) he studied it in connection with his family’s trade in eastern goods, Leonardo of Pisa (Fibonacci) is nominated sole ‘gatekeeper’ for the introduction of Hindu-Arabic numerals. The difference was that by producing a book about it in Latin, rather than in the vernaculars in which most ‘second-tier’ writings were produced, Leonardo’s ‘Liber abaci’ facilitated the establishment in Italy of specifically ‘commercial maths’ classes of the sort in which he had been trained elsewhere.

As one reviewer emphasised when reviewing an English translation of the Liber Abaci:

“Use of the advanced Hindu-Arabic system of numerals, [was] gained through Fibonacci’s commercial connections in North Africa and the Levant… It must be remembered that Fibonacci’s home city-state of Pisa had an extensive mercantile fleet operating in, and beyond, the Mediterranean to Byzantium.

A. F. Horadam [review of] “Fibonacci’s Liber Abaci”: a Translation into Modern English of Leonardo Pisano’s Book of Calculation by L.E. Sigler (Springer 2002).

If the hand which wrote that ‘4’ form in the Voynich manuscript was accustomed, already, to write the numeral in that way, the probability is strong that he (and ‘he’ is statistically more likely) is more likely than not to have belonged to a social and intellectual class beneath that of Latin Europe’s political and learned elites and to have had a direct link to the interests of those who were either engaged in the type of maritime trade that brought exotic goods (termed ‘spices’) from the Black Sea, Byzantium or ports of Egypt and North Africa, into Italy or, on the other hand, in naval service as was Michael of Rhodes.

In this context of multilingualism, sea-journeys, trade, exotics, favoured nation status and scripts, I think I should here again quote from a late fifteenth-century account that I quoted first some time ago when considering the possible implications of Baresch’s phrase “‘artis thesauros medicae Aegyptiacos”. In the present case it is especially relevant to note which maritime cities had favoured status in the ports of Egypt, and related issues of multilingualism and translation in such exchange. And, of course, resources for any possible alphabetic substitution cipher.

We have already seen how casually the author of one zibaldone refers to the trade in exotics from Alexandria as example for a problem using the ‘new math’ and Michael of Rhodes’ use of that simple ‘4’ shape for the numeral before 1440.

In Alexandria I saw four large fondaks [warehouses, Lat: thesauri], one for the Franks and another for the Genoese .. and two for the Venetians..

re: Misr [Cairo].I swear that if it were possible to place all the cities of Rome, Milan, Padua and Florence together with four other cities they would not, the whole lot of them, contain the wealth and population of Misr, and this is true…

In Misr there are many fondaks … a thousand and more warehouses in each fondak.. There is nothing in the world that you do not find in the fondaks of Misr…

If you ask how I could converse with the interpreter [when in Misr].. the interpreter is of Jewish descent and came to Misr to return to Judaism, because he is a Spaniard.. He knows seven languages – Hebrew, Italian, Turkish, Greek, Arabic, German and French.   ..

The Karaites’ script is different from all others, and they have not the letters ayin, he, aleph, or het, bet, tsade. .. {The Hebrew alphabet uses 22 letters; the Karaite thus only 16.]

from a Florentine ms. translated in  Elkan Nathan Adler, Jewish Travellers (801-1755), London: Routledge (1930) pp. 156- 208. cited passages p.162; 166-7; 171. First cited in connection with Voynich studies in D.N.O’Donovan, ‘ ” …thesauros Artis medicae Aegyptiacos” Pt1’, voynichimagery (blog), July 6th., 2013. The account is by Meshullam Ben R. Menahem of Volterra, in 1481 AD.

That account was given by a Jew of Volterra in 1481. The map below shows it in relation to Genoa, and to Florence, the cities with which the rest of this post will be concerned.

  • What is known from the records about the Jews of Volterra is reported in the Encyclopaedia Judaica under ‘Tuscany‘.

If indeed there is anything of Kabbalism in the Voynich manuscript, it is most likely to have come from the south-western Mediterranean and there is no necessity to explain its entering the Latins’ mental horizons by attributing any knowledge of Kabbalah to Ramon Llull. The reasonable explanation is that since Kabbalism was Jewish, knowledge of it was conveyed across the religious divide by Jews and was by them directly explained to a few Latins – willingly or otherwise – by refugees, corresponding scholars, Morescos and/or as newly-created converts serving as translators. The great wave of assaults against the southern, Sephardi Jews in 1391 finds a parallel increase in Jewish presence in Italy, Dalmatia and elsewhere.

An example may be in order before moving to consider the ‘commercial math’ classes in fourteenth- and early fifteenth century Italy and two Italians named Paolo, one of whom lived in the fourteenth and the other in the fifteenth century.

Example – Ha-Kohen and Lippomani, and a fifteenth-century hand.

We know, certainly, that one Italian ‘renaissance’ scholar living in Venice before 1430 wished to learn both classical Hebrew and the dialect of the Moriscos or Arabic-speaking Spanish Jews, the dialect known as Judeo-Arabic. We still have letters explaining the ‘grammar’ of Judeo-Arabic, the teacher being one Isaac ha-Kohen, a resident of Syracuse in Sicily and the student Marco Lippomani. A typically snide remark by Filelfo allows Kokin to date this exchange to a period before the 1430s; that is to the years in which the Voynich manuscript was made.

  • Daniel Stein Kokin, ‘Isaac ha-Kohen’s Letter to Marco Lippomano: Jewish-Christian Exchange and Arabic Learning in Renaissance Italy’, The Jewish Quarterly Review, Vol. 104, No. 2 (SPRING 2014), pp. 192-233.

Here again, I might mention that the form of the Voynich month-folios’ month-names was argued Judeo-Catalan by Artur Sixto. I quoted that comment which Sixto originally left at Nick Pelling’s ciphermysteries in an earlier post to this blog (here).

Script in a different fifteenth century Hebrew manuscript referenced by Kokin is shown below (n.114).

______________________

Fibonacci and Commercial maths.

To illustrate how the influence of commercial maths schools would expand in parallel with the rising importance of the merchant classes between the time when Leonardo of Pisa produced his ‘Liber abaci’ and when the Voynich manuscript was made, I take the works od two men named ‘Paolo’. Born a century apart. both were mathematicians whose careers flourished in Florence.

The first was born in Datini’s city of Prato about forty years after Leonardo (Fibonacci)’s death. The other was Florentine by birth. He, being born in 1397 and living through the period when the Voynich manuscript’s vellum was made (1404-1438) is of especial interest for us.

This second Paolo died a year after that description was written of the situation for traders in Alexandria and Cairo.

Two men named Paolo.

The first would be known best by his nicknames: “Paolo dell’Abbaco” and “Paolo the Surveyor” but his name was properly Paolo Dagomari.

In early adulthood he moved from Prato to Florence where for many years he taught ‘business math’ classes from the Trinity church in Florence.

In this case, as in many others where the term ‘school’ or even ‘academy’ is used, it is wrong to imagine a dedicated building like a modern school or college. We should think rather in terms of adult education classes where all that is needed is a person willing to teach and a group of voluntary students – or more-or-less voluntary depending on their age and the degree to which parental wishes were law.

Dagomari’s students were merchants and their sons. His basic text, as his nick-name suggests, was probably the Liber abaci, and by 1374-5, when Cresques’ world map was being created in Mallorca, Dagomari died in Florence having become by then a close friend of Boccacio and having seen 65,000 students pass through the course he offered in his Trinity Church ‘school’. We know that Francesco di Marco Datini, by then resident in Papal Avignon more than fifteen years, had also gained his education in commercial math in Florence, but there is no doubt at all that Dagomari taught the son of Dante Alighieri.

That connection to Dante is significant, for Dante also addressed himself to that ‘second tier’ in society, writing in the vernacular and not in Latin.

Nonetheless, Dante’s imagined journey though Hell, Purgatory and heaven in the Divine Comedy is a navigator’s sky-path along those “high roads of the sea” (to use Majid’s beautiful phase, which saw seas above and below the horizon).

As Gunter reports, Dante included in an early copy a parallel list of of Latin and of the increasingly-used ‘Arab’ star names in order that – in Dante’s words – those without Arab instruments might still follow the paths.

It is in an early copy of the Divine comedy, one probably made in Genoa, that we find certain characteristics unusual for formal art in medieval Latin Europe, but which come close to how the ‘ladies’ are represented in the Voynich calendar and ‘bathy-‘ sections.

(detail) from Bodleian Library, MS Holkham 48 p.4. Place of manufacture given as Genoa or Milan. Dated 1350–1375 AD. The text is described as written in a ’rounded Italian gothic hand’.

Points of similarity seen in this particular detail and figures in the ‘ladies’ folios of the Voynich ms include over-large heads, and relatively slender lower limbs. The ‘renaissance’ view of the human body was still unknown to that draftsman but occasionally, in the Voynich manuscript, its later date and probable Italian provenance is evidenced by a copyist’s slip which sees an occasional figure drawn little more shapely than the rest, and more shapely that the figure ought to be.

The differences between images in that copy of Dante’s Divine Comedy and the Voynich manuscript would make long and tedious list – too long to be included here – but with regard to these bodies, an obvious difference is that in the illustrations for the Divine Comedy, the figures are to be read as ‘people’ albeit souls, and while some effort is made to avoid emphasising male genitalia, they are drawn – whereas they are not in the Voynich manuscript’s images. And while in one sense the Voynich ms’ anthropoform figures might be regarded as ‘star-souls’ and/or as the soul of a given place, there is no evidence of intention to have them represent specific people. Of course, in this, if the labels are ever read, it may be that someone at some stage did associate each with some historical character. We shall have to wait and see.

Overall, too, we have very different vocabulary of gesture in these two work, and a very different approach to use of the ‘speaking gesture’.

As you’d expect, images in western Christian manuscripts are saturated with western Christian Europe’s two great pre-occupations (one might say obsessions) – organising everything in the universe into hierarchical rankings and then defining any person, thing, or quality according to whether its assigned ‘place’ is higher or lower than that accorded another. Ask a learned medieval scholar whether composing music was a ‘higher’ or ‘lower’ activity than designing a building and he’d surely have an answer. The disease of ‘class consciousness’ in Europe was not limited to the usual social classes, but it would allow a musician to look either up or down on any architect to whom he might be introduced.

That the Voynich manuscript is so glaringly devoid of such signals in its imagery is one among the many indications that the origin of its content is owed to persons and regions outside the Latin Christian domains.

There are no royal thrones, no horses, no military uniforms (save perhaps one Roman or Genoese ‘kilt’ on folio 80v). There are no figures of clerics, nor of kings. There is not a single chair to denote the teacher, nor any throne to denote royalty. Such costumes as are painted over the figures belong to a late stage of their evolution, as is also true for the cross-topped Byzantine style crown given one of the ‘ladies’. But the most resounding absences are the halo and the horse.

It is those details which are not there which have, for a century, reduced persons attempting to read the Voynich manuscript’s imagery to speculation, imagination and theory-driven narratives, attempting to assert the opposite of what any external and dispassionate scholar would say, and that many have said, viz: that they’ve seen nothing like it in the corpus of western Christian works, including the medical and alchemical texts.

In the detail shown above, the figures’ gestures are more limited in range than those in the Voynich manuscript but do (of course) speak directly to the conventions of medieval Latin Christian art, signalling such sentiments as pleading, despair, grief, remorse for sin and so on.

Gestures in the Voynich manuscript are more energetic, and the figures differently adorned with veils and classical headdress, their gestures so far outside the set of those employed in medieval western Europe’s Christian art that their meaning is still, most often, expounded only from a writer’s imagination, rather than from results of any wider horizon in their research.

One among the very few exceptions to the ‘theory-first’ approach was Koen Gheuens’ investigation of where and when we find other examples of the ‘deformed lobster’ in Europe after about the thirteenth century. He did not attempt to discover any earlier instances or define its time and place of first origin.

Despite such things, that detail from the early copy of Dante’s poem deserves our consideration, because it appears in manuscript made during the period of interest to us (1350–1375 AD); is attributed to northern Italy and probably to Genoa, one of the major maritime centres of Italy at that time.

I am NOT suggesting any direct or indirect connection between content in the Voynich manuscript and Dante’s poetry. Such a scenario was espoused, as I recall, in about 2008 or so, since when Dante’s name as been often invoked, and then dropped, and picked up anew, and dropped again in that peculiar parentless style of Voynich theories. If you’d like to re-create the lineage for that idea, you might begin from references in d’Imperio, then search ‘ciphermysteries’ and from there go through the archives of Jim Reeds’ mailing list. Unfortunately, though Rich Santacoloma promised a couple of years ago to do the same for that mailing list since the early 2000s, he has not yet found occasion to do so.

Paolo dal Pozzo Toscanelli

Generally believed indebted to Dagomari’s mathematics, the Florentine Paolo Toscanelli belonged to a family whose members were ( I’ll quote the wiki biography) “traders in eastern luxury goods (‘spices’) and who thus traded regularly with north Africa, Egypt and the eastern Mediterranean”.

He offers one of the clearest examples of a bridge between commercial maths, surveying, astronomy, cryptography and theology and, in terms of Italian society, between the ‘commercial class’ and the rulers’. For a time he collaborated with the Genoese Gian Battista Alberti, a figure of particular interest for cryptographers so I hope readers will forgive another digression, this time to consider Alberti.

Born in Genoa in 1404, Alberti moved to Florence but his career developed late being suppressed until 1446 by the fame of Brunelleschi. Alberti, like his elder, worked chiefly for what one writer has called the “high bourgeoisie” and brought to bear the same practical and commercial mathematics on which the ‘abacco’ schools focused.

That the range included problems of mapping is evident from the nickname given Dagomari as ‘Paul the Surveyor’ and though our later example, Paolo dal Pozzo Toscanelli is usually described as a ‘cosmographer’, his wiki biography quite rightly says that “astronomy was a close science to geography at that time”.

We know that Toscanelli was also a competent cartographer, because in 1474 he produced a map which argued for a fairly easy run westwards by sea from Spain or Portugal to ‘Cathay’. What is fascinating about that map and the accompanying letter (the map itself is now lost) is not so much their influence on the rulers of Spain and thus on Christopher Columbus, but that Toscanelli speaks of having had access not only to Ptolemy’s works but to those of the Phoenician ‘Marinus of Tyre’ – the original source which, as Ptolemy himself says plainly that he had simply edited and updated a little. Could Toscanelli have meant it? Was there still to be had a copy of the original work in Greek or in translation?

Practicalities.

Ever since Wilfrid Voynich presented the public with the manuscript and his own imaginative description and ‘history’ for it, the attitudes and assumptions of cryptographers have greatly influenced both how the manuscript was imagined and what approaches have been taken in attempting rightly to read both its written- and its pictorial text.

It is perfectly normal and understandable – part of standard method – ift a cryptographer should consider any text as a source from which to extract a body of quantifiable data, and then to engage in a process of creating a theory and considering nothing but that theory and how well it suits his or her data-base. It is natural for the cryptograper to presume a written text deliberately made opaque, and to presume that ‘underneath it all’ there should be a nice, clear literary ‘plaintext’.

Unfortunately, once the Friedmans had effectively co-opted the manuscript’s study and assumed all other sorts of research inferior and thus necessarily at the service of their own, they created a model which not only proved fruitless in their own case, and despite 30 years efforts, but has proven equally fruitless when adopted at large by Voynich theorists who were not concerned with the written text or issues around cryptography.

The lack of balance in Friedman’s attitudes – towards the manuscript and to the work of specialists in manuscript studies, as in the history of western art – continues to affect approaches to the Voynich manuscript to this day and is particularly noticeable within that ‘bible’ of the Voynich traditionalists, Mary d’Imperio’s Elegant Enigma.

It became the norm, from the early 2000s, to behave as if not only the written part of the Voynich text were ‘encrypted’ but as if everything in it were.

The cryptologists’ method was then generally adopted, that is, of first hitting on a ‘theory’ – a speculation as desired solution – and then hunting for ways in which to present that speculation as being sufficiently supported by evidence (evidence sought only within the parameters of that speculation) to deserve description as ‘plausible’ by persons who had no greater knowledge of medieval manuscripts, art, cryptography or scripts than did the person attempting to be voted ‘right’ as if by simple-headcount, social-media style.

It has not been so much as case of the blind leading the blind as of researchers first selecting a set of blinkers and then congregating according to the colour of those blinkers.

Much baseless ‘doctrine’ has resulted on the basis that it’s “widely accepted”, to the point where I was present when the edict went out (as the usual authorative-sounding but anonymous ‘meme’) that it was ‘unnecessary’ to consider any sources save fifteenth century German manuscripts.

On another occasion, the ‘meme’ asserted, in effect, that a scholar’s whole body of research might be ignored because they hadn’t made enough wisecracks.

In a very small way, I can see what that last meme was about.

Whatever their flaws and historically-inappropriate assumptions and limits, the cryptologists never treat the text casually, or produce theories to suit a popularity contest – tossing off nonsense with a gay grin and self-deprecating wisecrack.

On the other hand, I wish they would lighten up a little and give more thought to the ordinary experiences of people in the medieval world. You don’t need to find clues to Alberti’s creation of his cipher-wheel by turning to high society and theology, to Ramon Llull or Kabbalah, to understand how such an idea might have occurred to him. Why should he have had it from anyone else, or anyone in particular?

The underlying principle of such ‘revolutionary’ things as gridding maps ‘by the Rose’, diagrams associated with Kabbalism, developments in Italian architecture which brought fame first to Brunelleschi and then to Alberti, or indeed Alberti’s wheels are fairly simple and embodied in activities as old as human settlement. In this case, the construction of things formed as domes, or as globes.

Alberti didn’t have to know Llull, nor Llull to know anything of Kabbalah, though architects might well need to do, as Alberti did, and see how the dome-makers of Hagia Sophia work out problems of load-bearing and materials.

Dome, and domes composed of lattice work are still made today, just as they were in ancient and even in prehistoric times, but especially where it was important to keep watch over ripening crops.

This is how it’s done. In an area where some plant grows that produces long, flexible stems or branches, you cut and make a pile of them.

Then you trace a circle on the ground and, at regular intervals around one side, press a withy or ‘wand’ firmly into the ground. That’s the wand’s rising point.

Now, directly opposite each, around the other curve, you insert the free ends.

The lattice-pattern will appear as shadows on the ground so encompassed. At night, within the shelter, you will see the heavens ‘gridded’.

Of course, you can the cover the basic lattice, if you like, with whatever you like – fabric adorned with stars and stretched out as a tent, or something more substantial such a pise or plaster. It is not co-incidental, I think, that domes from China to the far west were customarily painted with images of the night sky.

The example shown below was made of willow wands in modern-day America.

If you need something placed at a given point around the ‘horizon’, you can nominate each space or each point with a letter, or a number, or the name of a real place on earth or (by the example of Majid’s compass-rose) by the names of stars.

But if such a dome is meant to evoke, or to represent the heavens as a dome, the question then naturally arises about how the points of that circuit actually connect with the initially matching points about the earthly horizon, when earth is imagined always stationary yet the sky perceived as wheeling over it, year by year.

As a mathematical and surveyor’s problem, that one is among the meanings embodied in this famous image of Roger Bacon.

Bust of Roger Bacon in the Oxford University Museum of Natural History. Photograph taken by Michael Reeve 2004.

Any such problem, expressed algebraically, must begin by having one specific unknown position defined as ‘x’. What Alberti’s wheel does, in effect, is have a series of circular points of correspondence defined not as a series of ‘x’s, but as a pair of alphabetic series.

I’m not saying that this was the original purpose of Alberti’s ‘wheels’ – I’d be more inclined to think that as a mathematician his interest in ‘unknowns’ had allowed his attention to shift from purely mathematical ‘unknowns’ to issues of encrypted documents.

My point is that there may be immediate and very practical observations, rather than reliance on important figures of European history, to explain his development of those cipher-wheels and much else impacting on ideas about the Voynich manuscript – such as that the imagery must be illegible in terms of western European conventions because deliberate made obscure rather than – as I hold to be the case – because it didn’t spring from those traditions in the first place.

By the middle of the fifteenth century, the construction of domes and the idea of ‘significant number’ were both hot topics in Florence but as you see, the basics of Brunelleschi’s famous dome in Florence are pretty much the same as the rural domed shelter which country-people everywhere have been making – and I’m speaking literally – from before the first cities were built. What made their monumental versions different was an ability to do the math.

The death of Brunelleschi in 1446 brought to the fore Leon Battista Alberti (1404-1472). Like Brunelleschi’s, Alberti’s career had long been delayed…

*Franklin, N.J., Borough schools, ‘Architecture of Brunelleschi and Alberti ..’ (pdf). I reecommend this as a very good first guide to works of Brunellleschi and Alberti online as a pdf, but one worth reading even if you’re well acquainted with their work.

And so, back to Toscanelli..

He appears in a Florentine fresco placed beside the ‘Greek-Syrian’ neo-Platonist Marsilio Ficino – one must always place close attention to the headwear given figures in Latin Christian art.

Here, Toscanelli wears the head-dress of a ‘Moresco’ and his facial features appear to have been painted so as to suggest to emphasise foreign and/or Jewish antecedents. ‘Moresco’ was a termed used, as said above, to describe those Spanish Jews who still spoke and read Arabic, and/or such dialects as Judeo-Catalan or Judeo-Occitan.

I can’t compress Toscanelli’s story better than did the author of one wiki article:

Thanks to his long life, his intelligence and his wide interests, Toscanelli was one of the central figures in the intellectual and cultural history of Renaissance Florence in its early years. His circle of friends included Filippo Brunelleschi, the architect of the Florence Cathedral, and the philosopher Marsilio Ficino. He knew the mathematician, writer and architect Leon Battista Alberti, and his closest friend was Cardinal Nicholas of Cusa—himself a wide-ranging intellect and early humanist, who dedicated two short mathematical works in 1445 to Toscanelli, and made himself and Toscanelli the interlocutors in a 1458 dialogue titled On Squaring the Circle (De quadratura circuli).

wiki article, ‘Paolo dal Pozzo Toscanelli’

In one sense, ‘squaring the circle’ was not a ‘modern-ancient’ problem but one long addressed and resolved in terms of Christian theology in the west.

It was precisely how Rhaban Maur had managed to introduce ‘pagan’ Euclid into an extremely conservative monastic environment during the ninth century. His copy of Euclid had probably arrived with a recent Nestorian embassy from Baghdad, the same eastern Christians having only recently translated it into Arabic.

In what follows, I’m not only quoting matter I’ve quoted in treating the Voynich manuscript, but which I’d quoted even longer ago in connection with other medieval European images, but since I can’t just collapse the text and make it optional, here it is.

—–

Maur began by formulating the quaestio, or problem by setting it as a problem about interpreting the Psalter correctly. Thus he begins,

“It is well that we should enquire what the Psalmist means by the circle of the earth and why, in several other places, he says that the earth is comprised of the same figure. On the other hand, in the 106th Psalm [Vulgate numbering: Ps cvii.3] he comprises the earth under four cardinal points… A very similar statement appear[ing] in the Gospel where it says: He shall send his angels with a great sound of a trumpet and they shall gather together from the four corners of the earth”

and so, having made Euclid an aid to theology, Maur continues:

Whence it is fitting to enquire how far the quadrate and circular shapes of the earth can agree, when the figures themselves, as geometricians maintain, are different. The Scriptures call the shape of the earth a circle for this reason: because to those who look at its extremity [i.e around the horizon] it always appears as a circle. This circle the Greeks call a horizon [a word meaning ‘belt’ or cord], signifying that it is formed by the four cardinal points; these four points signify the four corners of a square contained within the aforesaid circle of the earth.

Maur understands the horizon line as a knotted cord, resembling a monk’s waist-cord with its knots, and akin to the surveyor’s measuring-cord, also knotted at intervals and worn in the same way about the waist when not in use. But the geometric figure Maur has just begun to describe is the ancient figure for the world in microcosm. He suggests as much, speaking of the ‘Eye’ as simultaneously urbis (city) and orbis (orb/circle).

So then, taking East as his primary point, just as medieval Europe’s mappamundi did, but as Cresques’ map and the Voynich map do not, Maur locates the heart of the world as the microcosmic ‘city’ saying:

  • For if you draw two straight lines from the East, one to the south and one to the North, and in the same way also draw two straight lines from the Western point, one to each of the two aforesaid points, namely the south and north, you make a square of earth within the aforesaid circle. How this aforesaid square (demonstrativus quadrus) ought to be inscribed within the circle, Euclid clearly shows in the Fourth Book of the Elements.”

And since I’ve been once more obliged to quote from my own work I’ll add here two images which I included in a post published at voynichimagery in 2017. Details of my source, which was not speaking about the Voynich manuscript, were given at the time as seen in the images below. Today, however, that address is no longer current, though the second image (still dated October 25th., 2012) can be seen, with commentary, at luwanarch .wordpress com. This is how Alberti mapped Rome.

Again, about methodologies and Voynich research –

Between 1912 until 2012 or thereabouts, the most commonly seen approach, among those hoping to ‘solve’ the manuscript was to ignore the codicological evidence, the palaeographic evidence, the materials’ evidence, all earlier independent specialists’ opinions, and interpret the images only if and in a way compatible with their initial theory, often a theory naming some prominent European as ‘author’, effectively re-defining the manuscript as a slab of written text, which despite being lavishly illustrated, was all designed by one mind to deceive.

The common practice of ignoring the manuscript’s own testimony in favour of promoting a Voynichero’s pet theory reached its peak of absurdity about three or four years ago, when another of those borne-on-air sort of memes asserted that, since the written part of the text was also now to supposed a mere nonsense – a joke of some kind – that thus might also be treated as being quite as ‘irrelevant’ as the manuscript’s images, codicology and palaeography, so everyone should just adopt one of the most promoted theories as if it were of more substance than the manuscript itself.

It’s no wonder that Beinecke MS 408 needs its few friends. Who in the world would put up with such treatment being accorded the Vienna Dioscorides, or the Book of Kells, or any other important and apparently unique manuscript?

After which grumpy remark, I propose we adjourn for now.

Gd and the weather willing, I’ll be back in 3-4 weeks’ time.