Since posting about Voynichese’s strange single leg gallows behaviours a few days ago, I have continued to think about this topic. On the one hand, it’s clear to me how little of any genuine substance we actually know about how they work; and on the other, I’ve been wondering how I can start some broadening conversations about them (by which I mean ones that ask more questions than they answer).

As today’s experimental contribution, I’m going to write a post listing a load of the questions I have in my head to do with single leg gallows but without really trying to answer any of them. I can’t tell how this will work, but here goes regardless. 🙂

Incidentally, for anyone who wants to run their own statistical experiments on single leg gallows, I would strongly recommend using Herbal-B + Q13 + Q20 as their basic test corpus, because I’m acutely distrustful of any Voynich stats that combine Currier A and Currier B. Even though I’m basically doing the latter here. 😉

Questions: final flourish

Rather than finishing with a second vertical leg on the right hand side, single leg gallows instead cross over the left hand leg and finish with a slight flourish to the left. This final flourish can be (1) short, (2) long and straight, or (3) long and curved (i.e. finishing with something like an EVA c-shape).

1. Have the variations in the finishing flourish of single leg gallows been catalogued and/or transcribed?
2. Are these variations found uniformly throughout the manuscript, or are they strongly correlated with the various scribal hands (as recently proposed by Lisa Fagin Davis)?
3. If they have been transcribed, is each flourish type statistically associated with any neighbouring textual behaviours (e.g. contact tables, etc)?

Questions: followed by EVA e?

One huge difference between single leg gallows and double leg gallows is that non-struckthrough single leg gallows are very rarely followed by EVA e. If you count strikethrough gallows separately from normal gallows, the statistics are quite, umm, striking:

• k:ke = 9758:3809 = 39.03%
• t:te = 5802:1748 = 30.13%
• p:pe = 1383:5 = 0.36%
• f:fe = 416:3 = 0.72%
• ckh:ckhe = 876:242 = 27.63%
• cth:cthe = 905:190 = 20.99%
• cph:cphe = 212:64 = 30.19%
• cfh:cfhe = 73:14 = 19.18%

Moreover, looking at the eight instances in Takahashi’s transcription where EVA p and EVA f are followed by EVA e, I suspect that many of these may well be transcription errors (i.e. where Takahashi should have instead written EVA pch / fch).

Hence it seems to me that Voynichese has a secret internal rule that almost completely forbids following EVA p and EVA with EVA e. This is a massively different usage scenario from EVA t / EVA k (which are followed by EVA e 39.03% and 30.13% of the time respectively).

OK, I know I said I was only going to ask questions in this post, but looking at these numbers afresh, I can’t help but speculate: might it be that EVA p/f are nothing more complex than a way of writing EVA te/ke?

1. Has anyone looked closely at the eight places where pe/fe occur?
2. Why is there such a huge difference between pe/fe and the other six gallows?
3. Might this be because EVA p and EVA f are optional ways of writing EVA te and EVA ke?
4. Has anyone considered this specific possibility before?
5. How similar are the contact tables for EVA te/ke and EVA p/f?

Questions: Followed by EVA ch?

Similarly, comparing the stats for instances where gallows are followed by the (almost identical looking) EVA ch glyph reveals more differences:

• k:kch = 9758:1074 = 11.01%
• t:tch = 5802:975 = 16.80%
• p:pch = 1383:733 = 53.00%
• f:fch = 416:190 = 45.67%
• ckh:ckhch = 876:5 = 0.57%
• cth:cthch = 905:3 = 0.33%
• cph:cphch = 212:1 = 0.47%
• cfh:cfhch = 73:0 = 0.00%

Here, we can see that both p and f are followed by ch about half the time (53% and 45.67% respectively), which is significantly more than for k and t (11.01% and 16.80% respectively).

At the same time, the dwindlingly tiny number of places where strikethrough gallows are followed by ch (only nine in the whole manuscript) again raises the question of whether these too are either scribal error or a transcription error.

As an aside, I previously floated the idea here that c + gallows + h may have simply been a compact (and possibly even playful) way of writing gallows + ch, which would be broadly consistent with these stats.

1. Is there anything obviously different about Voynichese words containing EVA kch / tch and Voynichese words containing EVA pch / fch?
2. Has anyone looked in detail at the eight instances where strikethrough gallows are immediately followed by EVA ch?
3. If you remove paragraph-initial p- words from these stats, do the ratios for p:pch and f:fch settle down closer to the ratios for k:kch and t:tch?
4. How similar are the contact tables for EVA tch/kch and EVA cth/ckh?
5. How similar are the contact tables for EVA tech/kech and EVA pch/fch?

Questions: Double Leg Parallels?

Some researchers (perhaps most notably John Tiltman, if I remember correctly) have wondered whether EVA p / f might simply be scribal variations of (the much more common) EVA t / f.

1. Beyond mere visual similarity, is there any actual evidence that supports this view?
2. I would have thought that the pe/fe stats described above would have meant this was extremely unlikely, but am I missing something obvious here?

Questions: Paragraph-Initial?

Yes, single-leg gallows (mainly EVA p) are very often found as the first letter of the first word of paragraphs. But…

1. How often do single leg gallows (and/or strike-through single leg gallows) appear in the first word of a paragraph but not as the very first letter of the word?
2. Do these these paragraph-initial -p-/-f words show any pattern?
3. Are there structural similarities between paragraph-initial p-/f- words and other paragraph-initial?
4. Might there be some kind of numbering system embedded in paragraph-initial p- words (particularly in Q20)?

Questions: vs Double Gallows?

Yes, single-leg gallows are to be found mainly in the top line of paragraphs, but that’s imprecise and unscientific.

1. Are the number of gallows characters (whether single or double) per line roughly constant for both the first lines of paragraphs and for the other lines of paragraphs?
2. Do these statistics change between sections?

And Finally…

Please feel free to leave comments asking any other single leg gallows questions, I’m sure there are plenty more that could sensibly be added to this page. 🙂

All answers happily received too. 😉

Anyone who proposes that Voynichese works in a ‘flat’ (i.e. straightforward) way has a number of extremely basic problems to overcome.

For a start, there are the Voynichese’s ‘LAAFU’ (Emma May Smith’s acronym for Captain Prescott Currier’s phrase “Line As A Functional Unit”, though she now prefers to talk about “line patterns”) behaviours to account for. These relate to the curious ways that letters / words work both at the start of lines and at the end of lines, many of which are discussed by Emma May Smith here:

• Line-first words have a quite different first-letter distribution from the main body of words’ first-letter distribution
• Line-first words are slightly longer than expected
• Line-second words are slightly shorter than expected
• Line-final words frequently end in EVA ‘m’ / ‘am’

At the same time, there are also some odd PAAFU (“Paragraph As A Functional Unit”) behaviours to consider. The most famous of these is the way that the first letter of a paragraph (and even more so of the first paragraph on a page) has a significantly different distribution from elsewhere, one that strongly favours gallows characters (and in particular the single leg gallows EVA ‘p’).

But the other major PAAFU behaviour is that single leg gallows glyphs appear predominantly on the first line of paragraphs, and only rarely elsewhere (these are known as Tiltman lines, after my hero John Tiltman). You can see this throughout the Voynich Manuscript, right from Herbal A page f3r…

…to the Herbal B page f43r (which has an extra single leg gallows, but the remaining ones all sit on the first line of their respective paragraphs)…

…to the Q13 Balneo page f76v (where there are two extra single leg gallows, sure, but the rest of the page slavishly follows the pattern)…

So, even though the internal structure of Voynichese words changes significantly across the different sections (and that’s a separate topic entirely), this single-leg-gallows-mainly-on-top-lines-of-paragraphs Tiltman behaviour seems to remain essentially constant throughout them all.

This is an issue that has been floating round for decades, and I would be surprising if it had originated even from John Tiltman. More recently, Rene Zandbergen discussed it on voynich.ninja back in 2017, pointing out that this behaviour appeared – in his view – to be inconsistent with any model for Voynichese that was inherently uniform (which I call ‘flat’ here), whether linguistic, cryptographic or whatever.

So, the challenge to anyone trying to come up with some kind of theory for the Voynichese text is simply to explain why this unexpected behaviour is the way it is. What kind of mechanism could be behind it?

Q20 Paragraph-Initial Glyphs

For the rest of this post, I’m going to restrict my discussion to the twenty-three Voynich Q20 (‘Quire 20’) pages, simply because their lack of drawings make them particularly easy to work with.

The first thing to point out is that we have two single leg gallow behaviours (very frequent at paragraph starts, and very frequent on the top line of paragraphs) which overlap somewhat.

For example, f103r (the first bound page of Q20), has 19 starred paragraphs, of which 9 begin with the single leg gallow EVA ‘p’ (i.e. 47.3%). And if you count all the paragraph-initial p’s and f’s in Q20, you get:

Page	p	f	paras
f103r	9	0	18
f103v	7	0	14
f104r	5	0	13
f104v	7	0	13
f105r	7	0	10
f105v	7	0	10
f106r	11	0	15
f106v	6	1	15
f107r	9	1	15
f107v	10	0	15
f108r	6	2	16
f108v	7	0	8
f111r	4	0	6
f111v	7	0	8
f112r	8	1	12
f112v	8	0	13
f113r	7	3	17
f113v	10	4	15
f114r	5	2	13
f114v	5	0	12
f115r	4	2	13
f115v	6	0	13
f116r	6	0	8
Total	161	16	292

The values for Q20 as a whole are remarkably consistent: there is a 161/292 = 55.14% chance that a paragraph starts with EVA p, and 16/292 = 5.48% chance that a paragraph starts with EVA f.

Given that ‘p’ makes up 1.03% of the glyphs in Q20 (‘f’ makes up 0.19%), ‘p’ is ~55x more likely to appear as the first glyph of a Q20 paragraph than it is to appear in any other glyph position: even ‘f’ is 28x more likely to appear paragraph-initial than elsewhere. That’s striking, and not at all flat.

Q20 Tiltman Lines

Q20 contains about 10700 words across about 1100 lines (I don’t have the exact figures to hand): 643 of these contain a single leg gallow, i.e. the raw chance any given Q20 word contains a single leg gallow = 643/10700 = 6%.

But whatever the explanation for p being so strongly biased to this paragraph-initial position, I think we should try to separate the single-leg-paragraph-initial behaviour from the single-leg-top-line (Tiltman) behaviour.

So if we remove the 292 paragraph-initial words, the raw chance that any non-paragraph-initial Q20 word contains a single leg gallow goes down to (643-292)/(10700-292) = 3.3%, which is our baseline figure here.

But what of top-line-but-not-initial Q20 words? Given that Q20 has 292 paragraphs, each with a first line containing (say) ten words, and we are removing the first word, we have 292 x ~9 = ~2628 top-line words of interest. Of these (by my counting), 353 contain a ‘p’, and 80 contain an ‘f’. Hence the probability that any given Q20 paragraph-top-line-but-not-initial word contains a single leg gallows is 433/2628 = 16.5%.

Similarly, the probability that any given non-top-line Q20 word contains a single leg gallows is roughly (643-177-433)/(10700-292*10) = 0.4%. So if we discount all the paragraph-initial words, words containing single leg gallows are about 16.5%/0.4% = ~41x more likely to appear on the top lines of paragraphs than on the other lines.

Q20 Neal Keys

One of the interesting things that has been noted about these single leg gallows on the top line of paragraphs is that they seem to often appear in adjacent words. This is something that Voynich researcher Philip Neal first mentioned in a Voynich pub meet a fair few years ago that he had noticed: at the time, I christened them Neal keys.

But even though this is a visually striking thing, is it statistically significant, particularly if we remove all the paragraph-initial single leg gallows first?

For non-paragraph-initial-top-line words, the raw (expected) probability that a pair of adjacent words both contain a single leg gallows would seem to be 16.5% x 16.5% = 2.7%.

My counts for the actual number of pairs of adjacent non-paragraph-initial-top-line Q20 words both containing single leg gallows (i.e. ignoring all paragraph-initial words) were 5/5/6/1/8/12/7/6/7/4/5/0/8/6/3/5/9/4/12/5/1/5/2 = 126 instances out of (353 + 80) = 433.

So, of the 292 x (9-1) = ~2336 potential adjacent pairs (discounting the end word of each top line), 126 instances points to a chance of 126/2336 = 5.4%.

So my conclusion from this is therefore that the phenomenon of Neal keys (pairs of words containing single leg gallows on the top line of paragraphs) is, while visually striking, only 2x the expected value.

To be clear, the phenomenon is definitely there, but the main factor driving it appears to be the very strong tendency for single leg gallows to appear on the top line of paragraphs, rather than the adjacency pairing per se.

Verification

I’ve done a lot of this manually, because I didn’t have sufficient automated tools to hand. So can one or more other Voynich researchers please verify these figures?

• I used the Takahashi EVA transcription
• I counted ch / sh / ckh / cfh / cph / cth as individual glyphs
• I didn’t count space characters in the percentages

Paolo Guinigi was Lord of Lucca at the start of the 15th century: the Lucca archives hold the Governo da Paolo Guinigi (“GPG”), a substantial collection of his correspondence from 1400 to 1430 (he died in 1432). Of interest to cipher historians is that some of this correspondence may well be enciphered.

[Incidentally, thanks very much to Mark Knowles for flagging this a couple of years ago, many apologies for not following up sooner. 🙁 ]

Covering a multitude of subjects and situations, the letters (both to and from Guinigi) are in Latin and Tuscan (“Volg.”). Helpfully, a transcription / summary of the letters made by Luigi Fumi and Eugenio Lazzareschi is downloadable on the Archivio di Stato di Lucca’s website.

There, certain groups of transcriptions have sections (occasionally single words) that are rendered in italics, which are typically to or from specific correspondents. Fumi and Lazzareschi note:

Furono composti in corsivo i passi che nell’originale sono in cifra, oppure distinti da segni convenzionali; la quale decifrazione, fatta co ‘1 sussidio del registro ufficiale della cifra del Guinigi, é stata fatica più di pazienza che di diligenza, come generalmente ogni laborioso ordinamento d’ archivio.

…which I (freely) translate as…

The italicized passages were written using either cipher or unconventional signs; decrypting these (even with the help of Guinigi’s official cipher key) was less to do with patience than with diligence, as is generally the case with laborious archival work.

I couldn’t see in Fumi and Lazzareschi where Guinigi’s “official cipher key” was to be found, but perhaps this will become clear before too long. 🙂

It’s not obvious to me if there are any fully enciphered letters in the GPG. Typical cipher security practice was to destroy letters that had been deciphered (probably by burning, I’d expect), so my guess is that what saved these particular letters was that they were only partly enciphered.

Unfortunately, I can’t see a single scan of a (partly or fully) enciphered letter from the GPG anywhere on the web to verify this (the world of digitization has yet to knock on Lucca’s door, it would seem). Perhaps others will have more luck than me. 😉

List of enciphered letters

There are various series of GPG letters that have italicized sections:

• [1404] Jacobo de Faitinellis {Roma} – 15, 17-18, 20-29, 31
• [1405] Jacobo de Faitinellis {Roma} – 33
• [1406] D. Dino ser Paci {Roma} – 35
• [1410] Iohanello / Iohanni Thieri – 724-726, 731, 738
• [1413] Guido da Pietrasanta, Nicolao da Moncicoli, Nicolao Arnolfini – 966
• [1418] Guido da Pietrasanta, Nicolao da Moncicoli, Nicolao Arnolfini -970

There are also two received letters from 1413 (both from Guido da Pietrasanta, Nicolao da Moncicoli, and Nicolao Arnolfini) on pp.482-483.

Knowing Mark Knowles’ interest in the Barbavara family, I’m sure he’ll be pleased to know that there is correspondence with Gian Galeazzo Visconti’s chancellor Francesco Barbavara (2, 6, 10, 92, 112, 139, 140, 149, 166, 296, 819), and also with Manfredo Barbavara (173). (Though note I have no idea if those particular letters were enciphered.)

According to a news item I found just now, Mary D’Imperio died on 28th May 2020 in Springfield VA, at the age of 90. The details were relayed by her cousin Robert G. D’Imperio.

Voynich researcher Don Hoffman visited her a few times in December 2019 at the nursing home she was in. He put together these notes on her life:

Mary Evelyn D’Imperio
Father – Dominic D’Imperio, born Biccari, Foggia, Italy, 31 August 1888 – 29 July 1965, sculptor, came to America in 1905, settled in Philadelphia, PA.
Mother – Edith Brownback Roberts D’Imperio, born Philadelphia, PA, 1902 – 1977, artist.
Parents married 20 June 1928.
Mary Evelyn D’Imperio born in Germantown, PA on 13 January 1930, an only child
High School – Germantown Friends School, Germantown, PA
College – Radcliffe, majored in comparative philology and classics, graduated 1950, Phi Beta Kappa
             – University of Pennsylvania, for second degree, this time in structural linguistics
She was recruited at her home by the US Government and underwent three days of testing there for her first job – was told by testers that she was one in a million both before testing and after successfully completing it.
Jobs – only one for her entire career – started working for US Government at NSA in 1951 as linguist and cryptanalyst, but thought of herself as a computer programmer – she had thought she was doomed to be a secretary, clerk, teacher or nurse before the government came calling.
She originally worked with an ATLAS I computer and developed a program for text use on computers called Text Macro Compiler (TEMAC) from 1960 to 1962, but got nowhere with male bosses with it because they couldn’t see a use for it & didn’t think it was worthwhile (and she thinks also because she was female and not forceful).
I think she is more proud of her TEMAC work than her Voynich Manuscript work (which she admits she has mostly forgotten).
After retirement she worked as volunteer with entomologist Dave Nickle at the Smithsonian Institution.
1987 to 2006 – frequent contributor to North American Breeding Bird Survey.
Traveled extensively worldwide for pleasure (but only to safe countries), often to bird watch.

In the Voynich field, D’Imperio was a quiet giant, who will always be well remembered for her (1976) book “The Voynich Manuscript – An Elegant Enigma“. I’m sad to hear of her passing. My thoughts are with her family.

Apparently it’s Voynich Art History trivia weekend here at Cipher Mysteries. First up is this and this, both prints of Master E.S.’s “The Visitation” that I found recently:

Though classily executed, this is clearly (I think) in the same family as Diebold Lauber’s couples and the Voynich Manuscript Virgo roundel couple.

Ex Libris

I also stumbled upon this nice ex libris at the front of a book owned by Auxiliary Bishop Melchior Fattlin of Constance (1490-1548) (and show me a blogger who doesn’t get a guilty kick out of occasionally linking to catholic-hierarchy.org and I’ll show you a big fat liar):

While eerily reminding me of the Voynich Aries zodiac roundel, this also makes me wonder whether the surname “Fattlin” might have some goat- or sheep-related meaning etc.

Banderoles

The other thing I’m wondering about today is banderoles (aka “speech scrolls”). These started as ornate scrolls filled with text in drawings and paintings, more or less equivalent to modern speech bubbles (e.g. the former by the angel Gabriel, the latter by Garfield).

In the 15th century, these were a favourite of the Master of the Banderoles (active 1450-1475), who Wikipedia rather sniffily describes as a “crude” and “clumsy” copyist of Master E.S. and Rogier van der Weyden.

Here’s a much nicer example from Paris, BnF, lat. 11978, roughly 1450-1472:

Why am I interested in banderoles? Because I can’t see anything that better describes the lines of text spiralling out both from the inverted T-O map and the wolkenband on Voynich Manuscript f68v3.

Codicologically, my suspicion here is that the drawing f68v3 came from was itself derived from a French (specifically Parisian) original, but that that predecessor had only had the four seasons’ banderoles added. The extra four banderoles seem to have been added here as an additional construction layer. That is, I suspect that if you looked under a microscope at the boundaries where the extra four banderoles join on to the wolkenband, you would see the marks where the wolkenband was drawn but then erased to add in the extra four banderoles.

Having said that, I haven’t yet found a single fifteenth century astronomical drawing with banderole-style annotation. Perhaps this is something we should be looking for.

As should be clear from the last few posts here, my Voynich research focus has recently turned to the wave of astronomical instruments that appeared in the German-speaking lands in the first half of the fifteenth century.

The person behind much of this wave would appear to be John of Gmunden (AKA Johannes von Gmunden, Johannes de Gamundia, etc) (c.1380 – 1442), but I’ll return to him in more detail in a separate post.

Even though I’ve been looking mostly at theorice planetarum of late, I’m also interested in the nocturlabe / nocturnal / sternuhr (‘star clock’), which similarly appeared in the 15th century. Even though the earliest known description of the astronomical mechanism behind this was written by Raymond Llull, the first actual nocturnals started to be built in the fifteenth century.

Hence I’ve long wondered whether the curiously-repetitive circular diagram on the Voynich Manuscript’s page f57v might actually be describing a nocturnal in some way. Yet the practical problem with pursuing this further was that I was lacking a good reference for the very early (fifteenth century) history of the nocturlabe.

Ernst Zinner’s Sternuhr History

This was exactly the point where Ernst Zinner’s (1956) Deutsche und niederländische astronomische Instrumente des 11.-18. Jahrhunderts landed heavily on my doorstep. (Though I bought it second-hand, it was actually from the Adler Planetarium, which was a nice coincidence).

Zinner outlines the history of the Sternuhr on pp. 164-166, but given that our focus here is the fifteenth century, I’ll only transcribe (and lightly HTMLize) p.164.

You’ll need to know that Zinner refers to manuscripts and objects by their index number in Zinner (1925) “Verzeichnis der astronomischen Handschriften des deutschen Kulturgebietes“: astronomy historians typically call these ‘Zinner numbers’ (e.g. “Zi 3593”).

Oh, and you’ll also need to know the names of the stars in Ursa Major (despite having an Astronomy O-Level, I only knew Dubhe):

• α – Dubhe
• β – Merak
• γ – Phecda
• δ – Megraz
• ε – Alioth
• ζ – Mizar
• η – Alkaid = Benetnasch = Benenaz = the star right on the end of the plough handle

Finally: Dubhe and Merak were known as the ‘runners’ (Cursores) or ‘brothers’ (Fratres), that point towards Polaris, the Pole Star.

First paragraph…

Die Sternuhr, auch Nachtuhr = horologium noctis = noctilabium = nocturnalis gennant, wurde in Frankreich erfunden [149 d S.8] und von Raimondo Lullo in seiner Arte de navegar 1295 beschreiben [Opera omnia, Mainz 1721]. Er verwendete den Polstern und die Fratres genannten Sterne des Großen Bären. Das Gerät wurder im Kreise des Schülers Johanns von Gmunden in Wien verwendet; denn die 1438 beendete Abschrift von Gmundens Arbeit über das Astrolab [253 Nr. 3593] enthält einen Hinweis auf die Sternuhr mit der Verwendung von Polaris und Dubhe. Die Sternuhr besteht aus einer runden Scheibe mit einem Loch in der Mitte, um das sich einige Scheiben und ein über die Scheibe hinausreichender Zeiger bewegen lassen. Durch das Loch wird der Polstern beobachtet und der Zieger auf die beiden Hinterräder des Großen Bären, bezeichnet als die Läufer (cursores) oder Brüder (fratres), oder auf den letzten Deichselstern Benenaz des Großen Bären oder auf andere helle Sterne eingestellt. Wenn das Datum bekannt ist, so läßt sich dann die gleichlange Stunde bestimmen. Um die Stunden in der Nacht abzählen zu können, wurden an der Stundenscheibe Zacken oder Zähne oder Knöpfe der Stunden angebracht. Die Sternuhr wurde gelegentlich auf der Rückseite eines Sonnenquadranten oder einer Sonnenuhr angebracht. Bereits die 1445 bis 1450 auszugsweise abgeschriebene Arbeit [253 Nr. 7464a] zeigt, daß die Sternuhr auf ihrer Rückseite einen Sonnenquadranten für 51° Polhöhe hatte, ebenso 253 Nr 7464 d, e von 1458 und 1512, Nr 7470 b von 1512, Nr. 7465 a von 1492 und 7464, wo das ganzDie Sternuhr, auch Nachtuhr = horologium noctis = noctilabium = nocturnalis gennant, wurde in Frankreich erfunden [149 d S.8] und von Raimondo Lullo in seiner Arte de navegar 1295 beschreiben [Opera omnia, Mainz 1721]. Er verwendete den Polstern und die Fratres genannten Sterne des Großen Bären. Das Gerät wurder im Kreise des Schülers Johanns von Gmunden in Wien verwendet; denn die 1438 beendete Abschrift von Gmundens Arbeit über das Astrolab [253 Nr. 3593] enthält einen Hinweis auf die Sternuhr mit der Verwendung von Polaris und Dubhe. Die Sternuhr besteht aus einer runden Scheibe mit einem Loch in der Mitte, um das sich einige Scheiben und ein über die Scheibe hinausreichender Zeiger bewegen lassen. Durch das Loch wird der Polstern beobachtet und der Zieger auf die beiden Hinterräder des Großen Bären, bezeichnet als die Läufer (cursores) oder Brüder (fratres), oder auf den letzten Deichselstern Benenaz des Großen Bären oder auf andere helle Sterne eingestellt. Wenn das Datum bekannt ist, so läßt sich dann die gleichlange Stunde bestimmen. Um die Stunden in der Nacht abzählen zu können, wurden an der Stundenscheibe Zacken oder Zähne oder Knöpfe der Stunden angebracht. Die Sternuhr wurde gelegentlich auf der Rückseite eines Sonnenquadranten oder einer Sonnenuhr angebracht. Bereits die 1445 bis 1450 auszugsweise abgeschriebene Arbeit [253 Nr. 7464a] zeigt, daß die Sternuhr auf ihrer Rückseite einen Sonnenquadranten für 51° Polhöhe hatte, ebenso 253 Nr 7464 d, e von 1458 und 1512, Nr 7470 b von 1512, Nr. 7465 a von 1492 und 7464, wo das ganze Instrument « spera » genannt ist wie in 7464 a.e Instrument « spera » genannt ist wie in 7464 a.

The star clock (also called night clock = horologium noctis = noctilabium = nocturnalis) was invented in France [Henri Michel. Du Prisme méridien au Siun-ki (Ciel et Terre 1950 S. 1-13) p.8] and described by Raymond Llull in his (1295) Arte de navegar [Opera omnia, Mainz 1721]. Llull used the ‘Fratres’ pair of stars in the Ursa Major constellation. The device was used in the circle of the student Johannes von Gmunden in Vienna; a 1438 copy of Gmunden’s work on the Astrolabe [Zi 3593] describes a star clock using Polaris and Dubhe. The star clock consists of a round disc with a hole in the middle around which both a number of discs and a pointer extending beyond [the edge of] the disc can be rotated. Through the [central] hole, the [Pole Star] is observed and the pointer is then set to the two rear stars of Ursa Major [known as the runners (‘Cursores’) or brothers (‘Fratres’)], or to Benenaz [Eta Ursae Majoris, the ‘plough handle’ star of the Ursa Major constellation] or other bright stars. If the date is known, this device helps determine the hour of the night. To read the hour off in the dark, teeth or buttons (one for each hour) were attached to the hour disc. Star clocks were occasionally attached to the backs of quadrants or sundials. Already in 1445 to 1450 the partially copied work Zi 7464a demonstrates that the star clock on its back had a sun quadrant set for 51° latitude, likewise:

• Zi 7464d [1458]
• Zi 7464e [1512]
• Zi 7470b [1512]
• Zi 7465a [1492] and
• Zi 7464, where the whole instrument is called a «spera», as in Zi 7464a.

Second paragraph…

Zuerst wurde Dubhe (α Ursa) als der Richtstern des Zeigers genannt. Dieser Stern oder die beiden äußeren Rädersterne werden angegeben auch in den Arbeiten 253 Nr. 7468 b, geschrieben nach 1452, 253 Nr. 7468 nach 1457, 253 Nr. 7467 von 1459, 253 Nr. 7464 von 1461, 253 Nr. 7468 c um 1466, 253 Nr. 7463 a nach 1475. In 253 Nr. 7468 b ist als Leitstern außer Dubhe auch Benenaz genannt und dazu die Örter von Polaris, Dubhe und Benenaz für 1438 angegeben. Benenaz wird auch genannt in Wilhelms Arbeit [253 Nr. 11716] über die Herstellung und Verwendung der Sternuhr um 1471. Da Wilhelm Schüler Peurbachs war, so gehört auch seine Arbeit zu den Wiener Arbeiten.

Dubhe (α Ursa) was the first star to be mentioned in connection with the nocturnal’s pointer. This star or the two outermost stars of Ursa Major are also given in:

• Zi 7468b [after 1452]
• Zi 7468 [after 1457]
• Zi 7467 [from 1459]
• Zi 7464 [from 1461]
• Zi 7468c [1466]
• Zi 7463a [after 1475].

In Zi 7468b, Benenaz is also mentioned as the guiding star in addition to Dubhe and the locations of Polaris, Dubhe and Benenaz for 1438 are given. Benenaz is also mentioned in Wilhelm’s work Zi 11716 [around 1471] on the construction and use of the star clock. Since Wilhelm was a student of Peurbach, his work also belongs to the Viennese circle.

Third paragraph…

In der um 1460 entstandenen Arbeit [253 Nr. 7472] warden die Sterne β (Kochab) und γ des Kleinen Bären und zwar mit ihrem Ort für 1460 angegeben. Nun bilden diese Sterne mit Polaris nicht eine gerade Linie, so daß ein Irrtum vorliegen dürfte. Vielleicht war Kochab, der später auch von Köbel erwähnt wurde, allein gemeint.

In Zi 7472 [written around 1460], the stars β (Kochab) and γ of Ursa Minor are given, with their location for 1460. However these two stars do not form a straight line with Polaris, so there may be an error. Perhaps Kochab, which was later mentioned by Köbel, was meant to be used on its own.

Fourth paragraph…

Die Sternuhr wird so verwendet, daß zuerst die gezackte Stundenscheibe mit 12 Uhr auf den Monatstag gelegt wird ; dann gibt der auf die Hinterräder eingestellte Zeiger die Stunde an (Tafel 57, 1).

To use the star clock, once the jagged hour disc is placed on the day of the month at 12 o’clock, the pointer set on the rear wheels should indicate the hour (plate 57, 1).

Guards, Guards!

This is all very interesting, and helps to give an overall timeline. The ‘guards’ I mentioned previously (that point to Polaris) are another name for the same cursores / fratres first mentioned by Llull. I also didn’t know that Dubhe is the official star of the State of Utah. 🙂

The next step here will be to look more closely at the specific early 15th century manuscripts listed by Zinner, to see how they fit together into the overall nocturlabe timeline.

It turns out that the timeline of theoricae planetarum I previously put forward was missing three important entries:

• Theorica planetarum [antiqua] (misattributed to Gerard of Cremona)
• Theorica planetarum of Campanus of Novara
• Jean de Lignieres’ abbreviation of Campanus of Novara’s theorica
• Petrus Philomena de Dacia (Peter Nightingale)’s Equatorium
• Theorica novelle
• Theoricae novae planetarum of Georg von Peurbach

In Emmanuel Poulle’s (1100+-page) work on astronomical instruments and equatoria used to calculate planetary movements (“Les Instruments de la Théorie des Planètes selon Ptolemée: Équatoires et Horlogerie Planétaire du XIIIe au XVIe siècle”, 2 Bde, Genf/Paris 1980 (Centre de Recherches d’Histoire et de Philologie V: Hautes Études Médiévales et Modernes 42)), he named the instrument modelled in the 15th century theorice novelle as the ‘Erfurt-Leipziger instrument’, after two of its manuscripts. [pp.375-416]

(And no, I haven’t got my own copy of Poulle, much as I’d like to.)

So the first question is this: what specifically differentiated this theorice novelle from, say, Campanus of Novara’s theorica planetarum?

Equatoria vs volvelles

Carrying out the computations necessary to draw up a horoscope was fiddly and boring: it required the person doing to have not only access to tables of planetary positions (typically the Alfonsine Tables), but also the spherical trigonometry skills to do a load of tricksy interpolation to determine the planetary positions at times between the entries in the Tables.

Clearly, what was needed was some kind of physical instrument – broadly along the lines of an astrolabe – to do all the heavy lifting / maths for you. The ‘theoric’ (Latin: theorica) in all these titles is in fact not just a theory about the planets, but also a physical model that physically manifests a theory about the planets, and is therefore able to perform work.

What was initially devised was an equatorium. This was (in the case of Campanus of Novara, at least) an astrolabe-like backplate with a circular hole (a mater) into which a series of plugin disc devices (one per planet) was inserted. These plugin plates physically modelled the Ptolemaic deferents, epicycles, and equants that had been used to (numerically) model planetary movements for over a millennium.

Campanus of Novara’s theorica planetarum described exactly this kind of bulky equatorium, while its updated versions (such as that of Jean de Lignieres) tried to simply its mechanisms a little, with the aim of producing something a little more lightweight. Or at least, not quite so heavyweight.

The oldest known extant equatorium is in Merton College, Oxford (Merton SC/OB/AST/2), and dates to about 1350. Here is a photo of its back:

Somewhat extraordinarily, there is also a pair of (pretty much) contemporaneous manuscripts that specifically described this equatorium, which you can read about in Seb Falk’s fascinating (2016) “A Merton College Equatorium: Text, Translation, Commentary“.

• Cambridge University Library, Ms. Gg.6.3, ff. 217v–220v (c. 1348)
• Oxford, Bodleian Library, Ms. Digby 57, ff. 130r–132v (c. 1376)

According to the text, the Merton College equatorium was based on the equatoria of Campanus of Novara, Jean de Lignieres, and also that of Profatius Judaeus. However, Falk cautions (p.2) that this last attribution is incorrect (though widespread). Its third (and indeed closest) equatorium was in fact described in a family of manuscripts known as the Semissa, described in F. S. Pedersen’s (1983) “Petri Philomenae de Dacia et Petri de S. Audomaro opera quadrivialia“, Copenhagen. (Pedersen’s 1979 dissertation solely on the Semissa is online here.)

What united these 13th/14th century theoric tracts was that they described how to build a big, fat, brass equatorium – the Big Science of the day.

By comparison, the theorice novelle manuscripts were – as I understand it – completely different: the instrument they described (and indeed manifested) was a set of paper or parchment volvelles, one volvelle per planet. This was lithe, modern, exciting, lightweight science – much more like a tech startup.

Manuscripts in the equatorium genre were widely copied and disseminated through Europe’s astronomical / astrological communities – they were ‘open source’, effectively. But what of the theorice novelle mss?

Theorice novelle manuscripts

Of the three known manuscripts in this genre, the main two are from Erfurt and Leipzig (hence Poulle’s name). Even now, these two mss languish undigitized (and close to completely unknown) in local museums:

• Angermuseum Erfurt, Cod. 3153, [1458]
• Historisches Museum Frankfurt/M., Cod. X 16027 [1458-1464]

No prizes for guessing, however, that the third one is Gotha Chart A 472 (my current favourite volvelle-heavy mysterious manuscript), dated by Zinner to 1461. Scans for this are online courtesy of Jena.

However, I suspect – admittedly without proof – that the attribution of Gotha Chart A 472 to Profatius Judaeus will prove to be just as specious as the widespread attribution to him of Peter Nightingale’s Semissa manuscript.

All the same, it will take a very much closer reading of all three manuscripts to be able to trace the origins of the theorice novelle more accurately. What we really need is to find someone who has been looking at this for some years…

Theorica novelle researchers

So here’s where it gets interesting. Post-doc Samuel Gessner of SYRTE (at the Observatoire de Paris) is/was due to give a talk in Paris on 18th June 2020:

Between astronomical diagrams and instruments: spatializing numerical data of astronomical tables
Astronomers have connected their computational methods with geometrical representations in various ways. The ways these connections were elaborated on are not universal, but historically contingent of the local astronomical practice. Parchment instruments to graphically determine (approximate) positions of the planets, i.e. the family of planetary “equatoria” instruments, saw renewed developments in the 15th century. We will start with a European case study about a particular type of instrument that emerged in manuscripts from Erfurt and Leipzig termed “Theorice novelle”. In discussing this material the talk proposes to look into possible connections between the representation of computed data in tables and corresponding diagrammatic representations on the “Theorice novelle” and similar instruments. More generally, it raises the question of how the use of tables was preparing the minds for experimenting with new types of instruments and whether this trait can be used to characterise a specific astronomical practice.

In 2019, Gessner described his research focus here:

I focus on the diverse mathematical cultures in medieval and early modern Europe and how they communicate by studying the role of mathematical instruments as conceived by both theoreticians and practitioners. Using artefacts of material culture as primary sources along side with textual documents has become my favourite approach. I currently participate in a research project on Alfonsine astronomy lead by Matthieu Husson, Paris. My longer term goal is to understand the material and mechanical realisations of Ptolemaic theory in models, equatoria and planetary clocks and their role in history of astronomy. I was a co-organiser of the Oberwolfach Workshop “Mathematical Instruments Between Material Artifacts and Ideal Machines”, December 2017.

Unsurprisingly, I’ll be emailing Samuel Gessner shortly, and will let you know what I find out…