From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Saturday, March 18, 2006 9:59 PM
To : vms-list@voynich.net
Subject : RE: VMs: 1st, 2nd, 3rd...?
I'm not sure on what has been established so far for a transition period where the sought for
arabic number^suffix appears unequivocally, but this ms, dated around 1450, might be worth a look, especially if high bandwidth is available for downloading some excellent images available online:
HIGDEN, POLYCHRONICON, trans. John Trevisa; etc. England, s. XVmed-ex
The entry point for images online is: http://sunsite.berkeley.edu/Scriptorium/hehweb/HM28561.html#EETSos
(note the reference color spectrum) In several folios Trevisa looks to be very close to doing it, notably in folio 83.
Berj
From: Nick Pelling <>
Reply-To: vms-list@voynich.net
To: vms-list@voynich.net
Subject: RE: VMs: 1st, 2nd, 3rd...?
Date: Sat, 18 Mar 2006 18:03:49 +0000
Hi everyone,
At 15:43 17/03/2006 -0500, Berj N. Ensanian wrote: Perhaps the lady who has this website:
http://medievalwriting.50megs.com/whatis.htm
will have the answer, or be able to get it. She discusses the evolution of abbreviations in European script.
Though Dianne Tillotson is a very nice lady with a very nice website, it turns out that this '1st' query isn't really a thing she can help with. Oh well... onto the OWLS people next, probably. :-o
Cheers, .....Nick Pelling....
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Sunday, March 19, 2006 12:33 PM
To : vms-list@voynich.net
Subject : VMs: 1st etc. & Higden's Polychronicon & Trevisa in "1387"
According to at least once source Trevisa produced in 1387:
"John Trevisa's Translation of the 'Polychronicon' of Ranulph Higden, Book VI. An Edition Based on British Library MS Cotton Tiberius D.vii, ed. Ronald Waldron (Heidelberg: Universitatsverlag Winter, 2004), lix + 302 pp. ISBN 3-8253-1587-8. 64.00 [euro]. Ranulph Higden's Polychronicon, a Latin 'universal history' written in the mid-fourteenth century, was an instant best-seller; it survives in over 120 manuscripts from the fourteenth century, alone. John Trevisa produced his well known translation of it in 1387. Previously, scholars of Trevisa have had to rely on the Rolls Series ... "
That comes from the premium-content description of the paper here:
http://www.highbeam.com/doc/1G1:139755763/John+Trevisas+Translation+of+the+Polychronicon+of+Ranulph+Higden,+Book+VI~R~(John+Trevisas+Translation+of+the+Polychronicon+of+Ranulph+Higden,+Book+VI~C~+An+Edition+Based+on+British+Library+MS+Cotton+Tiberius+D~R~vii+)(Book+Review).html?refid=ency_topnm
So if correct, we are now down in the 14th century. Is Fr. Bacon beginning to stir again?
The Polychronicon of Fr. Higden itself looks very interesting - its famous map of the world has the familiar VMS-like moonfaces on its perimeter in one of the copies displayed here: http://www.henry-davis.com/MAPS/LMwebpages/232C.html
Much better images are needed of the Polychronicon.
Berj
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Sunday, March 19, 2006 4:38 PM
To : vms-list@voynich.net
Subject : VMs: proto-gallows in HM28561 ?
The HM28561 copy of Treviso's translation of Higby's Polychronicon becomes more interesting with every look. On the inside back cover, images:
http://dpg.lib.berkeley.edu/webdb/dsheh/heh_brf?Description=&CallNumber=HM+28561
the scribe appears to be practicing, or working the writing instrument. Some VMS symbols seem to be there without much doubt, the universal (see Capelli) EVA "l" (l as in letter) is one of the first ones I look for, and it is there. But there are other interesting symbols on the back cover that suggest EVA "ch" and GC "*". Most exciting of all is the possibility that a prototype EVA "t" or GC "k" gallows letter is not far from the scribe's calligraphic repertoire.
The most compelling example of this proto gallows occurs about midway down, at the right edge, and has to its immediate right a symbol resembling a triangled number 4.
We need opinions on this by those who have long studied gallows and near-gallows in non-VMS mss - please look. So, here is the current brief summary, (a bit hurried and lacking in organization), concerning the book HM28561:
0.) online entry point for study: http://sunsite.berkeley.edu/Scriptorium/hehweb/HM28561.html
1.) HM28561 is a 4-scribes copy of John Trevisa's translation of Higden's Polychronicon. The Poly was a medieval bestseller, written around 1350. The Poly contains a famous world-map, with some copies showing graphics elements reminiscent of those in some VMS cosmological illustrations (moon faces). High quality images of Poly map not yet found, but here are some:
http://www.henry-davis.com/MAPS/LMwebpages/232C.html
2.) Trevisa completed a translation of the Poly into "English" by 1387, and scribes started making copies from then on well into the next century. Trevisa is considered among the founders of English prose:
http://www.bartleby.com/212/0302.html
More on him specific to his Poly translation is here: http://www.bartleby.com/212/0304.html
These two links allow one to get a feel for Treviso's potential as a literary innovator, including how strict he may or may not have been in overseeing scribes during his lifetime.
3.) HM28561 seems to be associated with early, or proto usage of the Arnum^suf abbreviation, for example: 1st, 2nd, 3rd etc.
4.) HM28561 has no secure date. Someone, perhaps a scholar named "Rolls", or one of his students, placed it into 1450-1500 period, as the book's dating is currently regarded.
5.) HM28561 has some symbols in common with the VMS, and most interesting, on its back cover among practice scripting or nib working or both, possibly by the actual scribe(s) who wrote the book, are some executions, at least one of which suggests VMS gallows letter EVA "t" also known as GC "k".
Berj
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Monday, March 20, 2006 12:36 AM
To : vms-list@voynich.net
Subject : VMs: comments on HM28561
I've spent some time looking at the best images, and:
1.) fi is of obvious interest because it exhibits variations in scripting the letters.
2.) the partial verso to the left of f1, about a third of the way down, shows two lettes ligatured, apparently the beginning of the word "shalt", that are approaching EVA "k" or GC "h" - the idea being that that gallows is already "in the scribe's hand" and ready to come out VMS-like any moment.
3.) f104-105 looks like it has some bearing on the Arnum^suf. question.
4.) f324 the genealogy page, has bearing on the Arnum^suf. question; it may have some other things in there too.
5.) f326 seems to be the most in-your-face hint of the coming of the EVA "t" or GC "k" gallow - remove some flourish from the first big red letter (left column) and you have it. It is also in good company - that same line has other letters with high ascenders.
Alternatively one can look at it this way - ask the VMS scribe to render a flourished EVA "t" / GC "k", and who would be surprised if he came up with what is on this page? Atop the right column, in red, is a hint of EVA "k" / GC "h".
6.) inside back cover - the most puzzling thing about the very crude tentative "gallow" next to the "4" is that its ink seems to be related to the scribble underneath it that looks like "pipy", and pipy is contiguous with the nicely written "Welby". This is seen with the 6+ megabyte large image, especially when negatived. Further down the page there is another nicely rendered hinting at the coming of the gallows. The stuff on this inside back page looks to be difficult to sort out - scribes versus owners of the book. Is that a date = July 7, 1500 ?
So at this point I think HM28561, insofar as it is relevant, is in favor of the VMS dating to around its time, and that it would not have been a very big step at all for one of these scribes to innovate four gallows and put them together with already familiar symbols to make up the VMS alphabet.
Berj
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Monday, March 20, 2006 6:20 PM
To : vms-list@voynich.net
Subject : VMs: gallows in papal docs
The Vatican Secret Archives website is displaying some old diplomatic documents that sure look like they have gallows letters that appear no worse, if not even more VMS-like, than the Capelli Table 4 example: http://asv.vatican.va/en/dipl/1_papaldocuments.htm
Privilege by Alexander III, 23rd January 1168
ASV, Fondo Veneto I, 6559
Alexander IV Litterae Gratiose, 4th January 1261
ASV, Fondo Domenicani 331
http://asv.vatican.va/en/dipl/text.htm
Berj
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Wednesday, March 22, 2006 7:15 PM
To : vms-list@voynich.net
Subject : VMs: more gallows
proto/Gallows in last and 2nd last lines of an Athenian Curse Tablet: http://tropaion.blogspot.com/2006_01_01_tropaion_archive.html
The MS. Barocci 131 of Oxford's Bodleian Library seems to be a good hunting ground for gallows hunters:
f2*v:
Line 6, near right end, to the left of a dot mu: a letter similar to GC "h" / EVA "k"
Line 10, about 40% in from the start: a mu ligatured to a kneed GC "k" / EVA "t". The knee might be part of a ligature to a third letter in this group, perhaps another mu, or a zeta.
Line 13, appx. midway, to the right of a lambda dot: a letter similar to GC "k" / EVA "t"
4th Line up from the bottom, left starting side, the fifth letter in, inbetween nu and omicron: a letter similar to GC "k' / EVA "t".
Note also the first letter in that line: a taller, kneed GC "k" / EVA "t".
f3*v:
Line 20, about a thirdway in from the right end, inbetween an omega and iota, a letter similar to GC "u umlaut" i.e GC 252.
Line 28, third symbol in from the left start, following a kappa, something - perhaps a mu - ligatured to a GC "k" / EVA "t" .
These images and many more from the ms, are online here:
http://image.ox.ac.uk/show?collection=bodleian&manuscript=msbarocci131
Berj
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Wednesday, March 22, 2006 10:57 PM
To : vms-list@voynich.net
Subject : VMs: ~ GC "g" & GC 150 in ms Barocci 131
One of the following in ms Barocci 131 is truly thought-provoking:
f70v
Fist Line (red): to left of "9" space tau: a tau that almost makes it into a GC "g" / EVA "p"
Body Text:
Top triple-columns, Middle column, 2nd line, last letter: GC "g" / EVA "p"
Bottom dual-columns, right column, first line, toward the end: ~ GC 150
Bottom dual-columns, right column, line 16, end: ~ GC 150
Berj
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Friday, March 24, 2006 10:23 AM
To : vms-list@voynich.net
Subject : RE: VMs: Breakthroughs and breakdowns
It depends I think on the reading of the VMS text. If it is decoded according to an unambiguous procedure, AND it reads as a version of previously known literature, AND the reading is no farther in diverging from the previously known texts than, say, the various New Testament mss differ from one another, then the chances are excellent that the elegant mystery will generally be deemed penetrated. And if the reading is of general interest to the public, so much the more resuting public interest momentum.
In his comments on this thread Tony Mann refers to De Branges's proposed proof of the Riemann Hypothesis, and the difficulty De Branges has had getting a fair critical examination. The difficulty there is that only a few dozen living persons could act as fair examiners, and a final determination could take even them so much time that they are reluctant to take the time away from their own work on Riemann, if they are not immediately inclined to think that De Branges finally has it right this time. Unless the VMS concerns a discussion of the distribution of prime numbers written using a cipher that is based on a discovery about prime numbers that mathematics in general does not yet know about, then I think the VMS solution, when it comes, will be much easier accepted than things Riemann, where it isn't even clear yet if the basic proposition is even decidable. (1)
As I see it, the most valuable thing in the way of a unfied body of literature that the VMS research has produced is the vms-list archives. If not already, then once the VMS mystery is solved, the archives will be good for minting dozens of books and PhD theses around topics like why it took so long to figure it out, who got very close and then veered away and why, and so on. The solution to the VMS will be the beginning of a new phase of VMS research. Perhaps even a neo-literature using the VMS language might possibly arise, and with installed fonts we will see correspondence on this list in Voynichese.
(1) In the archives I saw that list member Jeff (George Boole Fan Club) was as recently as last December experimenting with primes numbers wrt VMS. I think Jeff, and whoever else is doing similarly, ought to be encouraged to continue that. There are places in the VMS where prime numbers seem to scream out from the page. Consider f6v:
on this page the "plant" illustration's ratio of macro-details to micro-details is overwhelmingly macro, and the macro-details just blast upon the observer the sequence of prime numbers 2, 3, 5, and 7, and numbers that possess them as their factors: at the top of this "number-plant" are 6 circular bulbs, and at its bottom are 14 roots, very plainly rendered, but 14 of them, and "roots", we also recall, are major ideas in mathematics. A circular bulb in the middle, without a stem, even suggests 1 as the orgin of the number-plant "graph". And there are 21 text lines. From one perspective at least, these textlines are presented in 2 paragraphs, and the number of "words" in the first paragraph's textlines proceed as 10, 9 , 8, 7, 6, in altogether 5 lines. Did the VMS's author intend the exhibition of these numbers and their relationships, or is it all just a "coincidence"?
Berj
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Friday, March 24, 2006 8:25 PM
To : vms-list@voynich.net
Subject : VMs: prime numbers in the VMS
Jeff wrote:
" >From: "J HALEY" <>
Reply-To: vms-list@voynich.net
To: <vms-list@voynich.net>
Subject: Re: VMs: Breakthroughs and breakdowns
Date: Sat, 25 Mar 2006 00:09:52 -0000
.......... I am still considering any mathematical connections in the VMS but have done no more investigations into prime numbers. The trail went very cold on that one.
...........
I would be interested in any observations that you make. If you wish to
discuss them off list I would be quite happy to help you in investigating them. At the moment I am running various analyses on GC's transcription. I may be at a point where I can plot possible cipher changes. The method changes at particular points in the VMS. In some cases these are subtle and in others quite dramatic. I would suggest, if you haven't already, that you investigate Currier's results. However, mathematics will play a very important role in the solution. For more than one reason........ "
Hi Jeff
My e-adr = ki3u@xxxxxx and I too will be very glad to discuss anything that comes up with primes on/offlist.
I haven't done much with prime numbers and the VMS except to notice they are in there (f6v example already noted), and wonder if they are in there intentionally. If yes, that would raise the VMS stakes considerably, as if they weren't high enough already.
My main observation so far is prefaced like this: because prime numbers investigation in the Voynich Manuscript means linking from the challenge of the world's most mysterious ms to the challenge of the most difficult problem in mathematics, then in order to prevent overload and risk spreading oneself too thin on current VMS focus (in my case the origin and meaning of gallows symbols), it seems to me that the gentle approach is:
peruse the first pages of the VMS, that are known/believed to be in correct foliation order, and see if there is a POGRESSIVE suggestion of a THEME concerned with prime NUMBERS, or at least arithmetic generally. This would not be too distracting from current favorite focus. And if indeed there appears a theme in progression, then, after same is fortified with further consistent observations on more pages in the book, one can make a decision on really rolling up the mathematical sleeves and getting down to serious calculating.
My preliminary observations on a possible-theme hypothesis in the first pages of the VMS, are very preliminary, and go like this:
The book's first page, f1r: by virtue of two unusual red colored symbols, suggests the numbers 1 and 2.
f1v: suggests the symmetry versus anti-symmetry properties of the even and odd number sequences.
f3v: the plant illustration along with its two paragraphs of 6 and 8 text-lines respectively, strongly suggests the concept of the sequence of even numbers.
f6v: as noted earlier in the Breakthroughs and breakdowns thread.
Preliminary as it is, if this is not just out of VMS-adventure-enthusiasm making patterns appear out of a rich potpouri, then it certainly has the feel of the beginning of an organized discourse on the theory of numbers.
I always find myself coming back to D'Imperio's belief, which is mine also, that the VMS illustrations involve a highly symbolic, artificial, and conventionalized graphic or mnemonic language.
So in the above preliminary theme, the "plants" are arithmetic graphs.
Berj
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Monday, March 27, 2006 1:06 AM
To : vms-list@voynich.net
Subject : VMs: prime number equations in f6r
On the assumption that f6v, with its telegraphic displays of prime numbers and their composites, does indeed suggest viewing some of the Voynich plant illustrations as symbolic mathematical graphs, then it is natural to wonder if some of the plants depict actual equations. With that in mind lets have a quick look at f6r.
What we see must be convincing - that is, given that we are thinking mathematically, nevertheless the equation that we "see" written as a plant must be logical, and not forced. We are motivated by a sensitivity to prime numbers in what we are looking at, but at this stage the Riemann level must be considered completely too complicated - we are looking for something that can be seen with only elementary mathematical curiosity for motivation. It must be obvious, or at least acceptable as quite reasonable.
I think it is reasonable, from this perspective, to say that we can entertain the possibility that the plant on f6r might depict one or more of this form:
Eqn. 1 : +/-4X * +/-3Y = +/- 8 where * = arithmetic operation
Lets pick the simplest: 4X + 3Y = 8 and re-write it:
Eqn. 3 : Y = ( 8 - 4X ) / 3
The reasoning in choosing this particular form goes like this:
The vertical axis of the plant has a below-ground-roots portion and an above-ground portion. Below ground we have an obvious constant, 8 roots. In the above-portion we have a cut-off shoot inbetween the radically different kinds of "leaves", so we will assume it means artithmetic operation, and take it as simple addition. And of course we have taken the interface between above and below to mean " = ". As above, so below, said the great Hermes T.
The leaves we will take to symbolize variables, X and Y - what other choice do we have? But we are justified some to assume they symbolize variables, because within type there is variation: the fern-leaves, especially the two lower ones, show considerable variation on the same basic plan. Likewise the 4 bulbs differ from each other.
So we've got an equation, and although there may well be believably depicted in that plant more complex equations, lets not press our credulity at this stage.
Now we need some numbers to plug in, and of course we would first harvest numbers from f6r itself, so lets get some numbers from this page, some of which will necessarily be transcription-sensitive:
f6r:
1 plant, 1 cut-off shoot on stem
2 kinds of "leaves"
3 fern-like leaves
4 bulb-like leaves
5 shoot-offs on right side of stem counting the cut-off
6 text-lines wrapped around plant
7 total "leaves", 7 above-roots branchings counting the cut-off
8 roots in 4 twisted pairs
8 unwrapped text-lines
14 total text-lines
16 intruding gallows letters
16 serrae in the fern-like leaves
13 (apparently) closed braid-ellipses in the roots
19 type EVA "t" / GC "k" gallows, counting both straight and intruding
23 straight gallows
39 total gallows letters
Counts of letters per line, with intruding gallows = 2 letters :
TABLE 1 :
line 1 : 34
line 2 : 32
line 3 : 37
line 4 : 27
line 5 : 28
line 6 : 29
line 7 : 25
line 8 : 21
line 9 : 26
line 10: 20
line 11: 18
line 12: 15
line 13: 16
line 14: 14
___________
total = 342 = 19 x 18 = 19 x 3 x 3 x 2
Other possibilities ? : 5 letters that touch the plant
11 and its composites conspicuously missing?
So we have a set of numbers, and we can plug them into the equation, and see if anything peculiar results. By dumb luck the first number I plugged in, gave a peculiar result, I thought, but it was just due to an arithmetic error. The luck was that it led me to plug in certain other numbers, and by the time I discovered my error, I was on this trail:
TABLE 2 Eqn. 3 with X = independent variable:
X = - 7 Y = 12
X = 11 Y = -12
X = -13 Y = 20
X = 17 Y = -20
X = -19 Y = 28
X = 23 Y = -28
How much further this absolute-magnitudes linking successive-prime-numbers will continue I don't know, but a well thought-out computer program could answer that. Such a program ought to be flexible enough to detect a transformation in the oscillations: in other words, detect a subtle continuation of an essential pattern when otherwise it looks like the peculiarity pattern has ceased with further X primes. Eqn. 3 is not just an interesting equation, but also its independent variables values, at least |19| and |23|, may be telling something, I don't know what, about major VMS objects: GALLOWS! - the trademark of the VMS. And even if the peculiar pattern stops with TABLE 2 at X = 23, it still makes me wonder about numbers, especially prime numbers, being a distinct signal to us from the author(s) of the Voynich manuscript, who probably figured we would all be irresistably tuning in to the mysterious book like would-be initiates in quest of the epiphany.
Any counting and arithmetic errors or typos are of course my embarrassment.
Is any of this significant, and a hint that the nine rosettes foldout is a mathematical masterpiece? Who knows. Perhaps even f6r has hidden in it more surprises with simultaneous equations derived as different forms of Eqn. 1. Logically, it would be good to next examine in like analytic fashion a "similar" non-VMS medieval herbal page, to see if it also just as readily produces peculiar numerical results, and therefore makes the above less remarkable. Or even more remarkable. Perhaps someone has suggestions for such a test herbal page.
Berj
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Wednesday, March 29, 2006 10:38 PM
To : vms-list@voynich.net
Subject : Re: VMs: prime number equations in f6r
From: Tony Mann <>
Reply-To: vms-list@voynich.net
To: vms-list@voynich.net
Subject: Re: VMs: prime number equations in f6r
Date: Wed, 29 Mar 2006 12:32:31 +0100
"Maurizio M. Gavioli" wrote:
>
> I also have a question: are we sure that, as the possible time where the
> manuscript was written, negative numbers already 'existed'? In the
> classical arithmetic, of geometric origin, they didn't and, in classical
> discussions of indeterminate equations like this, as appears mainly in
> Diophantus and later in Abu Kamil, negative solutions were discarded as
> 'impossible'.
I am afraid that, in certain senses, negative numbers didn't exist.
First, the idea of writing algebraic equations symbolically rather than
in words is quite late. Secondly, until the seventeenth century, my
understanding is that negative numbers were not used in equations, so
that the equation (writing xx for x squared, as was the practice at the
time)
xx -2x + 4 = 0
would be written as
xx + 4 = 2x
and would be regarded as different in form from
xx + 4x = 2
so that instead of a single form of quadratic equation with possibly
negative co-efficients, mathematicians of the sixteenth and early
seventeenth centuries would have had to consider four different forms.
This was discussed on the BBC Radio 4 "In our time" programme two weeks
ago, available at
http://www.bbc.co.uk/radio4/history/inourtime/inourtime_current.shtml
My conclusion would be that if the VM encodes this kind of equations,
then it certainly isn't as early as the fifteenth or sixteenth century.
Tony
Tony Mann
Head of Department, Mathematical Sciences,
School of Computing and Mathematical Sciences, The University of
Greenwich,
Maritime Greenwich University Campus, Old Royal Naval College,
Park Row, Greenwich, London SE10 9LS
Tel: +(44) (0)20 8331 8709, Fax: +(44) (0)20 8331 8665
Email: A.Mann@xxxxxxxx, website: www.gre.ac.uk/~A.Mann
I would like to address in this one posting the immediately preceding thread-points made by both Gavioli and Mann, because they are connected. I will make some general comments in addition to the specific ones, so please bear with me.
First, I appeal directly to the evidence in the vms-list archives - there we see over and over that everything under the sun has already been tried, and tried by people with very formidable intelligence, education, and motivation. That realization can have a potentially depressing impact upon enthusiastic VMS newcomers. And yet the unequivocal resolution of the mystery has not been achieved! And that, I say, makes a strong case for broadening the horizons - and that means relaxing some traditional assumptions, or even resurrecting long-ago-discarded assumptions about the authorship of the VMS.
Namely, he or she or they, may well have been far ahead of their time with their mathematical knowledge, AND may have practiced it secretly.
We have some history in that vein: the discovery and use of i = sqr(-1) was kept secret initially, and whoever it was that gave the Maya their calendar, had a superior calculating system to that of the pre-Renaissance Europeans, at least according to Beckmann in his book on the history of pi, if I recall correctly. Another example: recently it has been shown that Fra. Pacioli, who traditionally has been viewed as more of a transmitter of mathematics rather than a great creator, used and published, inconspicuously, the CONCEPT of logarithm before Napier, and the concept of logarithm is one of the turning-point ideas in mathematics. Pacioli is the central figure in mathematics' most dramatic, and also most mysterious painting - despite its obvious projection of Euclidean elements, the suggestion of "secret knowledge" just oozes out of it. Leonardo himself, Pacioli's buddy, may have been the painter.
There are many examples of big ideas, secret or not, taking time to get into the mainstream, and many examples of intentionally keeping knowledge secret for a while. And I'd be willing to bet that even here and today, some list members are sitting on some of their very interesting VMS discoveries, for one reason or another. If secret mathematics is the major theme in the VMS, then it is entirely reasonable: what in any age anywhere could possibly be more worthy of secrecy, than advanced mathematical knowledge? Pythagoras would likely agree.
To digress for just two seconds while I have your attention, I think it is not in-conceivable that a woman had a hand in the authorship of the VMS. In the archives Stolfi seems to consider the possibility more than casually. It would be useful to have a list of pros and cons regarding possible authorship by a woman or women.
Gavioli challenges the "peculiarity" of Equation 3, and its relationship to the gallows letters. Lets look at Table 2 again:
X = -7 Y = 12 12 = 2 x 2 x 3
X = 11 Y = -12
X = -13 Y = 20 20 = 2 x 2 x 5
X = 17 Y = -20
X = -19 Y = 28 28 = 2 x 2 x 7
X = 23 Y = -28
These numbers in the table resulted from transforming the f6r plant ILLUSTRATION into an equation, and then trying out numbers in the equation. Oscillation mathematics aside for the present, the X numbers show a sequence of prime numbers that are more-than-casually linked by the Y numbers. And the nature of the linking is such that it also exhibits special linkings: 7 and 11 are linked specially through |12|, 13 and 17 are linked specially, and 19 and 23 are linked specially. Lets consider 19 and 23.
Turning to the TEXT on f6r, what is the simplest and most immediate, and above all, most natural way to assign some numbers to (specifically) the mysterious VMS-trademark gallows letters? The answer is of course: counting their frequency, counting how many times the gallows appear in the text of f6r. And what do we find? We find that the f6r text contains 19 type EVA "t" / GC "k" gallows, and 23 straight (i.e. not intruding) gallows letters.
Gavioli does not say so explicitly, but he hints that equations that generate primes in succession for a while, are of themselves not terribly unusual. I believe he is right in that - I recall reading somewhere that someone said that somewhere there is a proof that Euler-type simple primes-in-succession equations exist in infinite multitude. But what is such an equation doing in f6r, AND BLATANTLY drawing attention to gallows letters? Euler was born in 1707!
Now, if all this is not peculiar, then I do not know what is peculiar about anything VMS!
Gavioli has, I believe, in his first response to this thread, conceded, in an indirect way, that the derivation of the equation could reflect a natural expression of the f6r plant picture. But in his latest post, pointing to the negative numbers in Table 2, he poses a very strong question: when did the concept of negative numbers as numbers in their own right enter mathematics? Mann provides the answer, and goes further - he addresses the issue of the history of algebraic forms, and concludes ((that if the f6r plant equation is for real and was so drawn intentionally, and the f6r text was accordingly planned to conform to the equation's number relationships, then)) the VMS would be post-15th or 16th century. And, given the current trend in VMS historical research, where a 15th century origin looks very good, that would be hard to swallow by many VMS students. Therefore, Gavioli and Mann have a strong argument that needs to be addressed.
Now of course, I began by appealing for broadening the assumptions-horizon, and here that would mean allowing for the possibility that the VMS author(s) were well ahead of their contemporaries mathematically, and calculated with negative numbers. Thus answering Gavioli and Mann. But we don't need to stretch the horizons far, if at all. And in his comments Mann implies the critical observation: it is the general equation, Equation 1, that is of prime importance, rather than its specific form of Eqn. 3 that I used to build the case in this thread.
Lets us recall Eqn. 1; it is, according to the arguments launching this thread, THE actual expression of an algebraic form, that in MODERN SCRIPTED expression we write:
Eqn. 1: +/- 4X * +/- 3Y = 8
The f6r plant actually invites writing an even more general form because of its braided roots, for example possibly:
Eqn. 4: +/- 4X * +/-3Y = +/-4Z*2
so that one could have:
Eqn. 5: 4X + 3Y = 4Z^2 = 8 where Z = sqr(2) = const.
but I will stay well away from that here because I must defend Eqn. 1 and 3, and because I am not now prepared to defend anything more general. And we don't need the more general form here; it is enough, and I feel not too soon, to invite attention to it.
Before addressing directly the point of Gavioli and Mann on the issue of negative numbers, we must first amplify how " * " , that is "mathematical/arithmetic operation", was logically translated from the plant illustration.
The plant in f6r has a feature that absolutely demands curiosity: the cut-off branch. Why was that drawn in there? On the assumption that it has a purpose, a symbol to convey, and thinking mathematically, we can only conclude that it must symbolize: mathematical operation. Why? Because the cut clearly implies that a person took a knife or a pair of scissors, and PERFORMED AN OPERATION on the plant - they cut it.
Our next first thought might well be that the operation must be subtraction:
4X - 3Y
because a part of the plant has been removed. But no! We are not justified to simply restrict the operation to subtraction. Why? Because, speaking as one who long has had good relationships with houseplants, I know that when I trim a plant, the operation can be one of two kinds: removing a wilted part of the plant that is sapping the plant of vitality, or removing healthy parts so as to rein in the growth of the plant. In the latter case I am definitely subtracting. But in the former case I am adding vitality to the plant, or at least I am operating so that vitality is added.
Now, because nowhere in f6r is there depicted the cut-off portion of the plant, so that we can judge whether it is healthy or wilted, the operation being symbolized in the illustration is ambiguous, and therefore allows, nay suggests, that we consider the general form of the equation to be +/-, and further (although not needed here) by viewing multiplication and division as repeated addition and repeated subtraction respectively, we should consider for f6r in its Eqn. 3 form, in general:
* : { + , - , x , / }
We are almost ready to answer Gavioli and Mann. We need one more procedural element, and to justify it we again invoke Hermes Trismegistus, whose "As above, so below" maxim gave us the means for identifying the plant equation's equal-sign. This time, Hermes encourages us to consider that the plant equation may be read either from the top toward the bottom, or from the bottom toward the top.
Now we are finally ready to deal with the negative numbers problem. We will choose * = - , that is subtraction, and we now write the two equations, which for future reference we will denote Eqn. 6, and 7 :
Table 3:
Read from top toward bottom Read from bottom toward top
4X - 3Y = 8 8 = 3Y - 4X
Now we plug in numbers, and Voila:
4(11) - 3(12) = 8 8 = 3(12) - 4(7)
4(17) - 3(20) = 8 8 = 3(20) - 4(13)
4(23) - 3(28) = 8 8 = 3(28) - 4(19)
We have reproduced all the relationships and ideas in Table 2, but without involving the concept of negative numbers. Even the idea of oscillation is reproduced, because this time, rather than a sign flipping back and forth, the sequence flips back and forth across the two coupled equations. Not only that, but also the previously alluded-to more subtle phase-shift phenomenon in the oscillation across prime numbers, that is observed as numbers outside of the range under consideration are explored, could also be observed from the coupled equations. But at this point, that is too close to over-stretching our credulity for VMS authorship, and as before, we'll just invite experimentation, and not cover it in more detail here. Q.E.D.
As before, any errors are my embarrassment.
And now comes what is perhaps the most important observation as concerns the entire hypothesis that at least some of the VMS botanical material is an unusual symbolically presented mathematics. And we must thank Gavioli and Mann for forcing the exhibit of it in Table 3 above. The observation is:
Everything in Table 3 can be done, and appreciated for what it signifies, with just the most elementary knowledge of arithmetic! All you need is to be able to add, subtract, and multiply, and know that some numbers, like 2,3,5,7,11,...etc. are prime numbers, and why they are prime numbers. It could have been done and appreciated thousands of years ago. All it would take is a mind that loves numbers and considers numbers their friend, and pays attention to their relationships in a couple of very simple operations. That, I believe, strengthens the case, that in some of those VMS pages, prime numbers and other mathematical ideas are being exhibited.
And this: a mind that conceived f6r, could very well be a mind appreciative of the grand idea of Platonic archetype, and therefore could well have realized that the exhibited number relationships held connections to very, very advanced ideas involving prime numbers, even though she or he may not have possessed, even secretly, the mathematical machinery for more advanced exploration of those ideas, that came later and then really took off with Euler, Gauss, Riemann, and others.
Perhaps even, the VMS author was saying: Study this, and invent mathematics for it!!
And in closing, to digress just once more for three seconds, it is only to be anticipated that if indeed the VMS turns out to be a far-advanced-for-its-time mathematics text, then the mystery of its origin may only worsen: was the author a plants-and-flowers-loving lady mathematician? Theoretical analytics alone, no matter how successful, cannot solve this mystery entirely, and confirmed unusual mathematical content in the book would just cry out stronger for VMS historical data. Therefore, if among the readers of this there is one who is wealthy, I ask you please to consider opening up your wallet and calling up the Beinecke and offering to fund some very-long-overdue spectroscopy on MS 408. Thank you.
Berj
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Tuesday, April 4, 2006 1:39 AM
To : vms-list@voynich.net
Subject : VMs: braided prime number equations in f6r
The previously considered oscillating pair of prime number equations in the thread "prime number equations in f6r", can be easily re-written as this pair:
Eq. 1: p1 + p2 = 6n
Eq. 2: p1 - p2 = 6n
where p = prime, p1 > p2, and n = 1,2,3,4,5,....
Apparently this pair of coupled equations is the symbolic meaning of the braided roots in the f6r plant drawing.
Here is what goes on with these equations that I've so far checked with hand calculation:
Table 1:
p1 p2 (p1+p2) (p1-p2) n 6n
7 5 12 2 12
11 7 18 3 18
13 11 24 4 24
17 13 30 5 30
19 17 36 6 36
23 19 42 7 42
29 23 6 1 6
31 29 60 10 60
37 31 6 1 6
41 37 78 13 78
43 41 84 14 84
47 43 90 15 90
53 47 6 1 6
59 53 6 1 6
61 59 120 20 120
67 61 6 1 6
71 67 138 23 138
73 71 144 24 144
79 73 6 1 6
83 79 162 27 162
89 83 6 1 6
97 89 186 31 186
101 97 198 33 198
I hope I copied these numbers accurately; also that the table will display well in the email without having to fuss with alignments. Needless to say, the Table speaks for itself. From 500 or more years ago?!
There is a lot more from where these equations came from, i.e. the general equation - it is a veritable gushing fountain of prime numbers in relationships. It is getting to the point where I have to crack some books on prime numbers and hunt for known theorems, and also write some computer programs.
It occurs to me that Table 1 above and its extension presents several possibilities for enciphering an alphabet. In other words, the possibility that f6r is not just symbolic prime number mathematics, but simultaneously is presenting a cipher key for some portion of Voynich text.
Berj
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Tuesday, April 4, 2006 5:30 PM
To : vms-list@voynich.net
Subject : RE: VMs: braided prime number equations in f6r
I would like to pose a historical question that I hope you will agree has a reasonable relevance to the historical aspects of the Voynich manuscript:
Is there a mystery somewhere back in the history of mathematics concerning a highest known prime number? That is, someone somewhere 500 or a 1000 years ago or so, was in possession of a prime number far higher than those possessed by contemporaries, and knew that it was a prime number, and nobody knows how the possessor got that prime?
To show the reasonable relevance to VMS history, let us attempt to use the prime oscillator equations from earlier in this thread to PREDICT prime numbers.
First I want to improve the notation, so the two equations are now written:
Eq. 3: pi - pj = 6n
Eq. 4: pi + pj = 6n
where we read "p sub i" and "p sub j", and "6 times n",
and pi > pj for i > j , and n = 1,2,3,4,5,.....
Now, let pj = p = a known prime number, and X > p and write:
Eq. 5: X - p = 6n or: X = 6n + p
Eq. 6: X + p = 6n or: X = 6n - p
1st attempt: let p = 123,457 a known prime, and n = 50,000
We get from Eq. 5: X = 423,457 it is prime!
and from Eq. 6: X = 176,543 it is not prime.
2nd attempt: let p = 423,457 and n = 100,002
From Eq. 5: X = 523,459 it is prime!
From Eq. 6: X = 176,555 cannot be prime because it divides by 5, and therein is a major significance we'll look at in a moment.
3rd attempt: let p = 523,459 and n = 211,610
From Eq. 5: X = 1,793,119 it is prime!
From Eq. 6: X = 746,201 it is not prime.
Now, I pulled the n values out of the air, and I began with a known prime, 123457, and of course I determined by calculation which of the predicted numbers, if any, were actually prime.
But, if the prime oscillation equations are good for generating primes some good percentage of the time - 50% would be wonderful, 100% would call for Cuban cigars, then the second attempt above suggests a strategy: just track along attempt-paths where one of the equations produces a number that is immediately recognized as not-prime. Thus the bag collecting the output numbers from the successive attempts has a filter resisting a blooming ratio of collected non-primes to primes.
Now lets imagine back in antiquity there was someone who claimed to have an ultra-prime, and later it turned out not so, and the claimant was relegated to the mathematical charlatan footnotes. But is the opposite somewhere in the annals of math? Was there someone who actually did have an ultra-prime, and to this day nobody knows where he/she got it?
If yes, surely we would want to have a closer look at that person, to see if anything about them rings VMS bells.
Berj
From: "Berj N. Ensanian" <>
Reply-To: vms-list@voynich.net
To: vms-list@voynich.net
Subject: VMs: braided prime number equations in f6r
Date: Tue, 04 Apr 2006 01:39:01 -0400
The previously considered oscillating pair of prime number equations in the thread "prime number equations in f6r", can be easily re-written as this pair:
Eq. 1: p1 + p2 = 6n
Eq. 2: p1 - p2 = 6n
where p = prime, p1 > p2, and n = 1,2,3,4,5,....
Apparently this pair of coupled equations is the symbolic meaning of the braided roots in the f6r plant drawing.
Here is what goes on with these equations that .....
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Wednesday, April 5, 2006 2:32 PM
To : vms-list@voynich.net
Subject : Re: VMs: braided prime number equations in f6r
Mark Hagerman wrote (1):
"It's only to say that any serious mathematician of the time would know that there was no "largest" prime."
and Jeff Haley wrote (2):
"Unless this was the workbook of a mathematician I do not see how the method
could have been encoded."
"Outside of this circle there could be others that have not come to light but earlier than the seventeenth century I would find it hard to see how these equations could have been encoded. For one thing there was no equals sign as such."
"As we have seen Euclid had already established the infinite nature of primes
so what you would need to establish is an alternative reason for the secrecy."
Answering Mark and Jeff:
Lets look at Table 1 again: the ideas in there rest on simple counting (pebbles for instance), and ordinary addition, subtraction, multiplication, and division. Aside from its purely mathematical implications, and wondering about ciphering, Table 1's startling preserve-the-sequence-continuity 6's invite us to notice: that a 6, when written upside down, looks like a 9, and there are so many of those GC-9 / EVA-y symbols in the VMS text, seemingly suggesting text terminals. And so much has been made of them being Latin abbreviations, but I have to wonder - are they sometimes Ekwall pointers for the text or some such thing? Don't we, for example, in the blatantly prime-numbers-riddled f6v, see a strong suggestion that the gallows letters are being pointed to? (3) It seems rather natural for a hypothetical mathematician author.
The equations can be expressed entirely verbally, for example:
31 pebbles increased by 29 pebbles comes to 60 pebbles. And 31 pebbles diminished by 29 pebbles comes to 2 pebbles. And as it is that 6 pebbles can be multiplied 10-fold to make 60 pebbles, and also that 6 pebbles cannot be multiplied to make 2 pebbles, so therefore the increasing of 31 pebbles by 29 pebbles belongs to the prime braid that weaves according to increase.
It even sounds a little poetic, like something one might read in an ancient Mesopotamian cuneiform tablet. Wonder what its entropy comes to. And I noticed a psychological effect: after reading it over a few times, I found myself grasping the concept verbally, and not even thinking visually in terms of modern algebraic equations. The modern notation is convenient and efficient for discussion of the ideas here, and I have already shown in the earlier big sister thread to this one, that controversies about when "modern" ideas like "the concept of negative numbers" came on the scene, can be avoided, if we take care. (But have you ever seen a stronger incentive for inventing negative numbers than the stuff that gushes out of the general equation?)
So my point with that is that the equations are expressible verbally in very simple language, and therefore in script as above, or as pictures of "plants".
The reasons for secrecy. The possibilities are many and could be explored in detail in another thread. There is still some society-type secrecy in mathematics today - "Nicolas Bourbaki" for instance. But mainly, I believe that there are some startling, well-ahead-of their-time prime number theorems encoded in the completely general f6r equation. That makes the most sense to me: someone had some really advanced prime number theorems, useful in many ways, including devising ciphers, and they kept them secret. (4) I think every detail of the f6r plant has a mathematical significance - for example, what does that cylinders-and-disk structure between the stem and braid-roots symbolize? I've got some notions, but it's too soon to crystallize them.
Concerning possession of a current-largest prime. It is some measure of mathematical power and prestige. And could have been more so in pre- calculating machine days: imagine that you suddenly convincingly exhibit an ultra-prime, one much, much higher than any possessed by others working with the common abacus and the Sieve of Eratosthenes. And everybody wants to know how you did it, but you keep it a secret to preserve your prestige, and therewith attract potential initiate candidates to your math school.
Berj
(1) >From: Mark Hagerman <>
Reply-To: vms-list@voynich.net
To: vms-list@voynich.net
Subject: Re: VMs: braided prime number equations in f6r
Date: Tue, 04 Apr 2006 22:29:50 -0600
(2) From : J HALEY <>
Reply-To : vms-list@voynich.net
Sent : Tuesday, April 4, 2006 8:02 PM
To : <vms-list@voynich.net>
(3) ref. my comments about prime numbers and specifically in f6v, in the vms-list thread "Breakthroughs and breakdowns", launched by Marke Fincher, March 2006.
(4) those secret theorems may well have been discovered and in the open mathematical mainstream by the time the VMS was written, written perhaps as a copy of older source material, with the purpose of preserving the old secret texts.
**************
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Friday, April 7, 2006 11:37 PM
To : vms-list@voynich.net
Subject : Re: VMs: braided prime number equations in f6r
From: "Robin Mackenzie" <>
Reply-To: vms-list@voynich.net
To: vms-list@voynich.net
Subject: Re: VMs: braided prime number equations in f6r
Date: Thu, 6 Apr 2006 12:08:34 -0700
Hi Berj,
Just quoting you on:
.............................................
Robin
Hi Robin
Well lets see what we can do. You have brought up a lot of points, but the main one is reference texts for analysis of the VMS text. At the end of this I'll list a couple of books that I think are good to have for this kind of work. (1)
First lets improve the poetry. Lets change one of the paragraph's words into a different (or possibly two) word(s), and thereby make the composed paragraph conform better to the IDEAS in Table 1, rather than just their turn-the-crank results:
31 pebbles increased by 29 pebbles comes to 60 pebbles. And 31 pebbles diminished by 29 pebbles comes to 2 pebbles. And as it is that 6 pebbles can be multiplied 10-fold to make 60 pebbles, and also that 6 pebbles cannot be multiplied to make 2 pebbles, so therefore the (simple) confrontation of 31 pebbles by 29 pebbles belongs to the prime braid that weaves according to increase.
Call that paragraph A.
Now lets compose a paragraph B, to express, generally, the theorem implied by the coupled prime oscillator equations and Table 1. Simultaneously lets take the liberty of changing the poetic theme from "pebbles", or the related and time-honored "heaps", to "kingdom", a concept more reflective of the royalty of prime numbers. Doing this will suggest more possibilities for poetic word and phrase substitutions, while not disturbing the main points we are developing and trying to illustrate here.
Paragraph B:
Let there be a/the first kingdom. And let it confront a/the second kingdom. (Or: Let them confront one another.) Be it favored by the 6-god that the kingdoms merge peacefully, in that the 6-god multiplies himself agreeably to the merger, then those kingdoms shall have shown that their common ancestry belongs to the strand of the holy braid that weaves according to peaceful brotherhood. But be it seen that the 6-god multiplies himself agreeably only to the remains of total war between the kingdoms, then it is known that their common ancestry is ruled by the strand of the holy braid that weaves according to the right of one, and only one Queen.
Poetically-inclined VMS students who also enjoy chess, might come up with verse modifications for the above, that while preserving the mathematics, diversify the reference versions available for testing.
For the time being, following in the sometime tradition of the great Pierre, we will leave it to others to prove or disprove interesting alegations/implications, conditionally or generally. We are justified in that because in these thread proceedings we are acting primarily as experimentalists, producing interesting, if not peculiar, data. And once more, here because the poetry hints it, I would like to suggest that we could use a list of pros and cons concerning a woman's mind, or the collective mind of a society of women, having a hand in the source material of the VMS. (2)
Now, we have two paragraphs A and B, so lets compose a couple of possible glue phrases:
C1: And here let us now see behind the veil the higher revelation.
C2: And here let us now cast before the veil a parable for common consumption.
So now we have two possible text-blocks:
TB-1: A -> C1 -> B
TB-1: B -> C2 -> A
This is reminiscent of the old friendly controversy in physics courses and textbooks of: is it better to start with Maxwell's equations and teach electromagnetism from the top down, or start with the historically discovered specifics and build up to Maxwell? A detective investigating that would soon be interested in: is the professor at heart an experimental physicist, or a theoretical physicist? It's not too soon I think to consider similar thoughts with regard to VMS authorship, especially if it becomes more likely that the book is not just doctrinal principles, but also a sort of textbook.
So we have TB-1 which goes from a specific example to the generalization, and TB-2 which goes from the general principle to a specific example. These reference textblocks, or other versions of them, are now ready for those who are well skilled at mathematical text analysis, that is mathematical what-sort-of-information-is-in-this-text? analysis. Of course there could still be an intermediate cipher transformation, of one class or another, to perform if it was assumed that cipher-text is the type of text of concern, as opposed to just plain text in an unknown language and alphabet. The analytics can then Zipf the text-blocks, zeta them, entropy them, Fourier them, Gauss-Jordan them, S-matrix them etc. etc., and finally come up with sophisticated mathematical objects that characterize the information content, and even more important, characterize the information waves, in these textblocks.
And we know that the characteristic information functions for these reference text-blocks ought to detect a pulse of some sort: with TB-1 they would detect the "specific-to-general pulse", and with TB-2 they would detect the "general-to-specific pulse". We hope! This could be tougher than it seems. We hope that the characteristic functions would discriminate with high confidence such particular pulses, and they might even be mirror images: pulse-front versus pulse-tail. And at the top of our wish-list would be detection of the unique pulse signature, invariant under cipher-transformations. And I would think also independent of source language, which for the poetry above is English.
Now if I were doing this, and at this point I still did not have a reliable detector function, but I was certain that there has to be one, I would try defining yet a new transform, one that takes text as input and produces a continuous but non-differentiable function. I'd find a way to invent a suitable transform. It seems reasonable to me that mysterious text of the VMS type, where you have no punctuation, no sure idea of what are true (cipher)-words and spaces, what are true logical lines and paragraphs, and so on, it is worth a shot mapping via Celerier and Weierstrass type functions - because, thinking poetically, isn't there a sort of analogy between such functions and an infinitely labrynthine text that always seems to cut itself loose from the anchors that we think we've established in it?
And then finally, with a working transform in hand, comes the real objective: are similar pulses evident in the VMS text? And that would mean scanning the entire VMS transcript over and over with all sorts of phase shifts, because we don't know for sure where in the VMS text, sentences actually start and so on, as just mentioned. But that much intense scanning and comparing is what computer programs are good at, so not to worry. As long as the programming is good, and the reference text blocks seem worthwhile to take a shot at it, perhaps we will see someone come forward with something striking: like a convincing TB-1 or TB-2 type pulse in the text of a VMS page, where other clues in the page also suggest that a TB-1 or TB-2 type discussion is going on in it.
Concerning reference texts drawn from unentscheidbare literature, I doubt it - not poetic enough, and I've been building a case upon a kind of poetry inspired by very down to earth math in ancient cuneiform tablets. Because, it seems to me that the minds that composed the VMS, are closer to the minds of poets than to the unentscheidbare minds. But I could be wrong and you might have a wonderful point: the VMS is a coherently written demonstration of unentscheidbar. I think Adam McLean suggested something very similar in the Big Sister thread to this one, and in his own way. But I hope not, and I really doubt that the VMS concerns itself with much quasi-mathematics, if any, because the book has too much art. Of course it might include things along the lines of Zeno's paradoxes, but Zeno is poetic when he teaches. In any case, the pulse analysis procedure would still be valid: just start with unentscheidbare literature for transformation input, after first having decided upon the pulses of interest in it. However, if I was going to use material like that as a reference text, I'd probably first decide to warm up the transforms and functions with text from Frege and Russell and Whitehead.
I myself am not entirely convinced that all, or most of the VMS is under the central plan of prime numbers. But I am convinced, beyond doubt, that at least some parts of it are, and definitely f6v and f6r are. I think the coupled prime oscillator equations were intentionally depicted by the f6r "plant", and they are only the tip of the iceberg. So many critical 6's, and you mentioned one of my favorite books, the Bible, wherein, for example, in Revelation, after we get some things about a "little book", comes in chapter 13:
"Here is wisdom. He that hath understanding, let him count the number of the beast; for it is the number of a man: and his number is Six hundred and sixty and six."
I hope we can we get an answer to the question I asked earlier: Is there a mystery somewhere back in the history of mathematics concerning a highest known prime number?
Berj
(1) The New Book of Prime Number Records, Paulo Ribenboim, Springer, 3rd ed. 1995/96. Full of detailed prime number theory. Also: Mathematics by Experiment, by Borwein and Bailey, A. K. Peters 2004. Lots of prime number goodies in this book.
(2) For a possible allegorical representation of Oannes the bearded Fishman conducting a balneological seminar for a society of female sages, see the vms thread "bearded man in f70r Pisces", 31 MAR 2006.
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Saturday, April 8, 2006 11:42 AM
To : vms-list@voynich.net
Subject : RE: VMs: Mathemetical symbols of the beast
Jeff
On the primes page you found, this looks interesting:
"Note: Some writers (e.g., [Picutti1989] and [BS96, p309]) include the primes 8191=213-1 (before 1458, Codice Palatino 573) and 131071=217-1 (1460, Codex Ottb. Lat 3307) in their list of records. But we omit them for lack of evidence that they were proven primes at that time, rather than just lucky guesses."
Clearly the writer does not allow the possibility that they might not have been just lucky guesses, but may have been calculated, and the method of calculation kept secret.
Codice Palatino of 1458, and Codex Ottb. Lat 3307 of 1460 might be worth a closer look.
Berj
From: "J HALEY" <>
Reply-To: vms-list@voynich.net
To: "Voynich Mailing List" <vms-list@voynich.net>
Subject: VMs: Mathemetical symbols of the beast
Date: Sat, 8 Apr 2006 15:51:54 +0100
OK I'll join in the debate. An interesting page is one from Robert Recorde's ...................
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Monday, April 10, 2006 9:23 AM
To : vms-list@voynich.net
Subject : RE: VMs: Codice Palatino 573
Jeff Haley wrote:
> Well if anyone is interested here is the reference for Codice Palatino 573.
>http://www2.math.unifi.it/~archimede/archimede/fibonacci/catalogo/ulivi.php
>[G. Arrighi, Nuovi contributi per la storia della matematica in Firenze
>nell'età di mezzo (il Codice Palatino 573 della Biblioteca Nazionale di
>Firenze), "Istituto Lombardo, Accademia di Scienze e Lettere, >Rendiconti, Classe di Scienze, A", 101 (1967), p. 401.]
Can someone read the commentary about Codice Palatino 573 and tell us what it says?
Berj
* * * * * * *
From ghost mirror at: http://www.gameszoo.org/voynichmonkeys
f58r prime numbers & zodiac 100 (Berj N. Ensanian) - 04-12-06 01:39
I have here some numbers and numbers relationships from f58r toward the hypothesis that at least some of the Voynich manuscript was planned with more than trivial mathematical thinking. (1)
The numbers results presented here come from just a first order attack on f58r, a rather simple attack, basically amounting to the completion of a small matrix. A big surprise to me came yesterday evening when the completion of the matrix resulted in the appearance of a pair of 100's. Since the matrix is directly linked to VMS stars, this pair of 100's is linked to VMS stars. This immediately arouses interest in the currently running thread launched a few days ago by Eddie Cottongim. (2)
Cottongim analyzed the VMS's astrology / zodiac section and identified a pair of coupled finite series of words, that to him indicate that their sums are each converging toward a total count of 100, alas impossible to verify directly because of missing VMS pages. Larry Roux corrected some input data for Cottongim's calculations, but the effect of that seems to be to actually
improve the Cottongim zodiac 100 (CZC) indication. Eddie is more sure that the two series are correlated in some way, than that they are sets of exactly 100 - the sums could be 99 or 101. Nevertheless, he chose 100 to call attention to.
In the following, when it helps make a point, I'll draw attention to prime numbers by appending a non-algebraic "p" to them, for example: 5p + 8 = 13p
As regards illustrations, f58r is about as bare bones as it gets in the VMS: Except for 3 drawn stars, the page is entirely text symbols, plus "58" in the upper right hand corner. The basic design of f58r is:
there are 3 paragraphs, each of which has 1 drawn star to the left (upper left with the 1st paragraph) of the opening word. All 3 paragraphs begin with a gallows letter, and are grossly distinguished as follows:
paragraph 1: 6-rayed star, with 1 dot? in its center, ligatured directly to the opening (gallows) letter of the 1st word on the 1st line. There is no mistaking this symbolism: the stars and the text are tightly linked!! The first 3 textlines are indented. 15 textlines total.
paragraph 2: 7-rayed star, with 1 circle in its center. 10 textlines
paragraph 3: 8-rayed star, with 1 circle in its center. 16 textlines.
Altogether the 3p paragraphs hold 41p textlines, of which line 29p, being the 4th line in the 3rd paragraph, is the ONLY textline having NO gallows letters.
Already from the gross structure, we can perceive a matrix or two, with all of its numbers being unambiguous, except for the dot and circles inside the stars.
We could assign: dot = 1, and circle = 2 because a circle has: 1 inside + 1 outside. Lets try that, with a matrix that exhibits ideas as well as numbers, and recall that only 1 of the gallows letters is symmetric, the EVA-t / GC-k:
TABLE / MATRIX 1: 3 rows x 5 columns
paragraph #, star rays, star internal, lines, starting gallows
1 odd, 6 even, 1 odd, 15 odd, assymetric
2 even, 7 odd, 2 even, 10 even, symmetric
3 odd, 8 even, 2 even, 16 even, assymetric
If nothing esle, this matrix strongly suggests: pay attention to symmetries versus asymmetries! Now consider the totally unambiguous sub-matrix:
TABLE / MATRIX 2:
6 15
7 10
8 16
Many interesting numerical results fall out of it, depending on path-through-matrix, and arithmetic operations at each path-node, for example:
Eq. 1: 6^2 + 7^2 + 8^2 = 149p = 7^2 + 10^2
which suggests an asymmetric T-cross, and further suggests finding a symmetric numbers cross in f58r. We do also get from MATRIX 2:
Eq. 2: (6^2 + 15^2) + (8^2 + 16^2) = 581 = 15^2 + 10^2 + 16^2
but for the moment the appeal of Eq. 1 is greater, so lets pursue it. First we note within MATRIX 2:
Eq. 3: 6 + 7 + 16 = 29p pointing? to the no-gallows line in paragraph 3.
To try explain the idea of pointing, here in this case, look at the matrix-path that gives the 41p textlines, and symbolically compare that path with the path of Eq. 3. Over and over, one gets the truly eerie feeling that the VMS author(s) was/were amazingly creative, to put it very mildly.
To continue our pursuit of the suspected fully symmetric numbers-cross, we now draw numbers from the fine structure of f58r: the symbols in the text-lines. The statistics of the symbols in the three paragraphs suggest many possibilities for constructing matrixes, and some interesting numbers relationships are quickly found. However we will here consider just a couple of observations on two of the non-intruding gallows. First, proceeding through the text from the beginning of paragraph 1 to the end of paragraph 3, we note that the symmetric EVA-t / GC-k gallows letter decreases in frequency, becoming rarer toward the bottom of the page, while the asymmetric gallows GC-h / EVA-k letter does the opposite.
For the total body of the text, the assymetric gallows letter occurs 73p times, but we must temporarily abandon that due to uncertainty: there are two occurrences of a letter resembling GC-u. If, because of scribe scripting variations, GC-u = GC-h, then the 73p is lost for special immediate interest.
To digress for a moment, and observe a possible transcription-problems resolution principle:
Differentiate ambiguously similar key VMS symbols according to their context-frequency = a prime number.
With the free (not intruding) symmetric gallows letter we find our unambiguous prime signal. It occurs 37p times in paragraph 1, 23p times in paragraph 2, and 19p times in paragraph 3. And:
Eq. 4: 37p + 23p + 19p = 79p
We now form TABLE / MATRIX 3:
star-rays, lines, no. of symm. gallows
6 15 37
7 10 23
8 16 19
__ __ __
21 41p 79p
This matrix has some interesting properties, but where is our numbers-cross? Also, that composite 21 stands out like a sore thumb in the sums row. Suppose we think along these lines: from the matrix proper, i.e. not including the sums row, we "borrow" the prime numbers, and put them together with 29p which has been itching to get in on the action. If we get somewhere with that, we will come back and find a way to repay our debt to MATRIX 3. Ok? Ok. Here's our prime numbers cross:
37
7 23 29
19
We see that:
Eq. 5: 7 + 23 + 29 = 59p
Eq. 6: 19 + 23 + 37 = 79p
It is a beautiful cross! So we are indebted to MATRIX 3, and we must repay it. And in doing so, we will suddenly confront the CZC indication. Now, to repay MATRIX 3, the best thing we could do would be to raise the status of the 21 in the sums row, that is, do something with lowly composite 21 so that it can hold its head up high down there next to its row-brethren primes
41p and 79p. And the best thing that 21 could accomplish would be a UNIFICATION of the borrowing of part of MATRIX 3 to join 29p in the resolution of the prime numbers cross. It turns out to be easy:
Eq. 7: 41p + 59p = 100
Eq. 8: 21 + 79p = 100
Symmetry and asymmetry, as we were prompted by the VMS author right in the beginning! So it turned out after all that there was a unifying thread in the manipulation of the sub-matrix, and the utilization of the unique 29p. And 2 occurrences of 100 of course make us sit up and wonder about the CZC indication. The CZC comes from the "zodiac" section. We already have in the
above:
2 coupled sums, each = 100, are unequivocally linked to VMS stars, at least on f58r.
Can we build further links between the 100's and the zodiac? Well, the 29p gallows-free line is also the 4th line in the 3rd paragraph, so:
4 x 3 = 12 -> zodiac!
Clearly, we have only barely begun to scratch the surface of possible intended mathematics in f58r. For example, the indented first 3 lines in paragraph 1 possibly suggests one or more, but not necessarily all of, the gross matrix elements being resolved into finer structure matrices, perhaps by starting from the special properties of 6: 6 = 2 x 3 = 1 x 2 x 3 = 1 + 2 + 3. And Steve Ekwall too, in a post this morning concerning one of his ideas, suggested something very much like that: a tic-tac-toe inside a
tic-tac-toe.
More globally, I suspect that f58r contains operations between matrices. That matrix and determinants algebra is a circa 19th century product, and that that mitigates against that idea, would seem a very weak arguement to me. For one thing, as the above shows, the matrices are being manipulated in uncoventional ways. But even more important, on the assumption that cryptography was involved in the VMS, we ought to allow the ancient cryptographers enough imagination to construct systems involving multiple parallel arithmetic operations, especially when their goal was not necessarily to produce mathematics suitable for systems of linear equations, vectors and tensors and so forth, but rather to come up with a very resistant cipher. The ancients knew magic squares, and why wouldn't they rub a couple of them together to make a cipher, and then take the next step to a better cipher with rectangular matrices? In this vein, the cipher creator is far less constrained than the mathematical physicist - the scheme just has to work to reversibly obscure input messages; it does not need to be at all suitable for solving equations arising out of powerful physical laws. And therefore, a type of matrix mathematics, call it cryptographic matrix
mechanics if you will, could well have arisen long, long before the 19th century. And remained secret.
Finally, let us recapitulate the hypothesis that at least some sections of the VMS are ultimately the product of a mathematician, or at least of a mind that included mathematician in its repertoire. Essentially, we have followed these two principles:
1. The mathematician author of the VMS employed mathematics and esoteric mathematical symbolism to both: encipher source material regardless of nature, and to exhibit or teach mathematics specifically.
2. To follow this VMS mathematician author: Navigate the prime numbers.
Immediately a MAJOR QUESTION arises. It is not a new question in VMS study, but in light of what we have just done with f58r, it arises in a modified form: Is some of the VMS text definitely NOT speech-language based, but rather the text symbols are there in their lines just to generate numerical and geometric relationships, and even without necessarily possessing
intrinsic numerical values of their own, for some mathematics that is being displayed? This is very different from the text being considered as possibly dummy, with no meaning at all. There are several possibilities, for example, is the text simultaneously carrying enciphered speech-language, and its symbols are serving to symbolically diagram mathematical ideas.
How then do we detect the attributes of the text? Perhaps the example of f58r suggests that in attacking mysterious text, the first thing to determine is: do its symbols display mathematics? If, with high confidence, definitely not, then, assuming other factors point to the text being meaningful, proceed by assuming speech-language.
In f58r we have discovered that the text symbols, at least some of them, are very meaningful, even if there is absolutely no speech-language content whatsoever riding on the text. So it remains only to answer if the text, or some remainder of it, is carrying any speech-language also. Perhaps by subtracting the demonstrated math-participating symbols from the text, the
entropy etc. calculations on the remainder will start to look more like real world language. Dennis Stallings, in his 1998 paper on the puzzling 2nd order entropy (3), anticipates this. One of his summary observations is:
"A verbose cipher, one which substitutes several ciphertext characters for one plaintext character, can produce the entropy profile of Voynich text."
And in footnote 6 he writes:
"The text was changed to the notation above. All numbers, English, Japanese, and other foreign words were removed until the character set (the number of characters MONKEY showed) matched the Hawaiian notation."
Therefore, it is only natural to ask: How do the 2nd-order entropies of f58r, with, and without, symmetric gallows letters compare? With suitable sample normalization of course.
No matter what the comparison shows, even if there is no significant difference, it is going to tell us something that we did not know before.
Berj
(1) see vms-list threads:
"prime numbers in the VMS", MAR 2006.
"prime number equations in f6r", MAR 2006.
"braided prime number equations in f6r", APR 2006.
"mathematical symbols of the beast", APR 2006.
(2) vms-list thread: "Astrology Pages: ok/ot labels", APR 2006.
(3) "Understanding the Second-Order Entropies of Voynich Text", by Dennis J. Stallings, May 11, 1998.
http://www.geocities.com/ctesibos/voynich/mbpaper.htm
* * * * * *
Re: f58r prime numbers & zodiac 100 (J HALEY) - 04-12-06 23:15
Hi Berj
Well that post certainly got my undivided attention. Some of the things I have been researching are starting to make sense. I will have to mull this over a little and get back to you. I am definitely interested in your ideas.
Jeff
Berj N. Ensanian wrote
>
>
> I have here some numbers and numbers relationships from f58r toward the
> hypothesis
* * * * * * *
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Monday, April 17, 2006 1:30 PM
To : vms-list@voynich.net
Subject : VMs: prime numbers and quadratic forms
Jeff Haley wrote:
From : J HALEY <>
Reply-To : vms-list@voynich.net
Sent : Thursday, April 13, 2006 9:19 PM
To : "Voynich Mailing List" <vms-list@voynich.net>
Subject : VMs: Prime numbers
To see why primes would be such a bitch to renaissance mathematicians could someone please factorise the following quadratic equation. Oh yes nearly forgot. You can't use any method developed later than around 1640. So at least you already have early algebra.
1.75x^2 + 4x + 4
Jeff
Hi Jeff
You've done, in my opinion, a great thing for VMS research with that post. I'd like here to consider your quadratic and run some ideas by you, and also those in the gang who are also open-minded to the hypothesis that prime numbers and prime number theorems may be central to parts of the Voynich Manuscript. (1)
Your quadratic is:
Expr. 1: 1.75X^2 + 4X + 4 = (7/4)X^2 + 4X + 4
You left room for interpretation of "factorization", which I think is key, but lets first consider a modern attack. We take the factorization specification to be:
Eq.1: AX^2 + BX + C = (DX + E)(FX + G)
and from B vs 2sqr(AC) or the curve:
Eq.2: Y(Z) = AZ + C/Z - B where Z = G/F
we see immediately what is coming with Eq.1, and indeed, we obtain, for example:
D = 1/4 ; F = 7 ; G = 8 + isqr(48) ; E = (2/7) - i[sqr(48)]/28
where i = sqr(-1) and we therefore immediately clash with the dignity of the current historical vms-and-mathematics view, because of the implication that complex variables may be intrinsic to vms design. But I think we can avoid that clash in what I'll present here.
Now, lets consider that "factoring" is analogous to "breaking" or "dividing". We break something up into parts, and if successful, we can recover the original by multiplying together those parts. Your quadratic does not want to be broken up into real parts, but it will break up into complex factors. Similarly, prime numbers do not want to be divided, but lets go ahead and break up a prime anyway, for example:
Eq.3: 3769p = (13 + i60)(13 - i60)
where we got the components of the complex factors by remembering the Pythagorean triangular equation:
Eq.4: 13^2 + 60^2 = 3769
and indeed there are some other primes satisfying:
Eq.5: P1^2 + N^2 = p2 for N : integer
and also some related varieties, for example:
Eq.6: P1^2 + N^2 = P2^2 e.g. 19^2 + 180^2 = 181^2
Now let us consider back in antiquity, that is in pre- complex numbers days, a mathematician, we'll call him Roger Rudolf, and he is upset because he can't factor Expr. 1. It occurs to him that there is some sort of analogy between the refusal of Expr.1 to factor, and the refusal of prime numbers to factor. He is aware of triangular relations like Eq.5 and Eq.6, and he thinks it could be interesting to find a way to solidify the analogy, and thereby build a link between at least some unfactorable quadratics and at least some prime numbers. And so he thinks about this.
And eventually Roger Rudolf reasons along these lines:
Some quadratics do not factor, but some do. No primes admit to factoring, but some primes, notably in pairs, do satisfy triangular equations. And there is a great similarity, a great tendency toward symmetry, between quadratics and triangulars:
Eq.7: (7/4)X^2 + 4X + 4 = N and X^2 + Y^2 = M^2 = N
Let, therefore, be suspected, and be worthy of discovery, the existence of links between the unfactorable and factorable through a higher symmetry between quadratics and triangulars.
We can see, in hindsight, that Roger Rudolf was reasoning in territory that was nearby the land of complex numbers. And what led him there? Prime numbers!
It is not just prime numbers that stand out in the VMS, but there are in places in it strong suggestions of quadratics. Indeed, the grand dramatic climax of the VMS is f86v, the great 9-rosettes foldout. And there, as sparkling as a sunny day, we behold the featured suggestion of the linking of prime numbers and quadratic forms.
Berj
(1) vms-list thread: "f58r prime numbers & zodiac 100", APR 2006
***************
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Saturday, April 22, 2006 1:29 AM
To : vms-list@voynich.net
Subject : VMs: Voynich f67v prime numbers, de Molay & Knights Templar
Hello All
While experimentally analyzing, according to the hypothesis and principles previously posted (1), the prime numbers structure of the f67v "cosmological" illustrations, I was surprised to see numbers and other symbols come up, that to me suggested the Order of the Knights Templar, and in particular suggested the life and violent execution of its Grand Master, Jacques de Molay, by Phillip IV, in 1314, marking the official end of Templar history. (2)
I have to add, that I had not the slightest thought about Templars in connection with the VMS until I examined f67v in detail. I have never, during my VMS studies made a conscious connection between the VMS and the Templars. I did not go looking for the Templars in the VMS. It's not that I think the idea is crazy, not at all, but rather it never occurred to me before. And I am surprised, in more ways than one.
I will here present, abbreviated, what I found. Then I'll make a few concluding comments. I will here consider primarily only the right hand side cosmological diagram of f67v.
Matters Knights Templar, and in particular Jaques de Molay, the 23rd and last (i.e. officially last) Templar Grand Master, are recognized by the travelling together of certain symbols and numbers. Among the numbers are 4, 14, 17p, 23p, 39, 54, 59p, 1307p, 1314. No matter the subset of Templar-symbolic numbers, 39 is usually among them, even if disguised. Lets have a look at this verso page 67p, and some of the numbers and symbols exhibited in its right hand illustration.
The illustration is a circular diagram suggesting, at first sight, an astronomical symbolism theme. The perimeter is a double-perimeter, containing alternating Voynich words, and designs suggesting the number 4, and/or a cross. At its center is a sun with a face, and from which radiate radially 18 fiery wavy rays. The tips of the rays meet lines of text that continue their respective radials straight toward the perimeter. At about the 2 o'clock position, two adjacent wavy rays meet at their tips, before meeting the text-ray that continues toward the perimeter. Thus altogether, there are 17 ray-tips, and extended by their text-rays, they create inbetween them 17 circumferential wedges. These wedges contain drawn-in stars, the number of them varying from 1 to 4 per wedge.
Noticably, the upper half of the sun-face is geometrically round, and the facial features are drawn realistically. In contrast, the lower half of the sun-face is far from round, and the facial features are hardly evident, but suggest distress. When one is thinking geometrically, this gives a strong impression of interlocked symmetry-antisymmetry. No doubt it also has more obvious symbolism along the lines of burdened-below vs perfection-above.
The orientation of the text-rays is such that a clockwise analysis of the diagram is suggested. The wedge at about the 4 o'clock position is the only one that contains just 1 star (3). In assigning index numbers to the wedges, I therefore chose it to be the #1 wedge. The remaining wedges are numbered 2-17, proceeding clockwise. Wedge 15 has, for its rightmost bounding text-ray, the one originating from the two meeting wavy rays. Wedge 15 is the only wedge bounded by: text-rays that begin with gallows symbols.
Wedges # 5,7, and 9, the only wedges containing green stars, are placed symmetrically opposite wedge # 15 (the one that is asymmetrically bounded).
The diagram is rendered in 3 colors: most of it in the brown color of the text, a single occurence of blue as the headband of the sun-face in the center, and three of the stars are colored green, or at least predominantly green.
The statistics on the stars are:
TABLE 1:
The Table is read as follows, for example: Wedge # 7 has 4 stars that between them total 28 star-rays. From closest to center, proceeding toward the perimeter, there are two brown stars each with 7 rays, followed by 1 green star with 7 rays, followed by 1 brown star with 7 rays. 4 columns:
Wedge #, No. of stars in wedge, No. of star-rays, star colors (going outward)
1,1,7, 1-brn-7
2,2,14, 2-brn-7
3,2,13, 1-brn-6
1-brn-7
4,2,13, 1-brn-6
1-brn-7
5,2,13, 1-brn-7
1-grn-6
6,3,21, 3-brn-7
7,4,28, 2-brn-7
1-grn-7
1-brn-7
8,2,14, 2-brn-7
9,2,14, 1-brn-7
1-grn-7
10,3,18, 3-brn-6
11,3,19, 1-brn-7
2-brn-6
12,2,14, 2-brn-7
13,2,14, 2-brn-7
14,2,13 1-brn-7
1-brn-6
15,2,13, 1-brn-6
1-brn-7
16,3,23, 1-brn-7
2-brn-8
17,2,14, 2-brn-7
Total 39 stars, 265 rays. 265 = 2x14 + 39 + 59 + 139
Looking over the table, I began to notice more and more the presence of strongly interlocked symmetry-antisymmetry. In particular, I suddenly noticed what was right in front of me all along: the circular diagram is built around a cross, a Templar cross, or Maltese cross, or Iron cross, either can be easily seen with the wedge pairs # 4 and 13, and 9 and 17. The pair 4-13 forms the vertical of the cross, and 9-17 forms the horizontal. I had not noticed this cross before, because of the optical effect of its slight clockwise offset. It was via looking at the symmetries and anti-symmetries suggested by the stars that I finally saw it. And once I saw the cross, and the type of cross it was, and how it emphasized the numbers 13 and 14 in its star counts, the prominence of 39 and 13 and 14 etc. etc. etc. began to ring Templar bells. I was definitely surprised by this!
On the more mathematical side, in line with the hypothesis that a mathematical author, one with a keen interest in prime numbers, is behind at least parts of the VMS regardless of the particular subject in page, f67v contains plenty. I'll just point out one example: there are a total of 10 6-rayed stars, 27 7-rayed stars, and 2 8-rayed stars, for 39 stars altogether. The numbers 10,27, and 2, form an interesting triangle of interlocked symmetry and anti-symmetry, that can be quickly visualized by drawing a triangle and placing these numbers at its vertices. The sides of the triangle can then be labeled with 3 new numbers, 37,29, and 17, generated from the original 3 as follows:
27 + 10 = 37p
27 + 2 = 29p
27 - 10 = 17p and 17 + 29 + 37 = 83p
Some concluding comments. First, I digress for just a moment to ask a question on the left-side diagram of f67v. At lower-left in that diagram is a circle containing four coupled balloon-heads set on a background of 3 colors. Am I seeing right - are those colors red, green, and blue, and in prominent ratio relationships? I ask because red, green, blue (RGB) are the primary colors of theoretical physiological color perception, and that theory, the Young-Helmholtz theory of color vision, has its formal start in 1807.
Some symbolic theme possibilities suggested by the above are, first, that the sun-face symbolizes de Molay's spirit as he burns at the stake. The two fiery sun-rays of Wedge 15 that meet, are de Molay cursing Phillip IV and Pope Clement V, this impression reinforced by the numbers 4 and 5 being displayed via the star counts of wedges 14-15, and 15-16. (Indeed, de Molay's two persecutors met their ends shortly after.) The 39 stars are the 39 Knights that were executed at the de Molay execution, and the three green stars are the three prominent Knights in addition to de Molay among the 39. Perhaps the blue headband symbolizes that de Molay was a man of truth in heaven.
If the f67v cosmological diagram was indeed designed to dramatize the 1314 Templars catastrophe, then the VMS cannot be earlier than 1314. And, it might be worthwhile to look at Scotland for VMS leads, since many Templar Knights were said to have escaped there when fleeing their persecution. The prominence of 17 suggests looking at Valencia, with its 17 Templar castles, among geographic possibilities, for VMS history clues.
If the VMS is a copy of an original, although scripted no earlier than 1314, it still remains entirely possible that it contains much older source material. And actually, if the Templars figure prominently in the VMS, then we might actually expect to see Byzantine and very ancient source material somewhere else in its pages, because of the complicated, and heavily-concerned-with-ancient-themes character of the Templars. How amazing it would be, if the legendary vanished Templar treasure was not earthly gold and such, but rather that "little book" mentioned by John in Revelation.
Berj
(1) see for example vms-list post: "VMs: prime numbers and quadratic forms", April 17, 2006.
(2) For Templars and de Molay history, see for example: A History of Secret Societies, by Arkon Daraul, Citadel Press, New York, 1962.
Online there are many Templars history resources, for example:
http://www.allcrusades.com/CHRONOLOGICAL/chrono-1300.html
http://www.gutenberg.org (search this site)
http://emotional-literacy-education.com/classic-books-online-b/2ppdl10.htm
http://oll.libertyfund.org/Home3/HTML.php?recordID=0011.02
http://www.templarhistory.com/index.html
http://www.freemasons-freemasonry.com/mackeyph07.html
(3) for another example where the 4 o'clock position is special in a VMS circular diagram, see vms-list post: "VMs: bearded man in f70r Pisces", March 31, 2006.
*******************************************
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Saturday, April 22, 2006 5:08 PM
To : vms-list@voynich.net
Subject : VMs: Knights Templar & f86v nine-rosettes
Concerning the possibility of Voynich Manuscript and Knights Templar connection (1), I just had a look at the manuscript's most dramatic foliation.
The Voynich f86v nine-rosettes foldout illustration seems to have rich possibilities for Knights Templar symbolism, both in its symbolically exhibited numbers, as well as its apparently primarily graphic symbols.
A few immediate observations:
There are 9 rosettes, the center one being larger than the others: the original 9 Templar Knights, of which only 8 are known, but the patron and lawgiver of the Templars, St. Bernard of Clairvaux, may have been more than just a patron, he may have been the conceiver of it all, and thus is symbolized by the central rosette. Then there are the mysterious first 9 years of the Order's existence.
1307, the year the crushing of the Templars began, is certainly easily seen from the middle top, and bottom rosettes.
In the lower-right rosette are strong suggestions of Templar founder Hughes de Payens' own seal, and also the Templum Domini, the Templar homebase (near the Dome of the Rock in Jerusalem).
The Templars had pointed-tower castles - we see those in f86v, as well as massive fortified walls.
f86v features "roads" - very significant in Templar history.
Berj
(1) for more on the possible VMS-Templars connection see vms-list post: "VMs: Voynich f67v prime numbers, de Molay & Knights Templar", April 22, 2006.
************
From : Berj N. Ensanian <>
Reply-To : vms-list@voynich.net
Sent : Sunday, April 23, 2006 6:04 PM
To : vms-list@voynich.net
Subject : VMs: Templars, Masons, and Voynich
Hello to all interested in the possibility of a Templar-Voynich connection,
I have some more data.
If we had to choose a letter from the Roman alphabet to symbolize the Templars, or their successors or sympathizers, we could do a lot worse than the choice of "M". This letter crops up with noticable frequency in everything relevant to Templar, from ancient to post 1307-1314, for example: Magister Militum Templi, Mary the Mother of God, Mandylion, manna, Moses, Maccabees, Merica, munk, monasterion, militia, monter, Mete, mysterium, martus, moneie, mathematika, .... many Templar important names and places.... and of course the Freres Maçons, the highly-mathematically-competent Masons, who had early and friendly relations with the Templars, and built all those grand buildings, castles, and walls.
M also has graphical and architectural affinities to symbols like temple, and the fundamental architectural element of the heavily Templar-influenced gothic style: coupled ogive arches.
We have seen that there is a ponderable possibility of Knights Templar symbolism in Voynich folios f67v [or f67v1 by some authorities], and also f86v, the great nine-rosettes foldout (1).
Therefore, if we were to find unmistakable featuring of "M" in f86v, we could consider the possibility of a Templar-Voynich connection becoming so much better. If we were to find a lot of M's, so much the better still. And, if we were to find that M is not only repetitively present in f86v, but it is in fact used to draw the very structure of walls, AND f86v's biggest castle, then we might consider that the Templar/Masons-Voynich trail is growing warmer. Lets see if it is.
The M's are there in f86v. With the Beinecke Library x8 image you are unlikely to spot them, unless you have first seen them in the 8.725 MB .sid image that they have for you to download. In the .sid image the M's are crystal-clear. Here is how to see them:
First, look at the connecting structure between the lower-middle and the lower-left rosettes. That structure is drawn across a folio folding. Toward the bottom of the structure you will see drawn, as a feature of the connecting structure, a long wall, and the top of that wall is drawn entirely with ligatured M's. At the left end, the wall and its M's are seen to originate by ligature with the gallows symbol GC-j.
Now look at the castle. It is at the 9 o'clock position of the upper-right rosette, therefore just about diagonally opposite the previous M's wall. You will see that the front wall of the castle has its top drawn just like the previously viewed wall - with the same kind of ligatured M's. There is at least one M in the back wall also.
Any theory, in my opinion, that attempts to explain Voynich ms f86v, will have to have a convincing explanation for these M's - they are intentional, they are meant to symbolize something very significant. Maybe it is something entirely different from Templars and Masons, but those M's must be explained. That castle is THE castle in THE folio of the VMS. If nothing else, the other M's of the wall connecting the lower-middle, and lower-left rosettes, and originating from one of the most important Voynich text-symbols, serve to say: yes, the castle M's are an important message - figure it out.
More speculatively, the castle is flanked by two square cross-section towers, the left one having a deep-blue flat-top. The right one also has a flat-top, but it is colored in the faintest yellow, only just enough to indicate that the tower has a closed flat top. Conceivably, these two flat tops are symbolizing the Templar Beauséant.
Next, on this trail, for an experimental exercise in symbolism, I'd like to borrow, partly, one of Steve Ekwall's highly emphasized ideas, and I'll state the partial loan-idea like this: Voynich manuscript folio FOLDING is important for understanding this book!!
Now let me utilize this borrowed capital: imagine before you the grand 9-rosettes foldout. Stand back a bit from it so that you can see the entire thing comfortably. Now, imagine all images and scripting removed from it - it is now a blank and unfolded parchment. What do you see? What you see is the best possible way of utilizing the parchment, the parchment intended for the f86v illustration, and with its pre-determined foldings, to exhibit one of the definite Knights Templar symbols: the double-bar cross.
Even assuming that the Templar-Voynich connection hypothesis is valid, I don't know if the VMS author was THAT SUBTLE in building-in symbolisms. But at this point, how can we put anything past this author's, or these authors' range?
Now, I stumbled across a couple of quotes on the web that reminded me of Harvard professor Barry Fell's work (2), and immediately had me wondering once more, but this time with Templars-Voynich in mind, about the famous VMS f93 botanical New World sunflower controversy, that started with the botanist Hugh O'Neill in 1944.(3)
Here are the quotes I found (4):
"The earliest dated portolan chart is the Opicinis de Canestris map of the Mediterranean of 1335 A.D. It demonstrates that maps of inexplicable accuracy began to appear in Europe less than 25 years after King Philippe's surprise raids against the Templars and the papal elimination of the Order under Clement V."
"...Is it mere coincidence that his flagship, the famous Santa Maria, bore Templar crosses on her sails when Columbus set sail from Palos? Is it mere coincidence that his voyage was financed, not by the sale of Isabella's jewelry as so commonly thought, but by a mysterious consortium of wealthy men which included Jews and other heretics? And is it only coincidence that Columbus weighed anchor on August 3, 1492 just a few hours before the deadline for all Jews to be out of Spain?"
- Michael Bradley, Holy Grail Across the Atlantic
So one has to wonder if there is a possibility after all, for some old and very controversial Voynich issues, the sunflower in particular, to be resolved, via the Knights Templar trail.
Finally, let me briefly summarize a few findings pointing toward a Templars, or Masons, and Voynich manuscript connection, the bulk of which are developed in detail in the posts given in ref. 1 below.
a) VMS f67v suggests, via its symbols and numbers, the going underground of elements of the Templars, subsequent to the catastrophe of 1307-1314.
b) VMS f86v, the nine-rosettes foldout, being the indisputable sensory zenith of the entire book, suggests multiple Templar symbols, beginning with the featuring of the number 9 that is so central in Templar history, for example: the 9 original Templar Knights, the mysterious first 9 years of the Order's existence, and the mysterious 9 Templars that suddenly confronted Pope Clement V during the latter's Council of Vienne on October 16, 1311, a council called to decide on the fate of the Templars.
c) f67v and f86v are both involved with the mysterious missing VMS folios.
d) everything remains consistent with the hypothesis that the VMS author was, among other talents, a mathematician.
Rosslyn Chapel in Scotland, the Knight Jean de Joinville, and the recently discovered Chinon document (5), are all well indicated for further VMS history exploration, I believe.
"It seems to me very likely, however, that there is some kinship between the philosophy underlying the manuscript and the Rosicrucian tradition. ....a serious student can scarcely afford to ignore any of this highly interesting material." - Mary D'Imperio, 1970's
Berj / KI3U
(1) see vms-list posts:
"VMs: Knights Templar & f86v nine-rosettes", April 22, 2006.
"VMs: Voynich f67v prime numbers, de Molay & Knights Templar", April 22, 2006.
(2) America B.C. : Ancient Settlers in the New World, by Barry Fell, Pocket Books, New York, 1976.
(3) see for example Mary D'Imperio's indispensible book: The Voynich Manuscript - An Elegant Enigma, Aegean Park Press, ISBN 0-89412-038-7.
(4) http://www.mystae.com/restricted/streams/masons/mysteries.html
(5) http://www.inrebus.com/chinon.html
***************