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 Early German Dozenal Pioneer, Dozenal proposal in 1800d
Shaun
Posted: Jan 26 2018, 03:04 PM


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by Johann Friedrich Werneburg (1777 - 1851).
Proposals for dozenal digits, number words etc.
I'd forgotten about this - found a card with reference from my student days (1962), when there was no possibiity of getting details as easily as here is today.
Werneburg referred to the dozenal system as 'Teliosadik' (Greek), and you can get details by googling teliosadik. More than just that, it is possible to get a reprint (from India), and I have ordered one for myself.

user posted image
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Kodegadulo
Posted: Jan 27 2018, 12:15 AM


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Here is the system of number names Werneburg "invented, designed, exhibited, and elaborated" (erfunden, entworfen, ausgestellt, und ausgearbeitet), in five European languages, including his native German:

Value
[z]
German
(Deutsche)
French
(Französisch)
English
(Englisch)
Dutch
(Holländisch)
Italian
(Italienisch)
1einsunoneeenun
2zweideuxtwotweedue
3dreitroisthreedrietre
4vierquatrefourvierquatro
5fünfcinqfivevyfcinque
6sechssixsixzessesei
7siebenseptsevenzevensette
8achthuiteightagtotto
9neunneufninenegennove
zehndixtentiendieci
Ɛmörmörmörmörmöri
10tauntauntauntauntauni
11tauneinstaununtaunonetauneentauni un
12taunzweitaundeuxtauntwotauntweetauni due
13taundreitauntroistaunthreetaundrietauni tre
14taunviertaunquatretaunfourtaunviertauni quatro
15taunfünftauncinqtaunfivetaunvyftauni cinque
16taunsechstaunsixtaunsixtaunzessetauni sei
17taunsiebentaunsepttaunseventaunzeventauni sette
18taunachttaunhuittauneighttaunagttauni otto
19taunneuntaunneuftaunninetaunnegentauni nove
1ᘔtaunzehntaundixtauntentauntientauni dieci
taunmörtaunmörtaunmörtaunmörtauni möri
20zweitaundeuxtaunstwotaun tweentaun duetauni
21zweitauneinsdeuxtaunsuntwotaun onetweentaun eenduetauni un
22zweitaunzweideuxtaunsdeuxtwotaun twotweentaun tweeduetauni due
23zweitaundreideuxtaunstroistwotaun threetweentaun drieduetauni tre
24zweitaunvierdeuxtaunsquatretwotaun fourtweentaun vierduetauni quatro
25zweitaunfünfdeuxtaunscinqtwotaun fivetweentaun vyfduetauni cinque
26zweitaunsechsdeuxtaunssixtwotaun sixtweentaun zesseduetauni sei
27zweitaunsiebendeuxtaunssepttwotaun seventweentaun zevenduetauni sette
28zweitaunachtdeuxtaunshuittwotaun eighttweentaun agtduetauni otto
29zweitaunneundeuxtaunsneuftwotaun ninetweentaun negenduetauni nove
2ᘔzweitaunzehndeuxtaunsdixtwotaun tentweentaun tienduetauni dieci
zweitaunmördeuxtaunsmörtwotaun mörtweentaun mörduetauni möri
30dreitauntroistaunsthreetaundrietauntretauni
40viertaunquatretaunsfourtaunviertaunquatrotauni
50fünftauncinqtaunsfivetaunvyftauncinquetauni
60sechstaunsixtaunssixtaunzessetaunseitauni
70siebentaunsepttaunsseventaunzeventaunsettetauni
80achttaunhuittaunseighttaunagttaunottotauni
90neuntaunneuftaunsninetaunnegentaunnovetauni
ᘔ0zehntaundixtaunstentauntientaundiecitauni
Ɛ0mörtaunmörtaunsmörtaunmörtaunmöritauni
100 = 102einardeunardeon'ardeeenardeunardi
10·100 = 103taun einardentaun unardestaun on'ardetaun eenardentauni unardis
1002 = 104zweinardedeuxardetwonardetweenardeduardi
10·1002 = 105taun zweinardentaun deuxardestaun twonardetaun tweenardetauni duardis
1003 = 106dreinardetroisardethreenardedrienardetreardi
10·1003 = 107taun dreinardentaun troisardestaun threenardetaun drienardentauni treardis
1004 = 108vierardequatrardefourardevierardequatrardi
10·1004 = 109taun vierardentaun quatrardestaun fourardetaun vierardentauni quatrardis
1005 = 10fünfardecinquardefiv'ardevyfardecinquardi
10·1005 = 10Ɛtaun fünfardentaun cinquardestaun fiv'ardetaun vyfardentauni cinquardis
1006 = 1010sechsardesixardesixardezessardeseinardi
10·1006 = 1011taun sechsardentaun sixardestaun sixardetaun zessardentauni seinardis
1007 = 1012siebenardeseptardesevenardezevenardesettardi
10·1007 = 1013taun siebenardentaun septardestaun sevenardetaun zevenardentauni settardis
1008 = 1014achtardehuitardeeightardeagtardeottardi
10·1008 = 1015taun achtardentaun huitardestaun eightardetaun agtardentauni ottardis
1009 = 1016neunardeneufardenin'ardenegenardenovardi
10·1009 = 1017taun neunardentaun neufardestaun nin'ardetaun negenardentauni novardis
100 = 1018zehnardedixardetenardetienardedieciardi
10·100 = 1019taun zehnardentaun dixardestaun tenardetaun tienardentauni dieciardis
100Ɛ = 101ᘔmörardemörardemörardemörardemöriardi
10·100Ɛ = 10taun mörardentaun mörardestaun mörardetaun mörardentauni möriardis
10010 = 1020einoideunoïdeon'oïdeeenoïdeunoidi
10020 = 1040zweinoidedeuxoïdetwonoïdetweenoideduoidi
10030 = 1060dreinoidetroisoïdethreenoïdedrienoidetreoidi
10040 = 1080vieroidequatroïdefouroïdevieroidequatroidi
10050 = 10ᘔ0fünfoidecinquoïdefiv'oïdevyfoidecinquoidi
10060 = 10100sechsoidesixoïdesixoïdezesseoideseinoidi
10070 = 10120siebenoideseptoïdesevenoïdezevenoidesettoidi
10080 = 10140achtoidehuitoïdeeightoïdeagtoideottoidi
10090 = 10160neunoideneufoïdenin'oïdenegenoidenovoidi
100ᘔ0 = 10180zehnoidedixoïdetenoïdetienoidediecioidi
100Ɛ0 = 101ᘔ0möroidemöroïdemöroïdemöroidemöroidi
100100 = 10200einiadeuniadeon'iadeeeniadeuniadi
100200 = 10400zweiniadedeuxiadetwoniadetweeniadeduiadi
100300 = 10600dreiniadetroisiadethreeniadedrieniadetreiadi
100400 = 10800vieriadequatriadefouriadevieriadequatriadi
100500 = 10ᘔ00fünfiadecinquiadefiviadevyfiadecinquiadi
100600 = 101000sechsiadesixiadesixiadezessiadeseiniadi
100700 = 101200siebeniadeseptiadeseveniadezeveniadesettiadi
100800 = 101400achtiadehuitiadeeightiadeagtiadeottiadi
100900 = 101600neuniadeneufiadenin'iadenegeniadenoviadi
100ᘔ00 = 101800zehniadedixiadeteniadetieniadedieciadi
100Ɛ00 = 101ᘔ00möriademöriademöriademöriademöriadi
1001000 = 102000tauniadetauniadetauniadetauniadetauniadi

Note that he cooked all this up in the year taun einarde sechstaun, or "taun on'arde sixtaun" (zen gross sixzy = 1060z = 1800d), when he was not yet even zweitaun (or "twotaun", or twenzy) years of age. Pretty good effort for a young man in that day and age. Apparently it was his annus mirabilis.

However, I think he was a little unrealistic about the suitability of that umlauted O in mör ("eleven") for all of those languages.

And I do find the nested layers of high powers a bit übertöten (overkill). smile.gif Although they apparently completely skirt the "apposition problem" (which has come to be rather notorious here due to a certain party's zealous jeremiads about it). Even though Werneburg uses the same German digit-words for both mantissa and exponent, he can distinguish zwei dreinarde (2,000,000z) from drei zweinarde (30,000z), because only the very last digit-word attached to a special ending is ever interpreted as an exponent. But I think we've already demonstrated less convoluted ways to solve that problem...
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Kodegadulo
Posted: Jan 27 2018, 04:36 PM


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In case the Fraktur is making any of you cross-eyed, here's a transliteration, and my attempt at a translation:

QUOTE (Johann Friedrich Werneburg)
Teliosadik,
oder
das allein vollkommene unter allen
Zahlen Systemen
und
das nach dessen GrundZahl bestimmte
allein vollkommene
Grad-, Zeit-, Maß-, Gewicht-, und Münz-
System
angewandt auf das bürgerlich Leben.
Zuerst erfunden, enworfen, aufgestellt und
ausgearbeitet
von
Johann Friedrich Christian Werneburg
der Welt Weisheit Doctor.
Ein ganz neues gemeinverständliches Hand Buch des
Rechnens für jeden StaatsBürger.
Erster Theil.
Das reine vollkommene
Zahlen System.
... Gehet hin in alle Länder un lehret allen Völkern der Erde -- Wahrheit! -- Verkündet sie, allen als das edelste, göttlichste, unveräusserlichste Kleinod! --
VerlagsHandlung für die neueste Litteratur. 1060 [1800]
Teliosadic,
or
the only perfect one among all
Number Systems
and
that which according to its Base Number determined
the only perfect
Degree-, Time-, Measure-, Weight-, and Money-
System
applied to civil Life.
First invented, designed, exhibited, and
elaborated
by
Johann Friedrich Christian Werneburg
the Doctor of World Wisdom.
A completely new generally-understandable HandBook on
Reckoning for every Citizen of the State.
First Part.
The pure perfect
System of Numbers.
... Go to all Lands and teach all the Peoples of the Earth -- the Truth! --  Proclaim it to all, as the noblest, most divine, most inalienable Treasure! --
Publishing Agency for the latest Literature. 1060z [1800d]


Well, I suppose in those days publishers hadn't yet gotten into the art of larding marketing hype on the dust-jacket in order to boost sales. (No dust jackets back then either.) So I suppose authors had to do all the hyping themselves, right on their cover pages.
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Kodegadulo
Posted: Jan 27 2018, 06:05 PM


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As to the title of this system, Teliosadik, it clearly derives from Greek τέλειος (teleios), "perfect", which jibes with Werneburg's self-promoting enthusiasm that he had devised the "perfect" (vollkommene) number system.

The -adik (or -adic) suffix also derives from Greek, appearing in words like monadic, dyadic, triadic, and is roughly the equivalent of the Latinate -ary, appearing in words like unary, binary, ternary.

In short, Werneburg's "Teliosadic" essentially means "Perfectary", implying dozenal is the "perfect base". Well, I won't argue with that much, at least. smile.gif
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Double sharp
Posted: Jan 28 2018, 03:29 AM


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Incidentally, Werneburg's transdecimals look like argam-14 and argam-15 respectively (turned 3 and reflected 9).
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Shaun
Posted: Jan 28 2018, 05:32 PM


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QUOTE (Double sharp @ Jan 28 2018, 03:29 AM)
Incidentally, Werneburg's transdecimals look like argam-14 and argam-15 respectively (turned 3 and reflected 9).

He may have chosen the capital letter Z (as printed in Fraktur), and rotated that for his 'ten' (zehn).
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Shaun
Posted: Jan 28 2018, 05:34 PM


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QUOTE (Kodegadulo @ Jan 27 2018, 04:36 PM)
In case the Fraktur is making any of you cross-eyed,

The Fraktur is fine, but I would appreciate numbers in your first column instead of the formulae my browser cannot interpret correctly.
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Kodegadulo
Posted: Jan 28 2018, 06:29 PM


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QUOTE (Shaun @ Jan 28 2018, 05:34 PM)
I would appreciate numbers in your first column instead of the formulae my browser cannot interpret correctly.

Done.
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Kodegadulo
Posted: Jan 28 2018, 06:50 PM


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By the way, the online reference I have is the free ebook here, which contains all of Volume 1 of Teliosadik. This appears to cover only the number system and a primer on arithmetic. Presumably there must be a Volume 2 that gets into his "only perfect Degree-, Time-, Measure-, Weight-, and Money- System", but I cannot locate an online resource for it.
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Kodegadulo
Posted: Jan 28 2018, 07:31 PM


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Note that Werneburg prescribes a highly-convoluted recursively-nested scheme for achieving large powers. For instance, to express a power like "dozen to the eleven-gross ten-dozen ninth power", 10Ɛᘔ9z, we must first peel off one power of dozen to make the rest a power of gross:

10Ɛᘔ9z = 10·10Ɛᘔ8z = 10·1005Ɛ4z

But then we must factor out each of the digits in the exponent, in reverse order:

10Ɛᘔ9z = 10·1004·100Ɛ0·100500z

in order to then render each factor as a word:

10Ɛᘔ9z = taun vierarde möroide fünfiaden

This is on the path towards the Byzantine complexity of Knuth's recursive "dozen-gross-myriad-myllion-byllion-..." system (though of course Knuth's was the most extreme possible expression of that idea).

Note: Additional commentary here.

EDIT: The additional commentary above used to be here, but it might be interpreted by some as off-topic. It apparently prompted DK to touch off a particularly snarky off-topic exchange which Shaun moved to a new Off Topic thread which he created for the purpose. However, in an apparent fit of pique, DK unexpectedly deleted that thread.

I've since moved the additional commentary elsewhere as a demonstration of technique (4) in this post. My hope is that this technique will lessen the likelihood of rancorous battles between members. And perhaps failure to apply this technique will reveal that the offender is acting in bad faith, or extreme laziness, or both.
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Shaun
Posted: Jan 29 2018, 09:28 AM


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QUOTE (Kodegadulo @ Jan 28 2018, 06:29 PM)
QUOTE (Shaun @ Jan 28 2018, 05:34 PM)
I would appreciate numbers in your first column instead of the formulae my browser cannot interpret correctly.

Done.

Thank you.
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Shaun
Posted: Jan 29 2018, 10:19 AM


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QUOTE (Kodegadulo @ Jan 28 2018, 06:50 PM)
By the way, the online reference I have is the free ebook here, which contains all of Volume 1 of Teliosadik. This appears to cover only the number system and a primer on arithmetic. Presumably there must be a Volume 2 that gets into his "only perfect Degree-, Time-, Measure-, Weight-, and Money- System", but I cannot locate an online resource for it.

There is also a shorter version available online, which includes his ideas on metrology and time.
It is dated 1798d and has a different symbol for the numeral eleven:

Teliosadik numerals
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Shaun
Posted: Jan 29 2018, 10:23 AM


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You can download the short version.


Short Summary
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Double sharp
Posted: Jan 29 2018, 11:02 AM


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{c} default dozenal

To be honest, I think that the main problem with the nested schemes like Werneburg's, Knuth's, and even the long scale without "milliards" and so on is that the nesting doesn't correspond to the base, and there's thus a lot of mental unpacking needed to figure out what power is being referred to. If Werneburg had decided to drop this ballast and call 10^ba9 neunarde zehnoide möriaden, I think his system would have gotten a lot stronger despite retaining this nesting. (Similarly, I find that Knuth's myllion system becomes a lot more palatable if you use it for octal or hexadecimal instead of decimal.) Of course, Kode's system is even better, since it recognises that the only words here that are actually providing useful information are those corresponding to the digits and the sign of the power, and economically leaves everything else while maximising the familiarity of these two things that do matter.
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Silvano
Posted: Jan 30 2018, 01:08 AM


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Mör and taun in French? Jamais de la vie!
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Paul Rapoport
Posted: Jan 30 2018, 02:38 PM


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Ni en anglais non plus.
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Shaun
Posted: Jan 30 2018, 08:26 PM


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Please note:
All I would like to see in this thread are discussions about Teliosadik, its symbols, nomenclature and metrological proposals (these latter are in the short version mentioned earlier and echo the metron - duor structures).

Arguments about other systems detract from the main topic and posts have been moved; see "Off Topic" thread in this forum.
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Kodegadulo
Posted: Jan 31 2018, 04:20 PM


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QUOTE (Shaun @ Jan 30 2018, 08:26 PM)
...and metrological proposals (these latter are in the short version mentioned earlier and echo the metron - duor structures).

Here is the actual link shown in Shaun's image: short version.

I started looking at that, but not being fluent in German, I haven't been able to locate the passage(s) about metrology. What page(s) did you find it on?

P.S. I've edited my post above to include courtesy links pointing into the posts Shaun moved to Off Topic. I also extend my post to include Pendlebury"s prefix scheme in the comparison against Werneburg's nomenclature.
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Shaun
Posted: Jan 31 2018, 09:14 PM


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QUOTE (Kodegadulo @ Jan 31 2018, 04:20 PM)

I haven't been able to locate the passage(s) about metrology. What page(s) did you find it on?


Page 17 onwards.


He divides the circle by powers of *10, and applying this to the day, comes up with the duor and suddivisions thereof; from the Earth's circumference he derives a unit in the same way, equivalent to the metron of *3.8 inches. (This is given in terms of the Parisian foot). From the metron we get the 'Stab' (i.e. dometron) of *10 metron.
Area is based on the square metron, volume, and weight, on the cubic metron.


He also suggests a unit of currency and gives the value in sous and deniers.


This is followed by a discussion of the application of twelves to music (which I haven't yet gone through in detail).


There are some printing errors here and there; he mentions the 3:4:5 triangle and the 5:*10:*11, but puts 1 for 5 in the latter and then gives its area as 6 when it should be *26. But these are minor points, considering that the whole text has been set by hand.

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Shaun
Posted: Feb 1 2018, 10:37 AM


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Please note that the OffTopic thread I opened, and to which Kode has created links, has been deleted by DK.

Members are entitled to delete their own posts, of course, but not those of other members; DK has been suspended for a month for deleting posts other than his own.
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Einmaleins
Posted: Feb 14 2018, 06:56 AM


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A strange structure; he uses names for powers of 100 (einarde) similar to our powers of a million. I will look at the original online to find his reasons out.
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