Early German Dozenal Pioneer

Jan 26 2018, 03:04 PM #1 2018-01-26T15:04

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.

Kodegadulo · Jan 27 2018, 12:15 AM #2 2018-01-27T00:15

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:

Note that he cooked all this up in the year taun einarde sechstaun, or "taun on'arde sixtaun" (zen gross sixzy = 1060_z = 1800_d), 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).

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 (2,000,000_z) from drei zweinarde (30,000_z), 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...

Kodegadulo · Jan 27 2018, 04:36 PM #3 2018-01-27T16:36

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

Johann Friedrich Werneburg wrote:

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.

Kodegadulo · Jan 27 2018, 06:05 PM #4 2018-01-27T18:05

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.

Double sharp · Jan 28 2018, 03:29 AM #5 2018-01-28T03:29

Incidentally, Werneburg's transdecimals look like argam-14 and argam-15 respectively (turned 3 and reflected 9).

Jan 28 2018, 05:32 PM #6 2018-01-28T17:32

Double sharp @ Jan 28 2018, 03:29 AM wrote: 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).

Jan 28 2018, 05:34 PM #7 2018-01-28T17:34

Kodegadulo @ Jan 27 2018, 04:36 PM wrote: 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.

Kodegadulo · Jan 28 2018, 06:29 PM #8 2018-01-28T18:29

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

Done.

Kodegadulo · Jan 28 2018, 06:50 PM #9 2018-01-28T18:50

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.

Kodegadulo · Jan 28 2018, 07:31 PM #10 2018-01-28T19:31

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", 10^{Ɛᘔ9}_z, we must first peel off one power of dozen to make the rest a power of gross:

10^{Ɛᘔ9}_z = 10·10^{Ɛᘔ8}_z = 10·100^5Ɛ4_z

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

10^{Ɛᘔ9}_z = 10·100⁴·100^Ɛ0·100⁵⁰⁰_z

in order to then render each factor as a word:

10^{Ɛᘔ9}_z = 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.

Jan 29 2018, 09:28 AM #11 2018-01-29T09:28

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

Thank you.

Jan 29 2018, 10:19 AM #12 2018-01-29T10:19

Kodegadulo @ Jan 28 2018, 06:50 PM wrote: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

Jan 29 2018, 10:23 AM #13 2018-01-29T10:23

You can download the short version.

Short Summary

Double sharp · Jan 29 2018, 11:02 AM #14 2018-01-29T11:02

{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.

Silvano · Jan 30 2018, 01:08 AM #15 2018-01-30T01:08

Mör and taun in French? Jamais de la vie!

Paul Rapoport · Jan 30 2018, 02:38 PM #16 2018-01-30T14:38

Ni en anglais non plus.

Jan 30 2018, 08:26 PM #17 2018-01-30T20:26

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.

Kodegadulo · Jan 31 2018, 04:20 PM #18 2018-01-31T16:20

Shaun @ Jan 30 2018, 08:26 PM wrote: ...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.

Jan 31 2018, 09:14 PM #19 2018-01-31T21:14

Kodegadulo @ Jan 31 2018, 04:20 PM wrote: 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.

Feb 1 2018, 10:37 AM #20 2018-02-01T10:37

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.

Einmaleins · Feb 14 2018, 06:56 AM #21 2018-02-14T06:56

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.