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 The Long Hundred To Purely Decimal, Did "we" move from "base twelfty" to 10?
icarus
Posted: Jun 20 2017, 01:25 PM


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This thread explores the "decay" of the use of the long hundred to purely decimal hundreds, or the short hundred, the breadth and depth of the use of the long hundred surely as a convenient large grouping like the score, and any usage of the long hundred as a number base. By number base I mean its harnessing as a common means of arithmetic. Was that usage of base "twelfty", if it indeed had existed, "societal", meaning the learned people or even most people (the hoi polloi) recokned in base twelfty for quotidian reasons?

The source I have on hand is "Zahlwort und Ziffern" or "Number Words and Number Symbols, a Cultural History of Numbers" by Karl Menninger, first published in English in 1969. I have the Dover version. It is a favorite book on the subject.

On page 154 of that book Menninger calls the "long hundred" the "great hundred". On the same page Menninger cites usage presumably in the mid-20th century of measure of fishes in northern Germany by great hundreds.

"The northern Germanic region, primarily Iceland, was the home of the great hundred; there hundraþ meant 120 in all monetary calculations and in designating military units, until the introduction of Christianity around 1000. Thereafter it came to stand for the small hundred of 100 in ecclesiastical and learned writings, and the hundreds were sometimes but not always, distinguished as the 10-(ti-roed) or the 12-(tolf-roed)."

Page 155-156:
"... Old Norse distinguished between hundreds made up of ten tens and those of ten twelves; in Anglo-Saxon 120 was expressed as hundtwelftig. The word tylft, the 'twelft,' to which the Mecklenburger term Tult is related, is the most widespread small measure of quantity: tvitylft = 24, þrennartylft = 36 items. 'Fifteen years old' in Norse poetry (the 12th-century Heimskringla) is:

gammal vetra tolf ok þriggja, '12 and 3 winters old'"

The text goes on to describe Charlemagne's monetary standard of 780 being based on twelves, via the libra / talentum of 20 solidus = 12 denarius, 1 pound of 20 shillings each of 12 pennies = 240 pennies, and other divisions that retain 240 pennies to the pound. By this Menninger establishes that 12 was the basic "subunit" of the long hundred.

On page 157 Menninger suggests that the native northern Germanic penchant for twelvefoldness corresponds but is not inherited from the Latin usage of uncial (fractional) twelve. The twelvefoldness might have (in Menninger's thoughts, "pointlessly") chose 12 as a highly divisible number but more rather for "excess", giving extra force.

It appears that Menninger's conception of the long hundred as composed of twelves is not the way I am accustomed to take his meaning. (cf. p 158). When he says this I am predisposed to say, 120 = 10 dozens. He means 120 = a dozen tens. Evidence is his construction of the long hundred as an "excess", i.e., 100 + 20 extra. This seems to conflict with citing 12 pennies as fundamental, of using a dozenal reckoning of ages, etc.

An assertion was made in this post that "the [long hundred]-to-decimal conversion of the 14th century, and the various aftershocks".

I'd like to explore these questions:

1. Evidence of a broad-based conversion from a place that used 120 as the principal basis to 10 as same. Menninger suggests that 120 was not a "base" as we know it but an over-grouping in the vein of "excess", giving extra. It was a secondary rank, not a number base. Now if it were a primary grouping it must've been pretty loose. We could have 10 dozens (pennies, first examples) or a dozen tens (page 157-158). There was no clearly organized subdivision of the long hundred and if there was it was not universal across the north. Also this was restricted to "northern Germanic" people, especially isolated "Iceland". I am not sure from Menninger or any other source I have read (which are not necessarily handy in my sub-library here at office) that the long hundred was a number base. Instead it seemed things were reckoned dozenally or decimally and interchangeably so.

2. Evidence of a deep popular exercise of the long hundred. Surely it was a cultural norm to group most goods by the long hundred either as ten dozens or a dozen tens. Was it a legal or state practice? Menninger cites legal documents on page 157 wherein the long hundred was used in legal code. So that is evidence. He gives around 1000 AD as when the Icelandic usage fell via the elite usage of ten-hundreds or "tiroed" per force. I don't have evidence of "state" use of the long hundred and "conversion" to decimal. I imagine it petered out in fits and starts and not in a fell swoop. It lost favor.

3. I would like links to web sources that cite a change from the long to the short hundred. Our wendy has weighed in at wikipedia here but contradicts Menninger on the dozens despite his evidence cited above. This source suggests the long hundred in force before the 15th century, perhaps primarily from the citations of legal code and records of mercantile exchange. It does not delve in depth on the breadth of the usage and any broad change to decimal. (wikipedia is subject to activist editors but does have good source links.)

It is my impression that the long hundred was a prominent grouping at a time when items in the hundreds or thousands were just about the biggest groupings people came in contact with. Most people in those days were not technical professionals. Therefore the usage of number was not comparable to our current usage but rather more applied to simple counting. Arithmetic of the time was not as broadly and constantly applied as it is now. Therefore the supplanting of the long hundred grouping by decimal arithmetic was not the "loss of base twelfty" with a rich arithmetic algorithm based on alternating digits or complementary divisor multiplication, but the acquisition of efficient arithmetic that happened to be decimal from practitioners outside the northern Germanic region. Curiously Menninger suggests that the schock was not the result of halving the long hundred, which is culturally counter to the northern Germanic tendencies; they apparently did not get on by duplation/mediation ("Russian" multiplication). They were enamored with high divisibility. Whether this was systemic, whether there was a rich use of "base twelfty" as we on this forum reckon it by influence of Wendy, does not seem answered by evidence. If there is evidence of such I would be very interested.
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Kodegadulo
Posted: Jun 21 2017, 12:13 AM


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QUOTE (icarus @ Jun 20 2017, 01:25 PM)
The twelvefoldness might have (in Menninger's thoughts, "pointlessly") chose 12 as a highly divisible number but more rather for "excess", giving extra force.

Uh, do you mean something like this? wink.gif
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Kodegadulo
Posted: Jun 21 2017, 12:13 AM


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Curiously, according to the Wiktionary entries for eleventy and twelfty, the following Old English words all began with the syllable "hund". I have hyperlinked each word to its entry in the Bosworth-Toller Anglo-Saxon Dictionary, to confirm this:

Old EnglishEnglish TranslationNumeric Value
hundseofontigseventy70d
hundeahtatigeighty80d
hundnigontigninety90d
hundtēontiga hundred100d
hundendlefontiga hundred ten110d
hundtwelftiga hundred twenty120d

This is rather odd. If hund had the supposed meaning of "a dozen tens", why would this appellation be applied to each of the decades below it all the way down to seventy? This is not what one would expect of an exactling "societal" system of arithmetic in unquadecimal base.

The entry for hund- is rather telling:

QUOTE (Bosworth-Toller)
hund- as a prefix to numerals from 70 to 120 is a shortened form of the word which appears, in Gothic as téhund, taihund [v. preceding word], and may be explained decade. O. Sax. prefixes ant [ = hund?], in O. Frs. the prefix is t, and a trace of such forms is yet left in the Modern Dutch t-achtig = 80. On these numerals March remarks 'Gothic has sibun-téhund. The Anglo-Saxon form was once hund-seofonta [decade seventh], like O. Sax. ant-sibunta. The -ta changed to -tig through conformation with the smaller numbers, and hund-, whose meaning had faded, was retained as a sign of the second half of the great hundred.' Grammar, p. 75. See also Helfenstein's Comparative Grammar, p. 229. For the great hundred [120] cf. Icel. tólfrætt hundrað as distinguished from tírætt hundrað. See Cl. and Vig. Dict. hundrað.

So it seems the original meanings of these may have been "decade-seventh" through "decade-twelfth", but the influence of "twenty" through "sixty" corrupted those phrases into "decade-seventy" through "decade-twelfty". So perhaps hund-raed actually derives from taihun-raed and simply meant "decade-count", i.e. some vaguely large grouping of tens. Menninger's theory of a twelve-decade count being just a grouping intending extra or exaggerated force would fit within these vague boundaries. And perhaps this overall looseness explains why the Germanics so readily adapted to the notion of "hundred" as "ten tens", when they found a compelling reason to do so, namely the very useful decimal arithmetic imported by the Romans and the Church.
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Oschkar
Posted: Jun 21 2017, 01:59 AM


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That hund is actually a remnant of the Proto-Germanic decades in a fairly roundabout way. (I’ll exclude ‘tenty’ and ‘eleventy’ from this analysis because it’s difficult to ascertain when exactly they came into being.)

Let’s start with the early PIE decades.

déḱm̥t, dwíh₁ dḱm̥tih₁, tríh₂ dḱomth₂, kʷétworh₂ dḱomth₂, pénkʷe dḱomth₂, sweḱs dḱomth₂, septḿ̥ dḱomth₂, h₃eḱtṓw dḱomth₂, h₁néwn̥ dḱomth₂

There was a strange alternation between the PIE consonants *d and *h₁ that has been dubbed the “Kortlandt effect”. The conditions under which PIE *d showed up as *h₁ are unclear, but they definitely included the “ten” suffix in the decades, which was originally, according to Donald Ringe, a neuter collective. As the multiplier and the suffix started to be thought of as a single word, the multiplier shifted to the combining form (*tri- rather than *trih₂; *kʷétwr̥- rather than *kʷétworh₂).

By late PIE, therefore, the decades looked like this:

déḱm̥t, h₁wíh₁ḱm̥tih₁, tríh₁ḱomth₂, kʷétwr̥h₁ḱomth₂, pénkʷeh₁ḱomth₂, sweḱsh₁ḱomth₂, septḿ̥h₁ḱomth₂, h₃eḱtṓwh₁ḱomth₂, h₁néwn̥h₁ḱomth₂

And with the loss of the laryngeals, this evolved into:

déḱm̥t, wī́ḱm̥tī, trīḱomtə, kʷétwr̥̄ḱomtə, pénkʷēḱomtə, sweḱsḱomtə, septm̥̄́ḱomtə, oḱtṓwḱomtə, néwn̥̄ḱomtə

This carried over into what would have been very early Proto-Germanic as the following, generalizing the zero grade of the decade suffix:

tehunt, wīhundī, þrīhunda, fedurhunda, fimfēhunda, sehshunda, sebunhunda, ahtāuhunda, newunhunda

Somewhere along the way, the numbers 7 and 9 acquired a final *t, being *sebunt and *newunt at this stage, due to influence from 10. The long *ē in *fimfēhunda spread to the higher decades by analogy, and a still unexplained *n popped up at the end of the decades from 30 onward.

tehunt, wīhundī, þrīhundan, fedurhundan, fimfēhundan, sehsēhundan, sebuntēhundan, ahtāutēhundan, newuntēhundan

By late Proto-Germanic, though, all this systematization of the number system got cut short by the development of a different suffix. There was a word *teguz, plural *tigiwiz, that meant “a set of ten”, and the inherited formations gave way to constructions like *twai tigiwiz for 20. For some reason this locution only spread to 60. (Perhaps it was already a half-hundred by then... Compare Gothic saíhstigjuz ‘60’ with sibuntehund ‘70’, or Old High German sehszug with sibunzo.) Final *n nasalized the previous vowel and then eroded away, and later final *t got deleted as well, reintroducing final *n into the language in third-person verbs and in the numerals 7, 9 and 10.

tehun, twain tigiwiz, þrīz tigiwiz, fedwōr tigiwiz, fimf tigiwiz, sehs tigiwiz, sibuntēhundą, ahtōutēhundą, newuntēhundą

This would have normally evolved into Old English as the following:

tīen, twentiġ, þrītiġ, fēowertiġ, fīftiġ, sixtiġ, sēofontǣhund, eachtatǣhund, nigontǣhund,

but the strange suffix was replaced with the more transparent tiġ, pushing the suffix hund to the beginning of the following word, which later spread to ‘seventy’ by analogy. Seofontig without the prefix is attested, but the usual word for 70 was hundseofontig. Something similar is responsible for the (unwritten) voiceless fricatives at the beginning of Dutch zestig, zeventig and the initial /t/ in tachtig.

tīen, twentiġ, þrītiġ, fēowertiġ, fīftiġ, sixtiġ, hundsēofontiġ, hundeachtatiġ, hundnigontiġ

Sources:

Donald Ringe, From Proto-Indo European to Proto-Germanic, 2006.
Andrew Sihler, New Comparative Grammar of Greek and Latin, 1995.
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wendy.krieger
Posted: Jun 21 2017, 08:57 AM


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Before I begin, i should like to make some opening remarks about sources.

A lot of the science here is more to do with people trying to relate what is in front of them to what they have learnt, or their world model.

As a simple example, we might note that English distinguishes between singular and plural. In fact there is a third number that is overlooked, one of /measure/. If you take the sentence fragment "____ __ comming from all over the place", for a singular, you would put "A lion is", but it makes no sense. You put "People are" which would make sense. But for things that are not countable, you could say 'water is' or 'data is'.

Then people start to decide things of whether data is singular or plural, since the latin form is plural. The prototype for the word is the block of givens at the beginning of a proof, the individual statement is called datum, the block itself as 'data'. A computer, seen as a mathematical process, takes a block of statements called 'data', and produces output which is also called 'data'. Data is here no longer the statements given, but the measure of the paper it is written on. It becomes a measure. But if you don't have the word for it, or the notion that it is a kind of number, then you argue over the single or plural nature of it.

Much of the historical evidence comes from the study of the past, the linguistics are used as a sequencing of events of this happened before that etc. Much of what Kode and Oschkar have written about the PG, does not give us clue that they even used base 10 or 120 or anything else. So if I were to attack a language, and come up with a nice printed list of ten, twenty, ..., ninety, there is nothing in my learning and nothing to suspect that teenty and elefty and other numbers exist, and that teenty and hundred might be different.

The people here are certainly learned, and you ought take a certain measure of weight with their opinions, but the weights are not even, and their opinions are based on what they have seen and inferred.

Icarus quotes Menninger on the long hundred and on tulft. The long hundred as an over-count, is out of line with the usual over-count even though an overcount of 1/5 is bigger than anything attested. Overcounts occur, but the ordinary count is there beside them: the baker's dozen of 13, the great score of 21, the great mandel of 16 are all one extra to the ordinary measures, the great shock of 64 and the great hundred of 124 are an overcount of 4. Of course, 'over-count' is a response if one were expecting 100, and supprised to find 120.

The long hundred is known all over the germanic world, including the goths.
http://www.wulfila.be/gothic/browse/text/?book=6&chapter=15

6: þaþroh gasaihvans ist managizam þau fimf hundam [taihuntewjam] broþre suns, þizeei þai managistans sind und hita, sumaiþ~þan gasaizlepun.
— After that, he was seen of above five hundred brethren at once; of whom the greater part remain unto this present, but some are fallen asleep.

In verse 6, the word taihunteweis leads to an out-gloss, the word is not in the greek bible, and word is taken to be a gloss describing the word hundram. It is 'teenty-wise'.

The normal word to designate a five-score hundred in West Germanic, including OE, is 'teentywise'. In the norse groups, it is 'ten-red' (ten-number). The word for the longer hundred is WG 'twelftywise' and NG 'tulftred', it is less frequent, because the thing is the normal meaning of the word.

Orrin W. Robinson (1992: Old English and its closest relatives: A survey of the earliest germanic languages) had little difficulty in finding the long hundred in the earliest of the allmenic and longbard groups. It simply disappeared as writing etc crept northward. This started in the south, and writing travelled faster: this is why the norse corpus on twelfty is larger than the west germanic.

We should also note that the norse word 'tulft' is a formation not found in the west, and is largely a contact word, ie evidence of contact. Stevinson, for example, mentions the norse word, but finds no examples of tulft that Menninger gives. It's a norse form, used in the same way that the US President A Lincoln used the score in 'four score and seven years ago'.

Stevinson gives also a translated quote of '144000' as 'one hundred and twenty four thousand', as if 'thousand' simply stood for host, crowd'. Oschkar gave the impression that Ringe (From PIE to PG, 2006, p206), as if the word thousand were not a compound of some stem þus+hund, giving no alternate, but the compound as 'folk etymology'. Of course, if you are looking at folk etymology that far back, it could be of current words, and hence etymology. At this time, we note that while there is a common word for 'hundred' in PIE, the words for thousand are reconstructable to particular branches only.

That the existance of the long hundred as the mainstream hundred, is a recent descovery, because the words were subsumed, and the people poking at the language are trying to map the words onto their current idioms.

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wendy.krieger
Posted: Jun 21 2017, 09:39 AM


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The main evidence for the existance of twelfty as a base, is more metrological rather than linguistic. None the same, there is clear proof in all of the germanic languages, that the decimal hundred needed to be qualified, and that the qualifier is not in PG but only appears in the earliest writings in quite different forms.

Goodare provides an argument that twelfty was not simply a grouping, but that calculations were done in it. The largest of the calculations do not go as high as the second power, and we need to look at somewhere else for this: Weights and Measures.

Apothecaries Weightshttps://en.wikipedia.org/wiki/Apothecaries%27_system

The apothecaries system preserves much of the roman system, because much of the text is derived from Roman imperial apothecaries. The thing to look for is that the ounce is divided to 576 grains in the latin countries, but 480 in the non-latin ones.

Weights are the closest thing we have to a large-scale fraction scheme: the romans actually used weight-fractions based on the pound.

If one suppose that the scruple is a count of 20 grains in the north, but a division into 24, then we see the alternation of 10, 12 in the north, where the latin system consistently divides into 12, 12.



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icarus
Posted: Jun 21 2017, 12:48 PM


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So no direct evidence of arithmetic in base "long hundred". We are reading tea leaves and imputing great weight to those who "argue" favorably for ideas we personally espouse. Then we are pushing it all over creation as a keen also-ran to be taught to seven year olds all over the falling West. We are resuscitating what we lost in halcyon days ushered away along with May poles and solstice excitement by monks and the Good News. With it we are bringing back lost letters that maybe had passed away peacefully before Shakespeare. We should rather abandon the manner by which we determine veracity by recasting how we take what we can learn from forefathers long passed, than to say it ain't so. Not that it couldn't be, but we can't fully know. Thus we should build upon what we guess. All well and good were it merely an interesting idea, but instead it is a hard line constantly sold to us all, without direct evidence.

I recognize that these things were done in the distant past. We get all excited when we see the remains of Egyptian homework scratch paper (recently found), we find evidence thousands of years old on account of the durability of the "paper" to support Babylonian maths, but the northern Germanic we are left "connecting dots".

I do not mind one's weaving a brilliant system out of the wisps of what we can impute from old Bible verses and merchants' logs. I do not detest a system brilliantly contrived. What I do mind is taking the wisp of "argument" in favor of one's own arrival at a concept and necessarily representing it as some societally systematic halcyon age wherein everyone did their sums and products using twelfty. There appears to be no direct evidence for that at all, just tea-leave reading and dot connecting. I also am bothered (as are others, even going back three years) that everyone is browbeaten into accepting this line of thought as if this system were practical by common folk. The insistence that this "ancient halcyon system" (no direct evidence of which existed?) is superior and ought to be adopted in place of "ten-like" systems is ludicrous. We know bases around the size of decimal have been used by societies; we know that sexagesimal similar to the construction of "twelfty" was used broadly (again, province of learned professionals and merchants, the basics of which maybe made it to the common person - talking about _arithmetic_ not the effects like measure and nomenclature, etc. that can be acquired without full understanding of arithmetic). It is not outside the realm of possibility that twelfty was not broadly used. Merely because something wonderful is _possible_ does not imply it was actually done.

True, any discussion of purported ancient language usage is stacking guess on guess however educated but it is marvelous to see (I am a fan of PIE with its plentiful asterisks, above thankfully elided for the profusion of bangs we'd see like the Fourth of July or ___ fill in your local celebration with fireworks). We know these are guesses. No one is selling it as concrete knowledge, thus acceptable. When you say "Proto Indo European" your mind goes to "so a wild ass guess stacked on less of a wild ass guess stacked upon knowledge of the way things work linguistically" and we know that they might be close to right, but no one sells PIE and expects folks to buy their claims wholesale. There is always an asterisk hovering around anything PIE. With twelfty, no asterisks as if it were _certain_ it was thus and so.

So no direct evidence of children's sums or laborer's or soldier's calculations in twelfty. Does not mean it did not happen, but also without direct evidence there is no detail as to how it actually was done. Northern Germanic culture was different than modern or even concurrent culture to the south. It was not urban. It was more clan/tribe than nation state (certainly). So no school system. How did an intricate system propagate? Sagas? If there were a saga about twelfty can we point to it? Where is the evidence? Some scrap or sign. But then we are back reading tea leaves if we are at the point of signs.

I do not intend to be inimical and this discourse is beneficial. I do desire to know where this constant injection of a model of what was intends to lead people. It is consistently injected into every thread visited upon by its purveyor and now let us see the goods we are to buy. Otherwise let us speak of it less broadly as a known value. "Anyone want banana hangers?" and we know what they are and can say, "sure" or "no way, Joe." No more mysticism. The purveyor made it all up quite brilliantly, more like a lovely tapestry than a useless fruit suspender. But lovely tapestries belong in museums or places of inspiration and not on my vestibule or mud room floor.

We have heard a lot about "twelfty".

Now I am going to read tea leaves. Believe me I am an expert. My family believes they can read tea leaves and that has led to a lot of strife haha. "I know what you're thinking" no you do not. An added benefit, I am part Swedish (Blekinge Lan) and Scots Irish (connect the dots Scotland was jutted right in the middle of all that friendly Viking activity...) so it's in my veins. Let's give it a go.

It is my tea-leave-reading that one needs a plus four standard deviation intellect to ride this ride, (to juggle alternating twelve-on-ten base 120), that shepherds and soldiers and housewives and royal court jesters and children did not wield it. If twelfty were actually used as we can surmise by connecting dots thusly, it was the province of a small group of elite professional "scribes" and a handful of other people who mostly needed calculation and not done in Kaupangen or Arus or Nyköping or anywhere by most common people.

Show me I am wrong in thinking this.
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Oschkar
Posted: Jun 21 2017, 11:24 PM


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QUOTE (wendy.krieger @ Jun 21 2017, 08:57 AM)
Stevinson gives also a translated quote of '144000' as 'one hundred and twenty four thousand', as if 'thousand' simply stood for host, crowd'.  Oschkar gave the impression that Ringe (From PIE to PG, 2006, p206), as if the word thousand were not a compound of some stem þus+hund, giving no alternate, but the compound as 'folk etymology'.  Of course, if you are looking at folk etymology that far back, it could be of current words, and hence etymology.  At this time, we note that while there is a common word for 'hundred' in PIE, the words for thousand are reconstructable to particular branches only.

I did give an alternative, though. I proposed that the word reflected in Gothic þūsundi, OCS tỳsęšti and Lith. tūkstantis was a feminine participle of the aorist stem of the PIE root *tewh₂- ‘to swell, to be strong’.

The root itself is very well attested as Latin tumeō ‘I am swollen’, Lith. taũkinas ‘fat’, OCS tyti ‘to get fat’, Skt. tavīti ‘he is strong’, etc. An /s/ after the root usually stands for the aorist tense: compare Latin dīcō ‘I say’ with dīxī ‘I said’, or Greek lúō ‘I am releasing’ with élusa ‘I released’. The alternation between the zero-grade implied in the Germanic ending -und- < PIE -n̥t- and the full grades implied in Balto-Slavic (OCS -ęšt- < PBSl. -ent-, Lith -ant- < PBSl. -ont-) is reminiscent of the endings of the PIE participles: -ont- in the strong cases of the masculine and neuter, -ent- in the strong cases of the feminine, and -n̥t- in the weak cases. Finally, participles formed ih₂-stem feminines in PIE, which is exactly what we have in the Germanic and Balto-Slavic words for ‘thousand’. Compare Gothic frijōndi ‘friend’ (fem.), OCS ljubęšti ‘loving’ (fem.), and Lithuanian ateinantis ‘coming’ (fem.) with the exact same endings.
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wendy.krieger
Posted: Jun 22 2017, 08:41 AM


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There are a number of different texts which indicate actual calculations in base 120. A lot of the calculations are done on the stone-table, and all we can really see is the results. Here is Menninger.

"One hundred men showed up as soldiers, eighty stayed and forty left".

Stevinsen and Goodare both give examples where the count is different, and that the count is consistant with base 120. Goodare converts during the conversion, a hundred of 40 in number being used.

Ronald Zupko's "dictionary of Weights and Measures of the British Isles" (1985), gives entries from the OED of a great number of units. The main dictionary runs to 445 pages, and a supplement follows the bibliography, is some 30 pages long.

Entries that I have discovered in it, that qualify the hundred, use C = 100 = hundred, and qualify the entry in terms of scores and units, and then gives the decimal. These are all from the XIV century. "A wey is 3 hundred li, The hundred being v score xii, this gives 336 li." (li = libra = pound, by weight: l = £).

Wendy's twelfty

I started using twelfty a year or so before any connection to the ancient germanics. My brother's comment at the time was 'between two stools ... '. There are a number of differences between the two versions.

My thousand is constructed on columns of the roman abacus. The actual diagrams are shown herehttp://z13.invisionfree.com/DozensOnline/i...post&p=22172386 .

We should further note that until the discovery of alternating arithmetic, the system was regarded as 120 digits. This changed when digital arithmetic was done at sub-base level. The meaning of digit was broken into three, and one would no doubt recall the base/staff/twistaff thing.

The early and later twelfty does colour how i see things, as does the things like the UES into the notion of dimension-theory.
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icarus
Posted: Jun 22 2017, 02:43 PM


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Menninger's comment does _not_ necessarily sustain calculation in base 120. Instead it illustrates _groupings_ of decades into a "hundred" of what we now see as 120.

"A number of different texts" is _not_ direct evidence. Which texts? There are no texts (my conjecture).

Weights and measure. I use feet and inches all day. Now I am a dozenalist so my _particular_ example is vanishingly vanishingly rare. Most people add inches and feet in decimal. Most people conduct manipulations of ounces and pounds in decimal. By most I mean pretty much all. Usage of a grouping does _not_ (necessarily) entail manipulation in that grouping's base.

Wendy's 120. That's fine.

What is unclear:

QUOTE
We should further note that until the discovery of alternating arithmetic, the system was regarded as 120 digits. This changed when digital arithmetic was done at sub-base level. The meaning of digit was broken into three, and one would no doubt recall the base/staff/twistaff thing.


_WHO_ "discovered" alternating arithmetic in that first sentence? Who regarded the system as 120 digits? If it was you, yes, then please say so. If it was a committee of folks or the country of Uzbekistan, then tell us that. The way it is written, and many many here have beef about this, makes it sound like an enormous congress of learned people or one exceedingly luminary person "discovered" and "regarded". This is what peeves everyone here. It comes off as if it were the practice of legions of people, perhaps a lost practice that has been wow!! Sham-Wow! brought back for us to consider now.

"You prolly think this thread is about you" but it is not. It is trying to ascertain the basis for the consistent insertion of "twelfty" in damn near every thread in this forum as if it were a thing beyond one bright gal in the Land Down Under.

By this thread I am not as much interested in what Wendy does (though it is interesting on its own accord) than the societal usage of base 120, if it ever occurred. It was used as a grouping or organizing unit. People measured things up to and by it. No evidence that it was used as a number base for arithmetic. There is evidence Wendy and a few others have done it fairly recently but certainly not in Carolingian days or Viking days. It is interesting it can be done with a lot of dancing and the dance is interesting but not for this particular topic.

Just trying to get to the bottom of it.
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wendy.krieger
Posted: Jun 23 2017, 08:59 AM


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Twelfty

The purpose of my comment here was to show the relation between my use of it, and my learning of the Germanic use. It's more that you can form a relation to what i am offering.

I found the alternating arithmetic method.

I know that the thread is not about Wendy's twelfty, but the loss of the ancient Germanic one.

Bases in General

The mathematicians that i read, have little desire to work on the problem of large bases. So what happens is that they say calculations are 'just done as in decimal'. So they can basically coopt the decimal names and use them for any bases (eg digits 0 - 59). People who read them think there is no solution, and take the mathematicians at face value. So you have this attempt to design glyphs that run into the hundreds.

This is not what we see in history. We see for even small bases like 20, the 'digits' are actually encoded in a lower system, such as 5 or 10. When I discovered the alternating arithmetic, it broke down the mathematician's digit into several different things. I have listed these elsewhere, but [59] is a place, 5 and 9 are digits. Digital arithmetic is done at column level, but repetition is done at a higher place.

I then reread some of the works with this new mindset, and things that were pussling or not even noticed, now stood out like a sore thumb. Of course they would do this, or whatever.

A hundred men

While Menninger's comment by itself do not illistrate a calculation in base 120, the word 'hundred' certainly represents this number, and not just a grouping. We know this from Gordon's 'Old Norse Grammar', §107, which lists without comment the numbers of old norse. 100 is given as teenty, 120 as a hundred, and 200 as one hundred and eighty.

There was no unit at five-score, and so the germanic hundred is neither an overcount or a unit, but of exactly the same metal as the decimal hundred: the unadorned number as spoken.

A hundred as today, can as readily be seen as an undercount, or a foreign unit, if the inclination of mind is to see otherwise. You really have to keep a close eye on what is fact, and what is opinion.
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Kodegadulo
Posted: Jun 23 2017, 11:22 AM


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QUOTE (wendy.krieger @ Jun 23 2017, 08:59 AM)
You really have to keep a close eye on what is fact, and what is opinion.

Indeed.

QUOTE
The mathematicians that i read, have little desire to work on the problem of large bases.  ...  People who read them think there is no solution, and take the mathematicians at face value ...

Are these "facts"? Or merely "opinions"? Are they not, in fact, little more than personal conjectures?
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wendy.krieger
Posted: Jun 23 2017, 12:23 PM


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QUOTE
The mathematicians that i read, have little desire to work on the problem of large bases.  ...  People who read them think there is no solution, and take the mathematicians at face value ...


It is a matter of observation that the words that are used closely follows the mathematicians, even where the pictures do not. For example, there are pictures that show the '59 digits of sexagesimal', when it can be clearly shown on stone tablets, that the symblols are separate characters in apposition. This we know because there are spaced separate glyphs for '40' and '2', which have been as part of the expression of the square root of 2.

In the present board, one sees a very close following that the base is 1,0, and that there must be single glyphs provided, or that an intent to do so, for all of the numbers less than 1,0. The idea that a base might run to several columns is not part of the standard idiom, and it was i who introduced it here.

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Kodegadulo
Posted: Jun 23 2017, 01:08 PM


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QUOTE (wendy.krieger @ Jun 23 2017, 12:23 PM)
For example, there are pictures that show the '59 digits of sexagesimal', when it can be clearly shown on stone tablets, that the symblols are separate characters in apposition.  This we know because there are spaced separate glyphs for '40' and '2', which have been as part of the expression of the square root of 2.

How many stone "tablets"? One? How many computational examples? Just this square root of two? A conclusion based on a single data point is nothing more than a hypothesis. It requires much more corroborating evidence to become an established theory. One would have to preclude other hypotheses, like perhaps this was just a kid who was a little nearsighted, or dyslexic, and needed to space out his tens from his ones just to see them clearly. To really know what was going on, you would have to get into the mind of a Babylonian doing arithmetic. Did they actually write anything about the algorithms they used? If not, this is nothing more than guesswork.

So why does any of this rise to the level of a societal imperative for our times, such that the topic must be "injected into every thread visited by its purveyor"?
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icarus
Posted: Jun 23 2017, 01:47 PM


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Okay.

We have thus established that _Wendy_ has devised a means of "alternating digits", that is, she has noted the Babylonian practice of writing a decade symbol and a unit symbol as is indeed evidenced throughout the Babylonian/Mesopotamian standard cuneiform sexagesimal notation. YBC 7289 is evidence and it is indeed a data point, but the system used to show sqrt(2) is just exercising the standard notation more or less, as if someone found a decimal etching of the same number. I don't take issue with any of that; Wendy's observation is correct and the fact she cited that single example - the single-ness of that example doesn't "unmake" her argument. It can be demonstrated that the Babylonian/Mesopotamian cuneiform sexagesimal system was profoundly systematic and the ancient mathematicians were keen in its use, as there are thousands of examples all over the world and down the street at the St. Louis Art Museum that I can visit today.

We know that there is a record that the grouping of the [long] hundred did often construe what we now see as 120, purely decimally. "Hundred" was more often deemed 120 than 100 for a few centuries around 1000 AD and was gradually supplanted over several centuries by the cultural and intellectual influence of the Church and associated clerics and learned people. Folks might have thought of grouping decades by dozens (or dozens by decades, etc.) or counting six score to the hundred; these are outcroppings of the high divisibility of 120 into highly divisible and useful smaller groups. We don't know and there is little (but probably no) direct evidence of any usage of base 120 as Wendy constructed twelfty or at all, really. This isn't to say that thinking about twelfty is useless, that there isn't anything to learn from it. It just means that we don't have a direct record of it ever being used.

Wendy is displeased with "mathematicians" who fail to consider large bases principally for the "symbology" or numeral-extension problem, the problem of how to convey the place values. I am not displeased with maths for this, but recognize and am kindred in the observation. My approach is "this attempt" of designing glyphs that run into the hundreds, pejoratively conveyed but nonetheless equally valid an approach as recreating the relative complexity and proneness to human fault that alternating "sub-bases" or coding presents. Indeed, when I encountered Wendy's rendition of "twelfty" but more that I saw it was the same general methodology long present in Ancient Mesopotamian sexagesimal, I took it as valid and weighed its merits. It does have merit. Questionable, however, that it completely renders "very large" or what we here in the Forum call "mid-scale" bases societally practical as a solution for today and beyond (i.e., not historically. Clearly, through the greater example of ancient Mesopotamia, the method _was_ a practical societal solution, but we no longer use the algorithms - complementary divisor method, largely - that they did).

I have been a rebel and was rightfully thrown out of high school. This said, I do not see the point in overmuch rebellion "just 'cause". Mathematicians are generally concerned with greater maths than arithmetic, higher or otherwise, and consider much deeper problems than number bases. I don't fault them for not caring to delve into the mechanics of number bases very much greater than hexadecimal, with the existence of vigesimal and sexagesimal being seen as eyebrow-raising exceptional instances. Purely because we want to wage a campaign against the way things are, is not a cause to inject the issue countless times.

One of the things I like about Wendy and really the only reason why I truck some of her writings instead of bouncing her as a troll (because a lot of her behavior in my opinion is trollish - derailing threads, sowing discontent, egging on reprisal and then pointing at it as an attack) is that she does come from a different angle. Too much rap or reggae in country music makes it not country and maybe not interesting to country music fans. It would be great if the mention of "twelfty" was better kept in its own discussion space (as indeed there are a few who see it as interesting in and of itself, including myself) but most like their peanut butter on one side and chocolate on the other. Here we are savoring chocolate and peanut butter does not belong in most cases.

We know and appreciate the example of "alternating digits" and the method Wendy has espoused. In many threads we are just plain not concerned with that particular question or facet. We have nothing to gain in conversation by it's being raised. It distracts from discourse and routes all attention to Wendy's pet issues and this is what roils this Forum. I would say that when Kode was an admin, aside from flying off handle he was undertaking his job when Wendy was first censured. The topic of twelfty and its mechanics is not an issue when discussing dozenal systems of measure. COF is not as of interest as evidenced by views and replies, as primel and TGM. Injecting lessons from conjectured northern Germanic uses of base 120 and the (brilliant! but immaterial to the topic) mechanics of "twelfty" devised by Wendy largely only derails the conversation and annoys the contributors to that topic. It interjects conflict needlessly.

I think the question of the OP has largely been resolved.

Please keep twelfty in its place. Thank you.
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Oschkar
Posted: Jun 23 2017, 05:10 PM


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There is at least one attested number system that almost counts as an instance of twelfty carried out to two places.

The proto-literate Babylonians had a great variety of numeral systems, possibly because, like Wendy, they thought of numbers as being determined by the sort of things they are counting. The oldest and most common system, which was the only one ever used to count people, was sexagesimal. However, there is a well-attested system for counting food rations, called Bisexagesimal System B, that used base 120. There were symbols for ½, 1, 10, 60, 1'00zd, and 10'00zd, as well as one that probably stood for 60'00zd. This would satisfy almost anyone, except for the fact that we don’t know whether there was a symbol for 1'00'00zd. (But then again, when would a Babylonian merchant ever be in possession of a long hundred long hundreds of fish or loaves of bread or whatever?)

Image taken from Jöran Friberg, A Remarkable Collection of Babylonian Mathematical Texts, 2007

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icarus
Posted: Jun 23 2017, 06:49 PM


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That's remarkable!

Were the system to exist and subtend lately, it would be fabulous.

Here's a catch-22.

We can presume since we have scant evidence of the use of base 120 per se in northern Europe (or anywhere for that matter) that it did not really exist, that it was used as a grouping or a milestone, but arithmetic was not carried out, at least not in the way Wendy brilliantly invented relatively recently. There is nothing that totally rules out someone sometime using such a system or something like it, but for everyday average Joe use, most likely it's a no.

Suppose we are wrong. Suppose there was, sometime, somewhere, certain elvish people (they need not be elvish but I need the handle) who did arithmetic in twelfty or some twelfty-like way. Short of a bloody conquering, or jihad/crusade/holy war, or cossack or hun or mongol horde, or meteor strike, or the eruption of Thera or the black death decimating the population, i.e., something we could point to, maybe even the 8.2 ka event, we have to conclude one thing. Twelfty did not compete with the finger-counting base. It was too complex. It was The Towering Inferno, it was the tower of Bab-El. Too lofty, too complex for minstrels and knights and cynings and slaves and countesses and knaves and mercenaries and shopkeepers and craftsmen and kids.

Substitute hypothetical elves for Germans. Lo, we still have Germans. They were not wiped out. We had a huge war over there with them last century. My home city is settled by a great number of them, evidenced by all the non-French street names (Eichelberger, Heidelberg, Weiss, Hoffmeister, Germania, Wachtel, Karlsruhe) and I myself married one; my kids are 45 twelftieths German, and if you count the Dutch as lowland Germans, heck I am 30 twelftieths as well (and they are half German). My son's name "Karl" is German. Ergo Germans still here, twelfty all gone, don't got a picture of 'em in her arms, now. If they had it they don't have it now; they gave it up. I am a quarter Swedish and the Swedes don't have it. Those are the northern folk that did use it.

Thus I conjecture it never was as we are often led to consider.

Either way, what we can conclude more or less, leaving the door just a tad open or maybe just unlocked should it come in late, is that twelfty either did not exist as presented often here, or it plain did not compete with "ten-like" finger counting bases or thereabouts. It was niftily crafted by our friend but the Vikings or anyone else, on a broad scale, did not compute the shopping list or the dimensions of a dragon-headed candy-stripe-sailed longboat with a hundraþ oars. It did not carry ten then carry twelve. Culture needed not the great big base, 120 storeys high, for the quantities of things they encountered were paltry and grokable. If these great numbers were needed by the state, the cyning or the weard, it was aggregated into hundraþs for the sake of record. But the invention of Wendy's was likely not used. Just finger-counting base or some other, simpler algorithm.

and if they did use it, they then likely soon sang:

(Billy, cue the bass. Arright le's git down!)

To my surprise, one hundraþ stories high /
People getting loose y'all, getting down on the roof /
Folks are screaming, out of control /
It was so entertaining when the
ti-roed started to explode /
I heard somebody say /

(Burn baby burn) disco inferno /
(Burn baby burn) burn that *** down, y'all
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Kodegadulo
Posted: Jun 23 2017, 07:21 PM


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Let me suggest a possible hypothesis, to be corroborated or refuted by whoever has _concrete_ evidence either way:

"The Babylonians/Sumerians/etc were good at counting things, and recording counts, and making durable receipts for bills of sale, etc; but they weren't particularly good at arithmetic as we know it today; and that's _why_ they gravitated toward using large, 5-smooth bases like sexagesimal."

Why would I suggest they picked sexagesimal because they were _bad_ at math? (Or at least, not as advanced in their understanding of arithmetic as we are now?)

If they were constrained in their abilities to only a few simple operations, they'd want a numbering system that optimized for those. What do sexagesimal et al optimize? Simple division or multiplication by single-digit-or-so factors: 2, 3, 4, 5, 6, (8). Multi-digit operations get complicated fast. But if you have a big shipment of grain in bushels counted in sixties or long hundreds and you need to distribute it to 4, 5, or 6 customers, splitting it up is a no brainer. But it doesn't mean you've mastered Stevins or Fibonacci.

I claim no knowledge of this topic, and only the mildest of curiosity about it. I am only posing the hypothesis. It is definitely a falsifiable hypothesis. All it would take is extensive cuneiform evidence of multi-digit multiplication and long division using arbitrary oddball factors other than simple subitizing numbers. But it would need to be eliminated as a hypothesis if one wished to make the case for a sexagesimal arithmetic golden age.
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Oschkar
Posted: Jun 23 2017, 08:38 PM


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QUOTE (Kodegadulo @ Jun 23 2017, 07:21 PM)
Why would I suggest they picked sexagesimal because they were _bad_ at math? (Or at least, not as advanced in their understanding of arithmetic as we are now?)

I think I’ve read somewhere that division as we know it wasn’t known to the Babylonians. They knew how to take the reciprocals of 5-smooth numbers, but when it came to dividing by 7 or 11 or anything else that was off-kilter, they mostly gave up with a note reading "it does not divide". When dealing with Problem 12 from tablet YBC 4698, which deals with buying two different types of wool, one costing 7 mana and the other costing 11 mana, the solver seems to have left everything in terms of 77ths just to avoid division. Figure 7b of this article appears to be a homework assignment: divide 1'01'01'01 by 13. The procedure seems to have been separating the dividend into digits, finding how many thirteens go into each digit, adding the partial quotients together and dividing the sum of the remainders by 13. And these are only the three smallest non-regulars...

As for multi-digit multiplication, the fact that the square root of 2 was calculated to three sexagesimal places indicates that they did know how to do it, because you'd need an enormous number of regulars calculated beforehand to find it otherwise. The comment "4'03 does not divide" in tablet BM 85210 cited in Neugebauer shows that at least that particular scribe wasn't familiar with richness-5 regulars, so it's extremely unlikely that anyone would have made a reciprocal table extensive enough to approximate the square root of 2 to any reasonable accuracy. Neugebauer does say that reciprocal tables beyond the standard 30-entry list have been found, but they’re from the later Seleucid period.

All in all, I don’t know if this supports your hypothesis or not, but it might point to something profitable.
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wendy.krieger
Posted: Jun 24 2017, 08:28 AM


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Among the interesting items cited by Neugebauer, is a document that deals with the 'seven brothers' problem, giving a solution that

8C4A6 < 1/7 < 8C4A8,

that is, 1/7 lies between 0;08 34 16 and 0;08 34 18.

This would suggest that there was no general algorithm for division at the time.

The existance of recriprocal tables, and multiples for the reciprocals, tells us that they probably interpolated their answers.
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Kodegadulo
Posted: Jun 24 2017, 04:01 PM


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So it's not unreasonable for me to suggest at this point, that resorting to the use of a large, 5-smooth base, is not necessarily a sign of mathematical brilliance, nor necessarily indicative of superior gifts at technical design of efficient systems (whether in an individual or a culture), but rather a krutch, possibly compensating for an underlying deficiency. A kludge that one must debug from a system before one can make true progress.

More often than not, brilliance is indicated by simplicity of design.

EDIT: Don't get me wrong, there are many cases of virtuoso compensation mechanisms that can be inspiring, and cases of krutches that represent brilliance in the design of krutches. Paraplegic atheletes competing in Olympic-calibre races and sports using high-tech wheelchairs and their amazingly buff arm muscles, spring to mind. But I don't know a single one of those who wouldn't give anything simply to stand up and walk.
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Double sharp
Posted: Jun 24 2017, 05:02 PM


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QUOTE (Kodegadulo @ Jun 23 2017, 07:21 PM)
Why would I suggest they picked sexagesimal because they were _bad_ at math? (Or at least, not as advanced in their understanding of arithmetic as we are now?)

If they were constrained in their abilities to only a few simple operations, they'd want a numbering system that optimized for those. What do sexagesimal et al optimize? Simple division or multiplication by single-digit-or-so factors: 2, 3, 4, 5, 6, (8).  Multi-digit operations get complicated fast. But if you have a big shipment of grain in bushels counted in sixties or long hundreds and you need to distribute it to 4, 5, or 6 customers, splitting it up is a no brainer. But it doesn't mean you've mastered Stevins or Fibonacci.

It strikes me that we would need to address the plausible "devil's advocacy" argument that the same is true for dozenal too, just to a lesser extent; since we are all familiar with Stevin by now, the main optimisation of changing from 10 to 12 is in fact simple division or multiplication by single-digit-or-so factors, while multi-digit operations are about the same as they are in decimal...

Sorry if this is off-topic. If it leads to any further discussion, I have no objections to splitting it out with a link to his old thread.
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Kodegadulo
Posted: Jun 24 2017, 06:11 PM


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QUOTE (Double sharp @ Jun 24 2017, 05:02 PM)
QUOTE (Kodegadulo @ Jun 23 2017, 07:21 PM)
Why would I suggest they picked sexagesimal because they were _bad_ at math? (Or at least, not as advanced in their understanding of arithmetic as we are now?)

If they were constrained in their abilities to only a few simple operations, they'd want a numbering system that optimized for those. What do sexagesimal et al optimize? Simple division or multiplication by single-digit-or-so factors: 2, 3, 4, 5, 6, (8).  Multi-digit operations get complicated fast. But if you have a big shipment of grain in bushels counted in sixties or long hundreds and you need to distribute it to 4, 5, or 6 customers, splitting it up is a no brainer. But it doesn't mean you've mastered Stevins or Fibonacci.

It strikes me that we would need to address the plausible "devil's advocacy" argument that the same is true for dozenal too, just to a lesser extent; since we are all familiar with Stevin by now, the main optimisation of changing from 10 to 12 is in fact simple division or multiplication by single-digit-or-so factors, while multi-digit operations are about the same as they are in decimal...

Sorry if this is off-topic. If it leads to any further discussion, I have no objections to splitting it out with a link to his old thread.

That's a fair enough question I suppose. Okay, let me make an analogy:

twos are like soft grass.
threes are like hard smooth linoleum.
fives are like hard spiky gravel.

Our ancestors evolved in an environment that had a lot of grass and spiky gravel. Nobody has trouble walking on soft grass. We all built up callouses on our feet to deal with the gravel, so we didn't seem to notice a problem. It's not very natural for us to walk on linoleum, because linoleum was hard to come by in the Paleolithic. But it turns out we're all pretty well pre-adapted to walking on hard smooth linoleum pretty easily -- who knew? So when we get the chance, we decide to pave linoleum over everything in our immediate environment that isn't soft grass. And then we discover we can do a lot more things a lot more efficiently. Sure, it means we don't all get those built-in callouses any more to deal with the hard lumpy gravel. It's a bit more painful to walk on that now, so we have to do it a lot more gingerly, or put on shoes (Stevins), which we're doing a lot of anyway these days. Shoes let us get over slippery ice (sevens), or hot asphalt (elevens), and other environments alien to our ancestors, without hurting ourselves. The shoes aren't as easy as going barefoot; they do take some effort to put on, and we have to learn to tie the laces right. But we get all that as kids so it's all right.

Are we compensating for a "deficiency" here? Or are we taking fuller advantage of a pre-adaptation? (3 is a subitizing number, 5 is not or at best borderline). We can artificially make our world more efficient for us, and ourselves more efficient within it. It may cost us a feature that we had purely as an evolutionary accident (the ability to grow callouses to deal with those lumpy fives), but are we really going to be hurting for that?

But I agree, if Icarus thinks these musings are off-topic, let's take them elsewhere.

----

EDIT: I can stretch the analogy a tad further: Walking barefoot on spiky gravel, you occasionally come across some sand. (Threes in omega relation with double-fives.) It sort of feels smooth like linoleum, but it's not quite the same. Can't really waltz or tango on it. You wind up kicking up dust and choking on it, or you get your feet caked in mud (repeating decimals). You really just have to give up the gravel, and put down some linoleum, if you actually want to hold a cotillion. And maybe invest in some dancing shoes...wink.gif
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wendy.krieger
Posted: Jun 25 2017, 08:58 AM


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The various examples in Ifrah, show that alternating systems are pretty much modular, in the sense that you add a division/multiplication alternation. This would make some sense if the division and multiplication processes are different entities.

The sexagesimal is different to nearly everything else, in that it is a division system, that is, subsequent digits go to the right end, and are finer divisions. Its precursors are, based on my use of twelfty, a division to six and a mutliple of ten. Neugebauer supposes a prototype in 'three scores', which is also possible, but the digits we inherit are for columns of 6 and 10.

Given that there are symbols for the sixths, (1/6 - 5/6), then the notion that a mina of 60 sheckels, can be thought of as 1/6 mina = 10 shekels, and 1/6 shekel = 10 berah, would bring the system into something that would give the desired form directly.

It goes without comment, that on Ruthe's avatar, the column of ounces is headed by a six-ounce measure, as if the prime fraction is to the head of the next column of abacus, and the fractions of ounce, are likewise done from the head of the column.

The mayan systems, from what I have learnt, has fractions in the greek style (1944 parts, where 2000 make the foot), but the divisions were done as far as integer and remainder. This is the same limit, if one supposes that the method of multiplication and division use the process of setting a number to the left on the abacus, and variously steps of +1 ; -x or (-1, +x) (at the level of each bead), until either the number on the left or right is exhausted (or left as remainder). This is the method i use for multiplication and division on the abacus.

Were one dividing 60 by 7, one can draw off a multiple of 35, and thrice 7, adding a multiple of 5, and thrice 1, the residue on the right is 4, and the accumulation in the left is 8, so 60 divides by 7, to give 8 remainder 4. But there are no more columns to the right, and calculation ceases.

The fact that we see in Neugebauer, that 0;8,34,16 < 1/7 < 0;8,34,18, is probably indicitive that one is looking for an entry for 1/7 into the reciprocal tables, and the best estimate is this. It would appear from this, that they did not get as far as the abacus method, which is further supported by the use of reckoners to make their tables. They probably did not have a very good multiplication method either.

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