I’m intrigued by in the traditional Welsh number system - the teen numbers change after 15 when the number words become, for 16 - ‘1 on 15’ and for 17- ‘2 on 15’.
Also the strange structure of 18 being 2 x 9 - in number words.
I have read in multiple places that this is the case but none ever explain the origins, the history of the etymology.
Does anyone know?
Well, to begin with it’s not only 16, 17 and 18, but traditional forms exist for all numbers between 11 and 100! They are most commonly used for stuff like telling time, dates and age:
11 - un ar ddeg, 12 - deuddeg, 13 - tri ar ddeg, 14 - pedwar ar ddeg, 15 - pumtheg, 16 - un ar bumtheg, 17 - dau ar bumtheg, 18 - deunaw, 19 - pedwar ar bumtheg, 20 - ugain, 21 - un ar ugain, … 30 - deg ar ugain, 31 - un ar ddeg ar ugain, … 40 - deugain
(Higher numbers do exist in traditional form, but they are rarely used, so I don’t see a lot of value in listing those)
I’m not sure if I can give you a satisfying answer, but the origins of this are still a matter of debate among historical linguists, but mathematically it boils down to a vegisimal representation of the numbers (using twenty as a base number instead of ten).
This feature can be found to varying degrees in some european languages, such as French and Danish, and of course in other Celtic languages, but like I said, it’s unclear when and where this started.
It’s also in Basque and still in current use, right through the number system - logical, but hard to get your head around when you need to produce a number quickly!
Intriguingly, the Basque for 20 is hogei where the ‘h’ isn’t pronounced… any relationship with ugain? I do wonder …
We wondered the same thing when we were on tour there with the band - the first 5 numbers also have a slight similarity if you count on your hand opening your fingers up as you go…
1 - bat > bawd (thumb)
2 - bi > bys (finger)
3 - hiru > hir (long - longest finger!)
4 - law > llaw (hand)
5 - bost > boch (cheek - the fleshy part at the base of the thumb is sometimes called the ‘butt’!)
… or the apparent similarity could just have been the amount of patxaran we were drinking over there! 
I hadn’t thought of that with the lower numbers. Hmmm…
I do come across other linguistic similarities from time to time and surely there would have been contact in the earlier centuries. I guess we’ll never know.
To begin with I apologise for the typo in my original question title - I meant ‘18’ when i wrote 8!! (And have now included the no. 19)
Thanks so much for replies I received to this question from a few people.
I think I need to expand on my question and give a little more context – here goes and I’ll try not to ramble too much!
My question is specifically about the apparent outlier nature of the numbers 16, 17 and 18 (and 19 I’ve just realised) in the traditional Vigesimal Welsh number system – but specifically I’m talking about the number words rather than the numbers.(E.G. 16 eggs are always 16 eggs numerically – but how you say this in words varies immensely (10+6, 1 over 15, 10 plus 5 and 1 over, 20-4 and so on).
So question 1 is simply re. the number 18 – why/how did this become worded as 2 x 9 (‘ Deunaw ’) and so completely out of step with the other Welsh teen numbers?
Question 2 requires a longer preamble (apologies!!) and follows below.
(First thing - as I understand it the Welsh number words system (not the numbers – the words for the numbers) has been modernised to make the teen numbers more uniform and standard. My question is about the older traditional system). I could be wrong about this.
So, assuming the above to be correct, with the traditional Welsh Vigesimal (counting by 20) system for teen numbers, it looks like 11-15 follow a standard system of combining the numbers 1-5 with the word for ten (or a variation of the word for ten) – e.g un(1) with derg (10) = Unarddeg (11) and this rule seems to apply up to no. 15.
But then, unlike other Vigesimal number systems there appears a new ‘rule whereby 16 becomes not 6 over 10 but, 1 over 15 ( Un ar bymtheg ). It’s the same method for no. 17 and no. 19 – from what I can see.
That is the crux of the 2nd question – why is this so? How did this originate? What is the etymology?
Background musings
The rule creation using no.15 does not seem to specifically be a component of other Indo-European number word systems.
For example Scottish Gaelic uses or used to use a Vigesimal numbering system (see the numbers 20,30.40 etc) but has the teen number words consistently using the ‘1 over 10, 2 over 10 method) ….I think.
I suspect that like Welsh, Scottish Gaelic may have had different ‘rules’ for forming some of the teen number words to the modern consistent ‘number over 10 system)….but so far cannot find anything attesting to that.
Does anyone know if an older Gaelic number word system did have different (non-uniform) number words system for the teen numbers?
A brief comparison with other Indo-European languages.
FRENCH
French, probably from shared Celtic/Gaul links has parts of the Vigesimal system (esp. numbers 80 (quatre vingt/4 x20) and 90 (quatre-vingt-dix (4 x 20 plus 10).
However French does not, with teen number words, have any significance with the number 15.
Although French teen number words do have two rules.
1st rule for French seems to be stemming from the Latin source for numbers 11-16 whereby the numbers for 1-6 are combined with a form of the number 10.(It’s hard to spot but I think time has just blurred the link with ten ( dix ) for numbers 11-16 e.g. onze, douze treize etc
Then French teen number words change rule for numbers 17-19 (becoming dix-sept (10 plus 7) dix-huit ( 10 plus 8) etc…
Interestingly the numbers 17-19 follow a different naming rule in French are almost the same numbers following a different naming rule in Welsh (apart from the number 16) BUT (intriguingly) in a totally different way.
To make the scenario even more etymologically intriguing – the Latin source for Romance languages like French, Italian, Spanish etc also has non uniform number words for the teen numbers.
LATIN
The teen number words for 11-17 follow a uniform system e.g. 11 is undecim from unus (1) and decem (10) and the pattern continues up to and inc. 17.
Then 18 and 19 number words are formed by a different system of subtracting from 20.
e.g. 18 is duedeviginti which means 2 from 20.
(apologies that I cannot type this with the appropriate diacritical marks)
INDIAN/HINDI (as an Indo-European language)
This is pure conjecture - but it strikes me that the Indian/Huindi number word for 16 shows no relation to the number 6 (e.g. so not ‘6 over 10’). Whereas the other numbers 13-19 seem to be comprised of the common 3 over 10, 4 over 10 etc method).
Which is, once again, intriguing…
I’ll end here and wait apprehensively to see what replies I get!!
Slainte Mhath
Colin
The Welsh mathematician Gareth Ffowc Roberts mentions this in his book Count Us In (and its Welsh version Mae Pawb yn Cyfrif). Unlike other vigesimal systems, traditional Welsh numbers have ‘stop-off’ points at 10 and 15 - that is why you get the anomalies between 10 and 15 - but no-one knows why. He hasn’t been able to find the reason and he’s our ‘top man’ when it comes to mathematics and its history.
Similarly, he can find no reason behind 18 becoming “two nines” although he does say that there is a theory that it is suspected (but not proved) to have come from agriculture - it could be linked to a practice of stacking sheaves of corn in nines so that a deunaw could refer to two such stacks, but he points out that there is no clear evidence for this or any other reason.
So I’m afraid the answer to all your questions is a frustrating “we don’t know”.
Very interesting. Would also like to know the former British non dezimal counting system with Schillings. May be it has nothing to do with the Welsh language. I am a Learner
It may not surprise you to know that Swizerland Belgium and Canada do not always follow the French vegisimal rule. I had to get used to saying “septante” instead of “soixante-dix”! C’est logique, non? 
Unfortunately it seems unlikely (unless Basque borrowed it at some point in time): the Welsh ugain is actually “the same word” as Latin viginti:
From Old Welsh uceint , from Proto-Celtic *wikantī , from Proto-Indo-European *h₁wih₁ḱm̥t (compare Latin vīgintī ) from *dwi(h₁)dḱm̥ti (“two-ten”).
There will always be random similarities between unrelated languages, I’m afraid… 
Hello @ursula-waldinger,
The British “old money” was
12 pennies (or pence) make a shilling,
20 shillings make a pound.
We also had florins (two shillings)
and half-crowns (two shillings and six pence).
Crowns ( five shillings) were not in common use in my lifetime, but were sometimes minted for special occasions.
A guinea was one pound and one shilling. I never saw a guinea coin, but invoices from professionals and grand businesses were sometimes still given in guineas.
The smallest coins in my lifetime were the farthings. These were worth one quarter of a penny. There were also half-penny coins. Then there were coins for one penny, three pennies, six pennies, one shilling, two shillings and half a crown. Larger amounts were in paper notes, ten shillings, a pound and so on.
The same system was used in Wales. I will leave it for someone who was there at the time to tell you about names for the various coins.
Sue
I have bought things for a farthing in my lifetime (born in 1945). It was an attractive bronze coin with a wren on one side. A sixpenny piece was a “tanner” and a shilling was a “bob”. Ten bob was “half a knicker”. Do I miss the old currency? Not a bit. 
I did wonder if pili-pala (pinpilinpauxa) and arth (artza) were just coincidence! Since - to my knowledge - bears haven’t existed in Wales, the word must’ve been borrowed from somewhere!
Bear in mind that lots of words in many languages have Latin or Greek origins which have been tweaked over time to fit the later language but still retain some similarities. Though not necessarily the case every time, it’s often the explanation.
Latin - butterfly - papilio
Greek - bear - άρκτος
Ah! I didn’t think of Greek for arth.
It’s said that pinpilinpauxa describes how the insect flies (onomatopoeic but for movement!), but speak to any Basque person and they have a different word for it, depending on their dialect! I’ve used pinpilinpauxa and been met with a blank look!
ooh - onomatopoeic for movement - excellent, I’ve not come across that before, now I definitely have to go look into that more! (you have piqued my not-so-inner-geek! 
)
As for different dialects with different words for the same thing and blank looks, yup, you’ll find Welsh has more than its fair share of those too! 