02 October 2014

Only Connect: The Strange Triangle

The Latin adjective triquetrus ‘triangular’ (neuter -um, feminine -a) is baffling. It’s obviously a compound, and it obviously contains the compositional form of the numeral ‘three’, *tri-. What else it contains is anything but obvious. Unfortunately, it’s the only specimen of its kind. The mysterious element -quetrus does not occur in any other Latin compound. It looks as if it could have something to do with quattuor ‘four’. When ‘four’ occurs as the first part of a compound, it has the shape quadru/i-. This form must somehow go back to *kʷətwr̥-, its metathetic variant *kʷətru-, or a hybrid combination of both, but the voicing of the *t is odd, not to say perverse, because its exact opposite, *dr > tr, was a regular change in the prehistory of Latin. The word ‘four’ is evidently such a fickle fellow that it just can’t resist breaking some established rules. For greater inconsistency, the adverbial numeral quater ‘four times’, which in other IE languages (and presumably in Latin as well) derives from *kʷ(e)twr̥-s ~ *kʷ(e)tru-s, shows no voicing. We see a voiced stop again, though, in the denominal verb quadrō ‘to square; put in order, arrange’ and a few related words such as quadra ‘square piece or slice, plinth, dining table, etc.’ and quadrātus ‘square (n. and adj.)’.

Some connections are impossible.
The second part of triquetrus doesn’t simply reflect *kʷetru- (or *kʷatru- < *kʷətru-), because the word is a second-declension o-stem, which means that its pre-form ended in *-tro- rather than *-tru-. The form *kʷetro- (or possibly *kʷatro-, since pre-Latin *a would have merged with *e in this position) does not otherwise occur as a variant of ‘4’ in Latin, but since we are dealing with a capricious word-family, it’s hard to rule out a connection. If it does mean ‘four’, however, why’s that? A triangle has three sides, it has three angles, but has it got three “fours”? It would not be strange if a word for the right angle had something to do with squares or rectangles, and therefore indirectly with the numeral ‘4’, but a triangle can have at most one right angle, certainly not as many as three (the Penrose tribar, shown on the right, would be an exception if it could exist in ordinary Euclidean space).

Can external cognates help? It’s tempting to compare triquetrus with Old English þrifeoþor (sometimes glossed as ‘triangular’ in reference books such as Bosworth and Toller’s Anglo-Saxon Dictionary). It has been suggested earlier by one of the commenters on this blog [Douglas G. Kilday] that the Old English word is a loan from (unattested) Gaulish *petros ‘corner’ (< *kʷetros), which became Germanic *feþra- after the operation of Grimm’s Law. This tantalising suggestion, however, can’t be correct. The word þrifeoþor appears in Old English glossaries (Corpus, Erfurt, and Épinal) three times (spelt ðrifeoðor, trifoedur, ðrifedor), and is translated into Latin as triquadrum. One might think that triquadrum is a distortion of triquetrum caused by “folk etymology” (the mistaken identification of the second part as the compositional form of ‘4’), but in fact it’s no such thing. Old English authors took the adjective triquadrus from Orosius, a Christian priest and scholar from the Roman province of Gallaecia (today’s Galicia, Spain). Orosius, active in the first decades of the 5th century, was the author of several enormously influential works, including  Historiae Adversus Paganos, with a chapter on the geography of the world. Here is the relevant passage (Book 1, Chapter 2; emphasis added):
Maiores nostri orbem totius terrae, oceani limbo circumsaeptum, triquadrum statuere eiusque tres partes Asiam Europam et Africam uocauerunt, quamuis aliqui duas hoc est Asiam ac deinde Africam in Europam accipiendam putarint.
[Our elders made a threefold division of the world, which is surrounded on its periphery by the Ocean. Its three parts they named Asia, Europe, and Africa. Some authorities, however, have considered them to be two, that is, Asia, and Africa and Europe, grouping the last two as one continent.]
The epithet triquadrus refers to “the circle of all the earth” (orbis totius terrae = the world). Orosius certainly doesn’t mean that the Earth is a triangular circle, or that it has three corners. He means that the landmass of the world (as he knew it) is tripartite, divided by most ancient geographers into three continents (in this context, quadra means ‘part, division, area’, not literally a square). Anglo-Saxon translators coined a calque, mechanically replacing Latin quadr- with feoþor- < *kʷetwr̥-, the compositional form of Old English fēower ‘four’. Þrifeoþor was never intended to mean ‘triangular’. Its second member is the same feoþor- (= Late West Saxon fiþer-, fyþer-) that we find as the first element in numerous Old English compounds, e.g. fiþerfēte ‘four-footed’ (= Latin quadrupēs).

External support for *kʷetro- thus evaporates, but triquetrus still has to be explained somehow. I would suggest that its second element is a derivative of *kʷet- ‘join pairwise’ with the instrumental suffix *-tro-. When the suffix was added to a root ending in a dental stop, the last segment of the root was dropped already in Proto-Indo-European (this process is known as “the metron rule”). Thus we get *métrom (Greek métron ‘measure’) from *méd-trom (*med- ‘allot, mete out’), and *h₁étrom (Vedic átra- ‘nourishment’) from *h₁éd-trom (*h₁éd- ‘eat’). The noun *kʷétrom < *kʷet-trom would be ‘something that holds a pair of things together’, hence ‘joint, connection’ or the like. There were several Proto-Indo-European roots with similar meanings, and accordingly several nearly synonymous nouns for things like woodworking joints; joint itself comes (via French) from Latin iunctus ‘connected’ (the root here is *jeug-, as in yoke). Tri-quetrus (< *tri-kʷetro-) is built exactly like tri-angulus (a noun is used as the second member of a compound adjective without altering its stem class), and its etymological meaning is ‘having three connections (between pairs of sides)’.

The next post, in which I shall return to the numeral ‘four’ itself, will be the last in this series.

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  1. Interesting and convincing! (And you should know better by now than to make promises/threats about "the last in this series"...)

    1. Too late! Enogh is enough, so I have already closed this thread, but of course any discussion in the comments will be appreciated.

    2. Bizarrely, I can't comment there. Let's try here:


      *meju-... *meju-... ah yes, there it is. Quote:


      18. [Hittite] miyu-/meyu- ‘4, four’, CLuw. [Cuneiform Luwian] mawa- ‘4, four’. From MS on.
      [would be P]IE **mei̯u-.
      [proposed to be cognate with] [Proto-]Alt[aic] *móju ‘all (totus), whole’: [Proto-]Tung[usic] *muja- ‘whole, all (totus)’, Jap. [Proto-Japonic] *múina ‘all (totus), all (omnis)’, [Proto-]Kor[ean] *măin ‘most, extremely, very’. [The accent represents the tone that yields the high tone of Japanese and the low tone of Korean.]
      Nostratic counting systems consisted of three members: “1, 2, >2/several/
      many”. Indo-Hittite expanded it to “1, 2, 3, >3/several/many”. For the next newly formed numeral, *meyu- ‘4’, Anatolian used the Nostratic stem that yielded Proto-Altaic *móju ‘all (totus)’. For details and typological discussion see Kassian 2009.
      Correspondences are regular (except for the secondary Indo-Hittite *o/e apophony).


      This is from p. 164–165 of this paper (pdf):
      Alexei Kassian (2009): Anatolian lexical isolates and their external Nostratic cognates. Aspects of Comparative Linguistics, v. 4. RSUH Publishers, pp. 152–186.

      The "Kassian 2009" it cites, and which I haven't seen, is:
      Alexei Kassian (2009): Anatolian *meyu- ‘4, four’ and its cognates. Journal of Language Relationship 2: 65–78.

    3. Yay! Too bad I overlooked that line break.

    4. Of course I may be completely wrong about the identification of the root of *kʷetwor-, but it's quite clear that the word is morphologically complex in Indo-European, and I have more confidence in the scenario of its introduction as a cardinal number that I sketch in the next post. If one wants to look for a cognate in other families, *meju- is a better candidate than something obviously formed by the IE morphological machinery. Note, however, that to the extent the phonetic shape of the higher cardinals can be reconstructed for Anatolian, they agree with the Core IE ones, so PIE must already have had a more sophisticated system, and the main Core innovation consisted in replacing '4' by a different lexeme rather than building a decimal counting system from scratch.

    5. David: The "Kassian 2009" it cites, and which I haven't seen...

      You can download it from Alexei's Academia.edu account:
      Anatolian *meyu- ‘4, four’ and its cognates

    6. Oh, neat, thanks! I can't download it because I don't have an academia.edu account myself, but I can still read it there. :-)

      It is quite interesting, though the lack of laryngeals in its PIE transcription is annoying.

    7. Hi, Piotr. I'm glad you retook the forum after a year of silence.

      Note, however, that to the extent the phonetic shape of the higher cardinals can be reconstructed for Anatolian, they agree with the Core IE ones, so PIE must already have had a more sophisticated system, and the main Core innovation consisted in replacing '4' by a different lexeme rather than building a decimal counting system from scratch.
      In my opinion, the entity called "PIE" never existed as a single language but it's rather a superposition of several diachronical and dialectal layers.

      As regarding numerals, it looks like '4' and '5' were coined in the Kurganic dialects (i.e. the ones spoken by nomadic shephers from the Pontic Steppes), replacing older numerals like Anatolian '4'. I also think Germanic *fedwar- '4' and *fimfe '5' are straightforward loanwords from respectively P-Celtic *petwar- and *pimpe.

      To be continued ...

    8. Why would anybody borrow "4" and "5" but not all higher numbers?

    9. Following up on this more than a year later... I found this in Ringe (2006: 116):

      It has also been suggested that PGmc *wulfaz ‘wolf’ < PIE *wĺ̥kwos (see 3.2.2 (i)) and/or its fem. nom. sg. *wulbī < *wlk̥ʷíh₂ (if the g of ON ylgr is really the result of leveling, see 3.2.4 (iii) ) and PGmc *twalib- ‘twelve’ (cf. Goth. twalif, twalib-) < *-likʷ- (cf. Lith. dvýlika) are evidence for a regular change of labiovelars to labials when *w preceded (cf. Seebold 1970: 531; Schaffner 2001: 62); such a sound change would also allow us to regard PGmc *werpaną ‘to throw’ (cf. Goth. waírpan, OE weorpan, etc.) as the regular reflex of the post-PIE *wergʷ- which appears in OCS vrĭgǫ ‘I throw’, inf. vrěšti (Seebold 1970: 559). But there are too few examples to create any confidence that the sound change was regular, and too many counterexamples: ON ylgr is a counterexample unless one particular relative chronology is right (see 3.2.4 (iii) ), and there are a handful of WGmc verb forms that appear ultimately to be derived from PIE *wokʷ- ‘voice’ but exhibit root-final -h- and -g- rather than a labial (ibid. 531). I think it prudent to suspend judgment.

      "4" and "5" don't have a *w preceding their *kʷ; but "4" does have a *w behind its *kʷ, and "5" at least has another labial, *p, preceding its *kʷ. So I wonder if they're actually regular or nearly so. Perhaps Germanic simply did the opposite of what Italo-Celtic did (which changed *p into *kʷ if another *kʷ followed).

      Are there any P-Celtic loanwords in Germanic?

      BTW, PGmc "5" was only *fimf. The *-e cannot be reconstructed, and its absence is the regular outcome of an apocope of final short vowels on the way to PGmc (which has also caused the lack of a vocative ending or of a distinction of 1sg and 3sg in the past tense).

  2. I must admit that OE _þrifeoþor_ looks like a sophomoric calque of Orosius' _triquadrum_ and cannot be used to support a Gaulish *petro- 'corner'. Nevertheless Gaul. *pettia: 'piece, fragment; plot of land' is well reflected in Romance (Meyer-Lübke, REW 6450). Matasovic', who does not recognize Stokes' Law, derives *pettia: along with OIr _cuit_ (fem. /i/-stem), OW _ped_, etc., from an isolated Proto-Celtic *kWezdi- 'piece, portion' (EDPC 179). The Insular forms can equally well be derived from *kWetti-, by Stokes' Law from PIE *kWet-ní-, from *kWet- 'to divide, cut' vel sim. From *kWet-nó- 'divided, cut' comes Gaul. *pettia: 'piece; plot' and *petto- 'cut (off), sharp, abrupt' (of landforms), *petto:n- 'crest, peak' > Galician _petón_.

    Latin derivatives are _cossus_ 'tapeworm' < *kWot-tó- 'cut, segmented' and *quetrum 'corner' (underlying _triquetrus_ 'triangular') < *kWét-ro- 'cut (off), sharp, pointed'. The latter is formally parallel to *h2ák^-ro- 'sharpened, pointy' appearing in Greek _ákros_ 'topmost', _ákron_ 'summit, peak'; Skt. _catur-as'ra-_ 'quadrangular'.

    I regret the early closure of this thread. I was eagerly awaiting the non-laryngeal explanation of the acute in Proto-Slavic *c^etýre. If this explanation is implausible, a root *kWeth1- could also account for the -g- in some of the Old Norse forms, by the same extension of Cowgill's Law which explains the Gmc. -g- in 'bridge'. De Saussure's Effect would have deleted *h1 from /o/-grade, *kWeth1-wor- > syllabic *kWet-h1wor- > *kWet-wor-. That *h1 served as a root-extension is indicated by *bHerh1- reflected in Grk. _phéretron_; Lat. _praefericulum_; Skt. _bharítra-_, _bhári:man-_; Russ. _berémja_. The sense of *kWeth1- was presumably 'to divide from the rest, point out' vel sim., and *kWét(h1)-wor-es meant 'the group which points out, the (four) fingers which together indicate (a direction)'.

    Original heteroclitic morphology of 'four' is very unlikely due to the complete absence of /n/-forms. No such absence is found with 'fire' and 'water' even though no non-Anatolian language preserves the original declension. I think the best parallel to 'four' is the class of in-law words discussed previously. The (wife's) brother-in-law was originally one of a group of men, the *dáh2(j)-wor-es, who physically separated the bride from her father's house and led her to her husband's house. And the (husband's) father-in-law was one of a group of elders, the *bHéndH-wor-es, who by their blessing bound the bride to her new household.

    1. I regret the early closure of this thread.

      I have finished the thread, but not the discussion. As I wrote in the last instalment, I am game if anyone wants to continue the topic and discuss the details in the Comments section. I will reply to your points soon under the "Two Is Company, Four is a Party" post.

    2. https://en.wiktionary.org/wiki/whet

  3. Tangentially related at best, but since you wrote about why "triangular" might involve a root meaning four . . .

    Certain sedges (Schoenoplectus pungens and relatives) have strongly triangular stems although their relatives have round stems. Schoenoplectus pungens and relatives have the common name "three-square." I like it. Somehow it does seem to express how prominent the corners are on the three-sided stems.

    1. Thanks! It makes sense psychologically. Perhaps we tend to regard "corners" (right angles) as prototypical angles (which would arguably make squares prototypical regular polygons).

      I have found this illustration of a S. pungens stalk (in cross-section): link

    2. Nice photo.