The UK (and several other English-speaking countries) use the word trapezium to mean the quadrilateral with one pair of opposite sides parallel. This shape is called a trapezoid in North America. The word trapezoid is rare in UK mathematics, but in general British use it is a synonym for the same quadrilateral with one pair of opposite sides parallel, or sometimes a 3-D shape having some faces with the same properties, such as a prism with trapezoidal cross-section, or a frustrum of a right square-based pyramid. Only France seems to have a usage exactly opposite to the US usage. German websites writing in English and using the word trapezoid have the US & modern British meaning. I am compiling a list of modern usage and categorising citations to further clarify the myth which seems to be common in US websites and old US dictionaries that UK usage is the exact opposite of US meanings for the words trapezoid and trapezium, when in fact they seem to have been synonyms in the UK and Australia for at least the last hundred years. I would be interested in modern usage (but not definitions from ancient dictionaries) from other English-speaking areas. Dbfirs 13:16, 23 January 2008 (UTC)
Could the usage note be incorrect?
- In German both usages (like in the US and like in the UK) exist and same is true for Latin. German dictionary www.zeno.org/Pierer-1857/A/Trapez states that German Trapez in the US sense in the more original and more common usage.
Thus, the differentation between an UK and an US sense could be modern, while old Englishmen in England (16th till 19th century) could have used both senses too. And also the Enlish US sense could have been the original sense in English.
- In ancient times Latin trapezium and mensula as well as Greek τραπέζιον could be a general term for trapezium (US) and trapezium (UK). See e.g. books.google.com/books?id=HxhEBgAAQBAJ&pg=PA188 (Euclid in Greek and with English translation). Paraphrased it is: "A trapezium is a quadrangle/quadrilateral which is not a rectangle (including square and oblong) and not a rhomboid (including rhombus)". Thus both trapeziums in the English sense (UK and US) are trapezia in the ancient sense (Euclid). Also it seems like there was no mathematical term like trapezoid in ancient times, even though LSJ has "τρᾰπεζο-ειδής, ές, trapezium-shaped, λόφος Str.14.6.3, cf. Placit. 3.10.3.".
- books.google.com/books?id=wwphAAAAcAAJ&pg=PA228 (W. Pickering, The Marrow of the Mathematicks, 1724, London, p. 83) has "a Trapezium consisting(?) of 4 unequal sides, and 4 unequal Angles". That should be the US sense before 1795 in a book from England, and this should prove that the currect usage note is not correct.
- books.google.com/books?id=U5RmAAAAcAAJ&pg=PA21 (German, from 1641 according to google) has the words Trapezium, Mensula, Tischlein in it, and no Trapezoides or Trapezoid. books.google.com/books?id=NfShXnGHVdAC&pg=PA184 (French/Latin, from 1667) has "Trapeses, Trapesia, (figurae sunt irregulares quadrilaterae)" in it and no trapesoid, trapezoid, trapesoides, trapezoides. Thus it seems like older trapezium was a hyperonym for the UK and the US sense of trapezium. This hyperonymic sense might once have also existed in English (16th and 17th century).
Side note 1: In the old German text Quadratum and Vierung do not mean square, but rather quadrangle. L&S translates quadratum, quadratus with "square, quadrate", while German dictionaries with "Viereck, Quadrat" or "Viereck; Quadrat", that is "quadrangle" and "square". So maybe ancient quadratum sometimes had a wider meaning than just square, and maybe English square once had a wider meaning too.
Side note 2: The German form of triangle (Tringulus/Triangulum, Triangul, Triangel) is sometimes neuter, sometimes masculine. While this might look like a gender-shift from Latin neuter (tringulum) to German neuter and masculine, there should be no gender-shift. Dictionaries also mention ancient Latin tringulus and quadratus.
Side note 3: As for Latin, there could be several more substantivations like isosceles for triangulum isosceles and aequilaterum for triangulum aequilaterum at least in New Latin.
-184.108.40.206 17:32, 24 August 2016 (UTC), 22:56, 24 August 2016 (UTC)