## Wednesday 19 October 2022

### #23 Trapezoid and Trapezium ...From old Math Terms notes

Trapezoid and Trapezium Both words come originally from the Greek word for table. Today, in the USA, the term trapezoid refers to a quadrilateral with one pair of sides parallel and a trapezium to one with NO parallel sides. Actually, the term for the case with no parallel sides is almost never used, so trapezium is an archaic term at best in the US. This is exactly the reverse of the original meanings and the meanings in some countries, particularly England, today. Here is a short comment on how this came about from Jeff Miller, a teacher at Gulf High School in New Port Richey, Florida, who maintains an excellent page on the first use of some common mathematical terms:

"TRAPEZIUM and TRAPEZOID. The early editions of Euclid 1482-1516 have the Arabic helmariphe; trapezium is in the Basle edition of 1546. Both trapezium and trapezoid were used by Proclus (c. 410-485). From the time of Proclus until the end of the 18th century, a trapezium was a quadrilateral with two sides parallel and a trapezoid was a quadrilateral with no sides parallel. However, in 1795 a Mathematical and Philosophical Dictionary by Charles Hutton (1737-1823) appeared with the definitions of the two terms reversed: Trapezium...a plane figure contained under four right lines, of which both the opposite pairs are not parallel. When this figure has two of its sides parallel to each other, it is sometimes called a trapezoid. No previous use of the words with Hutton's definitions is known. Nevertheless, the newer meanings of the two words now prevail in U. S. but not necessarily in Great Britain (OED2).
John Conway recently pointed out in a post on the use of the terms that;
What is true now is that the thing with two parallel sides is called a "trapezium" in England and a trapezoid in America, and that neither term is used in either country for the thing with no parallel sides. (The latest date for which I'vbe seen either of them so used was in a geometry book of 1912, which however, was a reprint of a 19th-century one.) Instead, to avoid confusion, the term "quadrilateral" is now standard for the general case. (This has a few earlier uses, dating back to about 1500, but was decidedly uncommon - like "trilateral" - before 1900.)
I also have an English textbook that uses trapezion (note the n ending) for the shape we more commonly call a kite. In A Junior Geometry by Noel S. Lydon published in 1903 the definition on page 55 states trapezion is a four-sided figure having two pairs of adjacent equal sides. It goes on to show the method of construction.

Some geometry textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides, so that a parallelogram is a type of trapezoid. Euclid did not define the shape we now call a trapezoid, and the "trapezia" is defined by default.... "let quadrilaterals other than these be called trapezia" [from the Heath translation]. Heath's translation states that the language used implies that Euclid may have been creating a new word, or using an existing one in a new way. Proclus seperated out trapeziums and trapezoids (backwards to what we now do as explained in the quote above from Jeff Miller's page) but it seems clear he meant that a trapezium had exactly one pair of sides parallel (an exclusive definition) rather than at least one pair of sides parallel(an inclusive defintion). Many mathematicians today prefer the inclusive definition so that a parallelogram is a special case of trapezoid. Apparently this has been a question in geometry for a while as I recently read a note from Neal Silverman which suggests that the inclusive definition has appeared in some books back at least to 1900.

" I recently acquired a copy of "New Plane & Solid Geometry" by Wooster Woodruff Beman and David Eugene Smith (Ginn 1900), a revision of their earlier 1895 work. ... But as to quadrilaterals, consider their definition of trapezoid: "A quadrilateral that has one pair of opposite sides parallel is called a trapezoid." They go on to state that "[b]y the definition of trapezoid here given it will be seen that the parallelogram may be considered a special form of the trapezoid. (section 97 at p. 59).

It seems that the inclusive definition was not well received as in the same post Neal adds, "D. E. Smith went on to write many more books on geometry, some of which were revisions of the old Wentworth books. I have never seen this statement in any of his later books."

John Conway recently posted a note about the etymology of trapezoid;

The etymology of "trapezoid" is quite interesting. It's a corruption of "tetra-pes-oid", whose three parts mean "four-leg-shaped", or perhaps more familiarly "table-shaped", since "tetra-pes" was a familiar Greek name for a small table.