Left Angles and Language Reversals

JD2718 commented on my last post and seemed surprised at my interest in the origin and development of mathematical language and terminology. He reminded me of one of the interesting stories that kindled my interest in the history of math language and symbols.

I first got interested in a very superficial way when my kids would ask those same questions year after year; "Why do we have "right" angles and not "left" angles?" "Who in the heck invented a word like Rhombus (and what does it have to do with Rodeo Clowns)?"

Over the years I developed an interest in the etymology of language in general, but my main focus has always been math words and symbols.

One of the surprising things I learned is that sometimes, the usage of a word completely reverses meaning over time. Words like "nice" for example, which once meant ignorant or stupid (it is actually a contraction of nescient, without knowledge, which is still in many dictionaries) "Silly" on the other hand, was a compliment (compliment is another interesting word, ( related to the math word complement). Silly was used much as we might call some one "bubbly" or effervescent today.

Often the usage by one influential person could determine the usage, or altered usage, of a term; note Shakespeare's influence on the meaning of weird. Just such a thing happened with the terms "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 A 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.

Functions, Parents, and Parent Functions

A few blogs ago, vlorbik sent a comment that included the suggestion that "mother function" might have been one of the victims of social censure and thus gave rise to the emergence of the (apparently) more socially acceptable "parent function."

From what I have been able to discover in a short period of researchs, the use of terms like "parent function" seems to have worked its way into mathematics from statistics, which seems to have gotten it from the anthropologists/sociologists.

Across America and Asia: Notes of a Five Years' Journey Around the World ...‎ - Page 250 by Raphael Pumpelly - Voyages around the world - 1870
"if it should bear the same relation to the parent population that.."

Prior to 1900 there are almost no listings of "parent function" in a mathematical usage. Around the end of the 19th century, statisticians began to talk about the distribution from which a sample was taken as the "parent distribution" of the sampling distribution (population of all samples of some size n). Some of these may be related to the study of eugenics in which the study was about the relations of some characteristic of the offspring to the actual parents, but the usage grew.

Data reduction and error analysis for the physical sciences‎ , by Philip R. Bevington - 1969 10-2 the F TEST As discussed in the previous section, the x2(chi square) test is somewhat ambiguous unless the form of the parent function is known because the statistic
x2 (chi-sq)... "

Occasionally I find the term "parent function" applied in this way when the distribution of the original sample was a normal distribution.
The Annals of mathematical statistics‎ - Page 179
by American Statistical Association, Institute of Mathematical Statistics, JSTOR (Organization) - 1948" Usually the parent function is the Type A or normal curve, as discussed by Gram "

There are also some early uses of the "parent function" in association with the use of inverses and derivatives in calculus and analysis texts back to about 1925. By 1970 the term had become commonly understood, but not abundantly used.

The teaching of secondary mathematics‎ - Page 521
by Charles Henry Butler, Frank Lynwood Wren - Education - 1965 - 613 pages
".. of an inverse function and its relation to the parent function or else in failure to attach clear meanings to the terminology and notation employed. ..."

It was the 1980's and the introduction of computers and graphing calculators into modern classrooms that seemed to make the term "parent function" ubiquitous. Any function that appeared on the calculator was a parent function, and the translations, rotations, shears, etc became the "children".
I think this use also led to the introduction of "mother function" rather than the other way around. I can only find a few examples of "mother function" and there does not seem to be any pattern to the frequency as one might expect if a term had arisen to replace this one as an "off-color" predecessor. In fact, it seems "mother function" is more commonly used by continental writers, often in conjuction with "daughter functions"; but admittedly the sample size I have to draw on was small.

If you are one of those people with access to old journals, or a collection of old texts, I would appreciate any references to the use of any of these terms and a source earlier than 1900; and if you have a way to make a digital copy and send it by email, I will have my students name their children after you.

Even the Science Magazines Can Blow It Sometimes

It is an in-joke among math types....
"What do you call a person who can't tell a coffee cup from a doughnut?"
answer.. A Topologist...

... Yesterday I got the following release from Nature.com on my news link...

"A team of three has solved a 45-year-old problem in the mathematics of topology.

The Kervaire invariant problem is 'one of the major outstanding problems in algebraic and geometric topology' says fellow mathematician Nick Kuhn, at the University of Virginia in Charlottesville."

Ok, I admit, I never heard of the Kervaire problem, but that doesn't mean I am topologically stupid, and so when they showed the picture at the top, my mind jerked.. ????

Sure enough, later in the article, they had the comparison I expected, right before a rather complicated description of the details of the Kervair invariant. It said, "Algebraic topology is a way of describing the properties that objects with the same topology have in common. Topologically equivalent objects are objects that can be converted into each other by deforming but not tearing them: a sphere and an eggshell, for example, or a doughnut and a coffee cup." (my emphasis).

But of course they mean a coffee cup with a handle, like this one, so that both shapes are really a form of torus (doughnut) like objects. Spheres punctured with a single hole.



Ok, I know they probably told some intern to get a file picture of a doughnut and a cup of coffee, and he did, but from top science mags... you just expect a little more.

Which brings me to my final question for someone who really knows topology.... is a Mobius Strip also a Genus one object like a torus?

Victorian Political Correctness, Math Terminology, and Urban Legends

I got a nice note from Vlorbik on my last post about the influence of political correctness (not his term) and its influence on math in the past. I don't know if that is a first name or a last name, and I actually tried a little to find out, but I do know he has an unusual collection of blogs, including one on Community College Calculus which is always a good read.

Vlorbik wrote, in part, "i've heard, though never verified, that victorian prudery also caused certain teachers to begin referring to the "arms" rather than the "legs" of a right triangle.."

I have a pretty extensive collection of old textbooks, including many British texts, and I don't remember ever seeing anyone use "arms" in that fashion, so my first thought was that, if it were true, it was only a very minor usage. Since the good lady ruled from 1819 to 1901, I thought I would search before and after her reign.

I pulled out my 1804 edition of Playfair's "Elements of Geometry", published in Edinburgh. He referred to the right triangles sides as..."sides"... His Book VI, prop. XXXI reads exactly like the Thomas Heath Translation. No help there, so I skipped forward 99 years to the other end of the Victorian period, 1903 and looked in "A Junior Geometry" by Noel S Lyndon, published in London, only to find he also used only the terms hypotenuse and "other two sides" in his statement of the Pythagorean Thm.

Perhaps neither term was common in the Victorian peridod, and these stories were a bit of urban legend. I went on to Google Books to see if I could find any examples of geometric usage such as Vlorbik had described..... I entered a search for "arms 'right triangle' geometry"

....Yikes", there they were. The first listing was "Plane Geometry" by Arthur Schultze, Frank Louis Sevenoak, Limond C. Stone, from 1901. It contained, "The sum of the squares of the arms of a right triangle is equal to ..." along with 388 other listings, some dated as late as 2008. "In a right triangle whose arms have lengths a and 6, find the length of the .." appears on page 451 of the fourth edition of Schaum's Outline of Geometry from that year. Ok, but that still did not mean it was the influence of the dreaded Victorian stuffed-shirts... I switched the cut-off to 1850... and there were NO results prior to that year... only one last check. Would there be examples with the use of "legs" prior to that year? There were indeed, including several by the famous American Mathematician, Benjamin Pierce. Another from 1734 was from the British Benjamin Martin.

So it appears that there was some pressure to use "arms of a right triangle" suggested by these dates; but there is still no smoking gun. Does anyone out there know of a document or statement of any kind in the math education literature that makes a clear suggestion to teachers? If you know of such a document, please share whatever level of information you have and I will pursue it.

And thanks again to Vlorbik for sharing this tidbit of math language history. I also followed up a second part of his comment, and learned a little more about the evolution of "parent functions".... but that will have to be another day.