Robert Recorde

I'm not sure what the force is behind certain blog posts popularity. In the last two months my second most popular "hit" has been "Math Symbols Are NOT All Created Equal". What makes that seem unusual to me is that I wrote it in the summer of 2008. The blog is about the equal sign and how it came to be the most ubiquitous of mathematical symbols. I suspect that it is the start of school year that has driven many of these searches by students driven to find the creator of the symbol. If that is true, then here is a little more interesting history for them (or their teachers) to share.

The symbol was, as you know if you read the blog, created by Robert Recorde in his "Whetstone of Witte", and published in London just before Recorde's death in 1558.

Recorde died in the King's Bench Prison at Southwark in London, most likely for being unable to pay his debts. Then, as now, teaching was not an economically rewarding position. Along the way, he had been a very successful writer of mathematics textbooks. His first arithmetic, "The Ground of Artes," was published around 1540-43. It became a very popular arithmetic and was reprinted over a dozen times. After his death, other mathematicians would use it as the base of their own books, making minor adaptations and keeping Recorde's well known name. One such edition by Edward Hatton bears the date 1699, almost 150 years after Recorde's death. Algebra, it seems, was a less popular topic. The "Whetstone of Witte", in which he introduced the "gemowye" lines of equality, never made it to a second edition.
If you remember that most publications in mathematics were in Latin, then the task of creating an English language Geometry or Algebra required the introduction of English terms for some of the technical terms needed. Recorde's "Pathway to Knowledge" was two decades ahead of Billingsly's translation of Euclid. In it he introduced many Saxon-English words for the Latin terms. A "poynt or prycke" was used for the point. One-hundred years later Newton would write that he used "pricked letters" to indicate a fluxion. "Sharp" and "blunt" corners were the translations from the Latin acute, and obtuse, to the detriment of many geometry students who would better remember and understand the Saxon terms. Recorde allowed that his "gemowe" or parallel lines (he used both terms) need not be straight. Those crooked copies of each other, like ss, he called "tortuous parallels".

Tangent lines were called "touche lynes" in an Anglo-Saxon translation. Vertical angles were referred to as "matche corners", and rectangles were "losenges or diamondes". If the parallelogram was oblique, it was called "diamonde-like".

None of these became popular terminology, most likely because the language of scholars continued to be Latin for over a century after his death. Although the terms "long square" for a rectangle and "diamonde" for rhombus appeared in the Billingsly translation of Euclid, little other evidence of Recorde's language inovations seem to appear.

Recorde was not totally adverse to using existing or foreign terms. He referred to "cooslike" numbers for variables, which had been introduced by Pacioli (1494) from the Latin "cosa" for thing, which had become very popular in Germany. The German algebraists were sometimes called cossists, and the study of algebra as "die cost."

Also, we should keep in mind that it was Recorde who first used the word sine in print in English.

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.

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.

We Just Don't Talk That Way Anymore

I love reading old journals and am often struck by the precision and beauty of language in old math and science journals.

Many brighter than I have commented on the seemingly inevitable reduction in rigor as schools require all students to take more advanced math. I think the same thing has happened across the board to language as everyone is expected to complete a high school education. I recently read a note that had two quotes, both saying essentially the same thing. The first is from Leonhard Euler and dates around the American revolution; the second is from George Box and dated around 1987Empirical Model-Building and Response Surfaces. Both are incredibly literate and knowledgable people, and yet..

"Although to penetrate into the intimate mysteries of nature and thence
to learn the true causes of phenomena is not allowed to us, nevertheless
it can happen that a certain fictive hypothesis may suffice for
explaining many phenomena"

All models are wrong... some models are useful