Monday, 24 August 2009

A is for Vector?.

Found a copy of “One Letter Words” by Craig Conley in a cheap resale shop the day before I flew back to England, so I bought it and read it on the plane…Some interesting references to mathematical symbols and constants got me thinking…somebody should put together a mathematical book of one letter words… I offer the title above for free… it comes from the quote in the book that said , noun (Mathematics) A vector… and then justifies with a quote from Marie Vitulli’s “A Brief History of Linear Algebra and Matrix Theory”… The use of a single letter A to represent a matrix was crucial to the development of matrix algebra. Early in the development the formula det(AB) = det(A)det(B) provided a connection between matrix algebra and determinants. [Italicized sentence added by me]
Ok, did he miss the point? Or did I… I assume Vitulli meant the use of a single letter (such as A) to represent a vector was a huge leap… but; I think the give-away was calling it a noun… variables like this are more like pronouns in my mind… using x in place of the number “four” or n for “some number not named”… correct me if you disagree…
Sadly, after all that, he left out the one that is known to almost every Algebra I student, “M is for slope”. Interestingly, that one has led to more mis-history in classrooms than any other topic, with the possible exception of the life of that Great American-Indian Mathematician, Chief Soh Cah Toa.

Here is what I have found about the slope, as it appears at my MathWords page; Slope is derived from the Latin root slupan for slip. The relation seems to be to the level or ground slipping away as you go forward. The root is also the progenitor of sleeve (the arm slips into it) and, by dropping the s in front we get lubricate and lubricious (a word describing a person who is "slick", or even "slimy").
Many variations of where the idea of M for slope originated seem to be mostly myth. One of the most common is that the letter was used by Descarte because it was the first letter of some French word or another that related. In a recent post to the AP Stats discussion list, Hector Hirigoyen shared the following story:
I was told by Mary Dolciani herself, that the SMSG group "decided "to use y=mx+b because of the French (Descartes, I presume)-"montant"; I found it strange because the "logical word" would be "pente"(which is slope (and the standard term in Spanish is pendiente, which matches this). However, several years ago, while visiting a French high school, I noticed the teacher used y=sx+b. I inquired, and she said because of the "American" word "slope." I believe they are using ax+b for the most part these days.
. Here are several other clips from postings about the topic on a discussion group about math history.
In his "Earliest Uses of Symbols from Geometry" web page, ... Jeff Miller gathered the following information: Slope. The earliest known use of m for slope is an 1844 British text by Matthew O'Brien entitled _A Treatise on Plane Co-Ordinate Geometry_ [V. Frederick Rickey]. George Salmon (1819-1904), an Irish mathematician, used y = mx + b in his _A Treatise on Conic Sections_, which was published in several editions beginning in 1848. Salmon referred in several places to O'Brien's Conic Sections and it may be that he adopted O'Brien's notation.

According to Erland Gadde, in Swedish textbooks the equation is usually written as y = kx + m. He writes that the technical Swedish word for "slope" is "riktningskoefficient", which literally means "direction coefficient," and he supposes k comes from "koefficient."
According to Dick Klingens, in the Netherlands the equation is usually written as y = ax + b or px + q or mx + n. He writes that the Dutch word for "slope" is "richtingscoefficient", which also means "direction coefficient." In Austria k is used for the slope, and d for the y-intercept. In Uruguay the equation is usually written as y = ax + b or y = mx + n, and the "slope" is called "pendiente", coeficiente angular", or "parametro de direccion".
It is not known why the letter m was chosen for slope; the choice may have been arbitrary. John Conway has suggested m could stand for "modulus of slope." One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is monter. However, there is no evidence to make any such connection. Descartes, who was French, did not use m. In _Mathematical Circles Revisited_ (1971) mathematics historian Howard W. Eves suggests "it just happened."
Jeff Miller's web site cited above now has updated the earliest use of m for slope.
The earliest known use of m for slope appears in Vincenzo Riccati?s memoir De methodo Hermanni ad locos geometricos resolvendos, which is chapter XII of the first part of his book Vincentii Riccati Opusculorum ad res Physica, & Mathematicas pertinentium (1757):
Propositio prima. Aequationes primi gradus construere. Ut Hermanni methodo utamor, danda est aequationi hujusmodi forma y = mx + n, quod semper fieri posse certum est. (p. 151)
The reference is to the Swiss mathematician Jacob Hermann (1678-1733). This use of m was found by Dr. Sandro Caparrini of the Department of Mathematics at the University of Torino.


Charles Wells said...

"slupan" is an Anglo-Saxon word, not a Latin one. The Latin word that gave us "lubricate" may very well have come from the same Indo-European root as "slupan" (I don't have a reference here at the lake cabin) but that is irrelevant.

Charles Wells said...

Addendum to my comment: Apparently "lubricate" and "slope" both come from the IndoEuropean root sleubh.

Anonymous said...

I tell the 'montant' story every year, go on to look at the root, eventually get to the RCMP and one of my favorite New Yorker cartoons, a stairway in Grand Central, with a sign saying "To Trains" painted with an arrow on the wall, and a mustachioed villain with a trussed up young lady over his shoulder walking down the stairs...

Then I tell the kids it's not true.


Pat's Blog said...

Professor Wells,

Thanks for the correction, and I have posted a correction on the web sites.