Wednesday, 20 February 2013

Why M for Slope

Over the many years I taught I developed a keen interest in the origin and use of the symbols and terminology of math. When the internet came around I jumped in pretty early with a web page on Mathwords, contributing on the Dr Math and Teacher to Teacher support Math Forum, and for several years, this blog. Over that time there are few questions that come up more often than the reason for the letter m as the symbol for slope in the standard American (and it does seem to be mostly American) linear slope intercept equation form, y = mx + b.
Almost no one seems to ask about the b, which is even more curious to me.

Interestingly, m for slope has led to more mis-history speculation 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 m for slope history.

Slope is derived from the proto European root slupan (another etymology site uses sleubh) 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 post to the AP Stats discussion list some years ago, 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.

A similar story was shared by Paul Foerster:
I can confirm Hector Hirigoyen's statement about the origin of "m" for slope. Hector was in the audience when I did a presentation at a math teachers' meeting in Florida about 1986 or 1987. His point was that "m" came from the French "montant," meaning "the rise." Because he taught AP French as well as mathematics, it seemed like a logical conclusion.
I was so thrilled with this new knowledge that I told it to my BC Calculus class when i returned to San Antonio. To add to the credibility, I asked Cassandra Stapfer, my exchange student from Paris, to confirm that this was correct. Cassandra said, "montant does mean 'the rise,' but in France we use 's' for 'slope'."So much for my brilliant discovery!

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.

I just checked Jeff's site again and found the following list of notes from his readers:
In 1830, Traite Elementaire D'Arithmetique, Al'Usage De L'Ecole Centrale des Quatre-Nations: Par S.F. LaCroix, Dix-Huitieme Edition has y = ax + b [Karen Dee Michalowicz].
Another use of m occurs in 1842 in An Elementary Treatise on the Differential Calculus by Rev. Matthew O'Brien, from the bottom of page 1: "Thus in the general equation to a right line, namely y = mx + c, if we suppose the line..." [Dave Cohen].
O'Brien used m for slope again in 1844 in 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. Salmon used a to denote the x-intercept, and gave the equation (x/a) + (y/b) = 1 [David Wilkins].
Karen Dee Michalowicz has found an 1848 British analytic geometry text which has y = mx + h.
The 1855 edition of Isaac Todhunter's Treatise on Plane Co-Ordinate Geometry has y = mx + c [Dave Cohen].
In 1891, Differential and Integral Calculus by George A. Osborne has y - y' = m(x - x').
In Webster's New International Dictionary (1909), the "slope form" is y = sx + b.
In 1921, in An Introduction to Mathematical Analysis by Frank Loxley Griffin, the equation is written y = lx + k.
In Analytic Geometry (1924) by Arthur M. Harding and George W. Mullins, the "slope-intercept form" is y = mx + b.
In A Brief Course in Advanced Algebra by Buchanan and others (1937), the "slope form" is y = mx + k.
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 "richtingscoëfficiënt", which literally means "direction coefficient."
In Austria k is used for the slope, and d for the y-intercept.
According to Julio González Cabillón, in Uruguay the equation is usually written as y = ax + b or y = mx + n, and slope is called "pendiente," "coeficiente angular," or "parametro de
direccion."
According to George Zeliger, "in Russian textbooks the equation was frequently written as y = kx + b, especially when plotting was involved. Since in Russian the slope is called 'the angle coefficient' and the word coefficient is spelled with k in the Cyrillic alphabet, usually nobody questioned the use of k. The use of b is less clear."

So why do we use m? Mostly habit... after all, you gotta' use something. I would hope some of my international readers would tell us what THEY use for the linear coefficient symbol in this slope intercept form of the equation; especially if they actually do use m.
Stay tuned.

ADDENDUM:

I got a couple of comments telling me that m is used for slope in Germany (y=mx+t) by Thony Christie, and one from Luke Robinson that in the UK they use  y = mx+c.
Maria Miller added that, "In Finland, it is y = kx + b, and the term for 'k' is 'kulmakerroin' - literally 'angle multiplier'. "

I also found a couple of older books  that show this was used in the UK at least into the mid 19th century.  Charles Joseph Hughes in his Useful formulæ, chiefly in the pure mathematics (1859) uses  y=mx+c in a book that says it is for the use of candidates preparing for Addiscomb, Sandhurst, Woolwich and the Universities  (with the note that it is "to be committed to memory").
An even older use in 1812 was in an American edition of Hutton's Course of Mathematics for the use of Academies.  Hutton (Adrian) uses x=ny+c and y=mx+d in the same line, so his use seems more along the line of letting middle letters (k,l,m,n) stand for pronumerals (a word I derived from pronouns to indicate numbers that represented some single unspecified value) and x,y,z to represent true variable quantities.    This is from Hutton's 5th and 6th edition and was "revised and corrected" by Robert Adrian.  Adrian was an Irish-American considered one of the brightest mathematicians in the early 19th century.  He independently found a method of least squares independent of Gauss (in fact published before Gauss' work was published).
It seems the use of middle letters in the alphabet for the linear coefficient of terms was pretty common near the beginning of the 19th century.  For example, in factoring a trinomial the 1804The mathematical correspondent: containing, new elucidations, discoveries, and improvements, in various branches of the mathematics says "will always be the product of two rational factors (mx+p)(nx+q)."

7 comments:

thonyc said...

In German textbooks the standard linear equation is:

y = mx + t

Belfry Bat said...

What may be ironic, I can't find any evidence for "slupan-" ever having been a Latin stem; OED pleads obscurity over the origin of "slope" older than "aslope" of 15C; it certainly does not look like a Latin word, with that "sl", but rather germanic or perhaps gaelic, like "slogan" and perhaps "slough". The most-Continental "sl-" in English I can think of would be "slovene", which seems to come from Slavonic "slovene".

Good Latin words for slope-like things seem to include the stem "clin-" remembered in the English "incline". The French "pent-" words are wholy mysterious to me. Latin "pend-" is about hanging and weighing; the "paen-" and "poen-" are related to English "pain" and "punnish", and perhaps "repent", but the similar Latin participle "repente" is about crawling or creeping, hence "reptile".

One might wildly conjecture that some misguided scholar-brittons post-Guillaume were talking with eachother about some "escline", eventually sluring it into something like "slip", as the Gauls and Franks gradually beat the grecco-latin "schola" into "escole" and eventually "école" which found its way into German as "Schüle", but there seems no textual record of such an evolution.

Luke Robinson said...

Here in the UK we use y=mx+c.

Rob said...

It may not be historically relevant, but since we use it, the reason I give my students is this: "B is for where we Begin, M is for how we Move." It makes some intuitive sense, which in turn assists in them remembering which is which.

Gregory Taylor said...

I found this website compiled from reader remarks:
http://www.mathsisfun.com/equation_of_line.html#Countrynote
...When I was posting up a rant about variables on my blog. My issue is mostly with the 'h' and 'k' in vertex form of the parabola, because while 'm' is pretty consistently slope in North America, 'k' is all over the map. (http://mathiex.blogspot.ca/2013/03/variables-may-vary-part-2.html)

I also suspect no one really asks about the 'b' because 'a' is often used for the x-intercept, so it sort of follows naturally. (Though 'a' has it's own issues.) Anyway, thanks for the history!

Pat B said...

Belfry Bat, I think the Latin was something more like lubricious, from which we get slippery (think "lubricated"). The Latin seems to come from the Proto indo Euro root slupan or sleubh- Here is from webpage "myetymology.com",
derived from the English word lubricous
derived from the Latin word lubricus (slippery; sinuous; inconstant)
derived from the Proto-Indo-European root *sleubh-

Perhaps my assertion to Latin was a little too direct. Thanks for the note.

Paul Foerster said...

Paul Foerster said...

I can confirm Hector Hirigoyen's statement about the origin of "m" for slope. Hector was in the audience when I did a presentation at a math teachers' meeting in Florida about 1986 or 1987. His point was that "m" came from the French "montant," meaning "the rise." Because he taught AP French as well as mathematics, it seemed like a logical conclusion.

I was so thrilled with this new knowledge that I told it to my BC Calculus class when i returned to San Antonio. To add to the credibility, I asked Cassandra Stapfer, my exchange student from Paris, to confirm that this was correct. Cassandra said, "montant does mean 'the rise,' but in France we use 's' for 'slope'."So much for my brilliant discovery!

I did figure out why it was logical to use "b" for the y-intercept. It seemed obvious that "a" was used for the x-intercept, as in the intercept form of the linear function equation, (x/a) + (y/b) = 1. Thanks for confirming my thought in this regard!