Why b for Intercept?

Date: 10/16/2003 at 12:07:04
From: Brian
Subject: In y = mx + b, why b?

In the slope-intercept formula

  y = mx + b
 
Why is the letter "b" used to represent the y-intercept?
 
It may be helpful to know that in class we established the reason 
for "m" representing the slope of the line.  A famous French 
mathematician, Rene Decartes, invented the x-y plane.  The French word 
meaning "to climb" is "monte".  
       
 slope = climb = monte = m
   
My guess is that whatever the "b" stands for, it has something to do 
with the French language.  

I know this is an obscure question, but any help you can offer would 
be greatly appreciated.

Brian C.

Date: 10/16/2003 at 13:08:28
From: Doctor Peterson
Subject: Re: In y = mx + b, why b?

Hi, Brian.

As we have said elsewhere in our site, we don't believe that the m 
really comes from a French word:

  Math terms
    http://mathforum.org/dr.math/faq/faq.terms.html 

  Why m for slope?
    http://mathforum.org/library/drmath/view/52477.html 

As far as we can tell, the form mx+b originated in some American 
textbook, and made enough sense to be copied by others. The letters 
used don't really matter; in other countries other forms such as ax+b 
and mx+c are common. Note that although Descartes did use x and y, he 
never used m!

My impression is that most likely these letters were chosen in much 
the same as x and y, which don't stand for words:

  We use letters near the end of the alphabet for the unknown 
  variables, x and y.

  We use letters near the beginning of the alphabet for the 
  intercepts, a and b; these are in the same order as the axes 
  they relate to, x and y.

  We use a letter in the middle of the alphabet, m, for the slope, 
  to keep it distinct.

Note that the intercepts appear prominently in the two-intercept 
form, which many students never see:

   x     y
  --- + --- = 1
   a     b

Using a and b makes this form very neat and memorable. And I think 
that is all that is behind this choice of letters.

Of course, that is just my impression; I never got to interview the 
first author to use this form and ask why he did it! If you have some 
historical evidence to support an alternative theory, I'd like to 
hear it.

According to Jeff Miller's page

    http://jeff560.tripod.com/geometry.html 

it appears that m was used before a and b, perhaps originating with 
mx+n; that may have become established first rather than after a and 
b as my summary implies, but the order doesn't matter. Note that he 
says

  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]. 

This lends support to my proposal.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/