A quantity which gives the inclination of a curve or line with respect to another curve or line. For a line in the 
-plane making an angle 
with the x-axis,
the slope 
is a constant given by
 |
(1)
|
where 
and 
are changes in the two coordinates over some distance.
For a plane curve specified as 
,
the slope is
 |
(2)
|
for a curve specified parametrically as 
, the slope is
 |
(3)
|
where 
and 
, for a curve specified as 
, the slope is
 |
(4)
|
and for a curve given in polar coordinates as 
, the slope is
 |
(5)
|
(Lawrence 1972, pp. 8-9).
It is meaningless to talk about the slope of a curve in three-dimensional space unless the slope with respect to what is specified.
J. Miller has undertaken a detailed study of the origin of the symbol 
to denote slope. The consensus seems to be that it is not
known why the letter 
was chosen. One high school algebra textbook says the reason for 
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. In fact, Descartes, who was French, did not use 
(Miller). Eves (1972) suggests "it
just happened."
The earliest known example of the symbol 
appearing in print is O'Brien (1844). Salmon (1960) subsequently
used the symbols commonly employed today to give the slope-intercept
form of a line
 |
(6)
|
in his famous treatise published in several editions beginning in 1848. Todhunter (1888) also employed the symbol 
,
writing the slope-intercept form
 |
(7)
|
However, Webster's New International Dictionary (1909) gives the "slope form" as
 |
(8)
|
(Miller).
In Swedish textbooks, the slope-intercept equation is usually written as
 |
(9)
|
where 
may derive from "koefficient"
in the Swedish word for slope, "riktningskoefficient." In the Netherlands,
the equation is commonly written as one of
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
In Austria, 
is used for the slope, and 
for the 
-intercept (Miller).
