Slope
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).
Contact the MathWorld Team
what is the slope
of a line
