Slope measures 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

where and are changes in the two coordinates over some distance.

For a plane curve specified as , the slope is

for a curve specified parametrically as , the slope is

where and , for a curve specified as , the slope is

and for a curve given in polar coordinates as , the slope is

(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

in his famous treatise published in several editions beginning in 1848. Todhunter (1888) also employed the symbol , writing the slope-intercept form

However, Webster's New International Dictionary (1909) gives the "slope form" as

(Miller).

In Swedish textbooks, the slope-intercept equation is usually written as

where may derive from "koefficient" in the Swedish word for slope, "riktningskoefficient." In the Netherlands, the equation is commonly written as one of

In Austria, is used for the slope, and for the -intercept (Miller).

See also

Gradient, Line, Slope Field, x-Intercept, y-Intercept

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References

Eves, H. W. Mathematical Circles Revisited: A Second Collection of Mathematical Stories and Anecdotes. Prindle, Weber, and Schmidt, 1972.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, 1972.Miller, J. "Earliest Uses of Symbols from Geometry." https://mathshistory.st-andrews.ac.uk/Miller/mathsym/geometry/.O'Brien, M. A Treatise on Plane Co-Ordinate Geometry, or, The Application of the Method of Co-Ordinates to the Solution of Problems in Plane Geometry. Cambridge, England: Deightons, 1844.Salmon, G. Conic Sections, 6th ed. New York: Chelsea, 1960.Todhunter, I. Treatise on Plane Co-Ordinate Geometry as Applied to the Straight Line and the Conic Sections. London, England: Macmillan, 1888.

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Slope

Cite this as:

Weisstein, Eric W. "Slope." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Slope.html

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