Why do we use m for slope?
Tina Hartley
Fred Rickey
West Point
A Perennial Question
• Does any reader know why or how “m”
became the symbol for slope?
– Arthur F. Smith, Mathematics Teacher, 1985.
• A logical explanation: In French, “monter” means to mount, to
climb, to slope up and Descartes and Fermat were French.
• A dictionary explanation: “modulus” is “the constant ratio
which is the coefficient of the variable in a linear equation.”
• BUT, the question needs a historical explanation.
BLUF
• We don’t know.
• The long answer follows.
Caveat
Detailed knowledge of the history of a few
mathematical topics leads to the view
common to historians that priority questions
are very elusive, seldom answerable and of
little import to historical understanding.
Kenneth O. May,
“Historiographic vices, II: Priority chasing,”
Historia Mathematica, 2 (1975), 315-317.
• Our goal: not to see who was first, but to
investigate the history of
– the equations of straight lines,
– how they are described, and
– their graphs.
• Our methodology:
– To examine literally thousands of textbooks
– Maybe even hundreds
Descartes, 1637
By Franz Hals
• No discussion of equations of lines.
• Therefore, no m.
Fermat 1630s
Published 1679
• Ad locos planos et
solidos isagoge.
• Every linear proportion
represents a line.
• Example:
Ut B ad D, ita R – A ad E.
Proportions, not equations
Johannes de Witt 1675 - 1672
The bodies of the brothers De Witt, by Jan de Baen
Elementa curvarum linearum, 1649
I.
y
bx
a
II . y 
bx
c
a
III . y 
bx
c
a
IV . y  
bx
a
c
a, b, c  0
Vincenzo Riccati 1707-1775
Opusculorum ad res physicas &
mathematicas pertinentium (1757)
Proposition One:
Building equations of the first
degree
By the method of Hermann it is
certain that such an equation
always has the form y = mx + n.
• This is the first use of m in the equation of a
line.
• Sadly, we have no evidence that it influenced
anyone.
• Thanks to Sandro Caparrini for finding this.
Maria Agnesi 1718 - 1799
We’ll look at this in the library.
• Mémoire sur la théorie des
déblais et des remblais (1784)
Si l’on veut exprimer que
cette droite passe par le
point M, dont les
coordonnées sont x’ & y’,
cette équation devient
y – y’ = a ( x – x’ ),
dans laquelle a est la
tangent de l’angle que cette
droit fait avec la ligne des x.
Gaspard Monge
1746 - 1818
Portrait by Naigeon in the
Musée de Beaune
Jean-Baptise Biot, 1774-1862
Traité analytique des courbes et des surfaces du second degré;
1st edition: 1802; subsequent editions in 1826 (French) and 1840 (English).
Y
PM
AP
M

A

x
sin(    )
By the law of sines.
y
P

sin 
X
So,
y x
sin 
sin(    )
First texts in English
The Principles of Analytical Geometry, by
H.P. Hamilton, 1826.
For in BZ take any point P, draw
PM parallel to AY, and BQ parallel
to AX, meeting MP in Q.
Y
Z
P
B
C
A
But, the triangles PQB, BAC being
similar, PQ
BA
Q
M
Let AM = x, MP = y.
Then y = MP = MQ + QP = b + QP……(1)

X
QB
 a ;
AC
 PQ  a . QB  ax
Hence, substituting in (1) this value of
PQ, there results
y  ax  b
Hamilton’s definition of a
Cor. 2. The constant quantity, a, denotes the
ratio of the ordinate to the abscissa, and may
be expressed in terms of the angles which the
straight line forms with the axes….then
Y
Z
P
B
C
A
a
AC
Q
M
BA
X
y

sin BCA
;
sin ABC
sin BCA
sin ABC
xb
The Elements of Analytical Geometry,
by John Radford Young, 1830.
Advertisement to the American Edition of Young’s text:
“Mr. Young, as will be seen by his preface, has drawn largely
from these sources [Biot and Bourdon]; and the eminent
superiority of his elementary treatises on the mathematical
sciences, is mainly to be attributed to the liberality of spirit
with which – casting off the trammels imposed upon
themselves by the countrymen of Newton – he has freely
availed himself of every discovery and improvement in
analysis, though such have been chiefly made on the French
side of the channel.”
- John D. Williams, editor
Equations of the Line (Young)
y   ax  b
y   ax  b
y   ax  b
y   ax  b
“It thus appears that when
both axes are intersected, the
proposed line may take four
different positions analytically
represented by four distinct
equations.”
Charles Davies, 1798-1876
Preface from Elements of Analytical
Geometry, by Charles Davies, 1836:
“The admirable treatises of Biot and
Bourdon have been freely
consulted…It has been the intention
to furnish a useful text-book, and no
attempt has been made to depart
from clear and satisfactory methods
adopted by others, merely for the
purpose of seeming to be original.”
- Charles Davies, 1836.
Finding the Equation of the Line (Davies)
Now, since the sides of a triangle are
to each other as the sines of their
opposite angles, we have,
Y
P
PD : AD :: sin  : sin(    )

A

D
X
But PD is to AD, as any ordinate y of
the line AP to the corresponding
abscissa x: therefore,
y : x :: sin  : sin(    )
which gives,
y  x
sin 
sin(    )
.
Finding the Equation of the Line (Davies)
• Presents simplified form as:
y  ax  b
• Defines a: “..the coefficient of x is equal to the
sine of the angle which the line makes with
the axis of X divided by the sine of the angle
which it makes with the axis of Y.”
First use of m…. (that we have found)
A Treatise on Conic Sections and the
Application of Algebra to Geometry,
by J. Hymers, 1837.
Let AB = c, and the tangent of angle PTN = m.
Draw BQ parallel to AX meeting PN in Q; then
PQ=BQ. tan PBQ = AN. tan PTN = mx,
and PN = PQ + QN = PQ + AB
 y  mx  c .
Y
John Hymers
1803-1887
P
B
T
A
Q
N
X
From Hymers
“The meanings of the constants m and c are to be
particularly observed; … m is the tangent of the
angle which that part of the line which falls above
the axis of x makes with the axis of x produced in
the positive direction.”
“The equation y=mx+c, which is the most
convenient form and the one commonly
employed,… m is a number or ratio, denoting the
tangent of the angle…”
Others begin to use m…
• A Treatise on Plane Coordinate Geometry, by Rev. M.
O’Brien, 1844.
– Begins with “general equation:” Ax  By  C .
– Algebraically finds form: y  mx  c .
– “m is the tangent of the angle which the line makes with the
axis of x; c is the portion cut off from the axis of y.”
– Uses “O” for origin instead of “A.”
• A Treatise on Plane Coordinate Geometry, by I. Todhunter,
1855.
– Geometrically finds form y  mx  c
– Defines c as “the intercept on the axis of y.”
• A Treatise on Conic Sections, by George Salmon, 1863.
– Uses y  mx  b .
The word slope appears…
Mathematical Dictionary and Cyclopedia of Mathematical
Science, by Charles Davies and William G. Peck, 1855.
SLōPE. Oblique direction. The slope of a plane is its
inclination to the horizontal. This slope is generally
given by its tangent…..
If, through any point of a curved surface, any number
of vertical planes is passed, they will cut out lines
of different slopes…
Naming is Important
• Naming Infinity: A True Story of Religious Mysticism
and Mathematical Creativity
Loren Graham and Jean-Michel Kantor, 2009
• This is more than a notational convenience. In
introducing this notion, Legendre reifies the concept,
making it into an object of independent study.
Steven H. Weintraub, AMM, March 2011, p. 211
• Dictionary: Since there was no such word until the
late seventeenth century, it follows that there was
essentially no such concept either.
Simon Winchester, The Professor and the Madman, 1998
Slope Appears in the Equation of the Line
A Treatise on Analytical Geometry,
by William G. Peck, 1873.
Proposition 2. – To find the inclination of the line joining two
given points.
The inclination of a line is the angle that it makes with the axis of x….
The tangent of the inclination is called the slope of the line. The
word slope, as here employed, is nearly synonymous with the term
grade in engineering.
Peck’s Equation of the Line
“…Substituting these values, we have,
y  ax  b
The quantity b…is called the intercept; the
quantity a, is the slope.”
The 1880’s: a decade of change
• Elements of Analytic Geometry, by Elias Loomis.
In 1851 he uses y  ax  b ; but in the 1881
edition he changes to y  mx  c .
• The Elements of Plane and Analytic Geometry, by
George Briggs, 1881. Uses y   x  b
and calls b the intercept and λ the slope.
• Elements of Analytic Geometry, by Simon
Newcomb, 1885. Uses y  mx  b and defines
the slope of a line as “the tangent of the angle
which it forms with the axis of abscissas.”
Naming the Lines – 1880’s-1900’s
• Most forms of the equation of the line were established by
this time.
• Authors begin assigning names to the different forms:
Form of Equation
x
a

y
Newcomb, Wentworth,
1885
1887
y  y1  m ( x  x1 )
Ashton,
1901
Symmetrical
Form
Intercept
Form
Symmetrical
Form
Intercept
Form
--
--
Slope Form
Slope Form
Slope Form
Normal
Form
Normal
Form
Normal
Form
Normal
Form
Normal
Form
--
--
--
--
Slope-Point
Form
1
x cos   y sin   p
Nichols,
1892
--
b
y  mx  b
Hardy,
1888
The Analytical Geometry of the Conic
Sections, by Rev. E.H. Askwith, 1908.
After presenting two forms of the equation of a line:
x
a

y
b
 1 B 
y  mx  b  D 
Askwith writes:
“It will be observed that b has the same meaning in (B) as in (D).
some writers use c for b in (D). It really does not matter what
letter is used. The advantage of using the same letter in (B)
and (D) is that attention is drawn to the fact that the same
thing is represented each time.”
An Elementary Treatise on Conic Sections,
by Charles Smith, 1906.
• Used as textbook at United States Military Academy
from 1898 – 1919.
• Copy in USMA library belonged to Wm. Cooper Foote,
USMA Class of 1913.
The Wrong Question
• We should be asking:
• Why do we use “slope” for “m”?
Howard Eves, 1911-2004
We designate the slope of a line
by m, because the word slope
starts with the letter m. I know
no better reason.