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 xb 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.