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The slope-intercept form of a non-vertical line is $y=mx+b$. I have been told that the slope is called $m$ because it is the first letter of the French word for mountain. But why is there the letter $b$? Wouldn't it be more logical to write $y=mx+n$ or perhaps $y=mx+a$? What is the history behind that convention?

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    $\begingroup$ It seems that there is no evidence for that explanation of the letter $m$. $\endgroup$ Jun 24 at 0:34
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    $\begingroup$ Really? I have always studied $y = mx+c$. $\endgroup$ Jun 24 at 0:41
  • $\begingroup$ George Albert Wentworth, Elements of Analytic Geometry, Boston: Ginn 1886, p. 36: " Notation. Throughout this chapter, and generally in equations of straight lines, $a =$ the intercept on the axis of $x$. $b =$ the intercept on the axis of $y$. $\gamma =$ the angle between the axis of $x$ and the line. $m =\tan \gamma$. " $\endgroup$
    – njuffa
    Jun 24 at 2:35
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    $\begingroup$ Earliest French source found so far: Joseph Carnoy, Cours de Géométrie Analytique, Paris: Gauthier-Villars 1876, p. 26: "Prenon ensin l'equation la plus générale du premier degré en $x$ et $y$ $Ay+Bx+C = 0.$ On en tire $y= -\frac{B}{A}x-\frac{C}{A}.$ Posons $m=-\frac{B}{A}, b=-\frac{C}{A};$ nous aurons à considérer l'équation $y = mx+b.$ Soit $OM_{2}$ une droite passant par l'origine et représentée par $y = mx.$ $\endgroup$
    – njuffa
    Jun 24 at 3:19
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    $\begingroup$ Jeff Miller's reference, in the section headed "slope", cites Riccati's 1757 memoir De methodo Hermanni ad locos geometricos resolvendos as the first use of $m$ for slope, writing $ y = mx + n$. The first use of $y = mx + c$ is found in O Brian, 1842 An Elementary Treatise on the Differential Calculus. The first use of $y = mx + b$ is found in Salmon, 1844 A Treatise on Conic Sections. See link for details. $\endgroup$
    – nwr
    Jun 24 at 5:07

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The first known use of $b$ to represent the y-intercept in the equation of straight line is in Gaspard Monge's "Mémoire sur la Théorie des Déblais et des Remblais", Histoire de l'Académie royale des sciences 1784 (Année 1781), p. 669.

Or, I'équation de la droite $Bd$, rapportée aux mêmes axes, eft généralement de la forme $y = ax + b$

However, the usage of $b$ along with $m$, the slope, dates back to George Salmon's A Treatise on Conic Sections, p.15, published in several editions beginning in 1848.

The equation, therefore, $y = mx + b$, being satisfied by every point of the line PQ, is said to be the equation of that line.

Nevertheless, there is no clear answer to the question of why the letter $b$, or even $m$, is used. In fact, many authors have used various other symbols.

Matthew O'Brien's An Elementary Treatise on the Differential Calculus (1842), p.1., uses $c$ to represent the y-intercept.

Thus in the general equation to a right line, namely, $y = mx + c$

Isaac Todhunter again made use of the letter $c$ in A Treatise on Plane Co-ordinate Geometry as Applied to the Straight Line and the Conic Sections.

Even there are different conventions in different countries. In Swedish textbooks, $k$ represents slope and $m$ represents the y-intercept. In Austria, $k$ is used for the slope and $d$ for the y-intercept.[Wolfram MathWorld]

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