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?
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]