Why is the slope-intercept form of the equation of a line often written $y=mx+b$? Why $m$ instead of $a$?

Question

After a quick google search, I read something about Conway suggesting the $m$ having to do with "modulus" ...

This seems odd to me, but perhaps there is some mathematical reason? I've heard of the uses of the word modulus in real/complex analysis and in number theory, but neither seem applicable here.

Answer 1

I found this page independent of vuur and found this question, so I might as well answer it. Specifically,

The earliest known use of m for slope appears in Vincenzo Riccati’s memoir De methodo Hermanni ad locos geometricos resolvendos, which is chapter XII of the first part of his book Vincentii Riccati Opusculorum ad res Physica, & Mathematicas pertinentium (1757):

Propositio prima. Aequationes primi gradus construere. Ut Hermanni methodo utamor, danda est aequationi hujusmodi forma y = mx + n, quod semper fieri posse certum est. (p. 151)

The Latin simply states that this is the form of a linear equation, and Riccati does not remark upon the use of $m$.

Interestingly enough, Traite Elementaire D'Arithmetique, Al'Usage De L'Ecole Centrale des Quatre-Nations: Par S.F. LaCroix, Dix-Huitieme Edition, published in 1830 does use $a$ instead of $m$ (although it obviously was not written by Riccati).

As DavidK pointed out, George Salmon also used $a$ in his work in the 19th century. I had omitted that from my original answer because he used a different form, $$\frac{x}{a}+\frac{y}{b}=1$$ but it is still a relevant usage.