After we have the intercept, the other thing we need is the slope. This is a very easy thing to start calculating and it’s extremely testable, but the idea weaves its way deep into all mathematics. It’s got an obvious physical interpretation. Imagine the x-coordinates are how far we are from some reference point in a horizontal direction, and the y-coordinates are how far we are from some reference point in the vertical direction. Then the slope is just the grade of the line: how much we move up or down for a given movement forward or back. It’s easy to calculate, it’s kind of obvious, so here’s what’s neat about it.
Here’s a subtler interpretation of the slope. Imagine we start from some particular point. Let me say it’s the point with x-coordinate 2, and y-coordinate 1, which I’ll imagine is on the line because I haven’t told you what line we really want and you’re willing to give me the benefit of the doubt. Imagine that we look for a new point which is awfully near this one, but which has a slightly different x-coordinate. Let’s say it has x-coordinate 2.1. To still be on the line, this new point probably doesn’t have y-coordinate 1 still. It had to change. Maybe the line is only a little bit off horizontal, and it had to change only to 1.01. Maybe the line is dropping dramatically, and it had to change to -14.75. Maybe the line drifts steadily upward, and the y-coordinate had to change to 1.15. The point is, if we want to stay on the line, and change one coordinate, we have to change the other. How big a change that has to be we could find by reading something off of an equation describing that line.
An enormous field of mathematics is based on measuring how quickly one variable has to change if we make a modest change in another, while keeping some other condition, such as that both variables have values which describe a point that’s on a particular line or curve, true. This leads us into differential equations, which are important not just because they chase out of mathematics majors everyone not already chased out by Calculus III, but also because pretty much any system in which the evolution of things depends on the current state of things, and the rate of change will vary as the current state changes, is a differential equations problem. Straight lines and their slopes are the first footsteps into this exciting forest.
(It’s also the first footsteps into a different but no less exciting forest if you refuse to go along with my imagining moving from one point to another and talking about changes in coordinate values. After all, if I’m talking about a line, I’m talking about something fixed in place and unchanging in time. It’s fair to call me on the question of what it even means to have an x-coordinate or a y-coordinate changing, or why this should measure anything of interest: my coordinates are a way of conveniently identifying points on a line, no more inherent to the line than, say, the Dewey decimal system numbers are inherent to a library’s books. I don’t expect to get anything meaningful by comparing the Dewey decimal numbers of books that happen to be physically nearby one another; why should I expect anything from comparing coordinates of points near one another? And that path leads to encounters with many fascinatingly eccentric people before satisfactory answers are fully established.)
In any case the slope describes in some way the steepness of the line. A slope near zero corresponds to a horizontal or nearly horizontal line. The larger a positive number the slope is, the more nearly vertical and rising as we go to the right the line is. The larger a negative number the slope is, the more nearly vertical and falling as we go to the right the line is.
For any particular line that’s drawn out on the paper where we can deal with it, we can tell the slope, if all else fails, by taking rulers to the line and measuring some. If we want to talk about a slope before we’ve got the line drawn out, though, or if we’re trying to find the slope based on information about the line, we need a symbol to represent this number we don’t yet know. In English, that’s pretty near universally settled on m as the symbol.
Unlike b for the y-intercept there’s no charmingly confused but ultimately plausible explanation for why we’ve settled on this letter. As best I can tell there’s no good reason for having settled on m to mean the slope. The earliest use of m for the slope seems to date to 1757 by Vincenzo Riccati, who claims to be following the pattern of Swiss mathematician Jacob Hermann. (Riccati says Herman used n for the y-intercept, by the way.) George Salmon crops up again with m for the slope, and it seems to be pretty popular from the 1840’s onward, with just a few eccentrics holding out for other symbols like s.
A popular bit of folklore says m is actually mnemonic, following “m” for the French “monter”, meaning, “to climb”. That sounds credible, although, Hermann was from the German-speaking side of Switzerland, so it seems like “steigen” and therefore s would be the more natural association; actual French-language textbooks seem to be using a for the slope (and b for the y-intercept) in 1784 (Histoire de l’Académie royale des sciences, page 669 if you’re curious) and 1830 (Traite Elementaire D’Arithmetique, Al’Usage De L’Ecole centrale des Quatre-Nations).
As far as I’m aware, nobody’s come up with a satisfactory explanation for why (English) mathematics settled on m for the slope. Probably it could as easily have settled on n or d or s or a and we’d be asking the same question about that letter.