The slope formula was likely discovered by Rene Descartes around the turn of the seventeenth century. Descartes is widely considered to be the father of analytical geometry, and is the inventor of the Cartesian grid, which is the most commonly used grid in mathematics today.
Though no concrete proof designates Descartes as the inventor of the slope formula, there is evidence to support that claim. Descartes introduced the world to both linear problems and methods for solving them. Since the solutions to linear problems center on the slope of a given line at a certain point, it follows that Descartes discovered that formula, as well.