2) In what ways did the geometric work of Nicole of Oresme prefigure modern analytic geometry? *

 

Nicole considered three forms of qualities, which he labeled uniform, uniformly difform, and difformly difform, which is now called as constant, linear, and nonlinear. He concluded the idea of representing a distance by an area rather than a line appeared in the case when the velocity changed during a motion. This is the case of constant acceleration. In that case, the distance traversed is what it would have been had the body moved the whole time with the velocity it had at the midpoint of the time of travel. This is the case now called uniformly accelerated motion.

He concluded that a difformly difform quality can be measured by adding its parts since it was composed of uniform or uniformly difform parts. He further emphasized that if the quality was difform but not made up of uniformly difform parts, say being represented by a curve, then it is necessary to have recourse to the mutual measurement of curved figures. This means that the distance traveled is the "area under the velocity curve" in all three cases.

In modern geometry, this is developed as the integration of velocity vs time curve to find the distance, since integration of a curve gives the area enclosed under the curve, thus generalizing it to for all curves.