Evidence mounted in the late 1600s that efforts to understand quadrature and attempts to quantify instantaneous velocity could be unified in a single theory. This link was confirmed by Isaac Newton of England and Gottfried Leibniz of Germany. For this achievement, we honor them as the discoverers of calculus.

Newton links quadrature to rate of change

We may view an object's velocity as the rate of change of its distance. Not all curves describe distance, but many curves allow for tangent lines. Thus, we speak of the rate of change of a curve from this point forward, unless the situation specifically describes motion.

We finally meet Isaac Newton (England, born 1642), a thinker whose broad interests could have allowed any of several earlier introductions. He approximated π, identified undiscovered series, pioneered work in interpolation, and determined the mathematics that underlies gravity. In exercise (6.5) we will see how he overcame the difficulty of expanding expressions like (a + e)1/2 using infinite sums. For that discovery alone, Newton would have earned a place in this story. His insights about quadrature and rate of change, however, promote him to central character, as reflected in the title given to his discovery: the fundamental theorem of calculus.

Here is one side of this story: in Figure 6.1, curve ABC increases as we move left to right. Corresponding to motion along this curve is motion along the horizontal from A to D to E and so on.