When Newton began to muse on the problem of the motion of the planets and what kept them in their orbits around the sun, he realized that the mathematics of the day werent sufficient to the task. Properties such as direction and speed, by their very nature, were in a continuous state of flux, constantly changing with time and exhibiting varying rates of change. So he invented a new branch of mathematics, which he called the fluxions (later known as calculus). Calculus allowed him to draw tangents to curves, determine the lengths of curves, and solve other problems that classical geometry could not help him solve. Interestingly, Newtons masterwork, the Principia, doesnt include the calculus in the form that hed invented years before, simply because he hadnt yet published anything about it. But he did combine related methods with a very high level of classical geometry, making no attempt to simplify it for his readers. The reason was, he said, to avoid being baited by little Smatterers in Mathematicks.