The Ghost of the Infinitesimal

"The Method of Exhaustion, as a precursor to calculus, was used to deal with smooth curved figures, including circles, parabolas, and so on."

"But Archimedes of Ancient Greece only applied this idea of infinite division and recombination to stationary objects."

"A circle was infinitely divided into lines with no area, then those infinitely many lines were rearranged into a rectangle to calculate its area."

"A sphere was infinitely divided into slices with no volume, then those infinitely many slices were rearranged to calculate the sphere's volume."

"Only by combining the Method of Exhaustion with Zeno's paradoxes concerning rest and motion could this idea of infinite division reveal its true power."

"Let us return to the question we first considered: crossing a street. The common-sense notion that a moving point forms a line, a moving line forms a surface, and a moving surface forms a solid."

"Clearly, this process of infinitely dividing and recombining stationary objects was the same as the continuous motion of an object. The path an object traced as time passed was like adding another spatial dimension."

"The greatest difference between calculus and the original Method of Exhaustion was that this new tool was used to describe the motion of all things in the world."

"This is the world we will be heading to next: the beginning of modern physics, the age when all things moved from stillness into motion, and also the era of the Second Mathematical Crisis."

Li Heng closed the old book on the shelf that recorded the Method of Exhaustion and Pi.

"Of course, before we go to that world, we first need to defeat Pythagoras."

The pinkish-white key on Achilles's chest, modified into the shape of a spiral drill bit, slowly floated upward. Its spiraling surface was covered in dense strings of numbers.