It is a number like no other. It is smaller than anything except zero, but It's not zero. It makes no logical sense, but it has endless uses. It's the infinitesimal, and it's back. Infinitesimally speaking, a circle is actually a polygon with infinitely many infinitesimal sides. A solid is in fact an infinite sandwich of infinitesimally thin slices. And velocity is an infinitesimal distance divided by an infinitesimal time. The whole world is made of these next-to-nothings. Yet 19th-century purists found these tiny slices of nothing just too much to swallow, and they were banished from mathematics for more than a century. Only recently has the concept been restored to respectability, and it's back with a vengeance. Infinitesimals are now giving us insights in physics and simplifying our understanding of key areas of pure mathematics. But how on earth can such a peculiar notion make sense? The idea of infinitesimals is an ancient one. Archimedes used them to calculate the volume of a sphere. He imagined cutting the sphere into infinitely many slices and hanging them on one arm of a balance. He rearranged them so that they exactly balanced a cone hanging on the other arm. Knowing how levers work, he related the volume of the sphere to that of the cone, and out popped his formula. It's a bizarre idea: if the slices are infinitely thin, how can they have any weight or volume? How can you add up a lot of zeros to get something substantial? Yet somehow it works.
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