Hermann Schwarz

Hermann Schwarz (1843–1921) was a German mathematician with a gift for finding sharp, exact inequalities and clever geometric tricks. He started out training to be a chemist before a couple of persuasive professors nudged him toward mathematics — a very good career change.

The idea that stuck

His most-used legacy is the Cauchy–Schwarz inequality, a rule that quietly powers almost every metric space and inner-product space by bounding how big a "dot product" can be. It's one of those small results that, once you have it, you reach for constantly. His name also lives on in the Schwarz lemma and the Schwarzian derivative in complex analysis.

Schwarz is also remembered for a beautiful trap. It seems obvious that if you approximate a curved surface with tinier and tinier flat triangles, their total area should close in on the true area. Schwarz built a fiendish concertina of triangles wrapped around a plain cylinder — the "Schwarz lantern" — whose area you can make blow up to infinity even as the triangles shrink to nothing. It was a splash of cold water on everyone's intuition and a warning that "obviously true" is not the same as "proved."