1. As a noun:
* "HESS-ee-uhn": This pronunciation is most common when referring to the Hessian soldiers, a group of German soldiers who fought for the British during the American Revolution.
2. As an adjective:
* "HESS-ee-uhn": This pronunciation is also used when referring to the state of Hesse in Germany, or anything related to it, like Hessian history or Hessian culture.
* "HESS-ee-an": This pronunciation is sometimes used for the adjective form, but it's less common.
In mathematics, when referring to the Hessian matrix (a matrix of second-order partial derivatives), it is usually pronounced as "HESS-ee-uhn".
Ultimately, the best way to pronounce "Hessian" is to consider the context and follow the pronunciation that is most common in that specific context.