Shaped like a silver sphere, it was used for stabilising discontinuous elements of spacetime, rationalising any region that was running in conflict with the established laws of the spiral politic. (PROSE: House Meddhoran Glossary) In other words, it searched for parts of the universe that had become unstable in some way, and it rebuilt them along a more rational line.
The first amaranth was created by maintenance engineers on Gallifrey to help counteract the warping effects caused by the black hole on the space time continuum. It contained an emergency function to transport its user to a more stable part of space. The Doctor once possessed an amaranth. (PROSE: Christmas on a Rational Planet)