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  Riemann first presented to the world his new idea in his doctoral dissertation of 1851, and elaborated its implications in his 1854 habilitation lecture, his 1857 treatises on Abelian and hypergeometric functions, and his posthumously published philosophical fragments. 
  Following Gauss, Riemann recognized that in the type of least action physical manifold exemplified by the catenoid or Gauss's potential surfaces, the curves of maximum and minimum curvature are harmonically related, which means that their mutual curvatures change at the same rate, in perpendicular directions. 
  Riemann showed, than Abel's extended class of higher transcendental functions, when expressed on Riemann's surface, express a type of transformation that increases the rate and the density at which singularities can be added. 
 www.wlym.com /antidummies/part60.html (3651 words) 
