Bernhard Riemann's inaugural lecture
Ueber die Hypothesen, welche der Geometrie zu Grunde liegenpublished by Richard Dedekind, after Riemann's death, in Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, vol. 13, 1867, is available here in the following formats:
William Kingdon Clifford translated Riemann's lecture into English. His translation
On the Hypotheses which lie at the Bases of Geometrypublished in Nature, vol 8, pp, 14-17, 36, 37, is also available here in the following formats:
Another Translation of Riemann's essay into English is to be found in Volume II of A Comprehensive Introduction to Differential Geometry by Michael Spivak (2nd Edition, Publish or Perish Inc, Berkeley, 1979, vol. II, pp. 135-153). This translation of Riemann's essay is followed by a discussion of its contents.
Some of the mathematical analysis underlying Riemann's lecture on the foundations of geometry is to be found in the second part of an essay submitted to the Paris Academy in competition for a prize relating to a question on heat conduction.