In Riemann's own words in his Habilitation thesis entitled "On the Hypotheses which lie at the Bases of Geometry'' we read (in Clifford's translation into English):

From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it. The reason of this is doubtless that the general notion of multiply-extended magnitudes (in which space magnitudes are included) remained entirely unworked. I have in the first place, therefore, set myself the task of constructing the notion of a multiply extended magnitude out of general notions of magnitude.

According to Weyl, in the introduction of his seminal book "The concept of a Riemann surface'':

The theory of Riemann surfaces came into existence about the middle of the nineteenth century somewhat like Minerva: a grown-up virgin, mailed in the shining armour of analysis, topology and algebra, she sprang forth from Riemann’s Jovian head.