In his inaugural lecture in 18541， Riemann introduced the concept of an \"n-fach ausgedehnte Grosse\"-roughly something that has \"n degrees of freedom\" and which we now would call an n-dimensional manifold.
　　Examples of manifolds are all around us and arise in many applications， but formulating the ideas in a satisfying way proved to be a challenge inspiring the creation of beautiful mathematics. As a matter of fact， much of the mathematical language of the twentieth century was created with manifolds in mind.
　　Modern texts often leave readers with the feeling that they are getting the answer before they know there is a problem. Taking the historical approach to this didactic problem has several disadvantages. The pioneers were brilliant mathematicians， but still they struggled for decades getting the concepts right. We must accept that we are standing on the shoulders of giants.