First I'd like to bring an example to make myself more clear. I know what the Jacobian matrix is and where, how and why it is used (some examples, at least) . But still I can't get it's geometrical interpretation. And I don't understand how and why it works.
I wonder how Jacobian matrix was discovered. What theoretical issues laid behind this discovery? What mathematical problems of the time did it solve?
Generally, is are there any books that describe the history of various mathematical discoveries, not [auto]biographies of the author, but rather chronicles of the discovery itself, but the demands of the time, the difficulties, approaches that didn't work, and eventually the discovery?