In addition to Sam's helpful comments, may I add that some traces
of the "analytical leap" into three dimensions can be found in the
Bernoulli & Leibniz correspondence (e.g. 04/12/1697; 26/08/1698).
Not having the book before my eyes, I cannot quote the key-pages
of Philippe de la Hire's "Nouveaux elemens des sections coniques,
les lieux geometriques, la construction ou effection des equations"
[Paris: A. Pralard, 1679], but definitely this is a good place to
start your research. Besides, I should not forget to mention the
tracts written by Antoine Parent (1666-1716), regarded as the first
systematic studies of analytic geometry in three dimensions.
Best regards, Julio