Eric Scerri has written the definitive history of Mendele’ev’s periodic table and how it came to be formulated. He also has a paper in which he proposes a new formulation, based on historical considerations of what it was that Mendele’ev was trying to do, and on more theoretical considerations of atomic number, not weight. Sciencebase discusses it here.
And of course, no post on the periodic table is complete without a link to Tom Lehrer’s “Elements” song. Here’s a fantastic version from YouTube:
[Hat tip: David Bradley]