A history of the tab by Ed Tenner in the MIT Technology Review. Apparently, it was the idea of James Newton Gunn who worked with Melvil Dewey.
The modern tab was an improvement on a momentous 19th-century innovation, the index card. Libraries had previously listed their books in bound ledgers. During the French Revolution, authorities divided the nationalized collections of monasteries and aristocrats among public institutions, using the backs of playing cards to record data about each volume. [Keeping Tabs]
