Bharata Natyam
The name was invented in the twentieth century by the great dancer Rukmini Devi and others.  It is supposed to indicate that this style was the original style described in a book on Natya by the sage Bharata, i.e. "Bharata's Dance".  The text describes poses but in the absence of illustration, the exact poses had to be re-imagined -- the quality of this re-imagination is a tribute to the intellectual and aesthetic abilities of Rukmini Devi and her teachers, but there is no other link between Bharatanatyam now and the dances described in ancient Tamil texts.