Madhyamaka divided over how far its method could be pushed in debate. Bhaviveka held that a Madhyamika should argue with formal independent inferences in the manner of the logicians, which requires both parties to agree about what the terms of the argument pick out. Buddhapalita, and after him Chandrakirti, held that this concedes too much, because accepting a commonly established object smuggles back in exactly the self-standing existence the argument is meant to deny. On their view a Madhyamika should only ever draw out the consequences of what the opponent already holds.
Tibetan scholastics later named the two positions Svatantrika and Prasangika, from the Sanskrit words for independent and for consequence. Those are Tibetan doxographical labels rather than names the Indian authors used of themselves, and modern scholarship generally treats the division as a real disagreement that was given a later and tidier shape than it had at the time.
The consequences of the classification were not academic in Tibet. Tsongkhapa built the Gelug curriculum on a strong Prasangika reading and made it the standard against which other positions were measured, and the argument about whether he had read Chandrakirti correctly has run in Tibetan monastic colleges ever since. Madhyamaka also travelled east. Kumarajiva translated the core treatises into Chinese, producing the Sanlun or Three Treatise school, and while that school did not survive as an institution its vocabulary was absorbed into Tiantai and Chan thought and stayed there.