How did buddha-nature thought develop?

The theory of buddha-nature (tathāgatagarbha, for the most part)--that all sentient beings somehow possess the innate buddhahood or potential to become buddhas and (sometimes by extension) that therefore all of reality shares in the same fundamental essence--seems to have appeared in India in as early as the Second Century CE before spreading to China and Tibet and beyond. Although it is interpreted in different ways, buddha-nature has become a central element of Buddhist doctrine in all Mahāyāna traditions. Since the early days of the religion Buddhists have been disagreeing about whether the true nature of reality can be described in positive terms or must always be couched in negations--Madhyamaka and Yogācāra are only the most famous examples of these two poles; Tibetans debate over "self-emptiness" or "other-emptiness," and the Chinese argue about Original Enlightenment and the Gradual Path, to name only a few such instances of the cataphatic/apophatic dialectic. Over the centuries buddha-nature has been interpreted as a definitive teaching, as a provisional teaching, or as not Buddhist (most recently in the form of Critical Buddhism in Japan). Scholars mostly now agree that buddha-nature theory developed alongside--rather than part of--the two main Indian Mayāyāna Buddhist doctrinal schools, and great thinkers of each have both embraced it and rejected it. This essay will briefly survey the historical development of buddha-nature, taking a Western critical approach which assumes that ideas and scriptures developed over time in conversation with other ideas and historical events.