Śaṅkara, the most famous teacher of Advaita Vedānta, held that ignorance, which makes one reality look like many things, has no beginning, and that searching for its origin was itself a work of ignorance. His successors could not leave it there. If Brahman, pure awareness, is all that finally exists, then ignorance must belong to something, and the answer divided later Advaita into two schools. One of them carries the name of a book by Vācaspati Miśra, a Brahmin scholar of Mithilā, in what is now northern Bihar: the Bhāmatī, his sub-commentary on Śaṅkara's commentary on the Brahma Sūtras.
Its answer is that ignorance belongs to the individual self, the jīva, and that there are as many ignorances as there are living beings. Brahman is never ignorant. It is what ignorance is about, and it governs ignorance as the Lord, īśvara: the individual is Brahman bounded by ignorance, and God is Brahman beyond it. The rival school, which grew from Prakāśātman's Vivaraṇa, a commentary on a work by Śaṅkara's pupil Padmapāda, placed ignorance in Brahman itself. Vācaspati also gave meditation the leading part on the way to liberation. On his account the seeker gives up ritual action and meditates without break on the meaning of the Upanishad's sentence "that you are", with study and reflection as aids.
Much of this he drew from Maṇḍana Miśra, the eighth-century Advaita author who until the tenth century was better known than Śaṅkara. Vācaspati wrote commentaries on two of Maṇḍana's books: the Brahmatattvasamīkṣā, now lost, on the Brahmasiddhi, and the Nyāyakaṇikā on the Vidhiviveka, a treatise on Vedic commands. The Bhāmatī sends its readers to both, and several of the positions that define the Bhāmatī school go back to the Brahmasiddhi. He harmonised Śaṅkara's thought with Maṇḍana's. On meaning, though, he parted from his model. Maṇḍana held the theory of the sphoṭa, Bhartṛhari's doctrine that a sentence is grasped as one indivisible whole; in his Tattvabindu Vācaspati weighed it against four rival theories, and in other works he follows the view of the Bhāṭṭa school of Mīmāṃsā, on which word meanings are grasped first and then joined.
He wrote for more than one school. The closing verses of the Bhāmatī list what he had written before: the three books just named, then works on Nyāya logic, on Sāṃkhya and on Yoga, and finally the Bhāmatī itself on Vedānta. His Tattvakaumudī, on the classic summary of Sāṃkhya, survives in more than ninety medieval manuscripts. His Tattvavaiśāradī explains the Yoga Sūtras. His long commentary on Uddyotakara's defence of Nyāya, the Tātparyaṭīkā, answers the Buddhist logicians, Dharmakīrti among them. Later scholars called him sarva-tantra-sva-tantra, one for whom every system is his own, and modern scholars still argue over whether he wrote each of these books as a partisan of its school or as one thinker moving between them.
His life has left few traces. The Nyāyasūcīnibandha, a work ascribed to him, gives its year as 898 of an era it does not name: 841 CE if the era is the Vikrama, 976 CE if it is the Śaka. Historians split accordingly between the ninth century and the tenth, and Yasutaka Muroya, a scholar of Nyāya, has questioned whether Vācaspati wrote the book at all. The Bhāmatī names only the king under whom it was written, Nṛga, whom no one has identified. The tradition says that the title is the name of his wife, and an Advaita tradition holds that Śaṅkara was reborn as Vācaspati to spread the doctrine through this book. The Bhāmatī gathered commentaries of its own: Amalānanda's Kalpataru around 1300, and in the sixteenth century Appayya Dīkṣita's Parimala on the Kalpataru. At Andhra Tharhi, the village in Madhubani district that claims him as its son, a statue and a library now stand on the site called Vachaspati Mishra Dih, and in 2019 the government of Bihar announced a state festival in his memory.