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SB 1.3 - Kalki, Hayagrīva, Haṁsa and Pṛśni-garbha

(22) At the conjunction of two yugas, Lord Kalki will appear as the son of Viṣṇu-yaśa. He is an āveśāvatāra. At this time the rulers of the earth will have degenerated into plunderers. This is another foretelling. Kalki will appear at the end of Kali yuga and beginning of Satya yuga. Vasudeva who previously appeared as Manu and Daśaratha will become Viṣṇu-yaśa, as the father of the Lord. Some say that Buddha and Kalki appear in every Kali yuga. '

(23) The Lord appeared as Hayagrīva in a sacrifice performed by Brahmā. He killed the demons Madhu and Kaiṭabha, who stole the Vedas from Brahmā, and the Lord again produced the Vedas from His nostrils.

(24) Haṁsa spoke bhakti yoga and knowledge concerning the Lord along with the svarūpa of the jīva to Nārada, Kumāras and others. He appeared from the water as a swan.

(25) Pṛśni-garbha or Dhruva-priya appeared before Dhruva and awarded him the Dhruva planet. Those dwelling above Dhruva-loka like Bhṛgu and those dwelling below such as the seven sages praise the planet. This avatāra is not mentioned by name in the second canto 7th chapter where a list of incarnations is given. But His activity of blessing Dhruva is mentioned in 2.7.8. and this cannot be attributed to any of the other avatāras listed there. Because Pṛśni-garbha’s actions are mentioned as taking place in the Svāyambhuva manvantara in tenth canto (10.3.32), the avatāra who gave benedictions to Dhruva should be equated with Pṛśni-garbha, according to Rūpa Gosvāmī.

The 13 avatāras: Kumāras, Nārada, Varāha, Matsya, Yajña, Nara-Nārāyaṇa, Kapila, Dattātreya, Hayagrīva, Haṁsa, Pṛśni-garbha, Ṛṣabha and Pṛthu, all appeared during Svāyambhuva manvantara. Varāha and Matsya appeared again in Cākṣuṣa manvantara, although Matsya appears after every manvantara. The eight avatāras: Vāmana, Paraśurāma, Rāma, Vyāsa, Balarāma, Kṛṣṇa, Buddha and Kalki appear in Vaivasvata manvantara.

The 25 avatāras listed are called kalpāvatāras because they generally appear once in every kalpa (day) of Brahmā. Of course some like Varāha, Matsya and others appear more than once.