|Canto 1: Creation||Chapter 3: Kṛṣṇa Is the Source of All Incarnations|
Bhaktivedanta VedaBase: Śrīmad Bhāgavatam 1.3.26
avatārā hy asańkhyeyā
hareḥ sattva-nidher dvijāḥ
sarasaḥ syuḥ sahasraśaḥ
avatārāḥ — incarnations; hi — certainly; asańkhyeyāḥ — innumerable; hareḥ — of Hari, the Lord; sattva-nidheḥ — of the ocean of goodness; dvijāḥ — the brāhmaṇas; yathā — as it is; avidāsinaḥ — inexhaustible; kulyāḥ — rivulets; sarasaḥ — of vast lakes; syuḥ — are; sahasraśaḥ — thousands of.
O brāhmaṇas, the incarnations of the Lord are innumerable, like rivulets flowing from inexhaustible sources of water.
The list of incarnations of the Personality of Godhead given herein is not complete. It is only a partial view of all the incarnations. There are many others, such as Śrī Hayagrīva, Hari, Haḿsa, Pṛśnigarbha, Vibhu, Satyasena, Vaikuṇṭha, Sārvabhauma, Viṣvaksena, Dharmasetu, Sudhāmā, Yogeśvara, Bṛhadbhānu and others of the bygone ages. Śrī Prahlāda Mahārāja said in his prayer, "My Lord, You manifest as many incarnations as there are species of life, namely the aquatics, the vegetables, the reptiles, the birds, the beasts, the men, the demigods, etc., just for the maintenance of the faithful and the annihilation of the unfaithful. You advent Yourself in this way in accordance with the necessity of the different yugas. In the Kali-yuga You have incarnated garbed as a devotee." This incarnation of the Lord in the Kali-yuga is Lord Caitanya Mahāprabhu. There are many other places, both in the Bhāgavatam and in other scriptures, in which the incarnation of the Lord as Śrī Caitanya Mahāprabhu is explicitly mentioned. In the Brahma-saḿhitā also it is said indirectly that although there are many incarnations of the Lord, such as Rāma, Nṛsiḿha, Varāha, Matsya, Kūrma and many others, the Lord Himself sometimes incarnates in person. Lord Kṛṣṇa and Lord Śrī Caitanya Mahāprabhu are not, therefore, incarnations, but the original source of all other incarnations. This will be clearly explained in the next śloka. So the Lord is the inexhaustible source for innumerable incarnations which are not always mentioned. But such incarnations are distinguished by specific extraordinary feats which are impossible to be performed by any living being. That is the general test to identify an incarnation of the Lord, directly and indirectly empowered. Some incarnations mentioned above are almost plenary portions. For instance, the Kumāras are empowered with transcendental knowledge. Śrī Nārada is empowered with devotional service. Mahārāja Pṛthu is an empowered incarnation with executive function. The Matsya incarnation is directly a plenary portion. So the innumerable incarnations of the Lord are manifested all over the universes constantly, without cessation, as water flows constantly from waterfalls.
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His Divine Grace A. C. Bhaktivedanta Swami Prabhupāda, Founder Ācārya of the International Society for Krishna Consciousness