Why did Bhishma fight with Lord Parashurama?
Bhishma recounts the story in the Udyoga Parva of the Mahabharata, but first some background. Bhishma was the son of Shantanu, king of Hastinapura, and the goddess Ganga. But Ganga abandoned Shantanu, and Shantanu later fell in love with a woman named Satyavati. But Satyavati's (adopted) father Matsyagandha, the chief of a fisherman tribe, wouldn't consent to the marriage, because he was afraid that Bhishma and his descendants would rule Hastinapura rather than Satyavati's children. So Bhishma made a vow that he would never become king, and that he would remain celibate so that his descendants would never claim the throne. (It's because of that terrible vow that he got the name "Bhishma" or terrible.) Then Matsyagandha agreed to the marriage, and in due course of time, Shantanu and Satyavati had two sons, Vichitravirya and Chitrangadaa. When Shantanu died Chitrangada became king, but he soon died after a confrontation with a Gandharva because the two of them had the same name (yes, really!).
So the throne now fell to Vichitravirya, who had no wives, but Bhishma heard that the King of Kashi was holding a Swayamvara for his daughters Amba, Ambika, and Ambilika. So Bhishma went to Kashi to win those princesses for Vichitravirya. He kidnapped the three princesses, fighting all the kings of the world when they tried to stop him, and brought them back to Hastinapura. But then the eldest princess Amba informed Bhishma that she was in love with a king named Shalva, so Bhishma allowed her to go back to him. But Shalva wasn't willing to take her back, because she had been abducted by another man.
Amba was furious that Bhisma ruined her life, so to take revenge she went to Parashurama and asked for his help in killing Bhishma. Parashurama was reluctant to do so, both because Bhishma was his former student and because he had made a vow that he wouldn't take up arms anymore. But the sage Akritavana told Parashurama that it's not proper to abandon someone whom you have promised to help, and then reminded him that he had also made an additional vow about taking up arms under certain circumstance:
This also was, O great Muni, the vow then made by thee, O Rama ... that thou wouldst slay in battle the person, be he a Brahmana, a Kshatriya, a Vaisya, or a Sudra, who would be a foe to the Brahmanas. Thou hadst further promised that as long as thou wouldst live thou wouldst not abandon those that would come to thee in fright and seek thy protection, and that thou wouldst ... slay that proud warrior who would vanquish in battle all the assembled Kshatriyas of the earth!
And since Bhishma, in the process of kidnapping the three princesses, had defeated all the kings of the world in battle, Parashurama was allowed to fight him. And so he said "Taking this maiden with me, I will repair myself to the place where Bhishma is. If Bhishma, proud of his achievements in battle, do not obey my behest, I will then slay that arrogant wight. Even this is my fixed resolve."
So Parashurama went to Kurukshetra, the future site of the Mahabharata war, and asked Bhishma to come there. Bhishma came immediately, eager to see his former guru. Then Parashurama asked Bhishma to marry Amba, since no one else would marry her after having been abducted. But Bhishma said that he couldn't marry her himself because of his vow of celibacy, and he couldn't give her to his brother Vichitravirya because he had already promised to let her go to Shalva:
O Brahmana, I cannot, by any means, bestow this girl on my brother. O thou of Bhrigu's race, it was to myself that she said, I am Salwa's! And it was by me that she was permitted to go to Salwa's city. As regards myself, even this is my firm vow that I cannot abandon Kshatriya practices from fear or pity, or avarice of wealth, or lust!
So Parashurama was left with no choice but to try to kill him. That is how guru and shishya were forced to fight. Now Bhishma was was wearing armor and sitting in a chariot, whereas Parashurama was just on foot, so Bhishma offered him weapons and a chariot as well, but Parashurama said that he had no need for such things, being an incarnation of Vishnu:
The Earth, O Bhishma, is my car, and Vedas, like good steeds, are the animals that carry me! The wind is my car-driver, and my coat of mail is constituted by those mothers in the Vedas (viz., Gayatri, Savitri and Saraswati). Well-covered by these in battle, O son of Kuru's race, I will fight!
And so the first great battle fought in Kurukshetra commenced. The more famous Kurukshetra war between the Pandavas and Kauravas took 18 days, but the war between these two heroes took 24 days. By the 23rd night, Bhishma had tried all the maneuvers and weapons he could think of, but none of them were able to defeat Parashurama, so he prayed that the gods would suggest a path to victory while he slept. (Like the Mahabharata war, this Kurukshetra battle was also fought between sunrise and sunset every day.) And sure enough, in a dream the gods told him that they would give him the knowledge of the Prashwapastra, a weapon formerly wielded by the god Dyaus Pita (because Bhishma is an incarnation of Dyaus Pita):
Rise, O Ganga's son, thou needst have no fear! We will protect thee, for thou art our own body! Rama, the son of Jamadagni, will never be able to vanquish thee in battle! Thou, O bull of Bharata's race, wilt be the conqueror of Rama in combat! This beloved weapon, O Bharata, called Praswapa, appertaining to the lord of all creatures, and forged by the divine artificer, will come to thy knowledge, for it was known to thee in thy former life! Neither Rama, nor any person on earth is acquainted with it. Recollect it, therefore, O thou of mighty arms, and apply it with strength! O king of kings, O sinless one, it will come to thee of itself! ... ! Afflicted by the force of this thy weapon, the son of Jamadagni, will fall asleep!
The next day, Bhishma suddenly gained the knowledge of the Pashwapastra, but when he tried to use it, Narada and other gods came and stopped him. Here is Bhishma's account of it:
When I had aimed it, Narada addressed me, saying, "Yonder, O Kauravya, stay the gods in the sky! Even they are forbidding thee today! Do not aim the Praswapa weapon! Rama is an ascetic possessed of Brahma merit, and he is, again, thy preceptor! Never, Kauravya, humiliate him." While Narada was telling me this, I beheld those eight utterers of Brahma stationed in the sky. Smilingly, O king, they said unto me slowly, "O chief of the Bharatas, do even what Narada sayeth. Even that, O best of Bharata's race, is highly beneficial to the world!"
And shortly thereafter Parashurama was approached by the spirits of his ancestors, who similarly told him to stop fighting, because Parashurama wasn't supposed to act like a Kshatriya anymore, and because Bhishma was destined to die at the hands of another:
O sire, never display such 'rashness again, the rashness, viz., of engaging in battle with Bhishma, or especially with any Kshatriya, O descendant of Bhrigu's race, to fight is the duty of a Kshatriya! Study (of the Vedas) and practice of vows are the highest wealth of Brahmanas! For some reason, before 'this, thou hadst been ordered by us to take up weapons.... Let this battle with Bhishma be thy very last, for enough of it thou hadst already.... Bhishma is one of the foremost of Vasus! ... -how can he be defeated by thee? Desist, therefore, O Bhargava! That foremost of the Pandavas, Arjuna, the mighty son of Indra, hath been ordained by the Self-create to be the slayer of Bhishma!
And so after some contretemps, Parashurama agreed to put down his weapons and surrender. He told Amba that he couldn't defeat Bhishma and that Bhishma was the one she should seek refuge in. But Amba wasn't interested in anything but revenge at this point and she said this:
As regards myself, I will not go a second time to Bhishma. I will, however, O perpetuator of Bhrigu's race, go thither, O thou endued with wealth of asceticism, where I may (obtain the means to) myself slay Bhishma in battle!
So she engaged in Tapasya (deep meditation) until Shiva appeared before her, and she asked that she be reborn as a man so that she could kill Bhishma. Shiva granted her the boon, and Amba immediately jumped into a fire. She was reborn as Draupadi's brother Shikhandi, who was instrumental in Bhishma's death, because Bhishma was unwilling to fight someone who had been a woman in a past life.
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