In ancient Greek times heroism was much different than it is now. Today, all you have to do to become a hero most of the time is rich, after you do this the media will take it from there. But in the time of the Trojans it took much more, thing's that Achilles thankfully, was very good at or things he thought was important. Achilles was not only a hero physically, but was more importantly, a hero for the Achian army's morale, also his chivalric properties were important with his being a hero.
Physically Achilles is superior to anyone that I have read of so far in the Iliad, other than Zeus and a few other gods. Even the best of the Trojan warriors and fighters cannot compare to Achilles' fighting skill or his strength.
Morale is something that the Achians are truly suffering from in the end of book six and into book nine. Something that the Achians need and Achilles provides when he is with the A.chians is a sort of a "father figure" if you would, a figure to look up to and to follow. Another way Achilles aggrandized the Achians morale was him just being on their side and not on the Trojan team.
Chivalry was a trait that Achilles saw to be very important, to strive hard to perform well in. He showed a few instances were he could have conducted himself differently but made a gallant decision. This includes when Agamemnon took away Briesies, Achilles' war prize. Achilles could have become very angry and could have killed him very easily but he refrained, he spared the Achians leader and left so no shame could soil his name.
Compared to most hero's in ancient Greece Achilles was a monster of a hero, for he showed multiple ways a hero can be the best hero, he also showed exactly how good a hero can be through his physical properties, his influence he had on the Achians army morale, and his desire to be the best warrior and hero he could be through his chivalrous acts.
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