Both Siddhartha and Gilgamesh believe in themselves, they do not let others define them or make decisions for them. Siddhartha demonstrates that he has strong will from the very beginning of the novel. He is taught by the Samana even though the teachings he received up to this point in his life say that the Samanas way is the wrong religion. It is not fitting for a Brahmin to speak angry and violent words/But indignation moves my heart/I do not wish to hear that request a second time from your lips(Hesse, 9). Siddhartha wants to make his own decision to study with the Samana, therefore, he needed to go against his fathers judgment and demand leave to travel with the Samana. This clearly shows Siddharthas strong will by wanting to study with the Samana strongly enough to disobey his father. Siddhartha also demonstrates determination when he does not fall prey to the temptations of Sublime ones teachings. Siddhartha believes you do not need a teacher or scriptures to be taught how to achieve Nirvana. He abandons the Buddha and also his friend with hope of finding the way to Nirvana on his own.
Gilgamesh on the other hand is the ruler of Uruk. In order to be able to keep his people alive he needs to be level headed and strong willed. Gilgamesh is so strong willed he seems arrogant, he believes he