According to Ellis’s explanation, why did Hamilton and Burr duel in the first place?

In 1804, after Burr had been defeated in the New York Gubernatorial race, he learned Hamilton had allegedly made some disparaging remarks about Burr's character. The particulars are unknown because Hamilton's insults were only implied, not quoted.

A man named Dr. Cooper wrote a private letter to an acquaintance, and the letter was somehow leaked to a newspaper. The part Burr found offensive:

"Genl. Hamilton and Judge Kent have declared in substance that they looked upon Mr. Burr to be a dangerous man, and one who ought not to be trusted with the reins of Government. I could detail to you a still more despicable opinion which General Hamilton has expressed of Mr. Burr."

Burr exchanged several letters with Alexander Hamilton requesting an apology, but Hamilton refused. Burr's initial approach seems to have been relatively straightforward and non-confrontational, but Hamilton's response was taunting and Burr's anger eventually escalated to the point of challenging Hamilton to a duel to defend his honor.