"Mighty” reflects how powerful and influential the speeches were to Douglass in a way that “great” would not.
<em>Mighty</em> means possessing power or authority. These elements are not exactly part of the definition of the word <em>great</em>, which simply means large, remarkable, or predominant. With <em>mighty</em>, there is a connotation of significance and impact. Indeed, the narrator tells us that these speeches fascinated him ("unabated interest") and influenced his thinking ("they gave tongue to interesting thoughts of my own soul").
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