The line, "In all things that are purely social we can be as separate as the five fingers, yet one as the hand in all things essential to mutual progress," came from an 1895 speech which catapulted whom to national fame?

Answer: Booker T. Washington.

Explanation:

The Cotton States and International Exposition Speech (September 18, 1895) had Booker T. Washington speaking about race relationships.

His hand metaphor became the framework for the Atlanta compromise, an arrangement that stated that Southern African-Americans would obey the white rule, as long as Southern whites provide them education and due process of law.