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Generally universities in the US, when they have records still in existence (doubtful after 100 years), will release the transcripts to the students/alumni only. You can contact Columbia at the following address to see if his records still exist or not, and if they will release them to you:
Student Service Center
205 Kent Hall
Amsterdam Ave., MC 9202
New York, NY 10027
Phone:(NNN) NNN-NNNNbr/>Office Hours: M-F, 9 a.m. -5 p.m.
The address to write to New York University is:
Mail: Office of the University Registrar
P.O. Box 910
New York, NYNNN-NN-NNNN
While there may be newspaper articles about your great grandfather opening a Chinese school in New York, they would most likely be found in the microfilm archives of libraries in New York City or at the New York State Library in Albany. http://www.nysl.nysed.gov/