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The Mellon Institute building was dedicated on May 6, 1937 to Andrew William Mellon (1855-1937) and Richard Beatty Mellon (1858-1933). It was built to house the Mellon Institute of Industrial Research, a nonprofit, independent research center. Robert Kennedy Duncan (1868-1914) first conceived of its principles of industrial research in 1906 while a Professor of Industrial Chemistry at the University of Kansas. In 1910 he moved and became the Director of Industrial Research at the University of Pittsburgh. In 1913, up until his death in 1914, he became the first director of the Mellon Institute of Industrial Research (which he had inspired).
Architects Benno Janssen and William York Cocken oversaw construction of the Mellon Institute building from 1931 to 1937. Janssen designed many buildings in the Pittsburgh area in the early 1900's, including the Pittsburgh Athletic Association (1911) and William Penn Hotel (1914-16 and 1927-1928).
Currently, the Mellon Institute building houses the Mellon College of Science (including the departments of Biology and Chemistry) and the Center for the Neural Basis of Cognition.
A brief history of the Mellon Institute building can also be found here at the Pittsburgh Supercomputing Center web page.
You can view a map of this location and get directions by following this link.