The New Hampshire Historical Society lost its chance to win a rare collection of Franklin Pierce correspondence at an auction on Tuesday. The society bid on the papers with the hope of adding to its collection of items related to Pierce, a Hillsborough, N.H., native who served as the 14th president of the United States from 1853 to 1857. But, alas, the society did not win. “No, unfortunately, we did not,’’ said Sarah Galligan, the library director for the society. “We learned a private buyer was successful in his bid.’’ The lot of papers — which included 90 autographed letters from Pierce to his secretary, Sidney Webster —was sold by Doyle, a New York-based auction house, for $62,500.