use of appendices at the close of certain chapters. Within the limits of a preface I may not enumerate all the libraries where I have been permitted to work or which have generously sent books—sometimes rare volumes—to Cleveland for my use, or all the librarians who have personally assisted my researches or courteously and carefully answered my written inquiries, or the other scholars who have aided or encouraged the preparation of this work, but I hope they may feel that their kindness has not been in vain. In library matters I have perhaps most frequently imposed upon the good nature of Mr. Frederic C. Erb of the Columbia University Library, Mr. Gordon W. Thayer, in charge of the John G. White collection in the Cleveland Public Library, and Mr. George F. Strong, librarian of Adelbert College, Western Reserve University; and I cannot forbear to mention the interest shown in my work by Dr. R. L. Poole at the Bodleian. For letters facilitating my studies abroad before the war or application for a passport immediately after the war I am indebted to the Hon. Philander C. Knox, then Secretary of State, to Frederick P. Keppel, then Assistant Secretary of War, to Drs. J. Franklin Jameson and Charles F. Thwing, and to Professors Henry E. Bourne and Henry Crew. Professors C. H. Haskins,[2] L. C. Karpinski, W. G. Leutner, W. A. Locy, D. B. Macdonald, L. J. Paetow, S. B. Platner, E. C. Richardson, James Harvey Robinson, David Eugene Smith, D’Arcy W. Thompson, A. H. Thorndike, E. L. Thorndike, T. Wingate Todd, and Hutton Webster, and Drs. Charles Singer and Se Boyar have kindly read various chapters in manuscript or proof and offered helpful suggestions. The burden of proof-reading has been generously shared with me by Professors B. P. Bourland, C. D. Lamberton, and Walter Libby, and especially by Professor Harold North Fowler who has corrected proof for practically the entire work. After receiving such expert aid and sound counsel I must assume all the deeper guilt for such faults and indiscretions as the book may display. ABBREVIATIONS Abhandl. Abhandlungen zur Geschichte der Mathematischen Wissenschaften, begründet von M. Cantor, Teubner, Leipzig. Addit. Additional Manuscripts in the British Museum. Amplon. Manuscript collection of Amplonius Ratinck at Erfurt. AN Ante-Nicene Fathers, American Reprint of the Edinburgh edition, in 9 vols., 1913. AS Acta sanctorum. Beiträge Beiträge zur Geschichte der Philosophie des Mittelalters, ed. by C. Baeumker, G. v. Hertling, M. Baumgartner, et al., Münster, 1891-. BL Bodleian Library, Oxford. BM British Museum, London. BN Bibliothèque Nationale, Paris. Borgnet Augustus Borgnet, ed. B. Alberti Magni Opera omnia, Paris, 1890-1899, in 38 vols.