PREFACE. XI If F(m)=S/(d), summed for all the divisors d of m, we can express /(m) in terms of F by an inversion formula given in Chapter XIX along with generalizations and related formulas. Bougaief called F(m) the numerical integral of/(m). The final Chapter XX gives many elementary results involving the digits of numbers mainly when written to the base 10. Since the history of each main topic is given separately, it has been possible without causing confusion to include reports on minor papers and isolated problems for the sake of completeness. In the cases of books and journals not usually accessible, the reports are quite full with some indication of the proofs. In other cases, proofs are given only when necessary to differentiate the paper from others deriving the same result. The references were selected mainly from the Subject Index of the Royal Society of London Catalogue of Scientific Papers, volume 1,1908 (with which also the proof-sheets were checked), and the supplementary annual volumes forming the International Catalogue of Scientific Literature, Jahrbuch liber die Fortschritte der Mathematik, Revue semestrielle des publications mathe*matiques, PoggendorfFs Handworterbuch, KlugePs Mathematische Worterbuch, Wolffing's Mathematischer Bucherschatz (a list of mathematical books and pamphlets of the nineteenth century), historical journals, such as Bulletino di bibliografia e di storia delle scienze matematiche e fisiche, Bolletino...., Bibliotheca Mathematica, Abhandlungen zur Geschichte der mathematischen Wissenschaften, various histories and encyclopedias, including the Enclyclope*die des sciences mathe*matiques. Further, the author made a direct examination at the stacks of books and old journals in the libraries of Chicago, California, and Cambridge Universities, and Trinity College, Cambridge, and the excellent John Crerar Library at Chicago. He made use of G. A. Plimpton's remarkable collection, in New York, of rare books and manuscripts. In 1912 the author made an extended investigation in the libraries of the British Museum, Kensington Museum, Royal Society, Cambridge Philosophical Society, BibliothSque Nationale, University de Paris, St. Genevieve, Tlnstitut de France, University of Gottingen, and the Konigliche Bibliothek of Berlin (where there is a separate index of the material on the theory of numbers). Many books have since been borrowed from various libraries; the Ladies' and other Diaries were loaned by R. C. Archibald. At the end of the volume is a separate index of authors for each of the twenty chapters, which will facilitate the tracing of the relation of a paper to kindred papers and hence will be of special service in the case of papers inaccessible to the reader. The concluding volume will have a combined index of authors from which will be omitted minor citations found in the chapter indices.