PREFACE. XI 
If F(m)=2/(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 
iiber die Fortschritte der Mathematik, Revue semestrielle des publications 
math^matiques, Poggendorff's Handworterbuch, Kliigel's Mathematische 
Worterbuch, Wolffing's Mathematischer Biicherschatz (a list of mathemat- 
ical books and pamphlets of the nineteenth century), historical journals, such 
as Bulletino di bibliografia e di storia delle scienze matematiche e fisiche, 
Bolletino . . . . , BibUotheca Mathematica, Abhandlungen zur Geschichte 
der mathematischen Wissenschaften, various histories and encyclopedias, 
including the Enclyclop^die des sciences mathematiques. 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 Chi- 
cago. 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, Bibliotheque 
Nationale, Universite de Paris, St. Genevieve, I'lnstitut de France, Uni- 
versity of Gottingen, and the Konigliche Bibhothek 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. 
