BIBLIOGRAPHY
The most complete treatment of the historical develop-
ment of number systems and elementary arithmetic pub-
lished in recent years is: K. Menninger, Zahlwort und Ziffer.
Eine Kulturgeschichte der Zahl, 2 vols., 2nd ed. (Göttingen,
1957-58); trans. P. Broneer as Number Words and Number
Symbols: A Cultural History of Numbers (Cambridge, Mass.
and London, 1969). This outstanding work contains an
extensive bibliography of primary literature. The same
subject is dealt with on a much more restricted scale in
the little book by D. Smeltzer, Man and Number (London,
1958). Good introductions are also the following: D. E.
Smith, Number Stories of Long Ago (Washington, 1919; repr.
1951); D. E. Smith and J. Ginsburg, Numbers and Numerals
(Washington, 1937); D. E. Smith and L. C. Karpinski, The
Hindu-Arabic Numerals (Boston, 1911).
T. Dantzig, Number, the Language of Science, 4th ed.
(New York and London, 1954) emphasizes the mathematical
development up to and including the Cantorian transfinite
numbers. C. J. Scriba,
The Concept of Number (Mannheim
and Zurich, 1968) was written as a text for a graduate course
offered at the Ontario College of Education; it deals with
the origins of number systems, the development of elemen-
tary arithmetic, algebra, and number theory, and includes
nineteenth-century contributions to the number concept.
Also recommended are Carl B. Boyer, A History of Math-
ematics (New York, 1968), and P. E. B. Jourdain, trans. and
ed., Contributions to the Theory of Transfinite Numbers
(Chicago and London, 1915; also reprint). The Diels refer-
ence is to H. Diels, Die Fragmente der Vorsokratiker...,
5th ed. (Berlin, 1934).
CHRISTOPH J. SCRIBA
[See also Axiomatization;
Infinity; Mathematical Rigor;
Pythagorean....]
