P. HISTORY OF ' AATHEMATICAL - School of Mathematics

A HISTORY OF MATHEMATICAL NOTATIONS
miatio prima." Add to the Hill collection the numeral forms, or supposedly
numeral forms, gathered from other than European sources,
and the material would fill a volume very much larger than that of
Hill. We are compelled, therefore, to confine ourselves to a few of the
more important and interesting forms of our numerals.'
75. One feels the more inclined to insert here only a few tables of
numeral forms because the detailed and minute study of these forms
has thus far been somewhat barren of positive results. With all the
painstaking study which has been given to the history of our numerals
we are at the present time obliged to admit that we have not even
settled the time and place of their origin. At the beginning of the
present century the Hindu origin of our numerals was supposed to
have been established beyond reasonable doubt. But at the preseat
time several earnest students of this perplexing question have expressed
grave doubts on this point. Three investigators+. R. Kaye
in India, Carra de Vaux in France, and Nicol. Bubnov in Russiaworking
independently of one another, have denied the Hindu origh2
However, their arguments are far from conclusive, and the hypothesis
of the Hindu origin of our numerals seems to the present writer to
explain the known facts more satisfactorily than any of the substitute
hypotheses thus far advan~ed.~
The reader who desires fuller information will consult Hill's book which is
very rich in bibliographical references, or David Eugene Smith and Louis Charles
Karpinski's The Hindu-Arabic Numerals (Boston and London, 1911). See also an
article on numerals in English archives by H. Jenkinson in Anliqmries J m d ,
Vol. VI (1926), p. 263-75. The valuable original researches due to F. Woepcke
should be consulted, particularly his great "MBmoire sur la propagation dea
chiffres indiens" published in the J m d midiqu (6th series; Paris, 1863), p. 27-
79, 234-90,442-529. Reference should be made also to a few other publications of
older date, such as G. Friedlein's Zahlzeichen und das elematare Rechm der
Griechen und Romer (Erlangen, 1869), which touches questions relating to our
numerals. The reader will consult with profit the well-known histories of mathematics
by H. Hankel and by Moritz Cantor.
G. R. Kaye, "Notee on Indian Mathematics," Journal and Proceedings of the
Asialic Society of Bengal (N.S., 1907), Vol. 111, p. 475-508; "The Use of the Abacus
in Ancient India," ibid., Val: IV (1908), p. 293-97; "References to Indian Mathematics
in Certain Mediaeval Works," ibid., Vol. VII (1911), p. 801-13; "A Brief
Bibliography of Hindu Mathematics," ibid., p. 679-86; Scienlia, Vol. XXIV
(1918), p. 54; "Influence grecque dans le dBveloppement des math6matiques
hindoues," ibid., Vol. XXV (1919), p. 1-14; Carra de Vaux, "Sur l'origine des
chiffres," ibid., Vol. XXI (1917), p. 273-82; Nicol. Buhnov, Arithmetische Selbst
sliindigkeit ak eu~opciischa Kdtur (Berlin, 1914) (trans. from Russian ed.; Kiev,
1908).
F. Cajori, "The Controversy on the Origin of Our Numerals," Sciatijic
Monthly, Vol. IX (1919), p. 458-64. See also B. Datta in A m. Math. Monthly,
Vol. XXXIII, p. 449; Proceed. Benares Mdh. Soc., Vol. VII.