502 
and to add results of their own research. In 
a few instances this history appears to us in- 
complete and defective. This we shall en- 
deavor to show in what follows. 
The authors very properly give much at- 
tention to the study of routes of commercial 
travel. There is every reason to believe that 
the migrations of the numerals took place 
along commercial routes. The authors con- 
sider the possibility of an early influence of 
China upon India; they speak of trade routes 
and the interchange of thought by land and 
sea, between countries bordering on the 
Mediterranean and far-off India. They even 
point out early relations of Greece with 
China. In view of these careful studies it is 
singular that practically nothing should be 
said on the intercourse which did or did not 
exist between Babylonia and India during 
the centuries immediately preceding and fol- 
lowing the beginning of the Christian era. 
They ignored a question which lies at the 
root of present-day speculations on the 
earliest traces of the principle of local value 
and the symbolism for zero. Of course, local 
value is considered by the authors in con- 
nection with the Hindu-Arabie numerals. 
Not to do so would be to examine the shell 
and ignore the kernel. Were these funda- 
mental notions wholly of Hindu origin or 
were the rudimentary ideas relating to them 
imported into India from neighboring terri- 
tory? Im the book under review this vital 
question is not adequately discussed. The 
authors are correct in stating that the pre- 
ponderance of authority has been in favor of 
the hypothesis that our numeral system, with 
its concept of local value and our symbol for 
zero has been of Hindu origin. But this con- 
clusion is coming to be recognized as unsafe. 
The change of opinion that is taking place is 
voiced by two German authors of brief his- 
tories, Tropfke and QGiinther. In 1902 
Tropfke said* 
Dass unser Positionssystem wth seinen Ziffern 
indischen Ursprunges ist, steht fest. 
1“¢Geschichte der Hlementar-Mathematik,’’ I., 
p. 10. 
SCIENCE 
[N.S. Vou. XXXV. No. 900 
In 1908 Giinther said’ 
Man kann... sich den Vorgang vielleicht so 
denken, dass Indien von Babylon her die ersten 
schwachen Andeutungen des Stellenwertes empfing, 
sie in seiner Weise um- und ausbildete und spater 
das reiche Geschenk des fertigen Positionssystemes 
den Nachkommen jener Geber zuriickerstattete. 
The evidence in favor of a possible Baby- 
lonian origin is even stronger than as stated 
by Giinther, for he was apparently unaware 
that symbols for zero had been found in 
Babylonia. These symbols are mentioned, 
but not adequately discussed by Smith and 
Karpinski, on page 51. The facts in our pos- 
session to-day are about as follows: 
1. Two early Babylonian tablets, one prob- 
ably dating from 1600 or 2300 B.c., use the 
sexagesimal system and the all-important 
principle of local value. It so happens that 
they contain no number in which there was 
occasion to use a zero. 
9. Babylonian records from the centuries 
immediately preceding the Christian era give 
a symbol for zero which was apparently “not 
used in calculation.” It consisted of two 
angular marks, one above the other, roughly 
resembling two dots, hastily written. 
3. About a.D. 180, Ptolemy in Alexandria 
used in his “ Almagest,” the Babylonian sexa- 
gesimal fractions and also the omicron O to 
represent blanks in the sexagesimal numbers. 
The symbol was not used as a regular zero. 
4. Strabo and others have described the 
trade routes by land and the trade between 
Babylonia and India, also the trade by sea. 
5. Sexagesimal fractions were used by 
Hindu astronomers. Historians do not deny 
that the Indian sexagesimal fractions were of 
Babylonian origin. 
6. The earliest form of the Indian symbol 
for zero was that of a dot, which, according to 
Biihler, was “commonly used in inscriptions 
and manuscripts in order to mark a blank.” 
This early use of the symbol resembles some- 
what the still earlier use made of symbols for 
zero by the Babylonians and by Ptolemy. 
Probably Aryabhata, in the fifth century 
2‘<Geschichte der Mathematik,’’ I., p. 17. 
