MarcH 29, 1912] 
A.D., knew our zero. The earliest undoubted 
occurrence of our zero in India is in 876 A.D. 
Were there overflows of Babylonian science 
into Greece and India? The question is per- 
tinent. The possibility of overflows into 
India has been recognized not only by 
Giinther, but by Nallino, who states that the 
Chaldeans of 100 B.c. (and even earlier) 
knew the sidereal year (estimated at 365 d. 
6 hr. 13 m. 48 sec.) and that this knowledge 
‘ probably passed from them to the Hindus 
and Persians. This statement is quoted by 
H. Suter with apparent approval.’ It seems 
to us that these facts point directly toward 
a summary of the case, somewhat as quoted 
above from Giinther. 
In recounting the earliest uses made of the 
Arabic numerals in Egypt and the Occident, 
reference is made to a trace on a pillar of a 
church in Egypt, giving the date 349 a.H. 
(=a.p. 961). Strange to say our authors 
completely ignore similar evidence, as given 
in the Philosophical Transactions of London. 
Why should this be? No less prominent a 
mathematician than John Wallis arrived at 
the conclusion that “their use in these parts 
was as old at least as . . . the middle of the 
eleventh century.”* Wallis refers to the 
“Mantle-tree of the Parlour Chimney at the 
dwelling House of Mr. Will. Richards, the 
Rector of Helmdon in Northampton-shire,” 
bearing an inscription with the date “ A° Do! 
M° 133” (=ap 1133). Thomas Luffkin’ 
names a building in Colchester bearing the 
date 1090. These dates are of interest to an 
Englishman or an American. If they are to 
be rejected, it would seem that the reasons 
therefor should be set forth in a publication 
aiming to weigh minutely all respectable evi- 
dence. Smith and Karpinski make no men- 
tion of an earlier use than 1539 of the nu- 
merals in Great Britain. 
In describing the different shapes of the 
*<¢Bibliotheca Mathematica,’’ Vol. 5,, 1904, 
p. 85. 
‘Philosophical Transactions, No. 154, p. 399 
(= Abridgment, Vol. I., 1705, p. 107). 
* Philosophical Trans. Abridged, Vol. I., 1705, 
p. 108. 
SCIENCE 
503 
zero in Europe the authors overlooked the 
curious use of theta @ to represent zero, 
found in the writings of Michel Rolle, and 
Enestrém’s note on this notation in “ L’Inter- 
médiaire des mathématiciens,” II., 1895, p. 
283. 
It has been said of early American geolo- 
gists that they crossed the western plains, 
eager to reach the Rocky Mountains, there to 
grapple with the problems relating to the 
geology of our land, that in so doing they neg- 
lected the geologic problems presented by 
the plains themselves. In the same way our 
authors hastened back half a thousand years 
to reach the conspicuously formative periods, 
and in so doing they forgot to take note of 
matters of interest connected with recent 
time, which for historical research is far from 
barren. Our authors note the different shapes 
which the numerals assumed among the 
Hindus, Arabs and Europeans of the Renais- 
sance. But if I mistake not, interesting 
forms, worthy of study, are found in seven- 
teenth and eighteenth century manuscripts, 
stored away in American libraries. It is of 
some interest that the figure 8 sometimes had 
the shape of our dollar mark written with a 
single downward stroke, thus, $. I remember 
mistaking such an eight for a dollar mark 
and recognizing my error only when the sum 
given in the manuscript would not come out 
as represented, except on the supposition that 
the mark stood for 8. The authors point out 
anomalous combinations of the Hindu-Arabic 
and the Roman numeral symbols which oc- 
curred more or less accidentally in the fif- 
teenth and sixteenth centuries, but they fail 
to notice a curious combination which occurs 
with surprising regularity in Spanish-Ameri- 
can manuscripts during the three centuries 
preceding the beginning of the nineteenth cen- 
tury. Of this notation we shall speak in a 
separate article. 
On page 28, line 10, the word “ vertical ” 
should be replaced by the word “ horizontal.” 
In the table of contents there is an omission 
such that the pronoun “their” in the line 
“arly ideas of their origin” refers to 
