JANUARY 5, 1912] 
Definitions: 
Genotype, the fundamental hereditary con- 
stitution or combination of genes of an or- 
ganism. 
Biotype, a group of individuals possessing 
the same genotype. 
Pure line, a group of individuals traceable 
through solely self-fertilized lines to a single 
homozygous ancestor. 
Clone, a group of individuals of like geno- 
typic constitution, traceable through asexual 
reproductions to a single ancestral zygote, or 
else perpetually asexual. 
Gro. H. SHULL 
HISTORY OF MATHEMATICS IN THE RECENT EDI- 
TION OF THE ENCYCLOPADIA BRITANNICA 
THE new edition of the Encyclopedia Brit- 
annica contains numerous articles which pur- 
port to deal with the history of various 
branches of mathematics. None of these have 
been written by specialists in this field and 
the articles bear abundant evidence of this 
fact. The history of mathematics may well 
ask of the editors of such an encyclopedia the 
same care in the selection of writers on these 
topics as that exercised in the Selection of 
writers in other fields, ably represented in 
general in the Britannica by the leading schol- 
ars of the world. 
In a recent issue of SctENcE (December 1, 
1911) Professor G. A. Miller has called atten- 
tion to certain inaccuracies and errors, espe- 
cially with reference to the theory of numbers 
and of groups. It seems to me unfortunate, 
in view of the general worthlessness of the 
historical passages, that Professor Miller has 
incidentally chosen for criticism one of the 
few correct statements. The passage in ques- 
tion occurs on page 867 in volume XIX., in 
the article on “ Numerals” in which the 
writer states that our present decimal system 
is of Indian origin. Attention is rightly 
called by Professor Miller to the fact that the 
zero appeared in Babylon long before it ap- 
peared in India, although the writer on “ Nu- 
merals” seems to be unaware of this. How- 
ever, the date is not 1700 B.c., as Professor 
SCIENCE 29) 
Miller states, but more than a thousand years 
later. Photographic reproduction of Baby- 
lonian tablets containing the zero appear in 
F. X. Kugler’s “ Die babylonische Mond-rech- 
nung,” Freiburg i. Br., 1900, and these tablets 
date from the centuries just before the Chris- 
tian era. Furthermore, no historian of math- 
ematics has made the claim that modern 
arithmetic is derived from the Babylonian 
arithmetic, as Professor Miller implies, but 
there is general agreement that our arithmetic 
comes to us from the Hindus through the 
Arabie writer (ec. 825 s.p.) Mohammed ben 
Musa Al-Khowarizmi. This subject is fully 
discussed in “ The Hindu-Arabic Numerals,” 
Smith and Karpinski, Boston, 1911. 
The article on “The History of Mathe- 
matics,” Vol. XVII., pp. 882-883, is too brief 
to invite comment. The incorrect statement 
is made: “The medieval Arabians invented 
our system of numeration.” Reference is 
given only to the works of Cantor (“1st Bd.,” 
“9d Bd.” and “3d Bd.”!) and to W. W. R. 
Ball’s “A Short History of Mathematics,” 
London, 1888, and subsequent editions. The 
latter work is in no sense an authority on the 
subject. 
The articles on “ Algebra, History,” Vol. L., 
pp. 616-620, and “Geometry, History,” Vol. 
XI., pp. 675-677, contain so many inaccura- 
cies and so much misinformation that selec- 
tion becomes difficult. I will devote myself 
more particularly to the longer article on the 
history of algebra. 
Some ridiculous statements made by Peter 
Ramus in his algebra of 1560 are quoted. 
Thus Ramus says: “There was a certain 
learned mathematician who sent his algebra, 
written in the Syriac language, to Alexander 
the Great, and he named it almucabala, that 
is, the book of dark or mysterious things, 
which others would rather call the doctrine of 
algebra ... and by the Indians... it is called 
aljabra and alboret.” This nonsense, evident 
on its face, as almucabala and aljabra are 
Arabie words, is taken somewhat seriously by 
this writer in the Britannica. “The uncer- 
tain authority,” he says, “ of these statements, 
and the plausibility of the preceding explana- 
