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March 29, 1912] 



their nidification presents nothing that may be 
deemed peculiar or even specially character- 
istic. In their flight and manner of procuring 
their food, however, they differ strikingly from 
all other birds, in these respects closely re- 
sembling certain insects, especially the cre- 
puscular hawkmoths (Sphingidse). Their 
food, consisting mainly of small insects, but 
in part also- of the nectar of flowers, is mostly 
gleaned from blossoms, before which they 
poise, with wings so rapidly vibrating as to 
be invisible except as a dim haze or halo 
partly surrounding the body and producing 
the humming sound from which these birds 
derive their vernacular name, the bill thrust 
inside the flower and the slender, semitubular 
tongue extended into the depths of the blos- 
som. Some species, instead of feeding from 
flowers, glean their insect food from the bark 
of forest trees, following along the branches 
in suspended flight in the same manner that 
the others pass from flower to flower. In their 
feeding from flower to flower, Humming Birds, 
like bees, butterflies, and moths, perform the 
same office in the economy of nature as insects 
by transferring pollen from one bloom to an- 
other, and thus assisting in the fertilization of 
plants. In flying from one point to another, 
the flight of Humming Birds, while essentially 
direct, is usually more or less undulating, and 
so extremely rapid that the eye can scarcely 
follow. Often this flight is accompanied (at 
least in the ease of males of some species) by 
a more or less remarkable screeching or 
grating sound, produced mechanically by some 
peculiarity of wing-structure. 

" Diminutiveness of size and metallic bril- 
liancy of coloring are the chief external char- 
acteristics of Humming Birds, though excep- 
tions to both occur ; and in these respects they, 
as a group, have no rivals. Unfortunately, 
stuffed specimens convey but a faint idea of 
their splendid coloring, for the perfection of 
their changeable refulgence can be fully real- 
ized only in the living bird, whose every 
change of position flashes to view a different 
hue — emerald green replacing ruby red, sap- 
phire blue succeeding fiery orange, or either 
becoming opaque velvety black — according to 

the angle at which the sun's rays touch the 
feathers, an effect which can only partially be 
imitated with the stuffed specimen by arti- 
ficially changing its position with reference to 
the light. Many species have a spot of the 
most luminous or brilliantly metallic color 
(usually green) that it is possible to imagine 
on the forehead at the base of the bill, this 
spot being surrounded by the most intense 
velvety black — evidently to enhance the bril- 
liancy of the ornament by contrast, just as a 
jeweler would, for the same purpose, display 
a diamond or other gem against a background 
of black velvet. Often there is a spot of bril- 
liant color and one of a contrasting hue just 
below it, the result being that first one color, 
then the other, is flashed forth as the bird 
changes slightly its position." 

The thirty-one plates give the structural 
details of bill, wings, tail and feet of each of 
the 121 genera, thus greatly facilitating iden- 
tification. It is hoped that Part V. may be 
followed in due time by the remaining volumes 
of this invaluable work, so indispensable to all 
students of American birds. 

J. A. Allen 

The Hindu- Arabic Numerals. By David 
Eugene Smith and Louis Charles Kar- 
pinski. Boston and London, Ginn and 
Company. 1911. Pp. vi + 160. 
This book gives in compact form a readable 
and carefully prepared account of the nu- 
merous researches which have been made in 
the endeavor to trace the origin and develop- 
ment of the Hindu-Arabic numerals. Teach- 
ers of mathematics will welcome it, while stu- 
dents specializing in the history of mathe- 
matics will derive great help from the many 
bibliographical references to other publica- 
tions on this subject. Like the arithmetician 
Tonstall the authors read everything in every 
language and spent much time in licking 
what they found into shape ad ursi exemplum, 
as the bear does her cubs. But it would not 
be a correct statement, were we to convey the 
idea that the book does not embody original 
research. In several cases the authors have 
been able to correct mistakes of earlier writers 



[N. S. Vol. XXXV. No. 900 

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-Arabic 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? In 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 Giinther. In 1902 
Tropfke said 1 

Dass vmser Positionssystem mit seinen Ziffern 
indisehen Ursprunges ist, stent fest. 

1 ' ' Geschichte der Elementar-Mathematik, ' ' I., 
p. 10. 

In 1908 Giinther said 2 

Man kann . . . sich den Vorgang vielleicht so 
denken, dass Indien von Babylon her die ersten 
schwachen Andeutungen des Stellenwertes empfing, 
sie in seiner Weise urn- und ausbildete und spater 
das reiche Geschenk des fertigen Positionssystemes 
den Naehkommen jener Geber zuriiekerstattete. 

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. 

2. 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. 130, Ptolemy in Alexandria 
used in his "Almagest," the Babylonian sexa- 
gesimal fractions and also the omieron 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. 

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 Nallirio, who states that the 
Chaldeans of 100 B.C. (and even earlier) 
knew the sidereal year (estimated at 365 d. 
6 hr. 13 m. 43 sec.) and that this knowledge 
probably passed from them to the Hindus 
and Persians. This statement is quoted by 
H. Suter with apparent approval. 8 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. 
0=a.d. 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 1 
M° 133" (=a.d 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 S , 1904, 
p. 85. 

* Philosophical Transactions, No. 154, p. 399 
(= Abridgment, Vol. I., 1705, p. 107). 

'Philosophical Trans. Abridged, Vol. I., 1705, 
p. 188. 

zero in Europe the authors overlooked the 
curious use of theta to represent zero, 
found in the writings of Michel Rolle, and 
Enestrom's note on this notation in " L'lnter- 
mediaire des mathematieiens," II., 1895, p. 

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 
" Early ideas of their origin " refers to 



[N. S. Vol. XXXV. No. 900 

" oriental names "; it should refer to " Hindu- 
Arabic numerals." The alphabetical index is 
not as complete as one might wish it to be. 

Florian Cajori 
Colorado College, 
Colorado Springs, Colo. 

A Laboratory Course in Physiology. By 
Walter B. Cannon, A.M., M.D., George 
Higginson professor of physiology in the 
Harvard Medical School. Second edition. 
Published by Harvard University. 1911. 
This is the set of loose-leaf laboratory notes 
and directions used in the course in physiol- 
ogy in the Harvard Medical School. It be- 
longs to a class of works which have only 
begun to appear in recent years. It is not a 
general laboratory manual like the well- 
known handbook of Burdon Sanderson, or 
that of Stirling. Its scope is much narrower. 
While these works aimed to give, within the 
limits of their size, accounts of all ordinary 
physiological methods, the work before us, on 
the contrary, is merely a precise description 
of a particular course. Accordingly, it is 
limited to such methods as the facilities of the 
Harvard School allow. Within these limita- 
tions, however, it is excellent. It has already 
been adopted as the basis of the physiological 
course in a number of other institutions and 
contains much that is valuable and suggestive 
for the teaching of physiology anywhere. 

The most striking defect of this "course" 
is that it contains far too much of the physiol- 
ogy of the frog and too little of the mammal. 
For the medical student direct personal ex- 
perience in working with the circulation in 
one living cat or dog is worth two or three 
experiments upon the frog's heart, and a 
dozen upon the frog's leg. It is most unfor- 
tunate that the limitations which misguided 
humanitarians and anti-vivisectionists place 
upon the supply of cats in Boston should 
make it necessary to have the circulation in 
this animal worked out by the students in 
groups of twelve. This certainly falls far 
short of the important educational principle 
urged by Pearce that " the students should do 
it themselves." The reviewer knows from 

personal experience that the largest number 
of students who can possibly take part in a 
blood-pressure experiment on one cat is five. 
If mammalian material were as abundant as 
it ought to be for such a course, the work on 
the frog here outlined could profitably be cut 
in half. Each group should number four or 
five students instead of twelve and should 
have, instead of one cat, six to ten. 

Much of the work on the frog here given 
could be profitably replaced by experiments on 
man. Simple sphygmomanometers can be pro- 
vided cheaply, and should be used for experi- 
ments on the students themselves on a much 
more extensive scale than is outlined in these 

The weakest point in the notes is the sec- 
tion on respiration. Only eight pages are de- 
voted to this subject, while muscle nerve 
physiology receives eighteen. The progress in 
knowledge of respiration within recent years, 
for which we are indebted principally to 
Haldane and his pupils, has been made largely 
by experiments upon man. These experi- 
ments are ideally suited to a laboratory 
course. Among them may be mentioned that 
of voluntary forced breathing and the suc- 
ceeding apnoea; that of the artificial produc- 
tion of Cheyne-Stokes breathing requiring for 
its demonstration merely a tin of soda lime 
and a long tube; and that of the duration of 
the voluntary holding of the breath without 
preparation, after forced breathing, after 
oxygen and after forced breathing and oxy- 

These, however, are merely criticisms of de- 
tail. In general this work is certainly by far 
the best of its kind that has yet appeared. 
No other educational institution in America, 
perhaps none in the world, in recent years has 
made so many valuable experimental contri- 
tributions to the theory and methods of teach- 
ing as has Harvard. Among these contribu- 
tions not the least valuable is the demonstra- 
tion that science in general and physiology in 
particular can be, and ought to be, taught by 
laboratory methods. Originally conceived by 
Huxley and first practised in this country by 
Newell Martin at Johns Hopkins and by the