NovEMBER 26, 1903] 
NATURE 
77 
- mathematics, and in this volume deals only with arith- 
metic and algebra; but his treatment is thorough, 
and his aim has been to give exact references to the 
original authorities for the statements in the text. 
The amount of labour that this has involved must 
have been very great; when the work is complete, 
with the indexes promised by the author, it will be a 
valuable repertory for those who wish to learn the 
facts at first hand. The number of bibliographical 
footnotes exceeds 1200, and since many of these give 
more than one reference, it will be seen how great a 
service the author has rendered to those who are in- 
clined for research. 
But the book is far from being a mere dry collection 
of facts and references. The style is concise, and there 
is no catchpenny rhetoric, but there is plenty to interest 
any intelligent reader. The arrangement allows us to 
trace in detail the development of methods and of 
notation; we are shown, with explanations, the actual 
symbols used and the processes employed by our pre- 
decessors; most important of all, there is an appendix 
with a selection of original examples ranging from 
Alchwarizmi to Leibniz and Newton. Few things are 
more instructive than an inspection of some of the 
older methods in arithmetic. Until the end of the 
fifteenth century, long after the decimal notation and 
the use of the ‘Arabic’? numerals had become 
familiar, and when arithmetical calculations were 
usually worked on paper, the rule for performing long 
division was of a most complicated character, with 
rows of figures above the dividend as well as below, 
and tedious cancellings and substitutions which must 
have made the operation both laborious and liable to 
error. It is almost certain that the process is of Indian 
origin, and it is probable that the figures which, in 
written examples, we find cancelled by a stroke drawn 
through them represent digits which were actually 
obliterated at an earlier period, when the calculation 
was performed with a stick on a layer of sand. 
A striking feature of early European books on 
arithmetic is the bewildering number of their so-called 
“‘rules.”’ One reason for this is simple enough. 
Many of these books were intended to help business 
men—bankers, merchants, and so on—in such calcu- 
lations as their calling obliged them to do. Their 
interest in arithmetic was purely practical, and all 
they wanted was a bundle of recipes for getting 
correct answers to questions of certain special types. 
Even in our own day we occasionally see such terms 
as “agricultural book-keeping ’’ or ‘‘ chemical arith- 
metic,’? which show that a demand for this sort of 
thing is not yet extinct. But even in treatises of a 
more theoretical kind duplatio and mediatio, in other 
words doubling and halving, were reckoned as 
separate rules. This is a historical survival, a sort of 
fossil relic of prehistoric times. It appears that the 
ancient Egyptians performed multiplication by a pro- 
cess practically equivalent to converting the multiplier 
into the binary scale; thus 
x XIZ=%4x8+x*% x44 x, 
where xx8 and xx4 were obtained by successive 
doubling. When an improved method of multipli- 
cation had been discovered, the older process became 
NO. 1778, VOL. 69] 
| 
obsolete ; but duplatio held its ground as a special rule, 
in recognition, so to speak, of its former importance. 
A considerable portion of this volume is naturally 
devoted to the theory of surds, and this cannot be 
separated from the Greek theory of geometrical 
irrationals. After all that has been written on the 
subject, lacunae remain which will probably never be 
filled up, unless new documents are discovered. Some 
undoubted facts are very puzzling when taken in com- 
bination. For instance, Euclid says in so many words 
that incommensurable quantities are not related to 
each other as numbers, and it really does seem that 
to a Greek geometer of Euclid’s time the relation, as 
to length, of the diagonal of a square to one side was 
something different in kind from the relation of two 
commensurable distances. At the same time the 
Greeks must have been practically acquainted with 
what we should call rational approximations to 2, 
and it is well known that the irrationalities considered 
in the tenth book of Euclid’s ‘‘ Elements,’’? when put 
into an algebraic form, correspond exactly to all the 
members of a particular group of surds, without 
omission or redundancy. Did the geometers, who pro- 
fessed to despise “‘ logistic ’’ in public, privately make 
use of it to help them in their researches ? 
Other subjects considered under the head of algebra 
are the development of the idea of number in general, 
the operations of algebra and their symbols, propor- 
tion, and equations. Under the last heading Diophan- 
tine analysis is included, and it may be noted as a fact 
not generally known that Diophantine equations of 
the form 
px*—qy?=r 
were actually discussed in India at least as early as. 
the time of Brahmagupta—that is to say, more than 
a thousand years before Fermat proposed the Pellian 
equation to the English mathematicians. G. B. M. 
OUR BOOK SHELF. 
La Lutte pour l’Existence et l’Evolution 
By J. L. de Lanessan. Pp. 277. 
Alcan, 1903.) Price 6 francs. 
Tur title of this book is most misleading. The reader 
naturally expects to find an account of the struggle 
for existence among primitive men and of the 
evolution which has resulted from the struggle. The 
first chapter has quotations from Buffon and Darwin 
which leave no doubt in one’s mind that this is 
the line which is to be followed. After this comes 
a description of primitive society or rather the social 
system which the author assumes to be primitive. 
The struggle for existence drops out, and is not 
mentioned. Society begins, he tells us, witha severely 
patriarchal régime. He seems not to have heard of 
an earlier polyandrous period. Out of the family bond 
arose the sense of duty. Speaking of the tribe, he 
lays it down that the chieftain was regarded as the 
owner of all the land which the tribe possessed. 
After this glance at primitive society, we plunge 
into French history. Many great questions are dealt 
with, and most of them with remarkable shrewdness. 
Our author discusses the origin of feudalism. He next 
decides that Christianity had nothing to do with 
the abolition of slavery. He traces the growth of the 
idea of liberty among the peasantry; it showed itself 
des Sociétés. 
(Paris: Félix 
