NATURE 
THURSDAY, ~FUEY: «9; 1974: 
HISTORY AND PHILOSOPHY “OF 
MATICS. 
(1) Le Scienze Esatte Nell’ Antica Grecia. By 
Prof. G. Loria. Second edition. Pp. xxiv+ 
MATHE- 
970. (Milan: Ulrico Hoepli, 1914.) Price 
g.50 lire. . 
(2) fst es wahr dass 2x 2=4 ist?» By Fred Bon. 
Vol. i. Pp. xxvili+523. (Leipzig : Emmanuel 
Reinicke, 1913.) 
1) |e 1889 Prof. Gino Loria’s’ notice was 
directed to a prize offered for the best 
history of mathematics. On turning his attention 
to this subject the author tells us that he became 
so interested in the study of the ancient Greek 
mathematicians that he decided to devote his atten- 
tion to this instead of the more general subject. 
His previous writings have been published in the 
transactions of the academies of Turin and 
-Modena, the latter between 1893 and 1902, and it 
is largely on these that the present volume is 
based. 
The work treats mainly of geometry and arith- 
metic, but applied mathematics is dealt with in 
one of the five books, in so far as it relates to 
astronomy, geodesy, and spherical geometry. 
Prof. Loria divides the history of Greek geometry 
into three periods—the pre-Euclidean period of 
Pythagoras, Socrates, and Plato; the ‘golden ”’ 
period of Euclid, Archimedes, Eratosthenes, and 
Apollonius, and a third period which is described 
as the “silver”? or Graeco-Roman age, of which 
Pappus of Alexandria forms. one of the central 
figures. 
In the section dealing with arithmetic and theory 
of numbers, great interest centres round the work 
of Diophantus, the discussion of which occupies 
eighty pages. The list of equations solved. 1- 
this remarkable mathematician, stated in the nota- 
tion of modern algebra, alone occupies twelve 
pages, and Prof. Loria has been throughout very | 
careful in connecting these old problems with their 
present-day equivalents. 
It is a great mistake that the Greek mathe- 
maticians in this book are only described by their 
modern Italian names. Such names as Erone, 
Tolomeo, Anassagora, Omero, will not convey 
much idea to foreign readers. The least the 
author should have done would have been to give 
the correct names in the index at the end, but this 
he has not done. 
For a treatise of this character the small-sized 
pages of the Manueli Hoepli are a serious dis- 
advantage. A pocket-book, the letterpress pages 
of which are a little larger than a quarter-plate 
negative, but smaller than “post-card” size may 
NOrn2 222, VOL. 93] 
475 
be \uitable Is this as a ne ee of publication for 
such Or yes buildings, diseases 
of pigs,~ Y rming, acetylene, or even 
calculus for engineers. But for a subject so teem- 
ing with points of historical and mathematical 
facts to be condensed into these tiny pages, closely 
printed in small type, renders the book very diffi- 
cult reading indeed. The strain involved in 
reading the letterpress greatly increases the diff- 
culty of assimilating the subject-matter. 
(2) The inquiring reader who wishes to ascertain 
the truth, or otherwise, of the statement that two 
and two make four will not find Bon’s attempts 
to enlighten him on this matter cramped by want 
of space. When he has come to the end of these 
520 octavo pages he will only have learnt what the 
author has to say regarding the nature and mean- 
ing of concept, judgment, and truth, and he will 
have seen that this is only the first volume of Bon’s 
work. He certainly will not yet have arrived at 
any definite conclusions as to whether two and 
two really make four or five, for that matter. 
This volume is divided into three parts, dealing 
with the nature and meaning of a concept, a 
decision, and of truth, with the object of examin- 
ing what these mean, and under what conditions 
it is possible to assert that a decision is true. 
In the chapter on the definition of a concept, 
the author starts with the statement that concepts 
are words, and arrives at the following kind of 
definition. 
By concept we understand a word which has 
a meaning for one or more individuals, or, by 
concept we understand a word which is understood 
by one or more individuals. — This attempt to 
identify a concept with a word will certainly not 
meet with unanimous acceptance, even in spite of 
the detailed discussions, extending over more than 
230 pages, which follow. It might surely be 
objected that a concept can exist independently 
of words, and that it is not the word itself, but 
its meaning, or something which is associated 
with the word, which constitutes the concept. Of 
course, the author has to examine what is under- 
stood by meaning, by understanding, by words, 
or by a definition, whether a concept is definable 
or not, and if so, how far this is possible; at the 
same time, it is evident from what has been said 
that the author’s will not meet with 
universal acceptance. 
In. the definition of a decision or judgment 
(Urteil) (p. 261) the author again uses language 
as the basis of his definition, regarding a decision 
as a sequence of words which has a_ meaning 
independent of the meanings of the separate 
words and is understood by one or more definite 
individuals. 
views 
U 
