DECEMBER 6, 1918] 
used for measuring the meteorological ele- 
ments, and while this is well written, it is a 
question if the space it occupies could not 
with advantage be utilized for a somewhat 
fuller discussion of other topics. The order 
of development of the subject proceeds from 
a discussion of temperatures, pressure, evap- 
oration and condensation to a consideration of 
fogs and clouds. This is followed by a brief 
and purely descriptive account of mirage, 
rainbows, halos and coronas, the chapter being 
labelled Atmospheric Optics. Two chapters 
are devoted to Atmospherie Circulation fol- 
lowed by what seem to be unduly abbreviated 
chapters on Forecasting and Climates. 
A well-selected list of reference works and 
the international symbols aré given in ap- 
pendices. M. 
A GREEK TRACT ON INDIVISIBLE 
LINES 
Tur development in recent years of the sub- 
ject of transfinite numbers, of point sets, and 
theories of the continuum is directing the in- 
terest of mathematicians to kindred specula- 
tions among the Greeks. Recent historians 
of Greek mathematics have paid due attention 
to Zeno’s arguments on motion as they are pre- 
sented in Aristotle’s “ Physics,” but thus far 
they have given no consideration to a kindred 
tract included among the works of Aristotle, 
namely, the “ Indivisible Lines ” or “ De lineis 
insecabilibus.” Perhaps the reason for this 
omission lies in the fact that the text as edited 
by Bekker was for the most part unintelligible. 
More recent collations of manuscripts, and the 
translation into English with careful annota- 
tions made by H. H. Joachim, of Oxford, 
render the tract of undoubted value in the 
history of mathematics.1 It reveals the argu- 
1 The Works of Aristotle translated into English 
under the editorship of J. A. Smith and W. D. Ross. 
Part 2: ‘‘De lineis insecabilibus,’’ by. H. H. 
Joachim, Oxford, 1908. We have not seen this 
tract used in any history of Greek mathematics, 
but H. Vogt referred to it in an article on the 
origin of the irrational, printed in the Bibliotheca 
mathematica, 3s., Vol. 10, 1909-10, pp. 146, 153. 
SCIENCE 
577 
ments on the existence and non-existence of 
indivisible lines, and on the possibility of con- 
structing a line out of points, as well as those 
exhibiting the interaction between physical 
speculation about atoms and the philosophy of 
geometry—arguments as they were presumably 
presented in the most celebrated academy of 
the most cultured city of antiquity. Who 
can doubt that the divergence of views then 
held and the perplexing paradoxes advanced 
discouraged Greek mathematicians from openly 
using in geometry the conceptions of the 
infinitesimal and the infinite? Euclid was 
about twenty years younger than Aristotle and 
no doubt was familiar with the trend of phi- ° 
losophie thought of his time. Rigor in geom- 
etry demanded the exclusison of paradox and 
mysticism. Notwithstanding Euclid’s total 
abstinence from controversial conceptions, it 
is evident that the infinitesimal, the indivis- 
ible and the infinite continued to command 
the attention of some mathematicians, as well 
as of philosophers, for more than two thou- 
sand years. We need only mention the title 
of Cavalieri’s famous work, “ Geometria in- 
divisibilibus continuorum nova quadam ratione 
promota,” 1635. 
The Aristotelean “De lineis insecabilibus ” 
contains five arguments current among the 
Greeks in favor of the existence of indivis- 
ibles; these are followed by twenty-six argu- 
ments supporting the contrary view, and 
twenty-four arguments intended to establish 
the impossibility of composing a line out of 
points. Some of these proofs are rigorous. 
Thus, it is argued that, if indivisible lines 
exist, they must be of equal length; an equi- 
lateral triangle each side of which is an in- 
divisible line has an altitude less than the 
indivisible. If a straight line composed of 
an odd number of indivisibles is bisected, one 
of the indivisibles will be divided. The Greek 
failure to build a satisfactory theory of the 
linear continuum as composed of points is 
due to their application of metrical ideas; the 
addition of points could never yield length. 
Aristotle’s failure to construct a satisfactory 
continuum by starting with a straight line 
