464 
here referred to, so far from this being rejected, 
it actually remains true. In the ‘elliptic’ or 
Clifford-Klein geometry its last clause becomes 
unmeaning, because the straight line does not 
divide the elliptic plane into two separated 
regions. Here we cannot distinguish two sides 
of a straight. Without crossing a given straight 
we can pass from any one point of the plane to 
any other point. In this elliptic geometry, the 
other assumption mentioned by Boyer, that two 
straight lines cannot meet at more than one 
point, is retained. Riemann’s epoch-making 
contribution was that the universe while un- 
bounded still may very well be finite. 
This gives us the assumption that a straight 
line has no point at infinity, that is, that every 
straight line is actually cut by every other 
straight line coplanar with it. 
Now dropping Euclid’s implicit, not explicit, 
assumption that the straight line is infinite, but 
retaining all his postulates and axioms, espe- 
cially Postulate 1 (Simon’s Euclid, 1901, p. 30), 
““ Let it be granted that one and only one sect 
can be drawn from any one point to any other 
point,’’ we have the elliptic geometry. 
On p. 245, line 23, is a misprint, 
for ‘ points.’ 
Peirce (C. S.), p. 247, is identified with his 
father in the index, his name being given as 
Benjamin and his dates as 1809-1880. 
Mr. C. S. Peirce is still alive, having an 
article in the January, 1901, number of the 
Popular Science Monthly, which contains the 
charming résumé by Professor Crawley en- 
titled, ‘Geometry: Ancient and Modern.’ 
Perhaps it could be wished that this article 
had more definitely emphasized Euclid. 
The advertisement of Boyer’s ‘Histoire’ calls 
mathematics the science of Euclid and New- 
ton. 
In writing of ‘The Wonderful Cenury,’ Al- 
fred Russel Wallace says of all time before the 
seventeenth century: ‘‘Then, going back- 
ward, we can find nothing of the first rank 
except Euclid’s wonderful system of geometry, 
perhaps the most remarkable mental product 
of the earliest civilizations.”’ 
The new departure, the non-Euclidean ge- 
ometry, is absolutely epoch-making, but fortu- 
nately it has intensified admiration for that 
‘joints’ 
SCIENCE. 
[N. 8. Vox. XIII. No. 325. 
imperishable model, already in dim antiquity 
a classic, the immortal Elements of Buclid. 
Professor Crawley’s exposition of the non- 
Euclidean geometry is exceedingly interesting. 
But as soon asit gets beyond two dimensions 
it becomes obscure. 
He says, p. 265, ‘‘ If we proceed beyond the 
domain of two dimensional geometry we merge 
the ideas of non-Euclidean and hyper-space.” 
_ If we do, we are very apt to blunder. Thus 
Professor Crawley says, p. 266, ‘‘ Professor 
Newcomb has deduced the actual dimensions of 
the visible universe in terms of the measure- 
ment of curvature in the fourth dimension.”’ 
This mistake of supposing that a non-Eucid- 
ean space requires or needs a space of higher 
dimensionality has often been publicly cor- 
rected. 
On page 293 of his ‘ Nicht-Euklidische Ge. 
ometrie, I.’ Felix Klein puts in pillory the 
unfortunate title of Newcomb’s contribution as 
follows: ‘‘ Elementary theorems relating to the 
geometry of a space of three dimensions and of 
uniform positive curvature in the fourth dimen- 
sion. (Die letzten Worte des Titels sind sehr 
merkwurdig und deuten auf ein Missverstand- 
niss.)?? 
After ‘we merge the ideas,’ Professor Craw- 
ley’s very next sentence is: ‘‘The ordinary 
triply-extended space of our experience is 
purely Euclidean.’’ This naive assertion not 
only Professor Crawley does not know and can- 
not prove, but, strangely enough, no one can 
ever know, no one can ever prove. For Kuclid- 
ean space the angle-sum of a rectilineal tri- 
angle must be exactly two right angles. Such 
absolutely exact metric results experience can 
never give. 
In connecting a geometry with experience 
there is involved a process which we find in the 
theoretical handling of any empirical data, and 
which therefore should be familiarly intelligible 
to any scientist. The results of any*observa- 
tions hold good, are valid, always only within 
definite limits of exactitude and under partic- 
ular conditions. When we set up the axioms, 
we put in place of these results statements of 
absolute precision and generality. In this 
idealization of the empirical data our addition 
is at first only restricted in its arbitrariness in 
