738 
It is explained in section VII. in what sense 
the metrical geometry of the material world can 
be considered to be determinate and not a matter 
of arbitrary choice. The scientific question as to 
the best available evidence concerning the nature 
of this geometry is one beset with difficulties of a 
peculiar kind. We are obstructed by the fact that 
all existing physical science assumes the Euclidean 
hypothesis. This hypothesis has been involved in 
all actual measurements of large distances, and 
in all the laws of astronomy and physics. The 
principle of simplicity would therefore lead us in 
general, where an observation conflicted with one 
or more of those laws, to ascribe this anomaly, 
not to the falsity of Euclidean geometry, but to 
the falsity of the laws in question. This applies 
especially to astronomy. . . . But astronomical dis- 
tances and triangles can only be measured by 
means of the received laws of astronomy and 
optics, all of which have been established by 
assuming the truth of the Euclidean hypothesis. 
It therefore remains possible that a large but 
finite space constant, with different laws of as- 
tronomy and optics, would have equally explained 
the phenomena. We can not, therefore, accept the 
measurements of stellar parallaxes, etc., as con- 
elusive evidence that the space constant is large 
as compared with stellar distances. 
Finally, it is of interest to note that, though it 
is theoretically possible to prove, by scientific 
methods, that our geometry is non-Huclidean, it 
is wholly impossible to prove by such methods 
that it is accurately Euclidean. For the unavoid- 
able errors of observation must always leave a 
slight margin in our measurements. A triangle 
might be found whose angles were certainly 
greater, or certainly less, than two right angles; 
but to prove them exactly equal to two right 
angles must always be beyond our powers. 
This I have been publishing for the past 35 
years in articles some 77 of which, not count- 
ing translations, Sommerville has registered 
in his Bibliography of non-euclidean geom- 
etry, 1911. But just here a former pupil of 
mine, Dr. R. L. Moore, has gone beyond his 
teacher. His results seem to be unknown to 
the Encyclopedia, though I called attention to 
them in Science, October 25, 1907, under the 
“scare” heading, “ HKven Perfect Measuring 
Tmpotent.” 
In the brief bibliography appended to this 
SCIENCE 
[N.S. Vou. XXXV. No. 906 
section VI., I notice a number of errors. Jn 
the title of Engel’s book the y should be 7. 
In the title of Dehn’s article, the word Legen- 
darischen should be Legendre’schen. In the 
title of Barbarin’s book the capital G@ and 
capital E should be lower case letters, and the 
hyphen should be omitted. 
In the title of Bonola’s book the capital E 
should be lower ease. 
In the title of the article by E. Study the 
nicht-Euklidische should be Nicht-Euklidis- 
che. This title upon a pamphlet of 97 pages 
(Greifswald, 1900] is Uber Nicht-Euklidis- 
che und Linien-Geometrie. 
In the title of Beltrami’s article given on 
page 728, note 3, the g should be a capital in 
Geometria and the E lower case in non- 
euclidea. In note 4, page 725, nicht-Euklid- 
ischen should be nichteuklidischen. In note 
1, page 727, nicht-Euklidische should be nicht- 
euklidische. 
The final heading, VII., is Axioms of Geom- 
etry, under which it is said: 
The second controversy is that between the view 
that the axioms applicable to space are known 
only from experience, and the view that in some 
sense these axioms are given a@ priori. 
Both these alternatives are wrong. These 
axioms are assumptions, belonging to what I 
have treated under the title “The Unverifiable 
Hypotheses of Science,” in The Monist, Oc- 
tober, 1910. 
The cruder forms of the a priori view have been 
made quite untenable by the modern mathematical 
discoveries. Geometers now profess ignorance in 
many respects of the exact axioms which apply to 
existent space, and it seems unlikely that a pro- 
found study of the question should thus obliterate 
@ priori intuitions. . . . The enumeration of the 
axioms is simply the enumeration of the hypotheses 
of which some at least occur in each of the sub- 
sequent propositions. 
On page 732, line 14, the comma after the 
word “ however” is a misprint, and should be 
deleted. 
Geometry with the assumption: Of any 
three points of a straight there is always one 
and only one which lies between the other two, 
Whitehead calls “descriptive geometry,” a 
