SEPTEMBER 24, 1897 ] 
Emory McClintock in his article ‘On the non- 
Euclidean geometry’ iu the Bulletin of the N. 
Y. (Amer.) Math. Soe., Vol. II., pp. 21-33, 
which reached the pitiful conclusion (p. 32): 
‘“¢The chief lesson to be obtained from all non- 
euclidian diversions (sic) is that the distinguish- 
ing mark of euclidian geometry is fixity of dis- 
tance—measurement.’’ 
Mr. Russell, with equal deftness, puts in pillory 
the gross blunder made by Andrew W. Phillips 
and Irving Fisher, professors in Yale Univer- 
sity, in the note on p. 23 of their Elements of 
Geometry, where they say: ‘‘ Lobatchewsky in 
1829 proved that we can never get rid of the 
parallel axiom without assuming the space in 
which we live to be very different from what we 
know it to be through experience.”’ 
By experience, of course, we can never know 
or prove our space to be other than a non- 
Euclidean space with a comparatively large 
constant. How unexpected, then, the error of 
Professor H. Schubert, of Hamburg, in the 
Monist, Vol. VI., No. 2, p. 295, where he says: 
“Let me recall the controversy which has been 
waged in this century regarding the eleventh axiom 
of Euclid, that only one line can be drawn through a 
point parallel to another straight line. The discus" 
sion merely touched the question whether the axiom 
was capable of demonstration solely by means of the 
other propositions, or whether it was not a special 
property, apprehensible only by sense-experience, of that 
space of three dimensions in which the organic world 
has been produced.’ 
After 20 years’ study of writers on the non- 
Euclidean geometry, the present reviewer can- 
not recall even one who was ever silly enough 
to think that the exact equality of the angle- 
sum of a rectilineal triangle to two right angles 
was apprehensible by sense-experience, or could 
ever be known through experience. 
This new Yale geometry also makes the old 
petitio principii of defining a straight line as the 
shortest distance between two points. This our 
author treats in his third chapter, p. 167: 
“We are accustomed to the definition of the 
straight line as the shortest distance between two 
points. * * * Unless we presuppose the straight 
line, we have no means of comparing the lengths of 
different curves and can, therefore, never discover the 
applicability of our definition.”’ 
SCIENCE. 
491 
In projective geometry any two points 
uniquely determine a line, the straight. But 
any two points and their straight are, in pure 
projective geometry, utterly indistinguishable 
from any other point-pair and their straight. 
It is of the essence of metric geometry that two 
points shall completely determine a spatial 
quantity, the sect. If our author had used for 
this fundamental spatial magnitude this name, 
introduced in 1881, his exposition would have 
gained wonderfully in clearness. 
Both the accepted popular and the accepted 
mathematical definition of ‘ distance’ make it 
always a number, as, e. g., the Cayley-Klein 
definition : ‘‘ The distance between two points 
is equal to a constant times the logarithm of the 
cross ratio in which the line joining the two 
points is divided by the fundamental quadric.”’ 
It is the misfortune of our author to use the 
already overworked and often misused word 
‘distance’ as a confounding and confusing 
designation for a sect itself and also the measures 
of that sect, whether by superposition, ordinary 
ratio, indeterminate as depending on the choice 
of a unit, or projective metrics, indeterminate 
as depending on the fixing of the two points to 
be taken as constant in the varying cross ratios. 
This whole book might be cited as an over- 
whelming vindication of the only American 
treatise on Projective Geometry against the 
attack on it made by a critic in SCIENCE, be- 
cause, forsooth, it was founded and developed 
as pure projective geometry, without any quan- 
titative ideas whatever. 
Into the fourth and last chapter, ‘ Philosoph- 
ical Consequences,’ we will not here go. Suffice 
it to say that Projective and Metric Geometry, 
though eternally separate in essence, and each 
unable ever to absorb the other, are happily 
wedded, and expand joyfully ever after. 
GEORGE BRUCE HALSTED. 
AUSTIN, TEXAS. 
Sight: An Exposition of the Principles of Monoe- 
ular and Binocular Vision. JosEPH LE 
Conte. New York, D. Appleton & Co. 
1897. Second edition, revised and enlarged. 
Pp. xvi+ 318. $1.50. 
A revised and enlarged edition of Professor 
