490 
what is here in question, and I have found only few 
people who received with special interest that which 
I communicated to them. In order for that one must 
first have felt right keenly what really is lacking, and 
about that most men are wholly indistinct. 
“*On the other hand, my intention was, with time, 
to put all on paper, so that it at least would not 
hereafter perish with me. 
‘“Therefore am I greatly pleased that this trouble 
can now be spared me, and it is most highly delightful 
to me that the son of just my old friend is he who in 
so remarkable a way has anticipated me. 
“‘T find his notations very pregnant and abridging: 
Yet I believe it would be good to establish for many 
chief ideas not merely symbols or letters, but definite 
names, and I have already since long thought of some 
such names. 
‘*So long as one thinks through the thing only in 
immediate intuition, one needs no name or symbol; 
these are first necessary, if one wishes to be compre- 
hensible to others. So, for example, the surface 
which your son calls F' could be called a Parasphere, 
the line Z a Paracykle: they are, in fact, spheres or 
circles of infinite radius. Hypereykle could be 
named the complex of all points which have like dis- 
tance from a straight with which they lie in a plane; 
even so Hypersphere. Yet those are all only unim- 
portant incidents; the main thing is the matter, not 
the form. 
* * * * * * 
“Just exactly in the impossibility to decide a priori 
between = and WS lies the clearest proof that Kant 
was wrong to maintain, Space is only Form of our 
intuition.”’ 
About the other independent discoverer of the 
non-Huclidean geometry, Lobachévski, Gauss 
writes to Schumacher, November 28, 1846, 
without a word of reference to Bolyai, as fol- 
lows: 
‘*T have lately had occasion to reread the opuscule 
of Lobatschewsky, intitled: Geometrische Untersuch- 
ungen zur Theorie der Parallellinien. This opuscle 
contains the elements of the geometry which would 
exist and development of which would form a 
rigorous chain, if the Huclidean geometry were 
not true. * * * You know that since fifty-four 
years (since 1792) I share the same convictions, with- 
out speaking here of certain developments which my 
ideas on this subject have since received. Therefore, 
I have not found in the work of Lobacheyski any fact 
new to me; but the exposition is wholly different 
from that which I had projected, and the author has 
treated the matter with a master hand and with the 
veritable geometric spirit.’’ 
SCIENCE. 
[N. S. Vou. VI. No. 143. 
How reconcile these letters with that of 1804 ? 
And since one says that Bolyai’s exposition is 
identical with that of Gauss, while the other 
declares Lobachévski’s wholly different from 
that, how reconcile them with the statement of 
Mr. Russell, p. 11 : ‘‘Very similar [to Lobachéy- 
ski’s] is the system of Johann Bolyat, so simi- 
lar, indeed, as to make the independence of the 
two works, though a well-authenticated fact, 
seem all but incredible ?”’ 
This letter of 1846 shows no hint of that other 
sort of non-Euclidean geometry which Riemann 
gave in his wonderful Probevorlesung, ‘ Ueber 
die Hypothesen welche der Geometrie zu Grunde 
liegen,’ June 10, 1854. 
But this dissertation was not published until 
1867, so that the waters of oblivion seemed to 
close over it as over the works of Bolyai and 
Lobachéyski. 
Mr. Russell should not have omitted in his. 
text all mention of Hotel, for Hotel it was who 
resurrected the non-Euclidean geometry, be- 
ginning with his own essay on the fundamental 
principles of geometry, published in 1863 at. 
Greifswald. (See his life in the Amer. Math. 
Monthly, April, 1897.) 
But not to give too much space to actual slips 
in history we must jump to the second of Mr. 
Russell’s four chapters, ‘Critical account of 
some previous philosophical theories of geom- 
etry :’ 
‘The importance of geometry in the theories 
of knowledge which have arisen in the past can 
scarcely be exaggerated.’’ 
The author believes that the usual forms of 
non-Euclidean geometry, the hyperbolic, the 
double elliptic and the single elliptic are the 
only logically self-consistent systems, and so 
says: ‘‘T shall contend that those axioms, 
which Euclid and Metageometry have in com- 
mon, coincide with those properties of any form 
of externality which are deducible, by the prin- 
ciple of contradiction, from the possibility of 
experience of an external world.’’ 
We see at once that pure projective geometry 
must be of supreme weight for him. 
It is a treat to see our author overwhelm the 
apparent subordination of the non-Euclidean 
spaces by the introduction of different measures 
of distance. This was the painful mistake of 
