546 SCIENCE. 
on page 175 of the second impression of 
these lectures, 1893, we still find Felix 
Klein saying, ‘Kein Zweifel bestehen 
kann, dass Lobatscheffsky sowohl wie Bol- 
yai die Fragestellung ihrer Untersuchungen 
der Gaussischen Anregung verdanken.”’ 
It isa privilege to begin my report by 
announcing the rigorous demonstration that 
this ungenerous legend is untrue. This 
point need not further delay us, since it has 
been treated by me at length in ScrENcE, 
N. S., Vol. IX., No. 232, pages 813-817, 
June 9, 1899. 
What a contrast to the pathetic neglect 
of its creators, Lobachévski dying blind, 
unrecognized, without a single follower, 
Bolyai Janos dying of disgust with himself 
and the world, lies in the fact that less 
than a year ago our American magazine, 
the Monist, secured from the famous Poin- 
earé, at great cost, a brilliant contribution 
to this now universally interesting subject, 
which I had the honor, through my friend 
T. J. McCormack, of reading in the orig- 
inal French manuscript. 
This extraordinary paper, published only 
in English translation, appears in the Mon- 
ist, Vol. 9, No. 1, Oct., 1898, pages 1-43. 
In the first section of his greatest work, 
Lobachévski says: ‘‘Juataposition (contact) 
is the distinctive characteristic of solids, 
and they owe to it the name geometric solids, 
when we retain this attribute, taking into 
consideration no others whether essential or 
accidental. 
““ Besides bodies, for example, also time, 
force, velocity are the object of our judg- 
ment; but the idea contained in the word 
juxtaposition does not apply thereto. In 
our mind we attribute it only to solids, in 
speaking of their composition or dissection 
into parts. 
‘This simple idea,which we have received 
directly in nature through the senses, comes 
from no other and consequently is subject 
to no further explanation. Two solids A 
[N. S. Von. X. No. 251. 
and B, touching one another, form a single 
geometric solid C, in which each of the 
component parts A, B appears separate 
without being lost in the whole C. In- 
versely, every solid Cis divided into two 
parts A and B by any section S. 
‘“‘ By the word section we understand here 
no new attribute of the solid, but again a 
juxtaposition, expressing thus the partition 
of the solid into two juxtaposed parts. 
‘“In this way we can represent to our- 
selves all solids in nature as parts of a 
single whole solid which we call space.” 
Poincaré starts off somewhat differently. 
He says: ‘‘ We at once perceive that our 
sensations vary, that our impressions are 
subject to change. The laws of these varia- 
tions were the cause of our creating geom- 
etry and the notion of geometrical space. 
‘‘ Among the changes which our impres- 
sions undergo, we distinguish two classes : 
““(1) The first are independent of our 
will and not accompanied by muscular sen- 
sations. These are external changes so-called. 
‘©(2) The others are voluntary and ac- 
companied by muscular sensations. We 
may call these internal changes. 
“We observe next that in certain cases 
when an external change has modified our 
impressions, we can, by voluntarily provok- 
ing an internal change, re-establish our 
primitive impressions. The external change, 
accordingly, can be corrected by an internal 
change. External changes may conse- 
quently be subdivided into the two follow- 
ing classes : 
“1, Changes which are susceptible of be- 
ing corrected by an internal change. These 
are displacements. 
‘9, Changes which are not so susceptible. 
These are alterations. An immovable being 
would be incapable of making this distinc- 
tion. Such a being, therefore, could never 
create geometry, even if his sensations were 
variable, and even if the objects surround- 
ing him were movable.” 
praesent 
St SEs 
