OcTOBER 20, 1899. ] 
fall into two main types; the limit surfaces 
form a transition case between these types. 
In elliptic space there is only one type of 
such a surface of revolution. 
“The same principles would enable the 
problem to be solved of the discovery in 
any kind of space of surfaces with their 
‘geodesic’ geometry identical with that of 
planes in any other kind of space.”’ 
So that which Macauly designated as ‘ un- 
doubtedly the most remarkable property of 
hyperbolic space’ has been by Whitehead 
not only generalized for hyperbolic space 
but extended to elliptic space. 
Bolyai Janos seemed fully to realize the 
weight, the scope, the possibilities, the 
meaning of his discovery. He returns to 
it in §37, where he uses the proportion- 
ality of similar triangles in F to solve an 
essential problem in S (hyperbolic space). 
Then he adds: ‘ Hence, easily appears 
(Z-lines being given by their extremities 
alone) also fourth and mean terms of a pro- 
portion can be found, and all geometric con- 
structions which are made in + in plano, in 
this mode can be accomplished in F’ apart 
from Axiom XI.” The italics are Bolyai’s, 
yet I find that they have not been repro- 
duced in my published translation (the only 
one in English), nor in Frischauf’s Ger- 
man, nor in Hotel’s French, nor in Fr. 
Schmidt’s Latin text, nor in Sutak’s Mag- 
yar. Whitehead’s researches will remind 
us all how great a thing it was to have 
reached the whole Euclidean system en- 
tirely apart from any parallel-postulate. 
It is a pleasure to be able to state that this 
was also done by Lobachévski. It is ex- 
plicitly given in his first published work 
‘O nachalah geometri’ (1829). ‘ Noviya 
nachala geometri’ (1835), devotes to it 
Chapter VIII. 
It is also at this point, so striking as 
pure mathematics, that general philosophy 
finds itself involved. Killing, Klein, and 
in general the German writers, distinctly 
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503 
draw back from any philosophical impli- 
cations. The whole matter, however, 
has been opened in ‘An Essay on the 
Foundations of Geometry,’ by Hon. Ber- 
trand A. W. Russell, Fellow of Trinity Col- 
lege, Cambridge (1897), who has had the 
good fortune tobe the very first to set forth 
the philosophical importance of von Staudt’s 
pure projective geometry, which in its 
foundation and dealing with the qualitative 
properties of space involves no reference to 
quantity. I discussed this point more than 
twenty years ago in the Popular Science 
Monthly, 4 propos of Spencer’s classification 
of the Abstract Sciences. 
In a note to the first edition of his clas- 
sification of the sciences (omitted in the 
second edition), Spencer says, ‘I was igno- 
rant of this as a separate division of mathe- 
matics, until it was described to me by Mr. 
Hirst. It was only when seeking to affiliate 
and define ‘ Descriptive Geometry’ that I 
reached the conclusion that there is a nega- 
tively-quantitative mathematics as well as 
a positively- quantitative mathematics.”’ As 
explanatory of what he wishes to mean by 
negatively-quantitative, we quote from his 
Table I.: ‘‘ Laws of Relations, that are 
Quantitative (Mathematics), Negatively : 
the terms of the relations being definitely- 
related sets of positions in space, and the 
facts predicted being the absence of certain 
quantities (‘Geometry of Position’).’’ He 
also says: ‘‘In explanation of the term 
‘negatively-quantitative,’ it will be suffi- 
cient to instance the proposition that cer- 
tain three lines will meet in a point, as a 
negatively-quantitative proposition, since 
it asserts the absence of any quantity of 
space between their intersections. Simi- 
larly, the assertion that certain three points 
would always fall in astraight line is ‘ neg- 
atively-quantitative,’ since the conception 
of a straight line implies the negation of 
any lateral quantity or deviation.” But 
Sylvester has said of this very proposition 
