554 &CIENCE., 
that it ‘‘ refers solely to position, and neither 
invokes nor involves the idea of quantity 
or magnitude.” 
“« Projective Geometry proper,’ says Rus- 
sell, ‘‘does not employ the conception of 
magnitude.” 
Now it is in metrical properties alone 
that non-Euclidean and Euclidean spaces 
differ. The distinction between Euclidean 
and non-Euclidean geometries, so important 
in metrical investigations, disappears in 
projective geometry proper. Therefore pro- 
jective geometry deals with a wider concep- 
tion, a conception which includes both, and 
neglects the attributes in which they differ. 
This conception Mr. Russell calls ‘a form 
of externality.’ It follows that the assump- 
tions of projective geometry must be the 
simplest expression of the indispensable 
requisites of all geometrical reasoning. 
Any two points uniquely determine a 
line, the straight. But any two points and 
their straight are, in pure projective geom- 
etry, 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 (German, strecke). If Mr. 
Russell had used for this fundamental spa- 
tial magnitude this name, or any name but 
‘ distance,’ his exposition would have gained 
wonderfully inclearness. It isa misfortune 
to use the already overworked and often: 
misused word ‘ distance’ as a confounding 
and confusing designation for a sect itself 
and also the measure of that sect, whether 
by superposition, ordinary ratio, indetermi- 
nate as depending on the choice of a unit; 
or by projective metrics, indeterminate as 
depending on the fixing of the two points 
to be taken as constant in the varying cross 
ratios. 
That Mr. Russell’s chapter ‘A Short His- 
tory of Metageometry,’ contains all the 
stock errors in particularly irritating form, 
and some others peculiarly grotesque, I 
[N. S. Von. X. No. 251. 
have pointed out in extenso, in ScrEncsz, 
Vol. VI., pp. 478-491. Nevertheless the 
book is epoch-making. It finds ‘‘ that pro- 
jective geometry, which has no reference to 
quantity, is necessarily true of any form of 
externality. In metrical geometry is an 
empirical element, arising out of the alter- 
natives of Euclidean and non-Kuclidean 
space.”’ 
One of the most pleasing aspects of the 
universal permanent progress in all things 
non-Euclidean is the making accessible of 
the original masterpieces. 
The marvellous ‘ Tentamen’ of Bolyai 
Farkas, as Appendix to which the ‘ Science 
Absolute ’ of Bolyai Janos appeared, a book 
so rare that except my own two copies, I 
know of no copy on the Western Continent, 
a book which has never been translated, a 
field which has lain fallow for sixty-five 
years, is now being re-issued in sumptuous 
quarto form by the Hungarian Academy 
of Sciences. The first volume appeared in 
1897, edited, with sixty-three pages of notes 
in Latin, by Konig and Réthy of Budapest. 
Professor Réthy, whom I had the pleasure 
of meeting in Kolozsvar, tells me the second 
volume is in press, and he is working on it 
this summer. 
Bolyai Farkas is the forerunner of Helm- 
holtz, Riemann, Lie, though one would 
scarcely expect it from the poetic exalta- 
tion with which he begins his great work. 
‘“‘ Lectori salutem! Scarce superficially im- 
bued with the rudiments of first principles, 
of my own accord, without any other end, 
but led by internal thirst for truth, seeking 
its very fount, as yet a beardless youth, I 
laid the foundations of this ‘ Tentamen.’ 
‘“Only fundamental principles is it pro- 
posed here so to present, that, Tyros, to 
whom it is not given to cross on light 
wings the abyss, and, pure spirits, glad of 
no original, to be borne up in airs scarce 
respirable, may, proceeding with firmer 
step, attain to the heights. 
