488 
number of unsatisfactory attempts to prove 
this postulate,’ and states that Sohncke gives 
a list of 92 authors on the subject before 1837, 
and that Perronet Thompson gave in English 
an account of like attempts before 1833, the 
very year our author cites for Legendre. 
Mr. Russell goes on to say : 
‘Parallels are defined by Legendre as lines in the 
same plane, such that, if a third line cut them, it 
makes the sumof the interior and opposite angles 
equal to two right angles. He proves without diffi- 
culty that such lines would not meet.’’ 
But so had every school boy in the subject, 
since this is part of Euclid, Book I, Prop. 28: 
“Similarly he can prove that the sum of the angles 
of a triangle cannot exceed two right angles, and that 
ifany one triangle has a sum equal to two right 
angles all triangles have the same sum.’? 
But these very demonstrations were published 
just a century before Mr. Russell’s ‘ first effort,’ 
in 1733, by Saccheri. 
Mr. Russell proceeds to speak of ‘ The origi- 
nator of the whole system, Gauss,’ and then 
says: ‘‘In 1799, writing to W. Bolyai, Gauss 
enunciates the important theorem that in hy- 
perbolic geometry there is a maximum to the 
area of a triangle.”’ 
How utterly misleading, nay, fantastic, is this 
statement will appear on quoting the letter 
from ‘ Halsted’s Science Absolute of Space,’ 4th 
edition, Austin, 1896, which our author cites. 
Gauss says: 
““T very much regret that I did not make use of 
our former proximity to find out more about your in- 
vestigations in regard to the first grounds of geometry; 
I should certainly thereby have spared myself much 
vain labor, and would have become more restful 
than any one, such as I, can be, so long as on such a 
subject there yet remains so much to be wished for. 
“In my own work thereon I myself have advanced 
far (though my other wholly heterogeneous employ- 
ments leave me little time therefor), but the way, 
which [ have hit upon, leads not so much to the goal 
which one wishes and which you assure me you have 
reached, as much more to making doubtful the truth 
of geometry. 
“Indeed, I have come upon much, which with most 
no doubt would pass for a proof, but which in my 
eyes proves as good as NOTHING. 
“For example, if one could prove that a rectilineal 
triangle is possible, whose content may be greater 
SCLENOE. 
(N.S. Vou. VI. No. 143. 
than any given surface, then I am in condition to 
prove with perfect rigor all geometry. 
‘Most would, indeed, let that pass as an axiom ; I 
not ; it might well be possible that, how far apart 
soever one took the three vertices of the triangle in 
space, yet the content was always under a given limit. 
“‘T have more such theorems, but in none do I find 
anything satisfying.’’ 
From this letter it is perfectly clear that in 
1799, so far from haying the remotest idea of a 
hyperbolic geometry, or any non-Huclidean 
geometry, Gauss was still trying to prove that 
Euclid’s is the only non-contradictory system 
of geometry and that it is the system of the ex- 
ternal space of our physical experience. The 
first is false; the second can never be proved. 
But that both Gauss and W. Bolyai continued 
for the next five years to pound away in at- 
tempts to do the impossible, we have now ob- 
tained demonstrative evidence, in recovering a 
treatise finished and sent to Gauss by W. Bol- 
yai in 1804. 
In a great casket at Maros-Vasarhely all the 
unpublished papers of Bolyai Janos are pre- 
served. All were placed freely at my disposal 
on my pilgrimage to this shrine of the non- 
Euclidean geometry. There, with extended 
researches anticipating the discoveries of Cay- 
ley and Klein in this subject, is an autobiog- 
raphy of Janos containing extracts from two 
letters written by Gauss to W. Bolyai (Farkas) 
and of transcendent importance as freeing Janos 
forever from the calumny again repeated by 
Mr. Russell where he says: 
p. 12. ‘‘Gauss was, as we have seen, the inspirer 
of Wolfgang Bolyai. Wolfgang appears to have left 
to his son, Johann, the detailed working out of the 
hyperbolic system.”’ 
Nothing could be more false. 
J&nos, wholly unaided, discovered by himself 
the non-Euclidean geometry and taught it to 
Wolfgang, who transmitted it to Gauss. The 
two letters quoted by Janos are one before and 
one after this transmission. 
This ery from the dead for tardy justice has 
since been shown exactly accurate by my friend, 
Fr. Schmidt, of Budapest, finding that the 
originals of these letters in the handwriting of 
Gauss still exist at Gottingen. The first is 
dated November 25, 1804, in answer to a letter 
