OCTOBER 20, 1899. ] 
“You may have pronounced this a thank- 
less task, since lofty genius, above the 
windings of the valleys, steps by the Alpine 
peaks; but truly everywhere are present 
gordian knots needing swords of giants. 
Nor for these was this written. 
‘“‘Forsooth Iswish the youth by my ex- 
ample warned, lest having attacked the 
labor of six thousand years, alone, they 
wear away life in seeking now what long 
ago was found. Gratefully learn first what 
predecessors teach, and after forethought 
build. Whatever of good comes, is ante- 
cedent term of an infinite series.” 
His analysis of space starts with the 
principle of continuity: spatium est quan- 
titas, est continuum (p. 442). This Euclid 
had used unconsciously, or at least without 
specific mention ; Riemann and Helmholtz 
consciously. Second comes what he calls 
the axiom of congruence, p. 444, § 3, ‘‘ corpus 
idem in alio quoque loco videnti, queestio 
succurrit: num loca ejusdem diversa equalia 
sint? Intuitus ostendit, eequalia esse.” 
Riemann: ‘‘Setzt man voraus, dass die 
Korper unabhangig von Ort existieren, so 
ist das Krimmungsmass wberall constant.”’ 
See also the second hypothesis of Helmholtz. 
Third, any point may be moved into any 
other ; the free mobility of rigid bodies. If 
any point remains at rest any region in 
which it is may be moved about it in in- 
numerable ways, and so that any point 
other than the one at rest may recur. If 
two points are fixed, motion is still possible 
in a specific way. Three fixed points not 
costraight prevent all motion (p. 446, § 5). 
Thus we have the third assumption of 
Helmholtz, combined with his celebrated 
principle of Monodromy. 
Bolyai Farkas deduces from these as- 
sumptions not only Euclid but the non- 
Euclidean systems of his son JAnos, refer- 
ring to the approximate measurements of 
astronomy as showing that the parallel 
postulate is not sufficiently in error to in- 
_ SCLENCE. 
555 
terfere with practice (p. 489). This is just 
what Riemann and Helmholtz afterward 
did, only by casting off also the assumption 
of the infinity of space they got also as a 
possibility for the universe an elliptic geom- 
etry, the existence of a case of which inde- 
pendently of parallels was first proven by 
Bolyai Janos when he proved spherics in- 
dependent of Euclid’s assumption. So if 
Sophus Lie, had ever seen the ‘ Tentamen,’ 
he might have called his great investiga- 
tion the Bolyai-Farkas Space Problem in- 
stead of the Riemann-Helmholtz Space 
Problem. 
The first volume of the ‘Tentamen’ as 
issued by the Hungarian Academy does not 
contain the famous appendix. But in 1897, 
Franz Schmidt, that heroic figure, ever the 
bridge between Janos and the world, issued 
at Budapest, the Latin text of the Science 
Absolute, with a biography of Bolyai Janos 
in Magyar, and a Magyar translation of the 
text by Sutak Jozsef. 
Strangely enough, though the Appendix 
had been translated into German, French, 
Italian, English, and even appeared in 
Japan, yet no Hungarian rendering had 
ever appeared. It was Franz Schmidt who 
placed the monument over the forgotten 
grave of Janos, only identified because there 
still lived a woman who had loved him. 
Now in this Magyar edition he rears a 
second monument. The introduction by 
Sutak is particularly able. 
The Russians have honored themselves 
by the great Lobachévski Prize ; why does 
not that glorious race, the Magyars, do 
tardy justice to their own genius in a great 
Bolyai Prize? 
One other noble thing the Hungarian 
Academy of Science has just achieved, the 
publication in splendid quarto form of the 
correspondence between Gauss and Bolyai 
Farkas : (Briefwechsel zwischen Carl Fried- 
rich Gauss und Wolfgang Bolyai). It was 
again Franz Schmidt who, after long en- 
