SEPTEMBER 24, 1897. ] 
from W. Bolyai of September 16, 1804, accom- 
panied by a Latin treatise, Theoria parallelarum. 
It read as follows: ‘‘Now * * * yetsomewhat 
about your geometric communication. I have read 
through your treatise with great interest and atten- 
tion, and am right delighted atits really profound 
keenness. But you do not wish my empty praise, 
which also might seem in a measure partial because 
your train of ideas has very great resemblance to the 
way I formerly sought the untieing of this Gordian 
knot and vainly seek till now. You wish only my 
candid, open judgment. And this is, that your pro- 
cedure does not give me satisfaction. I will seek, 
with as much clearness as I can, to bring to light the 
stone of stumbling which I still find therein (and 
which also again pertains to the same group of rocks 
wheron my attempts have hitherto been wrecked). 
“T have indeed yet ever the hope that those rocks 
some day, and even before my end, will grant a thor- 
oughfare. Meanwhile I have now so much other 
business on hand that I at present cannot think 
thereon, and, believe me, it will heartily delight me 
if you precede me and attain to overmaster all ob- 
stacles. I would, then, with inmost joy, do allin my 
power to make your service current and put it in the 
light. 
* * * * * * a * * 
Could you prove dke = ekf = fkg, etc., then would 
the thing be perfect. But this theorem is iniJeed true, 
only difficult to prove rigorously without already 
presupposing the theory of parallels. eo ee 
You have my candid judgment. I have given it, 
and I repeat that it would genuinely delight me if 
you overcome all difficulties.’’ 
Here we see, with startling clearness, that in 
1804 both Gauss and W. Bolyai (Bolyai Farkas) 
believe that Euclid’s Parallel-Postulate can be 
proven, and indeed are racing to demonstrate it. 
Before the next letter the unaided genius of 
the son, Bolyai Jénos, has created the new uni- 
verse, has found out all about it, mapped it, and 
proved Euclid’s Postulate forever indemon- 
strable. 
In transmitting in print to Gauss the im- 
mortal treatise of his son Janos, the most mar- 
vellous two dozen pages in the whole history of 
human thought, the father, Farkas, writes on 
June 20, 1831: 
““My son is already First Lieutenant in the Engi- 
gineering Corps, and will soon be Captain, a hand- 
some youth, a virtuoso on the violin, a fine fencer 
and brave, but has often dueled, and is still alto- 
gether too wild a soldier—but also very refined—light 
SCIENCE. 
489 
in darkness and darkness in light, and an impassioned 
mathematician with very rare gifts of mind. At 
present he is in the garrison at Lemberg—a great 
admirer of you—capable of understanding and appre- 
ciating you. At his desire, I send you this little 
work of his. Have the gooduess to judge it with 
your sharp, penetrating eye, and to write your high 
judgment unsparingly in your answer, which I ar- 
dently await. It is the first beginning of my work, 
which is under the press. I would gladly send with 
this the first volume, but it is not yet out. 
“ According to my view, is in the work of my son, 
u-(namely, where a first does not cut the b) geometri- 
cally constructed ; whence, however, is not deter- 
mined how great wis, from O on up to #& (that ex- 
cluded, this included). 
“Yet everything in geometry is either dependent on 
u or not; (e. g.) spherical trigonometry is in ¢ 26 set- 
tled as independent of it. * * * 
“At the end he also shows that if w not =F, then 
the circle can be squared.’’ 
Thus we see that the treatise sent to Gauss 
on June 20, 1831, was the immortal Appendix 
just as published. The Gordian knot, at which 
Gauss himself had for years tried in vain, was 
here forever gloriously untied. 
After waiting six months the anxious father 
tries again, and on January 16, 1832, once more 
sends Gauss the work of Jénos, saying in the 
accompanying letter : 
‘My son was not present when his little work was 
printed. He had printed the errata (which follow) ; 
in order to be less burdensome to you, I have cor- 
rected the most with a pen. 
“* He writes from Lemberg that he has since simpli- 
fied and made more elegant many things, and has 
proven the impossibility of determining a priori 
whether Axiom XI be true or not.’’ 
To this, on March 6, 1832, comes from Gauss 
at length an answer as follows: 
* * * “Now somewhat about the work of your 
son : 
“Tf [ thus begin ‘that I dare not praiseit’ you will 
a moment wonder ; but I cannot otherwise ; to praise 
it would mean to praise myself. For the whole con- 
tent of the book, the way your son has hit out and 
the results to which he is led, are identical almost 
throughout with my own meditations, made in part 
already 30-35 years ago. In fact, I am thereby ex- 
tremely delighted. My intention was to let nothing 
be known during my lifetime of my own work, of 
which moreover until now little has been put on 
paper. Most men have not at all the right sense for 
