MARCH 22, 1901.] 
pages) is certainly stimulating, and is adorned 
by a-full-page portrait of Lobachevski, taken 
from that given in the Kazan edition of his col- 
lected works. 
But there is extant another picture of Lo- 
bachevski far more impressive, showing him in 
the plenitude of his powers, which I first saw 
at Kazan, a daguerreotype from the life, a copy 
from which you may see as the frontispiece of 
Engel’s monumental ‘ Nikolaj Iwanowitsch Lo- 
batschefskij.’ 
As for Boyer’s account of what he rates so 
high, it begins as follows: ‘‘ From long ago it 
has been sought to demonstrate the famous 
axiom laid down twenty centuries ago by 
Euclid, to wit: through a point only one par- 
allel to a given straight can be drawn.”’ 
This of course is not Huclid’s postulatum, 
but rather a paraphrase of what is called (even 
by Cajori) Playfair’s axiom, though Playfair 
explicitly credits it to Ludlam, namely : ‘‘ That 
two straight lines which cut one another can 
not be both parallel to the same straight line.’’ 
Boyer continues: ‘‘ These attempts remained 
unfruitful. However, at the end of the 
eighteenth century, an Italian Jesuit, Saccheri, 
wished to found a geometry resting on a prin- 
ciple different from the celebrated postulate.’’ 
It was certainly not the end of the eighteenth 
century, for Saccheri died in 1733. Nor did he 
wish to set up any geometry different from 
Euclid, since the very title of his book was 
‘Euclid vindicated from every fleck.’ 
“Finally,’’ continues Boyer, ‘‘at the open- 
ing of the nineteenth century, a Russian, 
Lobachevski, and a Hungarian, John Bolyai, 
perceived at about the same time the impos- 
sibility of this demonstration.?’ In the index, 
citing to this page, Boyer gives as the dates of 
the birth and death of John Bolyai 1775-1856, 
the dates for his father Wolfgang Bolyai 
(Bolyai Farkas). Lombroso makes this same 
confusion and identification of father and son, 
and from it draws testimony for his thesis that 
great wits to madness are allied. 
John Bolyai (Bolyai J&nos) was born De- 
cember 15, 1802, and died January 27, 1860. 
At the celebration of his centenary next year 
in Hungary I hope to be present. Boyer 
continues: ‘‘ Their works published independ- 
SCIENCE. 
463 
ently of each other had without doubt been 
inspired by the doctrines of the philosopher 
Kant who, in a passage of his Critique of the 
pure reason, indicated a new consideration of 
space. For this latter, space existed a priorz, 
precedent to all experience, as completely 
subjective form of our intuition.”’ 
In regard to this bold attribution of influence, 
I may be allowed to say that not a particle of 
evidence has appeared to show that John Bolyai 
ever heard of even the existence of Kant. I 
examined Bolyai’s papers, his correspondence, 
his ‘ Nachlass’, at Maros-VAsarhely, and never 
found even the name of Kant. As for Loba- 
chevski, he might have had his attention 
called to Kant by Bronner the professor of 
physics at Kasan, at one time an admirer of the 
‘Kritik der reinen Vernunft,’ but that Kant 
influenced him is merest conjecture, and unnec- 
essary, since we sufficiently know the path of 
his mental on-going. Of Lobachevski’s doc- 
trine, Boyer says, p. 242, ‘‘ He declares at the 
beginning the following axiom: through a 
point can be drawn many parallels to a given 
straight.”’ 
What Lobachevski does assume is that 
through a given point can be drawn innumer- 
able distinct straight lines in a plane which will 
never meet a given straight in that same 
plane ; but of these, only those two are parallel 
to the given straight which approach it asymp- 
totically. 
Continuing, Boyer says of Lobachevski, p. 245, 
“When he died in 1856 he occupied the posi- 
tion of Rector of the University which he had 
entered as simple student.”’ 
Unfortunately Lobachevski had been deprived 
of his position of Rector for ten years before he 
died. 
Passing on to Riemann’s geometry, Boyer 
says: ‘'To construct this Geometry, its invent- 
or rejects the postulatum and the first axiom 
of Euclid : two points determine a straight.’ 
But the postulatum is: ‘‘ And if a straight 
cutting two straights makes with them interior 
angles lying on the same side, which together 
are less than two right angles, then the two 
straights intersect if continually produced on 
the side upon which these angles lie.”’ 
In the ‘spherical’ or Riemannian geometry 
