May 10, 1912] 
horrible piece of nomenclature, which no one 
should adopt, since this name belongs to the 
system of Monge, 1794, for representing solids 
in a plane, though also used by Sylvester for 
a geometry excluding all notions of quantity, 
such as my “Synthetic Projective Geometry.” 
The article proceeds to 
the simplest statement of all. Descriptive Geom- 
etry is then conceived as the investigation of an 
undefined fundamental relation between three 
terms (points); and when the relation holds be- 
tween three points A, B, C, the points are said to 
be ‘‘in the [linear] order ABC.’’ 
O. Veblen’s axioms and definitions, slightly 
modified, are as follows: 
1. If the points A, B, C are in the order ABC, 
they are in the order CBA. 
Dr. R. L. Moore (October 26, 1907) says this 
may be divided into parts, 1, inserting “ dis- 
tinct ” before “points”; and 1, inserting “ not 
all distinct,” after “ points.” 
2. If the points 4, B, C are in the order ABC, 
they are not in the order BCA. 
3. If the points A, B, C are in the order ABC, 
then 4 is distinct from C. 
4, If A and B are any two distinct points, there 
exists a point C such that A, B, C are in the order 
ABC. 
Dr. R. L. Moore modifies this to 4’ by in- 
serting “different from A and from B,” be- 
fore “such.” Then follow a definition, Def. 1, 
and axioms 5, 6, 7. Both in this definition, 
and in axiom 5 the shocking misprint occurs 
of using the symbol =, “plus or minus,” for 
the symbol =, “is not equal to.” 
Dr. R. L. Moore had already in 1907 sur- 
prisingly simplified this set of assumptions by 
proving that 1, is a consequence of 2 and 5 
and Def. 1, while 1, and 3 are both conse- 
quenees of 2, 4’, 5, 6, 7 and Def. 1.* 
Lobacheyski [or Bolyai] constructed the first 
explicit coherent theory of non-Euclidean geom- 
etry, and thus created a revolution in the philos- 
ophy of the subject. For many centuries the 
speculations of mathematicians on the foundations 
of geometry were almost confined to hopeless at- 
tempts to prove the ‘‘parallel axiom’’ without the 
introduction of some equivalent axiom. 
‘Trans. Amer. Math. Soc., Vol. XIII., No. 1, 
pp. 74-76. 
SCIENCE 
739 
In the Bibliography, Whitehead says of 
Lobachevski: 
His first publication was at Kazan in 1826. 
This is a mistake. In 1836 in his “ Intro- 
duction to New Elements of Geometry,” of 
which I was the first to publish a translation,’ 
he says: 
Believing myself to have completely solved the 
difficult question, I wrote a paper on it in the 
year 1826: ‘‘Exposition succinecte des principes de 
la Géométrie, avec une démonstration rigoureuse 
du théoréme des paralléles,’’ read February 12, 
1826, in the séance of the physico-mathematic 
faculty of the University of Kazan, but nowhere 
printed. 
No part of this French manuscript has ever 
been found. The latter half of the title is 
ominous. For centuries the world had been 
deluged with rigorous (!) demonstrations of 
the theorem of parallels.’ 
Saccheri’s book of 1733, containing a coher- 
ent treatise on non-euclidean geometry, of 
which I published the first translation, ended 
with another “demonstration rigoureuse du 
théoréme des paralléles.” If Saccheri had real- 
ized (as Father Hagen writes me he did) the 
pearl in his net, he could, with the new mean- 
ing, have retained his old title, Euclides ab 
omni naevo vindicatus, since the non-euclidean 
geometry is a perfect vindication and explana- 
tion of Euclid. 
But Lobachevski’s title is made wholly in- 
defensible. A new geometry, founded on the 
contradictory opposite of the theorem of paral- 
lels, and so proving every demonstration of 
that theorem fallacious, could not very well 
pose under Lobachevski’s old title. He him- 
self never tells what he meant by it, never 
tries to explain it. 
The title of Engel’s book already given 
erroneously in the Bibliography under VL., is 
now, under VII., given again with the former 
and two additional errors. 
After Riemann we see Gesamte Werke in- 
stead of gesammelte Werke. 
In the title of Poncelet’s work, on page 736, 
an accent is omitted which is given in the 
5¢‘Neomonie Series,’’ Vol. V., 1897. 
