gly 
clear that if /\ X YZ were to be given the rank of rt. 7 A” B’C” and a new one 
were to be formed from it, as it is formed from 2 A”B”’C”, then the point S” 
would fall upon H. ‘Therefore, since H is the in-center of /\ HaH»He, 8S” must 
be the in-center of /(\ MayMp Mev. i 
Therefore we see that the six Simson’s Jines, three with reference to one /\ 
and three with reference to the other, meet in the same point. 
32. This, at the same time, establishes another even more interesting propo- 
sition, namely: If the Simson’s lines of the vertices of a first /\ with reference to 
asecond /\ concur in a point 8”, then the Simson’s lines of the vertices of the 
second /\ with reference to the first /\ concur in the same point S”. 
“The broad scope covered by this proposition would enable me to double in 
number the points of concurrency of Simson’s lines, but there would be little 
benefit in merely pointing them out, as the interested reader can easily see them 
for himself. 
A BrBLioGRAPHY OF FOUNDATIONS OF GEOMETRY. By Morton CLARK BRADLEY. 
Euclid’s treatment of parallels and angles and his definitions and 
axioms—particularly his twelfth—are the points of controversy that cause 
the most discussion. For nearly twenty centuries Euclid’s work remained 
unquestioned. Since John Kepler’s day, however, there have been new 
theories constantly advanced, theories built on axioms and definitions, 
a part of which, at least, are different from those of Euclid. The most 
important of the non-Euclideans are John Bolyai, Lobatschevski, Helm- 
holtz, Riemann, Clifford, Henrici, Caley, Sylvester and Ball. The most 
prominent exponent of the non-Euclidean ideas in this country is Prof. 
Geo. Bruce Halsted, of Texas University. These mathematicians hold 
that Euclid’s twelfth axiom is not, strictly speaking, an axiom—that it is 
not “a self-evident and necessary truth,” but that it requires demonstra- 
tion. They claim, too, that his definitions are not sufficient nor necessarily 
intelligible. Some of these men have built up new theories upon their 
substituted axioms and definitions, retaining those of Euclid that fit their 
theories. A few of these “reform’’ works are mere quibbles on words, but 
others deserve the serious consideration of all interested in pure geometry. 
The list following is a complete list of English references to be found 
in the mathematical library of the University of Indiana or in the private 
