206 
STKCKER. 
there must be at least two equally valid conclusions. In 
brief, mathematicians have long since learned that there are 
several systems of geometry, each consistent with all the facts 
of experience. Euclid’s is the simplest of these systems, and 
we use it because it is the simplest, and for no other reason. 
Which of these systems is the true geometry of our space we 
cannot, in the nature of things, know. What more interest¬ 
ing topic than the study of this whole problem ? But we 
can touch here only on recent and important advances in this 
field of mathematics, giving such references, in their proper 
connection, as will enable the reader to refer to the past his¬ 
tory of the subject, should he so desire. 
FOUNDATIONS OF GEOMETRY. 
The question as to the necessary and sufficient premises 
upon which geometry may be constructed, and of their mu¬ 
tual independence and compatibility, is of vital importance. 
Various contributions to this subject culminated in 1899 in 
Hilbert’s classic* paper, “Grundlagm der Geometrie Fest¬ 
schrift zur Feier der Enthullung des Gauss-Weber-Denkmals 
in Gottingen. During the past year there has appeared an 
English f translation. It was translated into French in 1900. 
Out of this critical and fruitful piece of investigation has 
grown all the recent work in this field of mathematics. Two 
papers, one by Schur,J under the title “Uber die Grundlagen 
der Geometrie,” and one by Moore,§ on “ The projective 
axioms of Geometry ” have appeared in the last year. They 
have attempted (a) to modify the system of Hilbert with a 
view to simplicity; (b) Moore announced in his paper that 
he had found a redundancy in Hilbert’s system. In the cur¬ 
rent number of the Transactions of the American Mathe¬ 
matical Society, Moore has stated that he was in error, and 
that the redundancy does not exist. A careful comparison 
*See pp. 1, 6, 24, 48 for references to early literature, 
t Open Court Publishing Company, Chicago, Ill. 
t Math. Annalen, Bd. 55, p. 265. 
\ Trans, of the Ain. Math. Soc., Jan., 1902, vol. 3, p. 142. 
