128 TRANSACTIONS OF SCIENTIFIC SECTION. 
JANUARY 13, 1886—FORTY-SECOND REGULAR MEETING. 
Charles B. Warring, Ph.D., chairman, presiding. 
I’. Monteser, Ph.D., read a paper on ‘‘ The Axioms of 
Geometry,” of which the following is a brief abstract : 
He discussed the question whether the postulates 
which form the basis of our geometry are independent 
of experience or not. Kant assumed that they are, 
and made that assumption the cornerstone of his theory 
of knowledge avpriori. His argument was that the 
axioms of geometry have a character of true generality 
and necessity which no experience could possibly give 
to its results. Any assumption, contrary to the axioms, 
according to him, is not only false, but essentially in- 
conceivable. Modern mathematical investigations have 
overthrown that argument and taught us the real mean- 
ing of the axioms. It was shown by the researches of 
Gauss, Lobatschewsky, Beltrami, and especially Rie- 
mann and Helmholtz, how much of experimental facts 
there is reaily implied in those apparently so simple 
propositions, and how, starting from suppositions, con- 
trary to Euclid’s axioms, one could construct a per- 
fectly legitimate and logically consistent system of non- 
Kuclidian geometry. In other words, it was proved 
that our space is only a special form out of an infinite 
number of possible spaces, and that more extensive 
Measurements are needed before we can decidedly say 
that this actually existing space really has those proper- 
ties which we are accustomed to ascribe toit. Or to 
put the question in a very simple and practical form, 
we may state our point thus: If we have two straight 
lines, perpendicular to the same straight line, we know 
that they will not intersect—as far as we can prolong 
them. But we are not at all sure whether they would 
intersect or not when prolonged to a distance, compared 
with which the diameter of the earth’s orbit is exceed- 
ingly small. In this way geometry loses its trans- 
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