24 POPULAR SCIENCE MONTHLY 



suited a new edifice of science which, while different from the old and 

 containing many a strange proposition, yet never denied itself nor 

 violated any of the principles of logic. 



These results were obtained first by an Italian Jesuit priest named 

 Saccheri and timidly published in 1733. His work, however, has been 

 known to the modern world only very recently. The non-Euclidean 

 geometry was rediscovered by a Russian, Lobatchewsky (1826), and a 

 Hungarian, Bolyai (1832), though their work also remained unknown 

 to the world at large till 1866 when it fell under the notice of the 

 German mathematician Baltzer. The investigation of the parallel 

 axiom has been continued by Eiemann, Beltrami, Helmholtz, Sophus 

 Lie, Cayley, Klein, until it may fairly be said that, ten years ago, this 

 twelfth axiom of Euclid which had at first seemed such a stumbling- 

 block was better understood than any other of his definitions and 

 axioms. 



The next attempt after Euclid's to consider geometry as a whole 

 from a purely synthetic point of view was made by a German, Moritz 

 Pasch. His theory, delivered first in a course of lectures in 1873-4, 

 was published in a book called ( JSTeuere Geometrie ' in 1882. 



The advance of Pasch beyond Euclid consists essentially in the 

 clear perception of the notions undefined element and unproved proposi- 

 tion. In other words, he tries to state sharply just what concepts he 

 leaves undefined and does reduce the number of these much below 

 that of the elementary concepts employed by Euclid. He distinguishes 

 between his definitions and axioms. He aims to include in his axioms 

 every assumption that he makes. 



His undefined elements are ' point/ ' linear segment,' ' plane sur- 

 face.' These, according to the axioms, have relations such that a 

 point may be in a segment or a surface, a linear segment may be 

 between two points (called its end-points). There is also introduced 

 a relation called congruence (geometrical equality) of figures which 

 corresponds to the Euclidean idea of superposition. We will quote 

 only a few of Pasch's axioms, since they can not signify much apart 

 from the propositions developed out of them. 



1. Between two points there is always one and only one linear segment. 



2. In every linear segment there is a point. 



i i 3. If a point C lies in a segment AB, then 



y\ h in the point A does not lie in the segment BC. 



F 4. If a point C lies in a segment AB, then 



so do all the points of the segment AG. 

 5. If a point C lies in the segment AB, then no point can lie in AB which 

 does not lie in AC or CB. 



Out of these assumptions about the relations between points and 

 line segments, together with three other axioms, Pasch deduces the 



