Page:Popular Science Monthly Volume 68.djvu/28

24 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 Riemann, 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 'Neuere Geometrie' in 1882.

The advance of Pasch beyond Euclid consists essentially in the clear perception of the notions undefined element and unproved proposition. 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 surface.' 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.

3. If a point C lies in a segment AB, then the point A does not lie in the segment BC.

4. If a point C lies in a segment AB, then so do all the points of the segment AC.

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