MAIN LINES OF MATHEMATICS COOPER 327 



more difficult, to prove that other Euclidean properties of distance 

 such as that two sides of a triangle are greater than the third side 

 hold. 



To understand the definition of "angle," one must know that from 

 any "point" A two imaginary tangents can be drawn to the circle r. 

 The slopes of these tangents are complex numbers, say ^i and tz] if 

 AXf AY are any two lines through A, with slopes mi, rrh, then ^ 



"angle AXAY''=^. log (^i-^i) (^^-^^) 



where 



^=^f—l. 



Now a geometry as rich in content as that of Euclid can be con- 

 structed. Indeed all the theorems of Euclidean geometry which do 

 not need the axiom of parallels are true in it — the celebrated "Pons 

 Asinorum" about isosceles triangles, theorems on congruence of tri- 

 angles, for instance; but the sum of the angles of a triangle is less 

 than two right angles, Pythagoras's theorem, is false, and there are 

 no similar triangles which are not congruent. 



Another model for non-Euclidean geometry — Riemannian geom- 

 etry this time — can be obtained by considering a sphere S, and letting 

 "point" mean a pair of points on S lying at opposite ends of a diam- 

 eter, and "line" a great circle (i.e., a circle in which S is cut by a plane 

 through its center). Here again, the theorems of Euclid which do 

 not depend on the axiom of parallels are valid ; the reason for choosing 

 pairs of points to mean "point" is that we wish to ensure that any 

 two "points" lie on one and only one "line"; but now there are no 

 "parallel lines" (think of the lines of longitude on the earth, which 

 are great circles and, though apparently parallel at the Equator, meet 

 at the Poles) and the sum of the angles in a triangle is greater than 

 two right angles. 



Thus we have found sets of objects constructed in terms of concepts 

 of Euclidean geometry which obey non-Euclidean geometry. A set 

 of objects which obeys the axioms of a mathematical system is called 

 a model for that system. We have here models for both Lobachev- 

 skian geometry, in which through every point in a plane with a line 

 there is an infinite number of parallels to that line, and for Rieman- 

 nian geometry, in which there are no parallels to that line. 



» This definition of "angle" may be made clearer by the observation that In ordinary 

 Euclidean geometry, the angle between two lines with slopes nh and nij Is equal to 



1. (mi-i) (Tn»+t) 

 2« ^(m,+«) (mj-0 

 80 that in Euclidean geometry the two tangents to V are replaced by the two lines through 

 A of slopes i and —i; these are the lines joining A to the circular points at infinity. 



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