By P. K. Jain, Ahmed Khalid

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10. INDEPENDENCE OF THE AXIOM OF PARALLELS 33 into itself. By suitable interpretations it can be seen that in this "non-Euclidean " geometry all axioms except Euclid's Axiom IV are valid and since the existence of ordinary geometry has been proved in Section 9 the existence of non-Euclidean geometry follows now. , those which hold in Euclidean as well as in non-Euclidean geometries. As the most important examples two of Legendre's theorems will be given, the ftrst of which requires for its proof besides Axioms I through III also Archimedes' Axiom V, 1.

In the arithmetic of segments, the word "equal" will be used instead of "congruent" and the sign "=" instead of" ==". (--- b _ _ and b = BC is denoted by c = AC and is expressed as c = a + b. The segments a and b are said to be smaller than c. Symbolically, a

Denote a . the vertex of "2 by A. If e is constructed on the side AB in the interior of the angle 2\. BAC = : then by the assumed inequality the third side meets the segment BC at a point D (cf. p. 12). Archimedes' Axiom V, 1 amounts to the assertion that there exists a natural number n such that n • BD > Be. Construct now the angle e on the resulting third side towards the outside n times. There can be a case in which at the last n-th construction, the resulting third side no longer meets the ray BC, and, say, that the m oth construction is the first one at which this occurs.