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...An axiomatic basis for plane geometry by stewart s cairns the axioms fourth appendix of hubert grundlagen der geo metrie is devoted to foundation on three per taining transformations into itself object present paper attain same end quicker and simpler means simplifica tions are made possible using orientation reversing changing second third x y will mean set all distinct ordered pairs real numbers terms analytic which no geometric content need be given used modified prefix where ambiguity might arise thus we shall refer distance lines so general p any objects called points can put in one correspondence with convenience speak as if they were identical their images under such a following pertain t continuous trans formations transformation leaves two dis tinct fixed reverses reflection axiom form group b contains leaving let ta denote subset containing thereof leave carrying arbitrarily near pair c small neighborhood d e then curve our first establish theorem like lemma below similar res...