▁The ▁field ▁ Ω ( t ) ▁is ▁ordered ▁by ▁putting ▁x > y ▁if ▁the ▁function ▁x ▁is ▁larger ▁than ▁y ▁for ▁sufficiently ▁large ▁re als . ▁An ▁element ▁x ▁of ▁ Ω ( t ) ▁is ▁called ▁finite ▁if ▁m < x < n ▁for ▁some ▁integers ▁m , n , ▁and ▁is ▁called ▁infinite ▁otherwise . ▁ ▁De hn ' s ▁semi - E uclidean ▁geometry ▁ ▁The ▁set ▁of ▁all ▁pairs ▁( x ,   y ), ▁where ▁x ▁and ▁y ▁are ▁any ▁( poss ibly ▁infinite ) ▁elements ▁of ▁the ▁field ▁ Ω ( t ), ▁and ▁with ▁the ▁usual ▁metric ▁▁▁▁ ▁which ▁takes ▁values ▁in ▁ Ω ( t ), ▁gives ▁a ▁model ▁of ▁E uclidean ▁geometry . ▁The ▁parallel ▁post ulate ▁is ▁true ▁in ▁this ▁model , ▁but ▁if ▁the ▁deviation ▁from ▁the ▁per pend icular ▁is ▁inf init es imal ▁( mean ing ▁smaller ▁than ▁any ▁positive ▁rational ▁number ), ▁the ▁intersect ing ▁lines ▁intersect ▁at ▁a ▁point ▁that ▁is ▁not ▁in ▁the ▁finite ▁part ▁of ▁the ▁plane . ▁ ▁Hence , ▁if ▁the ▁model ▁is ▁restricted ▁to ▁the ▁finite ▁part ▁of ▁the ▁plane ▁( points ▁( x , y ) ▁with ▁x ▁and ▁y ▁finite ), ▁a ▁geometry ▁is ▁obtained ▁in ▁which ▁the ▁parallel ▁post ulate ▁fails ▁but ▁the ▁sum ▁of ▁the ▁angles ▁of ▁a ▁triangle ▁is ▁ π . ▁This ▁is ▁De hn ' s ▁semi - E uclidean ▁geometry . ▁It ▁is ▁discussed ▁in ▁. ▁ ▁De hn ' s ▁non -