CHAPTER 1
Metric Space s
1.1. Legendr e o n hyperboli c geometr y
The paralle l axio m o f plan e Euclidea n geometr y claim s tha t give n a poin t P ,
and a lin e t no t passin g throug h P , ther e exist s on e an d onl y on e lin e throug h
P tha t doe s no t mee t L Th e axio m wa s controversia l ove r man y centuries , an d
much effor t wa s devote d t o investigation s aime d a t provin g o r refutin g it . A s w e
know, the result of these investigations was the discovery of hyperbolic geometry by
Gauss, Bolya i and Lobatchevskii . Hyperboli c geometr y wil l provide a key exampl e
throughout thi s course, and w e will spend som e time reviewing its basic properties .
For ou r introduction , though , I wan t t o conside r a n argumen t give n b y Le -
gendre1
which was aimed at proving th e parallel axiom or the equivalent statemen t
that angl e su m o f eac h triangl e i s n radians . Thi s particula r argumen t aime d t o
show that th e 'acute-angl e hypothesis' , namel y tha t th e angl e sum o f each triangl e
is les s tha t n radians , woul d lea d t o a contradiction . Legendr e argue s a s follow s
(see figure 1 below).
Suppose that AB C is a triangle whose angle sum i s less than 71, and le t 6 be it s
defect, tha t i s the amount by which its angle sum falls short of n. Construc t anothe r
triangle CBA ' b y rotating AB C through n abou t th e midpoin t o f BC. (Notic e tha t
no theory o f parallels is used here , although o f course in our Euclidea n pictur e BA '
and CA ' look paralle l t o A C an d A B respectively. ) No w dra w an y lin e throug h
A' tha t meet s A B an d A C produce d a t B ' an d C respectively , an d conside r th e
triangle A B 'C I t i s made up of four sub-triangles , tw o of which (namel y ABC an d
the congruen t triangl e CBA' ) hav e defec t 6 , an d th e othe r tw o o f whic h (namel y
BB'A' an d CC'A' ) hav e a t leas t non-negativ e defect . I t i s easy t o show , however ,
that th e defec t i s additive: th e defec t o f AB'C i s the su m of the defect s o f the fou r
sub-triangles, an d is therefore a t least 26. Iteratin g this procedure we can construc t
triangles whos e defec t i s arbitraril y large . Bu t clearl y n o triangl e ca n hav e defec t
greater tha n n , an d thi s contradictio n 'proves ' th e paralle l postulate .
As Gra y [23 ] writes , 'I n spottin g th e fla w yo u wil l discove r mor e abou t th e
alien nature of non-Euclidean geometr y than by following an y texts.' Th e argumen t
depends o n th e hypothesi s tha t give n a n angl e (th e angl e BA C i n ou r figure ) an d
a poin t (suc h a s A' ) containe d betwee n th e tw o arm s o f th e angle , on e ca n fin d a
line through th e give n point tha t meet s bot h th e arm s o f the give n angle . Bu t thi s
hypothesis i s no t a geometrica l axio m an d i t i s eas y t o se e tha t i t fail s i n an y o f
For references , an d muc h mor e extensiv e historica l discussion , se e th e boo k o f Gra y [23] .
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http://dx.doi.org/10.1090/ulect/031/01
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