Edward Huntington - A Set of Postulates for Abstract Geometry, Expressed in Terms of the Simple Relation of Inclusion

7/28/2019 Edward Huntington - A Set of Postulates for Abstract Geometry, Expressed in Terms of the Simple Relation of Incl

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5 2 2 E .V . HWTJN~To~.

A set of postulates for abstract geometry, expressed in termsof the simple relation of inclusion.

ByE DW A R V V . H U n TIn GT ON o f C a m b r i d g e ( M a s s . ) , U . S . A .

Contents.P r e f a t o r y N o t e . Prec.

C h i e f p o i n t s o f d i~ e r e n e e b e t w e e n t h i s a n d e a r l i e r a rt i cl e s . I n d e b t e d n e s s o ft h e w r i t e r t o o t h e r a u t h o r s . . . . . . . . . . . . . . . . . . 5 2 2 - -5 2 5

I n t r o d u c t i o n , P l a n o f t h e s r t i e l e .S u m m a r y o f t h e p r i n c i p a l r e s u l t s . . . . . . . . . . . . . . . . . . . 525--"5 2 8 .

C h a p t e r I . D e f i n i t i o n s .F o r p l a n e g e o m e t r y , D efs . 1 - - 2 6 . . . . . . . . . . . . . . . . . . . . 5 2 9 ~ 5 3 5P e r g e o m e t r y o f t h r e e d i m e n s io n s , D e fs . 1 - - 3 2 . . . . . . . . . . . . 5 2 9 - -5 3 7

C h a p te r I I. T h e P o s t u l a t e s .G e n e ra l l a w s , P o s t u la t e s 1 - - 1 8 ; E x i s te n c e p o s t u l a t e s , E 1 - - E 7 . . . . . . 5 8 7 k 5 4 0C o n s is te n c y o f t h e p o s t u l a t e s . . . . . . . . . . . . . . . . . . . . . 540

C h a p t e r I lL T h e o r e m s .S u f fi ci e nc y o f t h e p o s t u l a t e s t o d e t e r m i n e s u n i q u e t y p e o f s y s t e m . . . . 5 4 1 - - 5 4 8

C h a p t e r I V . I n d e p e n d e n c e o f tl~ e p o s t u l a t e s .E x a m p le s o f p s e u d o -g e o m e t r i e s . . . . . . . . . . . . . . . . . . . . 5 ~ 8 - -5 5 ~

Appendix.P ro o fs o f t h e o re m s in c h a p t e r H I . . . . . . . . . . . . . . . . . . . 5 6 4 ~ DP r o f a t o r y N o t e .

(T h i s p re fa to ry n o te i s in ten d ed mere ly to p o in t o u t th o se f ea tu reso f th e a r t i c le w h ich a re l ik e ly to h e o f m o s t in tere s t to th e rea der w h ois pressed for t im e, and wh o i s a lready fam il iar w ith the l i terature o f th est~bjeet . The art ic le proper beg ins with the Introduct ion . )

T h e su b ject -ma t ter o f th e p resen t a r t ic l e i s s n ew se t o f p o s tu la te sfor ordinary ~i dean three-dimensional g~. T h e m e t h o d e m p l o y e d


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Postulates for abstra ct Geom etry. 5~ 3c a n b e r e a d i l y e x te n d e d t o E u c l i d e a n g e o m e t r y o f m o r e t h a n t h re ed im e n si on s~ b u t i s n o t s o r e ad i ly ~ a d a p t e d t o t h e s t u d y o f p r o j e c ti v eg e o m e ~ y .

T h e c h i e f p o i n t s o f d i ff e re n c e b e t w e e n t h e p r e s e n t s e t o f p o s t u l a t e sa n d t h e w e l l k n o w n se ~s* ) g i v e n b y P a s c h , Y e r o n e s e , P ea n o~ P i e r i , H i l b e r t ,V eb le n~ S e h w e i t z e r ~ a n d o t h e rs ~ a r e : 1 ) t h e u s e o f t h e s o l i d b o d y in s t e a do f t h e p o i n t a s a n u n d e f i n e d c o n c e p t ; ~ ) t h e e x t r e m e si m p l i c i t y o f t h eu n d e f i n e d r e l a t i o n o f i n c l u s i o n ; 3 ) t h e s y s t e m a t i c d e f i n i ti o n s o f t h e s t r a i g h tl in e , t h e p l a n e , a n d t h e 3 -s p a c e , w h i c h c a n b e r e a d i l y e x t e n d e d , i f d esire d~t o s p a c e o f n d i m e n s i o n s ; a n d ( 4 ) t h e a t t e m p t t o s e p a r a t e t h e ~ e xi st en c ep o s t u l a t e s ' f r o m t h e p o s t u l a t e s e x p r e s s i n g ' g e n e r a l l a w s ' .

I n r e g a r d t o t h i s l a s t p o in t~ a w o r d o f e x p l a n a t i o n m a y b e d e s ir ab l e.B y a n C e xisten ce p o s t u l a t e ' w e m e a n a p o s t u l a t e t h a t d e m a n d s t h e e x i s te n c eo f s o m e e l e m e n t s a t i s f y i n g c e r t a i n c o n d i t i o n s ; a s , f o r e x a m p I e , t h e p r o -p o s i t i o n t h a t a l in e p a s s in g t h r o u g h a v e r t e x o f a tr i a n g l e a n d a n yi n t e r i o r p o i n t m u s t i n t e rs e c t ~ he o p p o s i t e si d e ; o r t h e p r o p o s i t i o n t h a tt h r o u g h a n y p o i n t o u t s id e a g i v e n l i n e i t i s a l w a y s p o s s i b le t o d r a w a tl e a s t o n e p a r a l le l . B y a ' g e n e r a l l a w ' w e m e a n a p r o p o s i t io n o f t h e f o r m :~ i f s u c h a n d s u c h p o i n t s , li n e s , e t c. e x i s t , then s u c h a n d s u c h r e l a t i o n sw i l l h o l d b e t w e e n t h e m ' ; f o r e x a m p l e , t h e p r o p o s i t i o n t h a t i f B i s b e t w e e n~ [ a n d C , a n d X b e t w e e n X a n d B , t h e n X i s b e t w e e n A a n d C ; o r

*) M. Pasch , Ve rlesungen fiber neuere Ge om etrie, Leip zig 1882.G. Yeronese, Fondam enti eli Geom etrla a pi5 dimensioni~ 18 91 , translaf~d intoGerman by A. Seh epp , Grnndz~ge der Geometrle, 189~.G. Pean o, I Principii di Geom etrla~ Tu rin 18 89 ; also Sui fondamenti dellage om etria , in R ivista di Ma~em atica 4, p. 51--90, 1894.M. Pie ri, Della geom etria elementare com e sistema ipote~ico ded uttivo ; mon o-grap hia del pun to e del mote, Mem orie della Reale Accademia della Scienze di Torino,(2) 49, p. 173--222, 18 99 ; also Sur la g6omg~rie envlsagge co m m e un syst~m epurem ent logique, Bib l iot~q ue du Congr~s internat ion al de Philosophie, Par is 19003, p. 367--404.D. H ilbe rt, Grundlagen tier Geome~rie, Gaul]-Weber Festsehrift, 1899; tran slate dinto English by E. J . Townsend, The Foundations of Geometry, 1902; third Germanedition, as eel . 7 of th e serie s called Wissensc haft un d H ypo these , (Leipzig 1909).In this third edition, the numbering o f the a~io m s is s l ightly al tered, in vie w of anart icle by E. H. ~o ore , 1902; Ax iom II 4 of~h e original l is t is now om itted, and w hatwas origina lly Ax iom II 5 is now num bered II 4.O. Veblen, A system o f Axioms for Geomet~y~ Trans. Am . M ath. S ee. 5 ~1904),p. 843 --~8 4. Also, T he Foundations of G eometry, in a volum e called Monographson To pic s of Modern ]~[a~hematics relevant ~o the Elem entary F ield , edited byJ . W . A. Youn g, p . 1~ 51 , 1911.Another set o f postu lates , based on concepts n ot so closely connected w ith thepresen~ work, ha s been recently given by A. R. Sehweitzer, A ~ heo ry of geom etricalrelations, Am . ft. o f ~a~h .~ 31 (1909), p. 365--41 0.

34*


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5 2 4 E .V . HunTIngTon.th e p ro p o s i t io n th a t i f two d i st in c t l in es a re p a ra ll e l t o a th i r d l in e th e ya r e p a r a l l e l t o e a c h o t h er . N o w i n t h e u s u a l d e v e l o p m e n t o f t h e s u b j e c t ,t h e d e m o n s t r a t io n o f m a n y o f t h e g e n e ra l l aw s is m a d e t o de p e n d u p o nth e u se o f ex is t en ce p o s tu la t es ; fo r ex amp le , i n p ro v in g th e s imp les t l aw so f o rd e r fo r p o in t s o n a lin e , H i lb e r t an d V eb len u se a ' t r i an g le t r an sv er s e 'p o s t u l a t e , c o n c e r n i n g t h e p o i n t s o f i n t e r s e c t i o n o f t h e s i d e s o f a t r i a n g l ew i t h a l in e in t h e p la n e. I n t h e p r e s e n t t r e a t m e n t , o n t h e c o n t r a r y , t h ea t t e m p t h a s b e e n m a d e t o se p a ra te t h e g e ne r a l l aw s f ro m t h e e x i s t e n c ep o s tu la t es , an d to p ro v e a l l g en era l l aws , a s f a r a s p o ss ib le , w i th o u t th ea id o f au x i l i a r ly ' co n s t ru c t io n l in es '. Th i s r e s t r i c t io n ad d s co n s id e rab ly tot h e d i ff ic u l ty o f m a n y o f t h e p ro o fs , b u t t h e a t t e m p t , t h o u g h n o t c o m p l e t e l ys~uccessfu l , has a cer ta in log ical in teres t ; and the los t s impl ic i ty o f p roofcan b e a t o n ce r eg a in ed , i f d es i r ed , b y t r an sp o s in g th e ex i s t en ce p o s tu la t es~0 an ear l ier p lace in the l i s t .

A m o n g t h e definitions, t h e m o s t i m p o r t a n t i s t h e new definitio~ o f~inear segment (Def . 5 ) , fo r on th is def in i t ion , and the analogous def in i t ionso f t r i a n g l e a n d t e t r a h e d r o n , t h e w h o l e t h e o r y is ba se d . A t t e n t i o n m a yalso be ca l led to the def in i tion o f the mid-point o f a s e g m e n t ( D e f . 1 7 ) ,th e d e f in i t io n o f congruence (Def . 21) , and to t he fac t tha t a l l m e t r i c~ o p e r t i e s a r e o b t a i n e d d i r e c tl y i n t e r m s o f t h e f u n d a m e n t a l c o n c e p t s ,w i th o u t th e in t e rv en t io n o f Cay ley ' s ' ab so lu te '. A l so , th e w o rd ' sp h e re 'm a y b e r e p la c e d b y ' a n y c o n v e x so lid ', e x c e p t i n t h e p a r t s o f t h e p a p e rt h a t d e a l w i t h c p n g r u e n c e .

O n t h e consistency o f th e p o s tu la tes , see th e en d o f Ch ap te r II~ w h er ea n i n t e r e s t i n g g e o m e t r y o f points of finite size i s exh ib i ted .

I n r e g a r d t o t h e ind~endence o f the ~ostulates, t h e ' g e n e r a l l a w s ' a r es h o w h t o b e i n d e p e n d e n t o f e a c h o th e r , a n d t h e e x i s t e n c e p o s t u l a t e s a r es h o w n t o b e i n d e p e n d e n t o f e a c h o t h e r a n d o f t h e g e n e r a l l a w s . B ys l ig h t ch an g es in wo rd in g , it wo u ld b e easy to secu re ' ab so lu te ' i n d ep en -d en c e fo r th e co mb in ed l i s t o f g en era l laws an d ex i s t en ce p o s tu la t es ; b u ~su ch ch an g es wo u ld t en d to in t ro d u ce n eed less a r t i f i c i a l i t i e s , f ro m wh icht h e p o s t u l a t e s a s t h e y n o w s t a n d a r e e n t i r e l y f r e e .

M o r e i m p o r t a n t t h a n t h e q u e s ti o n o f i n d e p e n d e n c e is t h e p r o o f o ft h e suffcie~xy of the postulates to determine a unique type o f system ; or, ~ou se a p h rase o f Veb len 's , t h e p ro o f th a t th e p o s tu la t es fo rm a categoricalset. L i t t l e a t t en t io n seems to h av e b een p a id to th i s q u e s t io n ex cep t b yt h e p r e s e n t w r i t e r i n co n n e c t io n w i t h t h e f o u n d a t i o n s o f a l ge b r a* ) ~ a n d

*) E. V. Huntington, A complete set of postulates for the theory of absolutecontinuous magn itude, Trans. A m . ~fath. See., 8 (1902), p. 264--279 , and la ter p apers.Compare also the forthcoming Lehrbueh der A ]gebra by A. Loewy.


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Postulates for abstract Geometry. 5 2 5b y V e b l e n i n c o n n e c t i o n w i t h t h e fo u n d a t i o n s o f g e o m e t r y ; an d y e t t h e r ea p p e a r s t o b e n o o t h e r w a y o f p r o v i n g t h a t a ll t h e p r o p o s i t io n s o f as c ie n c e a r e d e d u c i b l e f r o m a g i v e n s e t o f p o s t u l a t e s , t h a n b y s h o w i n gt h a t t h e p o s t u l a t e s f o r m a ' s u f f i c i e n t ' o r ' c a t e g o r i c a l ' s e t . I n t h e p r e s e n ta r ti c le , t h e d e d u c t i o n s f r o m t h e p o s t u l a t e s a r e c a r r ie d j u s t f a r e n o u g h t oe s t a b l i s h t h i s ' t h e o r e m o f s u f f ic i en c y ', w h i c h f o r m s , i n f a c t , a n a t u r a ls t o p p i n g p l a c e i n a n y s t u d y o f ' f o u n d a t i o n s ' .

T h e p r e v i o u s a u t h o r s t o w h i c h t h e w r i t e r is c h i e f l y i n d e b t e d a r eP e a n o , R u s s e l l * ) , H i l b e r t , V e b l e n , a n d S c h u r * * ) .

T h e l o g ic a l s y m b o l i s m o f P e a n o , a l t h o u g h n o t e m p l o y e d i n th e p a p e ra s p r e p a r e d f o r t h e p r e s s , h a s b e e n o f a l m o s t i n d i s p e n s a b l e v a l u e i n w o r k i n go u t t h e d e ta il s o f t h e d e m o n s t r a ti o n s . W i t h o u t s o m e s u c h s y m b o l i s m , iti s a l m o s t i m p o s s i b l e , i n w o r k o f g his s o r t , t o a v o i d e r ro r s .

T h e t e r m ' p r o p o s i t i o n a l fu n c t i o n ', a n d t h e e m p h a s i s o n th e n o t i o no f t h e ' v a r i a b l e ' ( i n t h e I n t r o d u c t i o n ) a r e d u e c h i e f l y t o R u s s e l l . T h ec o n v e n i e n t n o t a t i o n f o r t h e p r o l o n g a t i o n s o f a s e g m e n t , t h e e x t e n s io n so f a ~ r i an g l e , e t c ., i s t a k e n f r o m P e a n o . T h e ' f o u r - p o i n g p o s t u l a t e '( P o s t u l a t e 1 1 ) is t h e f ir s t h a l f o f a p r o p o s i t i o n s u g g e s t e d b y S c h u r . T h ev e r y s i m p l e p r o o f o f t h e c o m m u t a t iv e la w o f m u l t ip l ic a t io n ~ b y m e a n so f t h e p r o p o s i t i o n a b o u t t h e t h r e e a l t it u d e s o f a tr i a n g l e , is a ls o d u e t oS c h u r . P o s t u l a t e 1 4 , t h e l as t o f t h e p o s t u l a t e s o n e o n g r u e n c e , . w a s s u g -g e s t e d b y o n e o f t h e a s s u m p t i o n s in Y e b l e n ' s l a t e r l is t . T h e c o n s t r u c t i o nf o r d r a w i n g a l i n e p e r p e n d i c u l a r t o a g i v e n l i n e i s d u e t o H i l b e r t .

O n a c c o u n t o f t h e u s e o f th e s o l id b o d y i n s te a d o f th e p o i n t a s t h ef u n d a m e n i m l v a r i a b l e , m o s ~ o f t h e p s e u d o - g e o m e t r ie s g i v e n i n t h e p r o o f so f in d e p e n d e n c e h a d t o b e n e w c o n s tr u c ti o n s. E x a m p l e E 5 , h o w e v e r , ist a k e n f r o m H i l b e r t .T h e w r i t e r is a ls o i n d e b t e d t o M r . P . E . B . J o u r d a i n , w h o h a s v e ri fi e dm a n y o f t h e d e m o n s t r a ti o n s .

I n t r o d u c t i o n . P l a n o f t h e a r t i c l e .T h i s i n t r o d u c t i o n i s i n t e n d e d to e x p l a in t h e p o i n t o f v i e w a d o p te d i n

t h e p r e s e n t a r t ic l e , a n d t h e n a t u r e o f t h e p r i n c i p a l r e s u lt s o b t a in e d .Fuedamental coneelots . W e a g r e e ~ o c o n s i d e r a c e r t a i n s e t o f p o s ~ a t e s( n a m e l y , t h e p o s t u l a t e s s t a te d i n C h a p t e r l I ) , i n v o l v i n g , b e s id e s t h e s y m -

* ) B . R u s s e l l , P r i n c i p le s o f M a t h e m a t l c s , 1 9 03 ; A . N . W h i t e h e a d a n d B . R u s s e l l ,Pr in c ip ia M ath e m at i ca , 1 , 1911 . S ee ~ .l so W h i teh ea d ' s exce l l en t p op u lar In trod u ctionto M ath em at i cs , ( 'Hom e Un ivers i ty L ib rary' , Lo n d en 1911) .

**) F . S ch u r , Zu ~ Prop ort ion s l eh re , Math . An n . 57 (190~) , p . 205M208.


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5 ~ 6 E .V . HUN~I~QTO~.bols w h i c h a r e n e c e s s a r y f o r a l l lo g i c al r e a s o n i n g , o n l y t h e f o l l o w i n gtw o variables:

1 ) t h e s y m b o l K , w h i ch m a y m e a n a n y c lass o f demen t s A , B , C , . . . ; a n d2 ) t h e s y m b o l R , w h i c h m a y m e a n a n y re la t ion A R B , b e t w e e n ~ w o

e lemen t s .Th ese p o s tu la t es a re n o t d e f in i te p ro p o s i t io n s - - t h a t i s, t h e y a re no ~

i n t h e m s e l v e s e i t h e r tr u e o r f a ls e . T h e i r t r u t h o r f a l s i t y is a f u n c t i o no f t h e l o g i c a l i n t e r p r e t a t i o n g i v e n to t h e v a r i a b l e s K a n d R , j u s t a s t h et r u t h o r f a l s i t y o f a c o n d i t i o n a l e q u a t i o n i n a l g e b r a i s a f u n c t i o n o f t h en u m e r i c a l v a l u e s g i v e n t o t h e v a ri a bl e s i u s u c h a n e q u a ti o n. T h e y m a yt h e r e f o r e b e c a l l e d '2ropositional functions' (to u s e a t e r m o f R u s s e l l' s) ,s in ce th e y b eco m e d ef in i te p ro p o s i t io n s ( t ru e o r f a l se ) o n ly w h en d e f in i t e'v a lu es ' a re ' g iv en to th e v a r i ab les K an d R .

F o r e x a m p l e , i f w e gi v e K a n d R t h e f o l l o w i n g v a lu e s :1 ) K == th e c l ass o f o rd in a ry sp h eres ( in c lu d in g th e n u l l sp h e res ) ;2 ) R - ~ t h e r e l a ti o n o f in c l u s io n ( so t h a t ' A R B ' m e a n s 'A i n s id eo f B ' ) ;

t h e n a l l t h e p o s t u l a t e s w i l l b e f o u n d t o b e t r u e ; b u t i f K a n d R h a v et h e v a l u e s a s s ig n e d t o t h e m i n a n y o n e o f t h e e x a m p l e s g i v e n in C h a p t e r I V ,as fo r in s t an ce :

1 ) K = a c lass co m p r i s in g n in e o rd in a ry n u m b ers , n am ely : 2 , 3 , 5 , 7 ,10, 14, 15, 21, 2 1 0 ;

2 ) R = t h e r e l a t io n 'f a c t o r o f ' ;t h e n a~ l e a s t o n e o f t h e p o s t u l a t e s ( h e r e P o s t u l a t e 4 ) w i ll b e f o u n d tob e f a l s e .

H a v i n g t h i s s e t o f p o s t u l a te s b e f o r e u s , w e a g r e e t o c o n s i d e r t h e i r~ogicat conseque nces, t h a t i s , t h e t h e o r e m s w h i c h c a n b e d e d u c e d f r o mt h e m b y t h e p r o c e ss e s o f f o r m a l l o g i c without reference to any ~art icu~ardeterrainat~ion of' the variab les K an d R .

T h e s e d e r i v e d t h e o r e m s , l i k e t h e o r i g i n a l p o s t u l a t e s , w i l l b e p r o -p o s i t io n a l fu n c t io n s~ wh o se t ru th o r f a l s i ty d ep en d s o n th e v a lu es g iv ento th e v a r i ab les K an d R . A l l t h a t ~he p ro cess o f lo g ica l d ed u c t io n t e l l sus~ i s tha t i f t h e o r ig in a l p o s tu la t es a re t r u e fo r ce l~ .a in v a lu es o f th ev ar i ab les , g h en a l l t h e d e r iv ed th eo rem s wi l l b e t ru e fo r th e sam e v a lu es :I n o t h e r w o r d s , i f a n y g i v e n s y s t e m ( K , R ) h a s a l l t h e p r o p e r t ie s s t a t e din th e p o s tu la t es , t h en i~ w i l l a l so h av e a l l t h e p ro p er t i e s s t a t ed in th et h e o r e m s .Defini tion o f abstract geom etry.*) T h e b o d y o f t h e o r e m s t h u s d e d u c e d

*) W e confine ourselves here to ordinary Euclidean three-dimensional geom etry;segs of pos~ulages for other varieties of geo m et~ m ay be obtalned by suilable modi-fics~ion~ cf t h e s e t h e r e g i v e n .


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Postulates for abstract Geometry. 527f ro m th e g iv en se t o f p o s tu la t es co n s t i tu t es an ab s t r ac t d ed u c tiv e th eo ry ,w h i c h , i n v i e w o f t h e fa m i l ia r e x a m p l e a l r e a d y m e n t i o n e d , m a y p r o p e r l ybe ca l led abstra~ geometry; a n d a n y g i v e n s y s te m (K ~ R ) - - f o r e x a m p l e,th e sys tem : K == the c lass o f sphe res , R--- -- the re la t ion o f inc lus ion mw h ic h sa t is f ie s a l l t h e p o s tu la t es m ay th en b e ca l led a concrete geor~ t~iealsystem, o r s imp ly ~ a concrete geometry.AFt~ied geometry. Ab st rac t g eo met ry , a s th u s d e f in ed ~ i s a p u re lym a t h e m a t i c a l t h e o r y , w h i c h w i l l a p p l y e q u a l l y w e l l t o a l l ' g e o m e t r i c a lsy s tems '~ wh erev er su ch sy s tems may b e d i sco v ered .

Fo r ex am p le , i t is co n ce iv ab le th a t in th e fie ld o f Eco n o mics , o r o fBotany~ a c lass K and a re la t io n R m igh t be d iscovered~ which wo uldsa t i s fy a l l t h e p o s tu la t es o f Ch a p te r I I , an d w h ich wo u ld th e re fo re fo rma g eo met r i ca l sy s t em.

I t i s imp o r tan t to n o t i e% h owever~ th a t th e q u es t io n w h e th er an yg iv e n sy s tem (K , R) d o es a c tu a l ly p o ssess th e p ro p er t i e s en u m era ted inth e p o s tu la t es~ i s n o t ( ex cep t in th e a r i th met i ca l case men t io n ed b e lo w)a q u e s ti o n o f p u r e m a t h e m a t i c s , b u t r a t h e r a m a t t e r f o r o b s e r v a t io n a n de x p e r i m e n t . T h u s , t h e a b s t r a c t t h e o r y o f g e o m e t r y a s s uc h g iv e s u s n oi n f o r m a t i o n w h a t e v e r a b o u t t h e n a t u r e o f p e r c e p tu a l s pa ce ~ a n y m o r et h a n i t d o es a b o u t t h e f a c t s o f E c o n o m i c s o r B o t a n y : a l l t h a t i t t e ll su s i s th a t i f a n y s y s t e m ( K , R ) , w h e r e v e r i t m a y b e fo u n d , i s r i g h t l yju d g ed to p o ssess th e p ro p er t i e s me n t io n ed in th e p o s tu la t es , t h e n i t wi l ln ecessa r i ly p o ssess a l so th e p ro p er t i e s men t io n ed in th e th eo rems .

F u r t he r m o r e ~ t h e o n l y s y s te m s ( K~ R ) a b o u t w h i c h s u c h a j u d g m e n tc a n b e m a d e w i t h a s s u re d a c c u ra c y a r e t h e s y st e m s w h o s e e l e m e n t s a r ep u re ly n u m er ica l , t h a t i s t o say , p u re ly lo g ica l; i n a ll o th e r cases~ o u ro b se rv a t io n s a re o n ly ap p ro x imate , an d in a l l su ch cases th e co n c lu s io n sr e a c h e d b y t h e a p p l ic a t io n o f g e o m e t r i c t h e o r y - - e v e n t h o u g h t h e p r o c es so f i n fe r e n ce is p e r f e c t ly r i g o r o u s - a r e o f co u rs e n o m o r e a c c u ra t e t h a nt h e p r em i s s e s o n w h i c h - t h e y a r e ba se d.Cons~stcnoy of the 1oos~ates. Th e f i r s t q u es t io n to b e ask ed co n -c e r n i n g a n y s e t o f p o s t u l a t e s s u c h a s t h a t w h i c h w e a r e h e r e c o n s i d e r i n g ,i s th i s : Do e s an y sy s tem ex i s t wh ich sat is fi e s a l l t h e p o s tu la t es? I f t h eex i s t en ce o f an y su ch sy s tem can b e es t ab l i sh ed , th en th e p o s tu la t es a resa id to b e cons i s~ t~ a n d n o t h e o r e m s d e d u c e d f r o m t h e m c a n e v e r le a dto co n t rad ic t io n .I n t h e p r e s e n t c a s e ~ w e h a v e a l r e a d y m e n t i o n e d o n e s y s t e m (K ~ R )as s a t i s fy in g a l l o u r p o s tu la t es , n amely ~ th e sy s tem : K ~ t h e c l ass o fs p h e r e s , a n d R ~ t h e r e l a t io n o f i n c lu s i on ; b u t i f w e m e a n b y 's p h e r e 'an d ' i n c lu s io n~ t h e c o m m o n n o t io n s g i v e n b y o b s e r v a ti o n o f p e r c e p t u a lspace~ th e n o u r ju d g men ~ th a t th i s sy s t em sa t is fi e s a l l t h e p o s tu la t es can


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5 2 8 E .V . HWZI~GTOLb e o n l y a p p r o x i m a t e , a n d t h e s y st em d oe s n o t p ro v i d e a s a t i s f a c t o r y p r o o fo f t h e c o n s i s t e n c y o f t h e p o s t u l a t e s . I f , h o w e v e r , w e u s e ~ s p h e r e ' a n d' i n c l u s i o n ' i n t h e s e n s e i n w h i c h ~h ese t e r m s a re u s e d i n a n a l y t i c a l g e o -m e t r y ( se e th e e n d o f C h a p t e r I I ) , th e n w e h a v e a s t r i c t l y n u m e r i c a ls y s t e m , w h i c h c e r t a i n l y e x i s t s , a n d w h i c h c a n b e s h o w n t o s a t i s f y a l lt h e p o s t u la t e s w i t h a b s o lu t e a c c u ra c y. T h e c o n s is te n c y o f t h e p o s t u l a t e si s t h u s c o m p l e t e ly e s t ab h s h e d .Separation of general an d existence postulates. T h e l i s t o f p o s t u l a t e sg i v e n i n C h a p t e r I I w i l l b e f o u n d t o b e d i v id e d i n to t w o g r o u p s , t h ef i r s t c o n t a i n i n g t h e ' g en e r a l l aw s ' ( P o s tu l a te s 1 - - 1 8 ) , a n d t h e s e c o n dc o n t a i n i n g t h e 'e x is te n c e p o s tu l a te s ' ( P o s t u l at e s E 1 - - E 7 ) .

T h e s e t w o g r o u p s o f p o s t u la t e s p l a y q u i t e d i s t i n c t r 6 1e s i n t h ed e v e l o p m e n t o f t h e s ub je ct , a n d th e a t t e m p t t o k e ep t h e m s e p a r a t e f r o mt h e s t a r t i s n o t w i t h o u t l og i ca l i nt er e st . ( C o m p a r e , h o w e v e r , t h e r e m a r k su n d e r T h e o r e m 15 .)

Indepen den ce o f the postula2ea. & s e c o n d q u e s t io n t o b e a s k e d i s : A r et h e p o s t u l a t e s i n d e p e n d e n t ? t h a t is , a re w e s u r e t h a t n o o n e o f t h e m ism e r e l y a c o n s eq u e n c e o f t h e r e s t ?

T o a n s w e r t h i s q u e s t io n , w e e x h ib i t, i n C h a p t e r I V , a l i s t o f ' p s e u d o -g e o m e t r i e s ' , b y w h i c h i t i s s h o w n t h a t t h e ' g e n er a l l a w s ' a r e i n d e p e n d e n to f e a c h o t h e r , a n d a l s o t h a t t h e ~ e x i s t e n c e p o s t u l a t e s ' a r e i n d e p e n d e n t o fe a c h o t h e r a n d o f t h e g e n e ra l la w s.

Sufficiency of the postulates to determine a unique type u f system. At h i r d a n d m o s t i m p o r t a n t p a r t o f o u r w o r k i s to s h o w t h a t a n y t w o s y s t e m s( K , R ) w h i c h s a t is f y a ll t h e p o s t u la t e s a r e formally equivalent, o r isomorphiv,w i t h r e s p e c t t o t h e v a ri ab le s K a n d R . T h i s m e a n s t h a t i f ( K ' , I t ' ) a n d( K " , R " ) a re a n y t w o ' g eo m e t r ic a l s y s te m s ' - - t h a t i s, a n y ~ -w o s y s t e m st h a t s a t is f y a ll t h e p o s t u l a t e s - - t h e n it i s p o s s i b l e t o s e t u p a o n e - t o ,

e t . .o n e c o r r es p o n d e n c e b e tw e e n t h e e l em e n t s A ' , B , C , 9 o f K ' a n d t h ee l e m e n t s A", B" , C" , . . . o f K " i n s uc h a w a y t h a t w h e n e v e r .A ' a n d /~ .i n o n e s y s te m s a ti sf y t h e r e la t io n A " R ' ~ ' , t h e n t h e c o r r e s p o n d i n g e l e m e n t sA " a n d B " i n t h e o t h e r s y s t e m w i l l s at is fy t h e r e l a t i o n A " R " B " . B yt h e e s t a b l i s h m e n t o f t h i s i s o m o r p h i s m , w e s h o w t h a t a ny t heo re m i ~ v o b ~on ly the two variables K and R, wh ich is true in one of the system s will: b~b 'ue in the o ther a ls o; h e n c e a / l t he sy s te m s ( K , R ) which satixfy=t]~~ s tu la tes may be sa id to belong to a s i ~ e type .

W i t h t h e p r o o f o f t h is t h e o r e m , t h e s e rie s o f d e d u c t i o n s w h i c h w ed r a w f r o m t h e p o s g ul at es is b r o u g h t t o a n a h a r al c o n c l u s i o n ; a n d i n v i ro f t h i s t h e o r e m , w e m a y t h e n d e f i n e a b s t r a c t geomebry a s t h e s tudy o f . thel~op erties of a particular type of system ( K , R ) , n a m e / y , that type which.~sco~/laletely dCz~rm~ed by the Postulates 1 - - 1 8 a n d E 1 - - E 7 .


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Postulates for abstract Geomet~T. 529

C h a p t e r I .D e f i n i t i o n s .

As ex p la in ed in th e in t ro d u c t io n , th e o n ly v a r i ab les th a t a re in v o lv edin ou r se t o f pos tu la tes are th e c lass K and the re la t ion R , and i t i sth eo re t i ca l ly p o s s ib le to ex p res s ev ery th eo rem o f g e o m et ry ex p l i c i tly inte rm s o f th es e v a r i ab les.

In o rd er , h o wev er , to av o id t ed io u s rep e t i t io n s o f ce r t a in co mb in a t io n so f th es e s y m b o ls , i t i s n eces s a ry to r ep lace th es e co mb in a t io n s b y s in g lete rm s , wh ich a re in t ro d u ced b y d e f in i tio n , fo r th e s o le p u rp o s e o f ab -brev ia t ion .

Fo r co n v en ien ce o f r e fe ren ce , a l l t h e d e f in i t io n s th a t we s h a l l r eq u i rea re co l lec ted to g e th e r in th e p res en t ch ap te r . In f r am in g th es e d ef in i tio n sw e h a v e f re e l y u se d t h e t e r m i n o l o g y s u g g e s te d b y t h e p a r t ic u l a r i n t e r -p re ta t io n o f th e v a r i ab les K an d R wi th w h ich w e a re mo s t f ami lia r~ n amely ~K ~ th e c l as s o f s p heres, an d R ~ th e re l a t io n o f in c lu s io n ; b u t i t m u s tb e c o n s t a n tl y b o r n e i n m i n d t h a t t h e m e a n i n g o f a ll w o r d s ( s u c h a s'po in t ', ' s egm ent ' , ' line ', e tc .) w hic h are here def ined in term s of K andR, w i l l v a ry w i th v a ry in g in te rp re ta t io n s o f K an d R , an d th a t a s f a r a sth e d e f in i t io n s a re co n cern ed , K an d R may s ~ .n d fo r an y c las s an d an yre la t io n th a t we p leas e .

( In th i s co n n e c t io n th e l i s t o f ' p seu d o -g eo met r i es ' i n Ch a p te r IV wi l lb e found ins t ruct ive . )

S p h e r e s , a n d t h e r e l a t i o n o f i n c l u s i o n .D e f i n i t i o n 1. I f 21 is a n e l e m e n t o f t h e c la ss K , t h e n A s h a ll b ecal led an abs t rac t sphere , o r s imply a sphere .D e f i n i t i o n 2. I f 21 R B , w e s ha ll s a y t h a t t h e s ph e r e A is withir~

t h e s p h e re S t , o r t h a t B contains _4.D e f i n i t i o n 3 . S u p po s e A R ~ a n d ~ R A a r e b o t h fa l se ; t w o c a se s

can th e n o ccu r . 1 ) I f th e re is n o s p h ere X s u ch th a t X R A an d X 1~ B ,t h e n A a n d B a r e c a l le d mutual ly exclusive , o r each outside t h e o t h e r ;w h i t e 2 ) i f th e r e i s s o m e s u c h sp h e r e X , t h e n A a n d B a r e s ai d t o

P o i n t s .D e f i n i t i o n 4 . I f A is a s p he re ~ a n d i f t h e r e i s n o o t h e r s p h er e X

suc h tha t X R A , the n A is ca l led an abs t ra c t point~ o r s im ply a po in t .T h a t is , a p o in t is an y s p h ere w h ich co n ta in s n o o th e r s p h ere w i th i~ it .

I t m ay b e n o t i ced th a t th e re is n o th in g in th i s d ef in itio n~ o r in an y


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5 3 0 E . V . H v s~ sQ T o~ .o f o u r ~ o r k , w h i c h r e q u i r e s o u r ' p o i n t s ' t o b e small; f o r e x a m p l e , a p e r -f e c t l y g o o d g e o m e t r y is p r e s e n t e d b y t h e c la ss o f a l l o r d i n a r y s p h e r e sw h o s e d i a m e t e r s a r e n o t l es s ~ a n o n e i n c h ; t h e ~p oin ts' o f t h i s s y s t e ma r e s i m p l y t h e i n ch - s ph e r e s. ( C o m p a r e t h e e x a m p l e g i v e n a t ~ h e e n d " o fC h a p t e r l l . )

S e g m e n t s , a n d t h e s t r a i g h t l i n e .T h e f o l l o w i n g d e f i n i t i o n i s o f c e n t r a l i m p o r t a n c e i n t h e e n t i r e t h e o r y .D e f i n i t i o n 5. L e t A a n d B b e a n y g i v en p o in ts . I f X i s a p o i n t

s u c h t h a t e v e r y s p h e r e w h i c h c o n ta i n s A a n d B a ls o c o n t a in s X~ t h e n Xis sa id to belong to the segment [AB ] o r [ B A ] .

T h e s e g m e n t [ A B ] i s fl f~s a c lass o f p o i n t ,u n i q u e l y d e t e rm i n e d b y A a n d B . T h e p o i n t s .Aan d B b e lo n g to th e s eg men t , an d a re ca l l ed i~sr~ . 1. end -points. I f w e e x c l u d e t h e e n d - p o i n t s , t h e c l a s s

th a t r emain s i s ca l l ed ~h e interior o f t h e s e g m e n t [ A B ] , a n d m a y b ed e n o t e d b y ( A B ) . I f ( A B ) i s t h e n u l l c l a s s , t h e s e g m e n t [ A B ] m a y b es a id to b e h d /o w.

D e f i n i t i o n 6 . B e s id e s t h e c la ss [ A B ] , t w o o t h e r c la sse s, c a l l e d t h e~rolongatio~s o f [AB] , a r e d e t e r m i n e d b y t h e p o i n t s A a n d B~ a c c o r d i n gtO th e fo l lo win g s ch eme.

If X is a point belo ngs to the then X is saidto belong to asuch tha t segment class denotedby

B [ .~ x ] [ :~ i]

and called the I~s bozen~ryprolongation of l consist~ of ~helAB] beyond point. . . . A AB B

(Th is n o ta t io n , wh ich i s d u e in i t s e s s en t ia l e l emen ts to Pe an o , i se a si ly r e m e m b e r e d i f w e o b se r ve t h a t [ A B ' ] , c o nt ai ns A , a n d [ B A ~ c o n -

t a i n s B ; t h a t i s , i n e a c h c a s e t h e l e t t e r w h i c h i s~4B]_[A~or[B_~~ n o t m o d i f ie d b y a n a c c e n t r e p r e s e n t s a p o i n t w h i c hA B~ . 3. be longs to the c lass in ques t ion . )I f w e e x c lu d e f r o m [ A B ' l a n d [BA'] t h e

b o u n d a r y p o i n t s A a n d B , t h e c la sse s t h a t r e m a i n m a y b e c a l le d ~ h e~ter iors o f t h e p r o lo n g a t io n s , a n d m a y b e d e n o t e d b y ( A B ' ) a n d ( / ~ A ' ) ,r e s p e c t i v e l y ; a n d i f e it h e r o f th e s e in t e r io r s i s t h e n u l l c l a ss , t h e c o r r e -s p o n d i n g p r o l o n g a t i o n m a y b e s a i d ~ o b e h o g o w .

Th e fo l lo win g s p ec ia l t e rm in o lo g y wi l l a ls o b e fo u n d to b e co n v en ien t~D e f i n i t i o n 7. I n a s et o f tw o o r m o r e c la sse s l ik e [ A ~ [ A B ' ] ,

e t c , i f n o ~wo o f ~he c las ses h av e an y p o in t in co m m o n (u n les s i t b e


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Postulates for abstract Geometa-y. 5 31c o m m o n b o u n d a r y p o i n t, r e p r e s e n t e d b y a n u n a c c e n te d l e t t e r t h a t a p p e a r se x p l i c it ly i n b o t h s y m b o l s ) , t h e n t h e c la s se s m a y b e sa id to f o r m a si/m~leset of non-overlapl~ing r eg i ons , and t he l og i ca l sum o f such a s e t m ay bec a r e d a simple s.um.

W e c a n n o w d e f i n e t h e l i n e A B as a c la s s o f po i n t s un i qu e l y de t e r -m i n e d b y A a n d B , a s f o ll o w s .

D e f i n i t i o n 8 . I f A a n d 13 a re t w o d i s ti n ct p o in ts , t h e l in e A B ist h e c l a ss o f a ll p o i n t s w h i c h b e l o n g t o t h e s e g m e n t [ A B ] or ~o ei thero f i t s t w o p ro l ong a t i ons . Th ree po i n t s a r e s a i d t o be collinear, i f a n y o n eo f t h e m b e l o n g s t o t h e l in e d e t e r m i n e d b y t h e o t h e r t w o .

r - ~ . B ' ] E ~ . B ] o ~ [ .B . A ] [ .B . A ' ]D e f i n i t i o n 9. T h e lin e A B i s s a i d t o be d i v i ded by t he po i n t A

i n t o ~ w o half-lines, o r raids one con t a i n i ng t he r eg i ons [ A . B ] a n d [ B A ' ]i n w h i ch B o ccu r s w i t hou t accen t , and t he o t he r con t a i n i ng khe r eg i onl A B ~j i n w h i c h B ' o c c u r s w i t h t h e a c c en t . T w o p o i n t s o f t h e I in e a r esa i d t ~ be on t he sam e s ide o f A o r o n opposi te s ide s of A acco rd i ng a sth ey b e long to t he sam e ha l f - l ine or to d i f fe ren t ha l f -l ines . In a s im i l a rw ay , t he l i ne i s s a i d t o be d i v i ded by t he po i n t B .

T r i a n g l e s , a n d t h e p l a n e .The fo l l ow i ng de f i n it ions , 10 - - 1 4 , fo r t he p l ane a re p rec i se ly ana -

l o g o u s t o th e d e f in i ti o n s 5 w 9 f o r th e s t r a ig h t lin e .D e f i n i t i o n 1 0. I f X i s a p o i n t s u c h th a t e v e r y s p h e re w h ic h c o n -

ra ins A , B~ and U, a l so con t a i n s X , t hen X i s sa i d to be l on g t o the trianyle[ A B C ] .The t r i ang l e i nc ludes t he po i n t s A , B , U w h i ch a re ca l led i ts vert/ces t

a n d t h e s e g m e n t s [ A B ] , [ A C ], a n d [ B C ] , w hich a re ca l l ed i t s s /des. T heinterior of t he ~ r iang le , ex c l ud i ng t he po i n t s o f t he s ide% is deno ted b y( A B C ) . I f t he c l a s s ( A B C ) i s a nu l l c l ass , t h e t r i angle i s sa id to behollow.

W e nex t d e f ine ce r ta i n ' ex tens i ons ' o f t he t r i ang l e , w h i ch a re t o bea n a l o g o u s t o t h e ' p r o lo n g a t i o n s ' o f a s e g m e n t . I n o rd e r t o e x h i b i t ~ i san a l og y i n it s c l ea re s t fo rm , w e f ir s t obse rve t h a t t he de i in i ti on o f ap r o l o n g a t i o n o f a s e g m e n t m a y b e w o r d e d a s f o l lo w s : cs X i s a p o i n ts u c h t h a t o n e e l e m e n t o f t h e b o u n d a r y is i n c id e n t w i th th e f ig u r e f o r m e db y j o i n i n g X w i t h t h e o p p o s i te e l em e n t o f t h e b o u n d a r y , t h e n X i s s a idt o be l ong t o a c l a ss deno t ed b y ' e tc . N ow i n t he ca se o f a t ri ang le , t hee l e m e n t s o f t h e b o u n d a r y a r e o f t w o k i n d s , t h e t h r e e v e rt ic e % A , B , C ,a n d t h e t h r e e s i d e s , [ A J 3 ] , [ A U ] , [BG ] ; hence t he des i r ed de f i n i t i ons w i l l


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532 E .V . HVNT~NQTO~.b e o b ta in ed b y as s o c ia t in g each v e r t ex wi th th e o p p o s i t e s id e , an d eachs id e w i th th e o p p o s i t e v e r t ex , a s in th e fo l lo win g s ch eme.

D e f i n i t i o n 1 1 .I f X i s i s inc i - thenX is said toa poin~ belong to a classsuch tha t dent with denoted by

a [ ~ c x ] [A ~ ' o ' ]~r EA c x ] [~ A 'o']o [ A ~ x ] E C a'~ ']

This class is calledthe extension o f[A.B C] beyond thevs~ex A

It s boundary consistsof the points of

[ A B ' ] , [AC'][ ~ a ' ] , [BC'][OX'], [C~'][~A~] [o.x '3 1 [ A ~ o' ][ x c l [ ~ x ] [ [A o 2 r3[ B e l [ A X ] [m O A ' ] ,, [AC~,, [~CS [ A ~ ], [ a t ' l , [ ~ c ' ][A el , [A ~ ' ] , [C~ '][ B e ] , [ ~ a ' ] , [ C ~ ']

Th e f i r s t th ree o f th es e c l as s es , [ A B ' C ~ ] , . . . , are ca l l ed th e vertivalex ten s io n s o f th e t r i an g le , an d th e l a s t th ree , [ A B C ~ , 9 t h e lateral e x -ten sio n s. Th e p o in t s n am ed as th e ' b o u n d ary ' o f each c las s w i l l t h em -

s e lv es b e lo n g to th a t c las s in an y s y s tem in w h ic h_ ~ A ] ~ P o s tu la te s 1 - - 2 a r e valid . T h e interiors o f t h es ev era l r eg io n s , ex c lu d in g . th e p o in t s o f th e b o u n d a ry ,

a r e d e n o t e d b y ( A B ' C ' ) , . . . , ( A C ' ) , . . . , r espeet ively .

~ / J / - ~ ] ~ ] B y a n im m e dia te e xte ns io n o f t he t er m in o lo g y~iB. s. ad op ted in D efini t ion 7 ,we ha ve:

D e f i n i t i o n 12. I n a s et o f t w o o r m o r e c la ss esl ike [ A B C ] , [ A B C ' ] , e tc . , i f n o two o f th e c l as ses h av e an y p o in t inc o m m o n (u n l e ss it b e a p o i n t o f a c o m m o n b o u n d a r y , r e p r e s e n t e d b yle t t e r s th a t ap p ea r ex p l i c i t ly in b o th s y mb o ls ) , t h en th e s e t o f c l as sesm a y b e c a l l e d a simple set an d th e i r lo g ica l s u m a simple s u m.

For example~ [BA'C'] a n d [BCA'] h a v e t h e c o m m o n b o u n d a r y [BA']~i f t h e y h a v e n o o t h e r p o i n t i n c o m m o n , t h e y f o r m a s im p l e s et .

W e n o w d e fi ne t h e p l a n e A ~ C as a c lass o f p o in t s u n iq u e ly d e te r -m i n e d b y A , B , a n d C, as fol lows.

D e f i n i t i o n 13. I f A , B , a n d C a re t h re e p o in ts n o t in t h e s a m eline, the ~ la ne A B C i s th e c las s o f a ll p o in t s th a t b e lo n g to th e t r i an g te[ A B C ] o r to an y o n e o f i t s s ix ex ten sio n s. Fo u r p o in t s a re s a id to b 8~ l a ~ a r , i f a n y o n e o f t h e m b e l o n g s t o t h e p l a ne d e t e r m i n e d b y t h eo ~ e r t h r e e .

D e f i n i t i o n 14. T he p la n s A B C i s s a id to b e d iv id ed b y th e l in eA . B i n t o t w o half-planes, o n e c o n t a i n i n g t h e f o u r r e g i o n s i n w h i c h Go ccu rs w i th o u t accen t , th e o th e r co n ta in in g th e th ree reg io n s in whic l~ C :


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Postulates for abs~rae~ Ge om etry. 53 3o c c u r s w i t h t h e a c c e n t. S i m i l a r l y , t h e p l a n e is d i v id e d b y t h e l i n e A U andb y t h e l i n e B C .Be fore pass ing to th e ana logo us de f in i ti ons o f a space ABU1), itwi l l be c onv en ien t t o in t rod uce a t t h i s p o in t de f in i ti ons conce rn ing pa ra l l e ll i nes and the con grue nce o f segm en t s . On th i s a r r ange m ent , a l l t he de ft-r a t ions r equ i red fo r t he spec ia l case o f geometry of two dimensions will b efound in consecu t ive o rde r .

P a r a l l e l l i n e s .D e f i n i t i o n 1 5. S u p p os e w e a r e d ea l in g w i t h a s y s te m (K , R ) i n

w h i c h t h e ' p l a n e s ' h a v e a l l t h e p r o p e r ti e s d e m a n d e d b y P o s t u l a t e s 6 - - 8 .T h e n i f t w o l in e s AB a n d GD l ie i n t h e s a m e p l a n e , a n d h a v e n o p o i n ti n c o m m o n , t h e y a r e sa id t o b e parallel, a n d w e w r i te : A B ][ CD.

T o i n d i c a t e t h a t t w o l i n e s AB and GD are e i the r pa ra l l e l o r co in -c iden t , we sha l l u se the no~a t ion AB ~ CD.D e f i n i t i o n 16. I f A B [ I UD and BC II DA, t h e n t h e f o u r p o i n t sA, B, C, D a re sa id to fo rm a paralldogram, o f w h i c h IX 0] and [BDJa r e t h e diagonals.By the a id o f pa ra l l e l l ines , we no w de f ine the m id-po in t o f a segm ent ,as follows.

D e f i n i t i o n 1 7 . Let [AB] b e a n y g i v e n s e g m e n t .pa ra l l e logram A X B Y o f w hic h [A.B] is one d iagonal~a n d i f t h e o t t m r d ia g o n a l i n te r s e c ts [ A B ] i n M , t h e n Mis ca ll ed a m idd le po in t o f ? .he segm ent [AB]. I f t h e r ei s on ly one such po in t M (as w i ll a lways be the case ine v e r y s y s t e m in w h i c h P o s t u l a t e s 1 - - 1 1 a r e v a l id ) , t h e nM is ca l led the mid-point o f [AB], a n d w e w r i t e :M ~- mid A B.

I f t he re i s a

:F ig . 4 .

I n t h is c as e t h e s e g m e n t lAB] i s sa id to be bisected at;M.T h e c e n t e r o f a s p h e r e .

In o rde r to de f ine the ' cen te r ' o f an ( abs t r ac t ) sphe re , w e f i r st de f ineth e po in t s Con the su r face ' o f a sphe re , a s a subc lass am ong the po in t st h a t a r e w i t h i n t h e s p h e r e .

D e f i n i t i o n 18 . I f A a n d B a re w i t h i n a s p h e r e S , a n d i f a ll p o i n t so f t h e p r o l o n g a ti o n s ( A B ' ) a n d ( B A ' ) a r e o u t s id e o f S , th e n t h e s e g m e n t[ A B ] i s c a l l e d a ctwrd o f t h e s p h e r e $ ,

D e f i n i t i o n 19. I f A is a n e n d- po h ~t o f a n yc h o r d o f a s p h e r e S , t h e n A is sa id t o l ie on thesurface o f t h e s p h e re .D e f i n i t i o n 2 0. I f 0 is a p o i n t w i t h in a s p h e re~q, and i f eve ry pa i r o f chord s w hich in t e r sec t a~ 0 ~ i g . 5 .


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5 3 4 E . V . H ~ T n~ T o~ .

a re t he d i agona ls o f a pa ra l l e l og ram , t hen 0 i s ca l led t he cen te r o f t hes p h e r e .

A ny cho rd t h ro ug h t he cen~e r i s cu l led a dia~neter t a n d i s b i s e c t e da t ~ he cen t e r. E i t h e r ha l f o f a d i am e t e r is ca ll ed a r a d i u s .

C o n g r u e n c e o f s e g m e n t s .B y th e a i d o f t h e t w o n o t io n s o f t h e m i d - p o i n t o f a se g m e n t a n d

t h e c e n t e r o f a s p h e r e , w e c a n n o w d ef in e t h e r e l a t i o n o f c o n g r u e n c ebe t w een t w o segm en t s~ a s fo l l ow s .

D e f i n i t i o n 2 1. T w o s eg m e nts [ A B J a nd [ 0 I ) ] a re ca l l ed c o n g r u e n ti n s y m b o l s, A / ~ ~ G D - - w h e n a nd o n l y w h e n o n e o f t h e f o l l o w i n gcond i t i ons i s s a t i s f i ed :

1 ) I f t h e t w o s e g m e n t s [ A B ] a n d [ C D ] a r e o n th e s a m e li ne , t h e n w em u s ~ h a v e e i t h e r [ A ~ ] ~ [ C D ] , o r m id A G = m i d B D , o r m i d A D = m id ~ G ;a n d f f t h e y l i e o n p a r a l le l l in e s , t h e n t h e y m u s t b e o p p o s i t e s i d e s o f apa ra l l e l og ram .

2 ) I f t h e y h a v e a c o m m o n e n d p o in t ( o r a c o m m o n m i d - p o in t) , b u ~d o n o t lie o n th e s a m e l in e , t h e n t h e y m u s t b e r a d i i ( o r d i a m e t e r s ) o ft h e s a m e s p h e r e .3 ) I f t h e y d o n o t li e o n t h e s a m e li n e o r o n p a r a l le l l in e s , a n d d on o t h a v e a c o m m o n e n d -p o in t o r a c o m m o n m i d - p o in t , t h e n t h e r e m u s tb e t w o s e g m e n t s [ O X ] a n d [ 0 Y ] w h i c h a r e c o n g r u e n t t o ~he g i v e n

s e g m e n t s a c c o r d in g t o 1 ) , a n d c o n g ru e n ~ t o e a c h o t h e r.I, ac co rd in g to 2)~A c cord i ng t o t h i s de f in i ti on , a l l t he r ad i i o f a g i ven

s p h e r e a r e o b v i o u s l y c o n g r u e n t ; h e n c e , a l l t h e p o i n t s o n t h es u r f a c e o f a s p h e r e m a y b e s a id to b e e q u i d i s t a n t f r o m t h el ~ i g . 6 . center .

I t w i ll b e n o t i c e d t h a t c o n g r u e n c e a c c o r d in g t o 1 ) i s c o n n e c t e dw i t h t h e i d e a o f trar~slation, a n d t h a t a c c o r d i n g t o 2 ) w i t h t h e i d e a o fro~a~ion ~ t h e s e t w o i d ea s b e i n g n e c e s s a ri ly i n v o l v e d in a n y a d e q u a t ede f i n i t i on o f congruenc e .

P e r p e n d i c u l a r l i n e s .D e f i n i t i o n 2 2 . I f t h e dia g on a ls o f a p a r a ll e lo g r a m a r e c o n g r u e n t ,

f ine par a l l e log ram i s ca l l ed a ~ectangle~ and a ~ riang le w h i ch fo rm s h a l fo f a r ec t ang l e i s ca l l ed a r ight t r iangle .

D e f i n i t i o n 2 3 . T w o lin e s ~ ha t i n te r se c t a t a p o i n t 0 a r e ca ll edt ~ T ~ d i v u Z a r , i f e v e r y s e g m e n t w h i c h j o i n s a p o i n t o f o n e l in e w i t h ap o i n t o f t h e o t h e r i s t h e d i a g o n a l o f a r e c t a n g l e w i t h o n e v e r t e x a t O .


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Postulates for abstract Geom etry. 5 3 5T h e n u m b e r l in e .

The fo l l ow i ng de f i n i t i ons w i l l be u se fu l w hen w e com e t o i n t roducecoord i na t e s i n t o ou r sys t em .

W e se l ec t an y f ixed l ine 0 U a s a spec i a l l ine t o w h i ch r e fe rencew i l l be m a de i n fu t u re ope ra t i ons , and w e ca l l t h i s l ine the number l ine,t h e p o i n t 0 t h e zero po i n t , and t he p o i n t /7 t he unit poin t .

I t i s t o b e u n d e r s t o o d t h a t a n y li n e w i l l a n s w e r t h e p u r p o s e o f th en u m b e r l i n e ; b u t w h e n o n c e c h o s e n i t m u s t r e m a i n f ix e d d u r i n g t h e c o u r s eo f an y pa r t i cu l a r i nves t iga t i on .

D e f i n i t i o n 2 4. I f A a n d B a re a n y t w o p oi ntso n t h e n u m b e r l i n e OU , a n d i f X i s a n o t h e r p o i n t o nt h a t l in e s u c h t h a t m i d 0 X ~ m i d AB~ t he n X i s ca l ledt h e s u m o f A a n d B , a n d w e w r i te X ~ A + B . l v i g . 7 .D e f i n i t i o n 2 5. L e t A a nd B b e a n y t w o p o in ts o nt h e n u m b e r l i n e 0 U , a n d l e t ~ b e a n y c o n v e n i e n t p o i n t n o t o n t h a t lin e .I f a l in e t h r o u g h B p a r a l le l t o U P m e e t s 0 P in Q , a n d i f a l in e t h r o u g hQ para l l e l t o P A m ee t s O U i n ~ and i f t h i s po i n tY is in d ep e nd e nt o f t h e p a rt ic u la r c ho ic e o f t h e o ~aux i l i a ry p o i n t P , t hen Y i s ca l led t he product of . .A a n d B ; a n d w e w r i te Y - - A >< B . I n p a r ti cu l ar , ww e w ri te A x A ---- A ~. ~ ' ~

D e f i n i t i o n 26 . I f A a n d B a re a n y tw o V - ~ - ~p o i n t s o n t h e n u m b e r l i n e 0 U , a n d i f ( a c c o r d in gt o a r e ad i ly u n d e r s to o d m o a n i n g ) t h e d ir e ct io n 1~f rom A t o B i s t he sam e a s t he d i r ec t i on f rom0 t o U , t h e n w e s a y t h a t A precede s .B, an d r i s . s.w r i t e : A < B .A po i n t X on t he nu m be r li ne i s ca ll ed negative o r positive a c c o r d i n gas i t p recedes o r fo l l ow s t he ze ro po i n t.

T e t r a h e d r a , a n d t h e s p ac e .W e n o w r e t u r n t o t h e D e f in i ti o n s 5 - - 9 f o r t h e s~ r ai gh t l in e , a n d

D e f i n i ti o n s 1 0 - - 1 4 f o r t h e p l a n e , a n d p r o c e e d t o c a r r y t h e a n a lo g y o ft hes e de f i n it ions one s t ep fu r~ he r~ i n to t h ree d im ens i ons . The fo l l ow i ngd e f in i ti o n s 2 7 - - 3 1 d o n o t i n v o l v e in a n y w a y t h e n o ti o n s i n tr o d u c e d i nd e f i n i t i o n s 1 5 - - 2 6 .

D e f i n i t i o n 2 7 . I f X is a p o i n t s u ch th a t e v e r y s p h e re w h i c h c o n -r a i n s A , B , C, a n d D , a ls o c o n ta i n s X , t h e n X i s s a id t o b e l o n g t o t h ete;krahedron [A B C1)].

T h e t e t r a h e d r o n i n c l u d e s t h e verbices A , Jg, C , .D, t he edges [A JB] , [A G ] ,[ A D ] , [ B C ] , [ B D ] , [CD], a n d t h e fa ce s [ A B C ] , [ A B D ] , [ A C D ] , a nd


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5 3 6 E . V . H U N T ~ aT O N .

[ B G D ] , w h i c h f o r m i t s boundary. T h e n o t a t i o n ( A B C D ) i s used as be-fo re tm denote the interior o f th e c l as s [ A B G D ] , ex c lu s iv e o f th e p o in t son the faces .

I n o r d e r to f o l l o w t h e a n a l o g y e x p la i ne d i n c o n n e c t i o n w i t h D e f t-n i t i o n 1 1, w e n o t ic e t h a t t h e b o u n d a r y o f a t e t r a h e d r o n c o n s i s ts o f 1 4 e le -ments~ na m ely : f ou r ver~ices~ s ix edges~ and fo ur fades. B y assoc ia t ingeac h o f th es e e l em en ts w i th i t s ' o p p o s i te ' e lemen t~ we o b ta in th e 1 4 ' ex -tens ions ' o f a te t rahe dron , as fo l lows .

D e f i n i t i o n 2 8 .[ t h e n X i s

If .~ is I I said ~o be- ] T h is classa uoint[ is inei- I . . [ . . . . Its botendaryconsists- . I I l O n g ~ o a I x s ca~eao n es u c h [ e n t w i t h I l a s s d e n o - I . . . o f t h e p o i n t s o f

~ O I I } n et e d b y

. . . . . . . . .i ~ ! :. [. [tensio nso fthe1 ~etrahedron[,4 B]

9 . .

[ABe]

[r [ i 2~c ' z r ] I 96 eewise ! [A 0 D ' ] , [B0"D'] , [ABO'] , [A/~D']extensions iof the tetra- Ihedron[ r x ]

9 , .[ ~ m o r a l 4 facial ex-[tensionsof hetotrahedron

[AB2~'], [.aOD'], [ ~ 0 D ' ]

T h e p o i n t s n a m e d a s t h e ' b o u n d a r y ' o f e a c h c l a ss w i ll t h e m s e l v e sb e lo n g to th a t c lass in an y s y s tem in w h ich Pos ~la la tes 1 - - 2 a re v a l id .

T h e n o t a t i o n s ( A B ' C ' D ' ) , . . . , ( A B C ' D ' ) , . . . , ( A B C D ' ) , . . .a re u s ed as b e fo re to d en o te th e interiors o f t h e s e v e r a lreg ions .

D e f i n i t i o n 29. A sim~le set o f n o n - o v e r l a p p i n g r e g i o n sis a s e t h a v in g p ro p er t i e s an a lo g o u s to th o s e g iv en in De-fini t ion 12.

D e f i n i t i o n 3 0. I f A~ B~ C , l ) a r e f o u r p o in t s n o t i nFig. ~. ~ae sa m e pl an e, th e s~ace A B G D i s the c lass o f a l l po in ts

w h i c h b e l on g t o t h e t e t r a h e d r o n [ A B C D ] o r to an y o f i ts fo u r tee n ex ten s io n s .D e f i n i t i o n 3 1. A s pa ce A B C D i s sa id to be d iv ided in to twohalfslgaces b y e a ch o f t h e p l a n e s A . B C , A B D , A C D , a n d B C D . ( Co m -

pare Def in i t ion 14 . )


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Postulates for abstract Geometry. 537P a r a l l e l p l a n e s a n d l i n e s .

D e f i n i t i o n 3 2 . A l in e a n d a p l a n e , o r t w o p la n e s , a r e s aid t o b ep w r ~ i f t h e y b e l o n g to th e s a m e s p ac e a n d ha v e n o p o i n t i n c om m o n .A l l t h e s e d e f in i ti o n s 1 - - 3 2 a r e e x p r e s s ib l e d i r e c tl y i n t e r m s o f t h ef u n d a m e n t a l v a r ia b le s K a n d R .

C h a p t e r I I .T h e P o s t u l a t e s .

I n t h i s c h a p t e r w e e n u m e r a t e t h e p o s t u la t e s w h i c h f o r m t h e s u b j e c to f d i s c u s s i o n i n t h e p r e s e n t a r ti cl e. P o s t u l a t e s l m 1 8 a r e ' g e n e r a l l a w s ';P o s t u l a t e s E l - - E 7 a re 'e x is te n ce p o s tu l at es ~.

A l l t h e p o s t u l a t e s a r e e x p r e s si b le i n t e r m s o f t h e t w o f u n d a m e n t ~ Iv a r i a b l e s K a n d ~ , b y D e f i ni ti o ns 1 - - 3 2 .

G e n e r a l l a w s f o r s p h e r e s a n d p o i n t s (see D efs . 1 - - 4 ) .P o s t u l a t e 1 . L e t `4 , B , C b e a n y ( a b s tr a c t) s p h er e s. I f . 4 is w i t h i n

B a n d ~ w i t h i n C ~ t h e n ` 4 i s w i t h i n C .P o s t u l a t e 2 . I f ~ f is w i t h i n ~ , t h e n `4 a n d B a r e d is t i n ct .P o s t u l a t e 3 . a ) I f t h e c la ss o f s p h e r e s w h i c h c o n ta i n t h e p o i n t `4

i s t h e s a m e a s t h e c la ss o f sp h e r e s w h i c h c o n t a i n t h e p o i n t B , t h e n . 4 ~ - B .b ) I f t h e c la s s o f p o i n t s w i t h i n a s p h e r e S i s t h e s a m e a s t h e c la s s o fp o i n t s w i t h i n a s p h e r e T, t h e n S - - T .

G e n e r a l l a w s f o r t h e s t r a i g h t l i n e (se e D efs . 5 - - 9 ) .P o s t u l a t e 4 . I f X i s a p o i n t o f t h e s e g m e n t [ ` 4 B ] , t h e n [ ` 4 ~ ] is

t h e ' s im p l e s u m ' o f t h e tw o s e g m e n t s [ ` 4 X ] a n d [ ~X - J.P o s t u l a t e 5 . I f tw o l in e s h a v e t w o d is t i n c t p o i n ts i n c o m m o n , t h e y

c o i n c i d e .G e n e r a l l a w s f o r t h e p l a n e (se e D efs . 1 0 - -1 4 ) .

P o s t u l a t e 6 . I f X is a p o i n t o f t h e t ri a ng l e [ ` 4 B 0 ] , t h e n [ ` 4 ~ C ~is t h e ' s i m p l e s u m ' o f t h e th r e e t r ia n g l e s [ ` 4 B X ] , [ ` 4 G X ] , a n d [BCX].

P o s t u l a t e 7 . I f t he s e g m e n t [ X I ~ in te rs ec ts t h e s e g m e n t [AJ~,t h e n t h e t r i a n g l e s [ `4BX] a n d [ `4BY] h a ve n o p o i n t i n c o m m o n e x c e ptthe points of [ ` 4 B ] .P o s t u l a t e 8 . I f t w o p la n e s h a v e t h re e n on - c ol li ne a r p o i n ts in c o m m o n ,

t h e y c o i n c i d e .Math ema t i a ch e A ~ n a l e n . L X X I I L ,~ 5


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538 E.V . HUSTI~GTO~.G e n e r a l l a w s f o r p a r a l l e l l i n e s (se e D efs. 1 5 -- 1 7) .

P o s t u l a t e 9. I f tw o l in e s a r e p ar a ll el t o a t h i rd l i n e , t h e y a r ee i th e r p a ra l l e l o r co in c id en t .

P o s t u l a t e 10. I f A B and C D a r e p a r a l l e l l i n e s, t h e n n o o n e o fthe f o u r p o i n t s A, B , C , D l i e s w i t h i n t h e t r i a n g l e f o r m e d b y t h e o t h e rth ree .P o s t u l a t e 11. ( ' F o u r - p o in t p o s tu l at e .') Let A, B , C, D b e a n y s e to f fo u r p o in t s , n o th ree o f wh ich a re co l l in ea r , an d A '~ :B ', C ' , D ' an yo t h e r s e t o f f o u r p o i n ts , n o t h r e e o f w h i c h a r e c o l li n e a r ; a n d c o n s i d e rth e two se ts o f s ix l ines ,

A B ~ A G , A D , .B C , B D , C D a u d A 'B ' , A 'C' , A 'D ' , B 'C ' , _B'D' , G'D ' ,w h ic h th es e p o in t s d e te rmin e . I f th e f i r s t f ive l in es o f o n e s e t , t a k e n ino rd er , a re p a ra l l e l to (o r c o in c id en t w i th ) th e f i r s t f iv e l in es o f th e o t h e rse ~, t a k e n in t h e s a m e o r d e r , t h e n t h e r e m a i n i n g s i x th l i n e o f t h e f i r s ts e t w i l l b e p a ra l le l to (o r co in c id en t w i th ) th e re m ain in g s ix th l in e o fthe o th er se t . T ha t is , i f A B ~ A ' . B ' , A C ~ A ' C ' , A D ~ A ' D ' , B C , ~ B 'D "a n d / ~ D , ~ B ' D ' , t h e n a l s o C D ~ C'D'.

Th is p o s tu la te was s u g g es ted b y a r em ark o f Sch u r ' s ( lo c. c i r. ) a n dt a k e s t h e p l a c e o f t h e s p e c i a l f o r m o f D e s a r g u e s ' T h e o r e m u s e d b y t t i l b e r t .

G e n e r a l l a w s f o r c o n g r u e n c e (see D efs. 1 8 -- 2 1) .P o s t u l a t e 12. I f A .B ~ CD a n d C / ) ---- E F , t h e n A . B ~ E E .P o s t u l a t e 13. I f t h e s u rf ac e s o f t w o c o n c en t ri c s p h er e s a r e

b y o n e rad iu s in A an d X , an d b y an o th er r ad iu s in B an d :IT,[ , ~ Z ] - - [ B 17].I n o t h e r w o r d s , th e p o r t io n s o f

s u r faces o f two co n cen t r i c s p h eres a reP o s t u l a t e 14. L et A , B , C , Xt rt h ree a re co Uin ear , an d l e t .,4", B , C ,

o f wh ich th e f i r s t th ree a re co Uin ear ;s eg m en ts d e~ermin ed b y th es e p o in t s.

cubt h e n

t w o r a d i i i n t e r c e p t e d b e t w e e n t h ec o n g r u e n t .b e f o u r p o i n t s o f w h i c h t h e f i r s ~X ' b e a n o t h e r s e t o f f o u r p o i n t san d co n s id e r th e two s e ts o f s i.~

T h e n i fA B ~ A ' B ' , A C ~ A ' C ' , B C ~ .B'C '~ A X -~ A ' X ' , a n d B X ---- B ' X ' ,

We shal l a lways have a lso C X ~ C ' X '.Th is p o s tu la te was s u g g e s ted b y As s u m p t io n 11 in Veb len 's l a t es ~

lis t (1 911 ); loc. cir.G e n e r a l l a w s f o r s p a c e ( se e D efs. 2 7 - - 3 2 ).

P o s t u l a t e 15. I f X i s a p o i n t o f t h e t e t r a h e d ro n [ A B G D ] , t h e n[ A B C D ] i s t h e ' s i m p l e s u m ' o f t h e f o u r t e t r a h e d r a [ A B C X ] , [ A B D X ] ~[AC,DX], and [BC X].


18/38

Postulates for abstract Geo metry. 58 9P o s t u l a t e 16 . I f t h e se g m e n t [ X Y ] i n te r se c ts t h e ~ ria ng le [AJBC],

t h e n t h e t e t r a h e d r a [A2~CX] a n d [A)3CY] h a v e n o p o i n t i n c o m m o ne x c e p t t h e p o i n t s o f [ABC].

T h e s e p o s t u l a t e s 1 5 a n d 1 6 a r e p r e c is e l y a n a l o g o u s t o P o s t u l a t e s 6a n d 7 f o r t h e p l a n e. I n s t e a d o f t h e e x a c t a n a l o g u e o f P o s t u l a t e 8 , h o w e v e r ,w e c h o o s e t h e f o l lo w i n g s t r o n g e r p o s t u l a t e , w h i c h l i m i t s t h e s y s t e m t ot h r e e d i m e n s i o n s .

P o s t u l a t e 17. I f A B C D i s a s p a c e , t h e n e v e r y p o i n t b e l o n g s t oth i s space .T h e f o l l o w i n g p o s t u l a t e is a n a l o g o u s t o P o s t u l a t e 1 0.P o s t u l a t e 1 8. I f a l in e X Y is p a ra l le l t o a p l a n e _4BC, t h e n n oo n e o f t h e f i v e p o i n t s A, B, C, X, Y b e l on g s t o t h e t e t r a h e d r o n f o r m e d

b y t h e o t h e r f o u r .E x i s t e n c e p o s t u l a t e s (see D efs. 1 - - 3 2 ) .

H a v i n g t h u s c o m p l e t e d t h e l is t o f ' g e n e r a l l aw s ', w e n o w g i v e t h e' e x i s t e n c e p o s t u l a t e s~.P o s t u l a t e E l . T h e r e a re , i n t h e c la ss K , a t le a s t t w o d i st in c t p o in ts .P o s t u l a t e E 2 . I f A B i s a l i ne , t he re i s a po in t X no t on tha t l i ne .P o s t u l a t e EB . I f A B i s a l i ne , and C a po in t no t on tha t l i ne ,t h e n t h e r e i s a p o i n t X s u c h t h a t C X i s para l le l to _4B.A s y s t e m i n w h i c h t h is P o s t u l a te E 3 i s s a ti sf ie d m a y b e c a l l ed a

sy s t e m in w hich pa ra ll e l l ines m a y be f ree ly d rawn . In gene ra l , i t i s on lyin suc h sys t em s tha t t he de f in it ions re l~Lting to con gru enc e (Defs . 18 - -2 3)h a v e a n y m e a n i ng .

P o s t u l a t e E 4 . I f l A B ] is a n y s e g m e n t i n a s y st em in w h i c hp a r a l l e ls c a n b e f r e e l y d r a w n , t h e n o n a n y h a l f l in e 0 P t h e r e is a p o i n tX s u c h t h a t t h e s e g m e n t [OX] =--[AB].

Tha t i s , any g iven segm ent can he ' l a id o f f ' on any g iven ha l f - l i ne .P o s t u l a t e E 5 . I f $1~ S s, S s , . . . i s an in f in i t e seque nce o f sphe res ,

e a c h o f w h i c h l i e s w i t h i n t h e p r e c e d i n g o n e , t h e n t h e r ei s a po in t X w hich lie s w i th in them a ll .Th i s is a s im ple m odi f ica t ion o f De dek ind ' s p os tu l a t e

conce rn ing c l a s ses o f po in t s on a l i ne .T h e f o l l o w i n g p o s t u l a t e i s m a d e n e c e s s a r y b y t h e f a c tt h a t w e h a v e t a k e n t h e s o li d s p h e r e i n s te a d o f t h e p o i n t ~ s . lo.a s o u r f u n d a m e n t a l v a ri ab l e.P o s t u l a t e E 6 . I f a n y s p h er e h a s a c e n t er , t h e n e v e ry s p h er e h a s

a c e n t e r ( s ee D e f s . 1 8 - - 2 0 ) , p r o v i d e d , o f c o u rs % t h a t i t i s n o~ i t s e lfa point .

85*


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F i n a l l y , t o g i v e t h e s y s t e m t h r e e d i m e n s i o n s w e m u s t h a v e :P o s t u l a t e E T . I f A B U i s a p l a n e , t h e r e i s a~ l e a s t o n e p o i n t n o t

i n t h a t p l a n e .A s w e s h a l l s h o w i n C h a p t e r I I I, thes e Postulates I ~ 1 8 a n d E l s E 7 ,a r e su ff ic ien t to d e ~ m ~ comtge te ly the abs tract theor y /o f ord inary Eucl ideanthre~-dimensionaI geom etry.

T o o b t a in a c o r r e s p o n d i n g s e t o f p o s t u l a t e s f o r two dimensions , w eh a v e s i m p l y t o on zi t Postulates 1 5 - - 1 8 , and replace Postulate E 7 b y i tsr~egative; i n t h i s c a s e t h e m o s t n a t u r a l i n t e r p r e t a t i o n o f 'a b s t r a c t s p he re '.w o u l d b e 'circlg i n s t e a d o f ' s p h e r e ' .

C o n s i s t e n c y o f t h e p o s t u l a t e s .I n o r d e r to g i v e a ri g o r o u s p r o o f o f t h e c o n s i s t e n c y o f t h e s e p o s tu -

l a t e s , w e m u s t c o n s t r u c t , out o f pure~ numer ica l mater ia l s , a s y s t e m ( K , R )w h i c h w i l l s ~ t i s fy t h e m a l l .

T o d o t h i s , l e t S(a , b , c , r ) d e n o t e t h e c l a s s o f a l l tr i a d s o f r e aln u m b e r s x , y , z , w h i c h s a ti sf y t h e e q u a t i o n

( x - a ) ' + + r 2,w h e r e a, b, c, r a r e r e a l n u m b e r s , an d r is not le ss tha n a certain f i,v , dn u m b e r g (p o s i t i v e o r z ero) .

W e t a k e a s o u r c l a ss K t h e t o t a l i t y o f a l l s u c h S ' s , and we de f ine ,t h e r e l a t i o n R b e t w e e n a n y t w o o f t h e s e S ' s b y a g r e e i n g t h a t

S(a ' , b ', c ', r ' ) R g( a" , b" , c ' , r ")w h e n a n d o n ly w h e n r ' ~ r " a n d e v e r y t r ia d x, y , z w h i c h s a t i s f i e s t h er e l a t i o n

( x - + ( y - b ' ) ' + - c ' ) ' = < r "sa t i s f i e s a l so the re l a t ion

( x - - a ")2 % ( y - - b " ) ' + ( z - - c " ) * ~ r '~.I n t h is s y s te m ( K , R ) , t h e ' p o i n t s ' a r e t h e e l e m e n t s o f th e fo r m

S( a, b, c, g ) ; a n d i t i s n o t h a r d t o s h o w t h a t a l l t h e p o s t u l a t e s a r e s a t i s f i e dI n t h e l a n g u a g e o f a n a l y t i c g e o m e t r y , t h i s s y s t e m i s s i m p l y ~he

system o f ~hores whose rad i i are no t l e ss than g , w h e r e , i n t h e m o s t f a m i la rc a se , g - 0 . I t i s i n t e r e s t i n g t o o b s e rv e , h o w e v e r , t h a t a r ty o t h e r v a l ~ :o f g i s e q u a l l y l e g i t i m a t e , s o t h a t w e m a y s p e a k o f a ~ e rf ec tZ y r ig o ro u s.geom 2try in whie.h the 'pointd, l ike the school-master's ch alk -m oz ks on t]wb.lackboard, are of definite, f inite size, a nd the 'H ne ] a n d 'pZanes' o f & fi ~ i~finite thick/hess.


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P o s tu l a t es f e z a b s t r a c t G e o m e t r y . 5 4 1

C h a p t e r H I .T h e o r e m s .

I n t h i s c h a p t e r w e g i v e s u c h t h e o r e m s a s a r e n e c e s s a r y f o r t h e p r o o fo f t h e s u f f i c i e n c y o f t h e p o s t u l a t e s t o d e t e r m i n e a u n i q u e t y p e o f s y s t e m .T o a v o i d i n t e r r u p t i o n i n r e a d i n g , t h e p r o o f s o f t h e t h e o r e m s ( w h e n a n yp r o o f s a r e n e e d e d ) a r e g iv e n s e p a r a t e ly i n t h e A p p e n d i x . A R e r e a cht h e o r e m , th e p o s t u l a t e s o n w h i c h t h e p r o o f d e p e n d s a r e s t a te d i n p a r e n -t h e s e s .

S p h e r e s a n d p o i n t s .T h e o r e m 1 . ( B y 1, 2 .) I f a s p h e r e A is w i t h i n a s p h e r e B , t h e n

B i s n o t w i t h i n A .T h e o r e m 2 . ( B y 1~ 2 .) I f A a n d B a r e a n y tw o d i s t in c t s ph ere s~

t h e n o n e a n d o n l y o n e o f th e f o l l o w i n g t h r e e r e la t io n s w i l l h o l d : 1 ) o n eo f t h e s p h e r e s i s w i t h i n t h e o t h e r ; o r 2 ) t h e y a r e m u t u a l l y e x c l u s i v e ;o r 3 ) t h e y o v e r l a p .

T h e o r e m 3 . ( B y 1 , 2 .) I f A is a p o i n t, an d S i s a n y s p h e r e w h i c h.is n o t a p o i n t , t h e n A i s e i t h e r w i t h i n S o r o u t s i d e o f ~q; t h a t i % t h ec a s e o f ~ o v e r la p p i n g~ c a n n o t o c c u r . F u r t h e r , i f . 4 a n d B a r e t w o d i s t i n c tp o i n t s ~ t h e y a r e m u t u a l l y e x c l u s i v e .

T h e s t r a i g h t l i n e .T h e o r e m 4 . ( B y 1 - -3 .) I f t h e e n d p o i n t s o f a s e g m e n t c o in c id e ,

t h e s e g m e n t c o n t a in s n o o t h e r p o i n t s ; t h a t is~ [AA ] ~= A.T h e o r e m 5 . ( B y 1 - - 4 . ) T h e s e g m e n t [ A ~ a n d i t s t w o p r o lo n -g a t i o n s [ A B ' ] a n d [ B A l i f o r m a 's i m p l e s e t o f n o n - o v e r l a p p in g r e g i o n s~.s o t h a t ~ t h e l in e A B is t h e ' s im p l e s u m ~ o f t h es e t h r e e r e gi on s .

T h e o r e m 6 . ( B y 1 - - 5 . ) I f X a n d I r a re tw o d i st in c t p o i n t s o f al i n e A B ~ t h e n t h e l i n e X I 7 w i l l c o n t a i n A a n d B , a n d h e n c e b d id e n -t ic a l ' w i t h t h e l i n e A B .

T h e o r e m 7. ( B y 1 - - 5 . ) I f t h re e p o i n ts a r e o n a l in e A B , t h e no n e o f t h e m is o n t h e s e g m e n t f o r m e d b y t h e o t h e r tw o .T h i s t h e o r e m e x p r e ss e s t h e n e c e s s a r y a n d s u f f ic i e n t c o n d i t io n t h a tt h r e e p o i n t s s h a l l b e c o l l i n e a r .

" F r o m t h e s e p o s t u l a te s a n d t h e o r e m s , a l l t h e 'g e n e r a l l a w s ' o f o r d e rf o r p o i n t s o n a s t r a i g h t l i n e c a n b e d e d u c e d ; t h a t is~ i n s o f a r a s p o i n t se x is ~ a t a l l o n a g i v e n l in e , t h e y w i l l h a v e t h e p r o p e r r e l a t io n s o f o r d e r .


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542 E .V . Hv~TmoTo~.T h e p l a n e .

T h e o re m 8 . (By 1 - -3 . ) I f t he ve rt ic e s o f a t r ia ng l e c o inc ide , t het r iangle conta ins no other points ; tha t i s [AAA] = A.

[ T h e o r e m 9 a. ( B y 1 - - 6 . ) T h e t ri a ng l e [ABC] and i ts three verticalextensions form a s imp le set of non -ov erlap ping regions.]T h e o r e m 9 . ( B y 1 - - 7 . ) T h e t ri an g l e [ABC] and all i ts s ix ex-

tensions, [AB'C'], .. ., [ABC~, . .. , form a s imple se t of non-overlappingregions , so tha t the plane ABC i s the s imple sum of these seven regions .(Pr oo f by con sidering six possible cases.)

T h e o r e m 10. (By 1 - -8 . ) I f X , :Y, a nd Z a re t h r e e non -c o l l i ne a rpo in t s o f a p lane A BU, t he n t he p l a ne X YZ wil l con ta in A, B, and C,a nd he nc e be i de n t ic a l w i th t he p l a ne AB C .

T h e o r e m 11. (By 1 - -8 . ) I f f ou r po in ts a r e i n a p l a ne ABC, t he ne i the r : 1 ) one of them be longs to the t r i ang le fo rm ed by the o the r th ree ;

or 2 ) the segment jo in ing tw o of them in te rsec t s the~ ~ A f / ~ c s e gm e n t fo rm e d bY the r e m a in ing two .This theorem expresses the necessary and suf-

~ . 1~ f ic ient condi t ion tha t fou r poin ts sha l l be coplanar .(Peano, loc. cir.)T h e o r e m 12. (By 1 - -8 . ) I f X a nd Y a re two d i s ti nc t po in t s i n

a plane ABC, t he n e ve ry po in t o f t he . l i ne X Y wi l l be long t o the p l a neABG.Fro m these pos tu la tes 1 - - 8 the fo l lowing fur the r theorem s can be

deduced, wi thout the a id of any of the exis tence pos tula tes .T h e o r e m 1 8. ( B y 1 - - 8 . ) I n t h e t ri a n g le [ABC], i f X is on theside opposite A, and Y on the s ide opposi te B , then thec

A v / ~~ segm ents [A X ~ and [BY] wil l have a common poin t .To es tabl ish this theorem 13, we cons ider the point Xin re la t ion to the t r i ang ie lAB Y], and show, by a process ofexclus ion~ tha t X m ust l ie in the la tera l extens ion ~CBYA'];hence , by Def in it ion 11 , the segments [AX] and [BY] mus t in te rsec t .

T h e o re m 14. (By 1 - -8 . ) I f X a nd Y are . po int s i n the in t e r io r o fan y one of the seven co mp artments of a~ plane ABC, t he n e ve ry po in tof the segment [X Y ] is in the in te r io r o f the same compar tment .

P a r a l l e l l i n e s .T h e o re m 15. (By 1 - -9 . ) T h roug h a g iven po in t, t he re i s no t m ore

than one l ine para l le l to a given l ine .This theorem shows tha t Pos tu la te 9 , a l though s ta ted in the form

o f a ~general law ' , ne verthe less enables us to infer th e 'existence ' of poin tsof in tersect ion of m an y l ines. I t m igh t perhap s b e ca l led an existence


22/38

Postulates for abstract Geometry. 5 4 3pos tu l a t e i n d i sgu i se. Th e same rem ark app l i es t o severa l o the r pos tu-l a t e s , a s, f o r e xa m pl e , t o P os t u l a t e I 0 , w h i c h g ive s u s t he f o l low i ngt he o r e m .

T h e o r e m 1 6 .A, B a re any twot h e n e i t h e r [ A X ]c o m m o n p o i n t .

T he f o l l ow i ngT h e o r e m 1 7 .

(By 1~1 .0 . ) I f two para l le l l ines a re g iven , and i fpo in t s o f one , and X, :Y a ny t w o po i n t s o f t he o t he r ,a n d [ B Y ] , o r e ls e l A Y ] an d [BX] , wil l have a

t he o r e m s de pe nd upon t he ~ ou r - po i n t pos t u l a t e ' .( B y 1 - - 11 . ) A s e gm e n t c a nno t ha ve m or e t ha n one

m idd le po in t . Hence , t he d i agona l s o f every pa ra l le logram b i sec t each o the r .T h e o r e m 18. ( By 1 - -1 1 . ) I f m i d A X = m i d A Y , t hen X- - - - Y .Th a t is , i f one end of a segm ent is chan ged wh i l e the o the r end i she l d f a s t , - t he n t he m i dd l e po i n t w i l l he c ha nge d .T h e p r oo f o f t h i s t he o r e m 18 m a y be m a d e t o de pe nd on t he f o l low i ngl e m m a .L a m i na f o r T he o r e m 18 . ( B y 1 - - 11 . ) I f a g i ve n p l a ne c on t a in s a t

l eas t one pa ra l l e logram A.BCD, w i t h i ts m i dd l e po i n t ~ f , a nd a t le a s ton e o~her po in t E d i s t inc t f rom A, B , C, 1), a n d M , t h e n e v e r y s e g m e n tin th e pMne wi l l be the d i agona l o f a pa ra l l e logram in the p l ane , an d. he nc e w i l l ha ve a m i dd l e po i n t ; a nd f u r t he r , t h r o ug ho u t t he p l a ne , apa r a l l e l c a n a lw a ys be d r a w n t o a ny g i ve n li ne t h r oug h a ny g i ve n po i n tno t on t ha t l i ne .

A l l t he p r e c e d ing t he o r e m s ha ve be e n ob t a ine d f r om P os tu l a te s 1 - - 11 ,w i th ou t t he use o f any of t he ex i st ence pos tu l a tes . I f now we add Po s tu-l a te s E l - - E 3 , w e h a ve th e f o ll ow i n g t h eo r e m s c o n c e rn i n gexis tences .

T h e o r e m 19. ( By l m l l , E l - - E 3 . ) I f a p o in t P i s inth e in t e r io r o f a t r iang le [ABC], t hen the l i ne A P in t e r sec t s ~g . 18.the opposi te s ide ( .BC) .Th i s Theo rem 197 tog e the r wi th Th eore m 13 , w ere used as pos tu l a t esb y P e a n o .T h e o r e m 20. (B y 1 - -1 1 , E l - - E 3 . ) I f [ABC] i s a t r i ang le , andl i : is on (BC), a nd D on t he p r o l onga t i on (BA'), t h e n t h el i ne D ~ w i l l m e e t t he s ide ( A C ) i n a po i n t F . . A cThi s i s t h e C t ri ang le t r ansver se ax iom ' i n the fo rmus e d by V e b l e n .T h e o r e m 21. (B y 1 - - 1 1 , E l - - E 3 . ) I f [AB] is as e gm e n t , t he r e a r e po i n t s on t he p r o l onga t ions ( ,4 B ' ) a nd (BA '). ~ . 1~.T h e o r e m 2 2. (B y 1 - - 1 1 , E l - - E 3 . ) T h e f o ll ow i n g c on stru eb io ns a rea l w a ys pos s i b l e :


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544 E.V. HwTI~o~o~.1 ) to d r a w a li n e p a r a l le l t o a g i v e n l in e t h r o u g h a n y p o i n t n o t o n

t h a t l i n e ;2 ) t o b i s e c t a g i v e n s e g m e n t ;3 ) t o e x t e n d a g i v e n s e g m e n t t o d o u b l e i t ~ l e n ~ h .

C o n g r u e n c e , a n d p e r p e n d i c u l a r i t y .T h e o r e m 2 3 . ( B y 1 - - 1 2 . ) T h e r e la t io n o f c o n g r u e n c e is r e fl ex iv e ,

s y m m e t r i c a l , a n d t r a n s i t i v e .T h e o r e m 2 4. ( B y 1 - - 1 2 . ) i f Z is o n [ O A ] , a n d [ 0 X ] ~ [OA],

t h e n X ~= A .T h a t i s, o n a g i v e n h a lf - li n e OA~ n o t m o r e t h a n o n e s e g m e n t [OX]c a n b e l a i d o f f c o n g r u e n t t o a g i v e n s e g m e n t .T h e o r e m 2 5 . ( B y 1 - - 1 3 . ) I f A , B~ C ar e c o il ; ne a r, a n d i f A', 13", 0"

a r e c o ] li n ea r a n d i n t h e s a m e o r d e r a s A , B , C~ t h e n w h e n e v e r A B - ~ A ' - B 'and A C ~ A ' C ' ~ w e s h a l l a l w a y s h a v e B C - ~ B ' C ' .B r i e fl y s t a t e d , t h i s t h e o r e m t e ll s u s t h a t t h e s u m s o f c o n g r u e n t

s e g m e n t s ar e c o n g r u e n t .T h e o r e m 2 6. ( B y l m 1 4 . ) L e t t w o g i v en l in es m e e t i n O . I f a n y

s e g m e n t [ X Y ] , j o i n i n g a p o i n t o f o n e o f t h e l in e s w i th a p o i n t o f t h eo t h e r , i s t h e d i a g o n a l o f a r e c t a n g l e h a v i n g o n e v e r t ex a t 0 , t h e n e v e rys u c h s e g m e n t w i l l h a v e t h e s a m e p r o p e r ~ y , a n d t h e l i n e s w i l l b e p e r -p e n d i c u l a r .

T h e o r e m 2 7. ( B y 1 - - 1 4 . ) I f t w o l in e s i n t h e s am e p l an e a re p e r -p e n d i c u l a r t o t h e s a m e l i n e , t h e n t h e y a r e e i t h e r p a r a l le l o r c o in c i d e n t .A l s o , i f , i n a n y p la ne ~ a l i n e i s p e r p e n d i c u l a r t o o n e o f t w o p a r a l l e l l i n e s ,i t w i l l b e p e r p e n d i c u l a r m t h e o t h e r a l s o .

T h e o r e m 2 8 . ( B y 1 -- 1 4 . ) I f X i s i n a l in e p e r p e n di c u la r t o [ A B ]a t i t s m i d d l e p o i n t M , t h e n X i s e q u i d i s t a n t f r o m A a n d -B .

T h e o r e m 2 9 . ( B y 1 - - 1 4 . ) I f X i s e q u i d is t a n t f r o m A a n d -B , a n dM ~ m i d A.B , t h e n X M i s p e r p e n d i c u l a r t o A B .

T h e s e t w o t h e o r e m s g i v e u s t h e i m p o r t a n t p ro p e r t ie s o f t h e is o s ce le st r i a n g l e .

T h e o r e m 3 0 . ( B y ~ 1 --1 4.) I f th r o u g h t h e v e r ti ce s o f a ~ r ia ~ g lel i n e s a r e d r a w n p e r p e n d i c u l a r t o t h e o p p o s i t e s id e s~ t h e s e t h r o e l in e s w i l lh a v e a c o m m o n p o i n t . ( O r ~ h o c e n ~ r. )

T h e o r e m 3 i . ( B y 1 - - 1 4 .) I f t he s ur fa ce s o f t w oc o n c e n t r ic s p h e r e s a r e c u t b y o n e ra d i u s i n A a n d X ,a n d b y a n o t h e r r a d i u s i n -B a n d Y , t h e n t h e c r o s s s e g -m e n t s , IX 1~ and [ -BX ], w il l be c o n g ru e n t~ a n d t h e c h o r d s

~ ig. I~. [ A ~ ] a n d [ X Y ] w i l l b e p a r a l l e l


24/38

P o s tu l a t es f o r a b s t z a c t e o m e t r y . 5 4 5B y th e a id o f th e f i r s t fo u r ex i s t en ce p o s tu la tes ~ we h av e a l so th ef o l l o w i n g t h e o r e m .T h e o r e m 32 . ( B y 1 - - 1 4 , E l - - E 4 . ) T h e f ol lo w in g c o ns tr uc tio n sa re a lway s p o s s ib le :1 ) to d raw a s p h ere w i th an y g iv en p o in t 0 as cen te r , an d an y g iv ens e g m e n t [ 0 2 1 ] a s r a d i u s ;~ ) to f in d th e p o in t o f in te r s ec t io n o f th e s u r face o f a s p h ere w i th

a n y l i n e t h r o u g h i t s c e n t e r ;3 ) t o q a y o ff ' ;o n a n y g i v e n h a l f - l i n e a s e g m e n t c o n g r u e n t t o a n yg i v e n s e g m e n t ;4 ) t o d r a w a p e r p e n d i c u l a r t o a n y g i v e n l i n e t h r o u g h a n y g i v e n p o in t.

T h e n u m b e r l i n e .T h e o r e m 33 . (B y 1 - -5 . ) Let , A , B , C b e a n y p o i n t s o n t h e n u m b e r

l i n e ( s e e D e f . 2 6 ). T h e n w e h a v e :I ) I f A a n d B a r e d is ti nc t, t h e n e i t h e r A < ~ o r B < A .2 ) I f < B , t h e n 21 a n d B a r e d is ti nc t.~3) I f ~ i < ~ a n d B < C , t h e n X < 0 .

T h e p o i m i m o n t h e n u m b e r l i n e t h e r e f o re f o r m a s er ie s* ), o r o r d e r e d c l a s s ,w i t h r e ~ p e c t t o t h e r e l a t i o n < .T h e o r e m 3 4. ( B y 1 - - 1 1 . ) L e t A , .B~ C be a n y p o i nt s o n t h e n u m b e r

l in e . Th en , in so fa r a s th e s u m s in q u es t io n ex i s t ( s ee Def . 2 4 ) , wes h a l l h av e th e fo l lo win g l aws o f ad d i t io n * * ) :

1 ) ( .4 + B) + 0 = A + (B + C) . (Associa t ive law. )2) A + B == B + A. (Co m m utat ive law. )3) i f 2 1+ X== A + Y , then X=~ Y .4 ) I f X is n o t z e r o , t h e . i t + 2 1 + . . 9 + a i s n o t z e r o .W e s h a ll al so h a v e t h e f o ll o w i n g l a w c o n n e c t i n g + a n d < :5 ) I f X < I7~ t he n 2 1 + X < 2 1 + 1 .T h e o r e m 3 5. ( B y 1 - - 1 1 .) T h e p r o d u c t o f t w o p o in ts o n t h e n u m b e r 'l i n e i s i n d e p e n d e n t o f t h e p o s it io n o f t h e a u x i l i a r y p o i n t u s e d i n t h e

cons t ruct ion (see Def . 25) .T h e o r e m 3 6 . ( B y I - - i i . ) L e t A , B , 0 b e a n y p o i n t s o n t h e n u m b e r*) G. Yailati, Sui prlneipt fondamen tall della Geom etria della te tra , Rivista diM atem atica, 2 (1892), p.'71--75; E. V. Hu ntington, The Con tinuum as a Type o f Order,reprinted from the A nn als of M athematics, 1905 (Publication Office of HarvardUnivarsity).**) E. Y. Huntington, The Fundam ental La w s of Addition and M ultiplication inElementary Alge bra, reprinted fro m the A nna ls of Ma~em stics, 1906 (PublicationOffiee of Harvard University). Also, The Fundam ental Propositions of Algebra, in thevolume o f M athematical M onographs edited by J. W . A. Young , p. 150- -207, 1911.


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5 4 6 E .V . HU~T~r~GTO~.l i n e , an d l e t i t b e p o s s ib le to d raw p ara ll e l l in es a t p leas u re ( so th a tp r o d u c t s o f s u c h p o i n ts c a n a l w a y s b e o b t a in e d ). T h e n w e h a v e t h e f o l lo w i n gl a w s o f m u l t i p l i c a t i o n * ) :

1 ) ( A C-- -- A ( B 0 ) . (As s o c ia t iv e l aw .)2) A A x C , an d3) ( B + C) x A -----B x A + C x A . ( D i st ri b ut iv e l aw s .)4 ) I f A x X ~ A o r X Y x A , a nd A n o t ze ro , t h e n

X = Y .A l so w e h a v e t h e fo l l o w i n g l a w c o n n e c t i n g a n d < :

5 ) I f X < 1 an d A p o si tiv e , th e n A x X < A Y an d X x A < 1 7 A .T h e o r e m 37. ( B y 1 - - 1 4 . ) . I f A a nd B a re p oi nt s o n t h e n u m b e rl in e , a s in T h e o r e m 3 6, w e h a v e t h e c o m m u t a t iv e l a w f o r m u l t i p l ic a t i o n :

A 2 1 5O f t h e s e t h e o re m s , 3 3 - - 3 7 , t h e f i r st f o u r a r e o b t a i ne d b y t h e a i d

o f Po s tu la te s 1 - -1 1 ; fo r th e l a st , h o wev e r , we as s u me a l s o th e p o s tu la teso f c o n g r u e n c e .

B

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