,an ,elem5t>y ,c\rse 9 ,syn!tic ,projective ,geometry by ,lehm]1 ,d]rick ,norman ,edition #a1 7,novemb] #d1 #bjje7 ,,preface ,! foll[+ c\rse is int5d$ to give1 9 z simple a way z possible1 ! ess5tials ( syn!tic projective geometry4 ,:ile1 9 ! ma91 ! !ory is develop$ along ! well-beat5 track laid out by ! great ma/]s ( ! subject1 x is believ$ t !re has be5 a slight smoo?+ ( ! road 9 "s places4 ,especially w ?is be obs]v$ 9 ! *apt] on ,involution4 ,! au?or has nev] felt satisfi$ ) ! usual treatm5t ( t subject by means ( circles & anh>monic ratios4 ,a purely projective notion ought n to be bas$ on metrical f\ndations4 ,metrical developm5ts %\ld be made !re1 z else:]e 9 ! !ory1 by ! introduction ( inf9itely di/ant elem5ts4 ,! au?or has dep>t$ f ! c5tury-old cu/om ( writ+ 9 p>allel columns ea* !orem & its dual4 ,he has n f\nd t x conduces to %>pness ( vision to try to focus 8 eyes on two ?+s at once4 ,^? :o pref] ! usual me?od ( proc$ure c1 ( c\rse1 develop ! two sets ( !orems side by side2 ! au?or has n f\nd ?is ! bett] plan 9 actual tea*+4 ,z reg>ds nom5clature1 ! au?or has foll[$ ! lead ( ! e>li] writ]s 9 ,engli%1 & has call$ ! sy/em ( l9es 9 a plane :i* all pass ?r\gh a po9t a p5cil ( rays in/ead ( a bundle ( rays1 z lat] writ]s seem incl9$ to d4 ,= a po9t consid]$ z made up ( all ! l9es & planes ?r\gh x he has v5tur$ to use ! t]m po9t sy/em1 z be+ ! natural dualization ( ! usual t]m plane sy/em4 ,he has also reject$ ! t]m foci ( an involution1 & has n us$ ! cu/om>y t]ms = classify+ involutions--hyp]bolic involution1 elliptic involution & p>abolic involution4 ,he has f\nd t all ^! t]ms are v confus+ to ! /ud5t1 :o inevitably tries to connect !m 9 "s way ) ! conic sections4 ,en\gh examples h be5 provid$ to give ! /ud5t a cle> grasp ( ! !ory4 ,_m are ( suffici5t g5]ality to s]ve z a basis = individual inve/igation on ! "p ( ! /ud5t4 ,?us1 ! ?ird example at ! end ( ! fir/ *apt] w be f\nd to be v fruitful 9 int]e/+ results4 ,a correspond5ce is !re indicat$ betwe5 l9es 9 space & circles ?r\gh a fix$ po9t 9 space4 ,if ! /ud5t w trace a few ( ! consequ5ces ( t correspond5ce1 & det]m9e :at configurations ( circles correspond to int]sect+ l9es1 to l9es 9 a plane1 to l9es ( a plane p5cil1 to l9es cutt+ ?ree skew l9es1 etc41 he w h acquir$ no little practice 9 pictur+ to himself figures 9 space4 ,! writ] has n foll[$ ! usual practice ( ins]t+ hi/orical notes at ! foot ( ! page1 & has tri$ in/ead1 9 ! la/ *apt]1 to give a consecutive acc\nt ( ! hi/ory ( pure geometry1 or1 at lea/1 ( z mu* ( x z ! /ud5t w be able to appreciate :o has ma/]$ ! c\rse z giv5 9 ! prec$+ *apt]s4 ,"o is n apt to get a v wide view ( ! hi/ory ( a subject by read+ a hundr$ biographical footnotes1 arrang$ 9 no sort ( sequ5ce4 ,! writ]1 moreov]1 feels t ! prop] "t to le>n ! hi/ory ( a subject is aft] ! /ud5t has "s g5]al ideas ( ! subject itself4 ,! c\rse is n int5d$ to furni% an illu/ration ( h[ a subject may be develop$1 f ! smalle/ possible numb] ( fundam5tal assumptions4 ,! au?or is aw>e ( ! importance ( "w ( ?is sort1 b he does n believe x is possible at ! pres5t "t to write a book along su* l9es :i* %all be ( mu* use = elem5t>y /ud5ts4 ,= ! purposes ( ?is c\rse ! /ud5t %\ld h a ?or\gh gr\nd+ 9 ord9>y elem5t>y geometry s f> z to include ! /udy ( ! circle & ( simil> triangles4 ,no solid geometry is ne$$ beyond ! little us$ 9 ! pro( ( ,des>gues' !orem 7#be71 &1 except 9 c]ta9 metrical developm5ts ( ! g5]al !ory1 !re w be no call = a k ( trigonometry or analytical geometry4 ,naturally ! /ud5t :o is equipp$ ) ^! subjects z well z ) ! calculus w be a little m mature1 & may be expect$ to foll[ ! c\rse all ! m easily4 ,! au?or has _h no difficulty1 h[ev]1 9 pres5t+ x to /ud5ts 9 ! fre%man class at ! ,univ]sity ( ,cali=nia4 ,! subject ( syn!tic projective geometry is1 9 ! op9ion ( ! writ]1 de/9$ %ortly to =ce its way d[n into ! second>y s*ools2 & if ?is little book helps to accel]ate ! movem5t1 he w feel amply repaid = ! task ( work+ ! mat]ials into a =m available = su* s*ools z well z = ! l[] classes 9 ! univ]sity4 ,! mat]ial = ! c\rse has be5 drawn f _m s\rces4 ,! au?or is *iefly indebt$ to ! classical works ( ,reye1 ,cremona1 ,/e9]1 ,poncelet1 & ,von ,/audt4 ,ackn[l$gm5ts & ?anks are also due to ,pr(essor ,walt] ,c4 ,eells1 ( ! ,u4,s4 ,naval ,academy at ,annapolis1 = 8 se>*+ exam9ation & ke5 criticism ( ! manuscript2 also to ,pr(essor ,h]b]t ,ellswor? ,slaught1 ( ,! ,univ]sity ( ,*icago1 = 8 _m valuable sugge/ions1 & to ,pr(essor ,b4 ,m4 ,woods & ,dr4 ,h4 ,n4 ,wright1 ( ! ,univ]sity ( ,cali=nia1 :o h tri$ out ! me?ods ( pres5tation1 9 _! own classes4 ,d4 ,n4 ,,lehm] ,,b]keley1 ,,cali=nia ,,cont5ts ,preface ,cont5ts ,,*apt] ,i - ,,"o-,,to-,,"o ,,correspond5ce #a4 ,def9ition ( "o-to-"o correspond5ce #b4 ,consequ5ces ( "o-to-"o correspond5ce #c4 ,applications 9 ma!matics #d4 ,"o-to-"o correspond5ce & enum]ation #e4 ,correspond5ce betwe5 a "p & ! :ole #f4 ,inf9itely di/ant po9t #g4 ,axial p5cil2 fundam5tal =ms #h4 ,p]spective position #i4 ,projective relation #aj4 ,inf9ity-to-"o correspond5ce #aa4 ,inf9itudes ( diff]5t ord]s #ab4 ,po9ts 9 a plane #ac4 ,l9es ?r\gh a po9t #ad4 ,planes ?r\gh a po9t #ae4 ,l9es 9 a plane #af4 ,plane sy/em & po9t sy/em #ag4 ,planes 9 space #ah4 ,po9ts ( space #ai4 ,space sy/em #bj4 ,l9es 9 space #ba4 ,correspond5ce betwe5 po9ts & numb]s #bb4 ,elem5ts at inf9ity ,,problems ,,*apt] ,,ii - ,,relations ,,betwe5 ,,fundam5tal ,,=ms ,,9 ,,"o-,,to-,,"o ,,correspond5ce ,,) ,,ea* ,,o!r #bc4 ,sev5 fundam5tal =ms #bd4 ,projective prop]ties #be4 ,des>gues's !orem #bf4 ,fundam5tal !orem conc]n+ two complete quadrangles #bg4 ,importance ( ! !orem #bh4 ,re/atem5t ( ! !orem #bi4 ,f\r h>monic po9ts #cj4 ,h>monic conjugates #ca4 ,importance ( ! notion ( f\r h>monic po9ts #cb4 ,projective inv>iance ( f\r h>monic po9ts #cc4 ,f\r h>monic l9es #cd4 ,f\r h>monic planes #ce4 ,summ>y ( results #cf4 ,def9ition ( projectivity #cg4 ,correspond5ce betwe5 h>monic conjugates #ch4 ,sep>ation ( h>monic conjugates #ci4 ,h>monic conjugate ( ! po9t at inf9ity #dj4 ,projective !orems & metrical !orems4 ,l9e> con/ruction #da4 ,p>allels & mid-po9ts #db4 ,division ( segm5t into equal p>ts #dc4 ,num]ical relations #dd4 ,algebraic =mula connect+ f\r h>monic po9ts #de4 ,fur!r =mulae #df4 ,anh>monic ratio ,,problems ,,*apt] ,,iii - ,,comb9ation ,,( ,,two ,,projectively ,,relat$ ,,fundam5tal ,,=ms #dg4 ,sup]pos$ fundam5tal =ms4 ,self-correspond+ elem5ts #dh4 ,special case #di4 ,fundam5tal !orem4 ,po/ulate ( cont9uity #ej4 ,ext5sion ( !orem to p5cils ( rays & planes #ea4 ,projective po9t-r[s hav+ a self-correspond+ po9t 9 common #eb4 ,po9t-r[s 9 p]spective position #ec4 ,p5cils 9 p]spective position #ed4 ,axial p5cils 9 p]spective position #ee4 ,po9t-r[ ( ! second ord] #ef4 ,deg5]ation ( locus #eg4 ,p5cils ( rays ( ! second ord] #eh4 ,deg5]ate case #ei4 ,cone ( ! second ord] ,,problems ,,*apt] ,,iv - ,,po9t-,,r[s ,,( ,,! ,,second ,,ord] #fj4 ,po9t-r[ ( ! second ord] def9$ #fa4 ,tang5t l9e #fb4 ,det]m9ation ( ! locus #fc4 ,re/atem5t ( ! problem #fd4 ,solution ( ! fundam5tal problem #fe4 ,diff]5t con/ructions = ! figure #ff4 ,l9es jo9+ f\r po9ts ( ! locus to a fif? #fg4 ,re/atem5t ( ! !orem #fh4 ,fur!r important !orem #fi4 ,pascal's !orem #gj4 ,p]mutation ( po9ts 9 ,pascal's !orem #ga4 ,h>monic po9ts on a po9t-r[ ( ! second ord] #gb4 ,det]m9ation ( ! locus #gc4 ,circles & conics z po9t-r[s ( ! second ord] #gd4 ,conic ?r\gh five po9ts #ge4 ,tang5t to a conic #gf4 ,inscrib$ quadrangle #gg4 ,inscrib$ triangle #gh4 ,deg5]ate conic ,,problems ,,*apt] ,v - ,,p5cils ,,( ,,rays ,,( ,,! ,,second ,,ord] #gi4 ,p5cil ( rays ( ! second ord] def9$ #hj4 ,tang5ts to a circle #ha4 ,tang5ts to a conic #hb4 ,g5]at+ po9t-r[s l9es ( ! sy/em #hc4 ,det]m9ation ( ! p5cil #hd4 ,brian*on's !orem #he4 ,p]mutations ( l9es 9 ,brian*on's !orem #hf4 ,con/ruction ( ! p5vil by ,brian*on's !orem #hg4 ,po9t ( contact ( a tang5t to a conic #hh4 ,circumscrib$ quadrilat]al #hi4 ,circumscrib$ triangle #ij4 ,use ( ,brian*on's !orem #ia4 ,h>monic tang5ts #ib4 ,projectivity & p]spectivity #ic4 ,deg5]ate case #id4 ,law ( duality ,,problems ,,*apt] ,,vi - ,,poles ,,& ,,pol>s #ie4 ,inscrib$ & circumscrib$ quadrilat]als #if4 ,def9ition ( ! pol> l9e ( a po9t #ig4 ,fur!r def9+ prop]ties #ih4 ,def9ition ( ! pole ( a l9e #ii4 ,fundam5tal !orem ( poles & pol>s #ajj4 ,conjugate po9ts & l9es #aja4 ,con/ruction ( ! pol> l9e ( a giv5 po9t #ajb4 ,self-pol> triangle #ajc4 ,pole & pol> projectively relat$ #ajd4 ,duality #aje4 ,self-dual !orems #ajf4 ,o!r correspond5ces ,,problems ,,*apt] ,,vii - ,,metrical ,,prop]ties ,,( ,,! ,,conic ,,sections #ajg4 ,diamet]s4 ,c5t] #ajh4 ,v>i\s !orems #aji4 ,conjugate diamet]s #aaj4 ,classification ( conics #aaa4 ,asymptotes #aab4 ,v>i\s !orems #aac4 ,!orems conc]n+ asymptotes #aad4 ,asymptotes & conjugate diamet]s #aae4 ,segm5ts cut (f on a *ord by hyp]bola & its asymptotes #aaf4 ,application ( ! !orem #aag4 ,triangle =m$ by ! two asymptotes & a tang5t #aah4 ,equation ( hyp]bola ref]r$ to ! asymptotes #aai4 ,equation ( p>abola #abj4 ,equation ( c5tral conics ref]r$ to conjugate diamet]s ,,problems ,,*apt] ,,viii - ,,involution #aba4 ,fundam5tal !orem #abb4 ,l9e> con/ruction #abc4 ,def9ition ( involution ( po9ts on a l9e #abd4 ,d\ble-po9ts 9 an involution #abe4 ,des>gues's !orem conc]n+ conics ?r\gh f\r po9ts #abf4 ,deg5]ate conics ( ! sy/em #abg4 ,conics ?r\gh f\r po9ts t\*+ a giv5 l9e #abh4 ,d\ble correspond5ce #abi4 ,/e9]'s con/ruction #acj4 ,application ( ,/e9]'s con/ruction to d\ble correspond5ce #aca4 ,involution ( po9ts on a po9t-r[ ( ! second ord]4 #acb4 ,involution ( rays #acc4 ,d\ble rays #acd4 ,conic ?r\gh a fix$ po9t t\*+ f\r l9es #ace4 ,d\ble correspond5ce #acf4 ,p5cils ( rays ( ! second ord] 9 involution #acg4 ,!orem conc]n+ p5cils ( ! second ord] 9 involution #ach4 ,involution ( rays det]m9$ by a conic #aci4 ,/atem5t ( !orem #adj4 ,dual ( ! !orem ,,problems ,,*apt] ,,ix - ,,metrical ,,prop]ties ,,( ,,involutions #ada4 ,introduction ( inf9ite po9t2 c5t] ( involution #adb4 ,fundam5tal metrical !orem #adc4 ,exi/5ce ( d\ble po9ts #add4 ,exi/5ce ( d\ble rays #ade4 ,con/ruction ( an involution by means ( circles #adf4 ,circul> po9ts #adg4 ,pairs 9 an involution ( rays :i* are at "r angles4 ,circul> involution #adh4 ,axes ( conics #adi4 ,po9ts at :i* ! involution det]m9$ by a conic is circul> #aej4 ,prop]ties ( su* a po9t #aea4 ,position ( su* a po9t #aeb4 ,discov]y ( ! foci ( ! conic #aec4 ,! circle & ! p>abola #aed4 ,focal prop]ties ( conics #aee4 ,case ( ! p>abola #aef4 ,p>abolic reflector #aeg4 ,directrix4 ,pr9cipal axis4 ,v]tex #aeh4 ,ano!r def9ition ( a conic #aei4 ,ecc5tricity #afj4 ,sum or diff]5ce ( focal di/ances ,,problems ,,*apt] ,x - ,,on ,,! ,,hi/ory ,,( ,,syn!tic ,,projective ,,geometry #afa4 ,anci5t results #afb4 ,unify+ pr9ciples #afc4 ,des>gues #afd4 ,poles & pol>s #afe4 ,des>gues's !orem conc]n+ conics ?r\gh f\r po9ts #aff4 ,ext5sion ( ! !ory ( poles & pol>s to space #afg4 ,des>gues's me?od ( describ+ a conic #afh4 ,reception ( ,des>gues's "w #afi4 ,cons]vatism 9 ,des>gues's "t #agj4 ,des>gues's /yle ( writ+ #aga4 ,lack ( appreciation ( ,des>gues #agb4 ,pascal & 8 !orem #agc4 ,pascal's essay #agd4 ,pascal's orig9ality #age4 ,de la ,hire & 8 "w #agf4 ,desc>tes & 8 influ5ce #agg4 ,newton & ,maclaur9 #agh4 ,maclaur9's con/ruction #agi4 ,descriptive geometry & ! second revival #ahj4 ,duality1 homology1 cont9uity1 cont+5t relations #aha4 ,poncelet & ,cau*y #ahb4 ,! "w ( ,poncelet #ahc4 ,! debt :i* analytic geometry owes to syn!tic geometry #ahd4 ,/e9] & 8 "w #ahe4 ,von ,/audt & 8 "w #ahf4 ,rec5t developm5ts ,,index ,,*apt] ,i - ,,"o-,,to-,,"o ,,correspond5ce 99#a4 ,def9ition ( "o-to-"o correspond5ce499 ,giv5 any two sets ( individuals1 if x is possible to set up su* a correspond5ce betwe5 ! two sets t to any individual 9 "o set corresponds "o & only "o individual 9 ! o!r1 !n ! two sets are said to be 9 "o-to-"o correspond5ce ) ea* o!r4 ,?is notion1 simple z x is1 is ( fundam5tal importance 9 all bran*es ( sci5ce4 ,! process ( c\nt+ is no?+ b a sett+ up ( a "o-to-"o correspond5ce betwe5 ! objects to be c\nt$ & c]ta9 words1 '"o1' 'two1' '?ree1' etc41 9 ! m9d4 ,_m savage peoples h discov]$ no bett] me?od ( c\nt+ ?an by sett+ up a "o-to-"o correspond5ce betwe5 ! objects to be c\nt$ & _! f+]s4 ,! sci5ti/ :o busies himself ) nam+ & classify+ ! objects ( nature is only sett+ up a "o-to-"o correspond5ce betwe5 ! objects & c]ta9 words :i* s]ve1 n z a means ( c\nt+ ! objects1 b ( li/+ !m 9 a conv5i5t way4 ,?us he may be able to m>%al & array 8 mat]ial 9 su* a way z to br+ to light relations t may exi/ betwe5 ! objects !mselves4 ,inde$1 ! :ole notion ( language spr+s f ?is idea ( "o-to-"o correspond5ce4 99#b4 ,consequ5ces ( "o-to-"o correspond5ce499 ,! mo/ useful & int]e/+ problem t may arise 9 connection ) any "o-to-"o correspond5ce is to det]m9e j :at relations exi/+ betwe5 ! individuals ( "o assemblage may be c>ri$ ov] to ano!r assemblage 9 "o-to-"o correspond5ce ) x4 ,x is a favorite error to assume t :atev] holds = "o set mu/ also hold = ! o!r4 ,magicians are apt to assign magic prop]ties to _m ( ! words & symbols :i* !y are 9 ! habit ( us+1 & sci5ti/s are con/antly confus+ objective ?+s ) ! subjective =mulas = !m4 ,aft] ! physici/ has set up correspond5ces betwe5 physical facts & ma!matical =mulas1 ! int]pretation ( ^! =mulas is 8 mo/ important & difficult task4 99#c499 ,9 ma!matics1 ef=t is con/antly be+ made to set up "o-to-"o correspond5ces betwe5 simple notions & m complicat$ ones1 or betwe5 ! well-explor$ fields ( rese>* & fields less kn[n4 ,?us1 by means ( ! me*anism employ$ 9 analytic geometry1 algebraic !orems are made to yield geometric ones1 & vice v]sa4 ,9 geometry we get at ! prop]ties ( ! conic sections by means ( ! prop]ties ( ! /raight l9e1 & cubic surfaces are /udi$ by means ( ! plane4 ,figure #a ,,fig4 #a ,figure #b ,,fig4 #b 99#d4 ,"o-to-"o correspond5ce & enum]ation499 ,if a "o-to-"o correspond5ce has be5 set up betwe5 ! objects ( "o set & ! objects ( ano!r set1 !n ! inf]5ce may usually be drawn t !y h ! same numb] ( elem5ts4 ,if1 h[ev]1 !re is an inf9ite numb] ( individuals 9 ea* ( ! two sets1 ! notion ( c\nt+ is necess>ily rul$ out4 ,x may be possible1 nev]!less1 to set up a "o-to-"o correspond5ce betwe5 ! elem5ts ( two sets ev5 :5 ! numb] is inf9ite4 ,?us1 x is easy to set up su* a correspond5ce betwe5 ! po9ts ( a l9e an in* long & ! po9ts ( a l9e two in*es long4 ,= let ! l9es 7,fig4 #a7 be ,,ab & ,a',b'4 ,jo9 ,,aa' & ,,bb'1 & let ^! jo9+ l9es meet 9 ,s4 ,= e po9t ,c on ,,ab a po9t ,c' may be f\nd on ,a',b' by jo9+ ,c to ,s & not+ ! po9t ,c' :]e ,,cs meets ,a',b'4 ,simil>ly1 a po9t ,c may be f\nd on ,,ab = any po9t ,c' on ,a',b'4 ,! correspond5ce is cle>ly "o-to-"o1 b x w\ld be absurd to inf] f ?is t !re 7 j z _m po9ts on ,,ab z on ,a',b'4 ,9 fact1 x w\ld be j z reasonable to inf] t !re 7 twice z _m po9ts on ,a',b' z on ,,ab4 ,= if we b5d ,a',b' into a circle ) c5t] at ,s 7,fig4 #b71 we see t = e po9t ,c on ,,ab !re are two po9ts on ,a',b'4 ,?us x is se5 t ! notion ( "o-to-"o correspond5ce is m ext5sive ?an ! notion ( c\nt+1 & includes ! notion ( c\nt+ only :5 appli$ to f9ite assemblages4 99#e4 ,correspond5ce betwe5 a "p & ! :ole ( an inf9ite assemblage499 ,9 ! discussion ( ! la/ p>agraph ! rem>kable fact 0 br\ght to light t x is sometimes possible to set ! elem5ts ( an assemblage into "o-to-"o correspond5ce ) a "p ( ^? elem5ts4 ,a mom5t's reflection w conv9ce "o t ?is is nev] possible :5 !re is a f9ite numb] ( elem5ts 9 ! assemblage4--,inde$1 we may take ?is prop]ty z our def9ition ( an inf9ite assemblage1 & say t an inf9ite assemblage is "o t may be put into "o-to-"o correspond5ce ) "p ( itself4 ,?is has ! advantage ( be+ a positive def9ition1 z oppos$ to ! usual negative def9ition ( an inf9ite assemblage z "o t _c be c\nt$4 99#f4 ,inf9itely di/ant po9t499 ,we h illu/rat$ above a simple me?od ( sett+ ! po9ts ( two l9es into "o-to-"o correspond5ce4 ,! same illu/ration w s]ve also to %[ h[ x is possible to set ! po9ts on a l9e into "o-to-"o correspond5ce ) ! l9es ?r\gh a po9t4 ,?us1 = any po9t ,c on ! l9e ,,ab !re is a l9e ,,sc ?r\gh ,s4 ,we mu/ assume ! l9e ,,ab ext5d$ indef9itely 9 bo? directions1 h[ev]1 if we are to h a po9t on x = e l9e ?r\gh ,s2 & ev5 ) ?is ext5sion !re is "o l9e ?r\gh ,s1 accord+ to ,euclid's po/ulate1 :i* does n meet ! l9e ,,ab & :i* !re=e has no po9t on ,,ab to correspond to x4 ,9 ord] to smoo? out ?is discrepancy we are accu/om$ to assume ! exi/5ce ( an inf9itely di/ant po9t on ! l9e ,,ab & to assign ?is po9t z ! correspond+ po9t ( ! exceptional l9e ( ,s4 ,) ?is und]/&+1 !n1 we may say t we h set ! l9es ?r\gh a po9t & ! po9ts on a l9e into "o-to-"o correspond5ce4 ,?is correspond5ce is ( su* fundam5tal importance 9 ! /udy ( projective geometry t a special "n is giv5 to x4 ,call+ ! totality ( po9ts on a l9e a po9t-r[1 & ! totality ( l9es ?r\gh a po9t a p5cil ( rays1 we say t ! po9t-r[ & ! p5cil relat$ z above are 9 p]spective position1 or t !y are p]spectively relat$4 99#g4 ,axial p5cil2 fundam5tal =ms499 ,a simil> correspond5ce may be set up betwe5 ! po9ts on a l9e & ! planes ?r\gh ano!r l9e :i* does n meet ! fir/4 ,su* a sy/em ( planes is call$ an axial p5cil1 & ! ?ree assemblages--! po9t-r[1 ! p5cil ( rays1 & ! axial p5cil--are call$ fundam5tal =ms4 ,! fact t !y may all be set into "o-to-"o correspond5ce ) ea* o!r is express$ by say+ t !y are ( ! same ord]4 ,x is usual also to speak ( !m z ( ! fir/ ord]4 ,we %all see pres5tly t !re are o!r assemblages :i* _c be put into ?is sort ( "o-to-"o correspond5ce ) ! po9ts on a l9e1 & t !y w v reasonably be said to be ( a high] ord]4 99#h4 ,p]spective position499 ,we h said t a po9t-r[ & a p5cil ( rays are 9 p]spective position if ea* ray ( ! p5cil goes ?r\gh ! po9t ( ! po9t-r[ :i* corresponds to x4 ,two p5cils ( rays are also said to be 9 p]spective position if correspond+ rays meet on a /raight l9e :i* is call$ ! axis ( p]spectivity4 ,also1 two po9t-r[s are said to be 9 p]spective position if correspond+ po9ts lie on /raight l9es ?r\gh a po9t :i* is call$ ! c5t] ( p]spectivity4 ,a po9t-r[ & an axial p5cil are 9 p]spective position if ea* plane ( ! p5cil goes ?r\gh ! po9t on ! po9t-r[ :i* corresponds to x1 & an axial p5cil & a p5cil ( rays are 9 p]spective position if ea* ray lies 9 ! plane :i* corresponds to x2 &1 f9ally1 two axial p5cils are p]spectively relat$ if correspond+ planes meet 9 a plane4 99#i4 ,projective relation499 ,x is easy to imag9e a m g5]al correspond5ce betwe5 ! po9ts ( two po9t-r[s ?an ! "o j describ$4 ,if we take two p]spective p5cils1 ,a & ,s1 !n a po9t-r[ a p]spective to ,a w be 9 "o-to-"o correspond5ce ) a po9t-r[ b p]spective to ,b1 b correspond+ po9ts w n1 9 g5]al1 lie on l9es :i* all pass ?r\gh a po9t4 ,two su* po9t-r[s are said to be projectively relat$1 or simply projective to ea* o!r4 ,simil>ly1 two p5cils ( rays1 or ( planes1 are projectively relat$ to ea* o!r if !y are p]spective to two p]spective po9t-r[s4 ,?is idea w be g5]aliz$ lat] on4 ,x is important to note t betwe5 ! elem5ts ( two projective fundam5tal =ms !re is a "o-to-"o correspond5ce1 & also t ?is correspond5ce is 9 g5]al cont9u\s2 t is1 by tak+ two elem5ts ( "o =m suffici5tly close to ea* o!r1 ! two correspond+ elem5ts 9 ! o!r =m may be made to approa* ea* o!r arbitr>ily close4 ,9 ! case ( po9t-r[s ?is cont9uity is subject to exception 9 ! neighborhood ( ! po9t at inf9ity4 99#aj4 ,inf9ity-to-"o correspond5ce499 ,x might be inf]r$ t any inf9ite assemblage c\ld be put into "o-to-"o correspond5ce ) any o!r4 ,su* is n ! case1 h[ev]1 if ! correspond5ce is to be cont9u\s1 betwe5 ! po9ts on a l9e & ! po9ts on a plane4 ,consid] two l9es :i* lie 9 diff]5t planes1 & take m po9ts on "o & n po9ts on ! o!r4 ,! numb] ( l9es jo9+ ! m po9ts ( "o to ! n po9ts j( ! o!r is cle>ly mn4 ,if we symbolize ! totality ( po9ts on a l9e by inf9ity1 !n a reasonable symbol = ! totality ( l9es drawn to cut two l9es w\ld be inf9ity#b4 ,cle>ly1 = e po9t on "o l9e !re are inf9ity l9es cutt+ across ! o!r1 s t ! correspond5ce might be call$ inf9ity-to-"o4 ,?us ! assemblage ( l9es cutt+ across two l9es is ( high] ord] ?an ! assemblage ( po9ts on a l9e2 & z we h call$ ! po9t-r[ an assemblage ( ! fir/ ord]1 ! sy/em ( l9es cutt+ across two l9es ought to be call$ ( ! second ord]4 99#aa4 ,inf9itudes ( diff]5t ord]s499 ,n[ x is easy to set up a "o-to-"o correspond5ce betwe5 ! po9ts 9 a plane & ! sy/em ( l9es cutt+ across two l9es :i* lie 9 diff]5t planes4 ,9 fact1 ea* l9e ( ! sy/em ( l9es meets ! plane 9 "o po9t1 & ea* po9t 9 ! plane det]m9es "o & only "o l9e cutt+ across ! two giv5 l9es--namely1 ! l9e ( int]section ( ! two planes det]m9$ by ! giv5 po9t ) ea* ( ! giv5 l9es4 ,! assemblage ( po9ts 9 ! plane is ?us ( ! same ord] z t ( ! l9es cutt+ across two l9es :i* lie 9 diff]5t planes1 & ought !re=e to be spok5 ( z ( ! second ord]4 ,we express all ^! results z foll[s3 99#ab499 ,if ! inf9itude ( po9ts on a l9e is tak5 z ! inf9itude ( ! fir/ ord]1 !n ! inf9itude ( l9es 9 a p5cil ( rays & ! inf9itude ( planes 9 an axial p5cil are also ( ! fir/ ord]1 :ile ! inf9itude ( l9es cutt+ across two skew l9es1 z well z ! inf9itude ( po9ts 9 a plane1 are ( ! second ord]4 99#ac499 ,if we jo9 ea* ( ! po9ts ( a plane to a po9t n 9 t plane1 we set up a "o-to-"o correspond5ce betwe5 ! po9ts 9 a plane & ! l9es ?r\gh a po9t 9 space4 ,?us ! inf9itude ( l9es ?r\gh a po9t 9 space is ( ! second ord]4 99#ad499 ,if to ea* l9e ?r\gh a po9t 9 space we make correspond t plane at "r angles to x & pass+ ?r\gh ! same po9t1 we see t ! inf9itude ( planes ?r\gh a po9t 9 space is ( ! second ord]4 99#ae499 ,if to ea* plane ?r\gh a po9t 9 space we make correspond ! l9e 9 :i* x int]sects a giv5 plane1 we see t ! inf9itude ( l9es 9 a plane is ( ! second ord]4 ,?is may also be se5 by sett+ up a "o-to-"o correspond5ce betwe5 ! po9ts on a plane & ! l9es ( t plane4 ,?us1 take a po9t ,s n 9 ! plane4 ,jo9 any po9t ,m ( ! plane to ,s4 ,?r\gh ,s draw a plane at "r angles to ,,ms4 ,?is meets ! giv5 plane 9 a l9e m :i* may be tak5 z correspond+ to ! po9t ,m4 ,ano!r v important me?od ( sett+ up a "o-to-"o correspond5ce betwe5 l9es & po9ts 9 a plane w be giv5 lat]1 & _m weighty consequ5ces w be d]iv$ f x4 99#af4 ,plane sy/em & po9t sy/em499 ,! plane1 consid]$ z made up ( ! po9ts & l9es 9 x1 is call$ a plane sy/em & is a fundam5tal =m ( ! second ord]4 ,! po9t1 consid]$ z made up ( all ! l9es & planes pass+ ?r\gh x1 is call$ a po9t sy/em & is also a fundam5tal =m ( ! second ord]4 99#ag499 ,if n[ we take ?ree l9es 9 space all ly+ 9 diff]5t planes1 & select l po9ts on ! fir/1 m po9ts on ! second1 & n po9ts on ! ?ird1 !n ! total numb] ( planes pass+ ?r\gh "o ( ! select$ po9ts on ea* l9e w be lmn4 ,x is reasonable1 !re=e1 to symbolize ! totality ( planes t are det]m9$ by ! inf9ity po9ts on ea* ( ! ?ree l9es by inf9ity#c1 & to call x an inf9itude ( ! ?ird ord]4 ,b x is easily se5 t e plane 9 space is includ$ 9 ?is totality1 s t ! totality ( planes 9 space is an inf9itude ( ! ?ird ord]4 99#ah499 ,consid] n[ ! planes p]p5dicul> to ^! ?ree l9es4 ,e set ( ?ree planes s drawn w det]m9e a po9t 9 space1 &1 conv]sely1 ?r\gh e po9t 9 space may be drawn "o & only "o set ( ?ree planes at "r angles to ! ?ree giv5 l9es4 ,x foll[s1 !re=e1 t ! totality ( po9ts 9 space is an inf9itude ( ! ?ird ord]4 99#ai4 ,space sy/em499 ,space ( ?ree dim5sions1 consid]$ z made up ( all its planes & po9ts1 is !n a fundam5tal =m ( ! ?ird ord]1 :i* we %all call a space sy/em4 99#bj4 ,l9es 9 space499 ,if we jo9 ! tw(old inf9ity ( po9ts 9 "o plane ) ! tw(old inf9ity ( po9ts 9 ano!r plane1 we get a totality ( l9es ( space :i* is ( ! f\r? ord] ( inf9ity4 ,! totality ( l9es 9 space gives1 !n1 a fundam5tal =m ( ! f\r? ord]4 99#ba4 ,correspond5ce betwe5 po9ts & numb]s499 ,9 ! !ory ( analytic geometry a "o-to-"o correspond5ce is assum$ to exi/ betwe5 po9ts on a l9e & numb]s4 ,9 ord] to ju/ify ?is assumption a v ext5d$ def9ition ( numb] mu/ be made use (4 ,a "o-to-"o correspond5ce is !n set up betwe5 po9ts 9 ! plane & pairs ( numb]s1 & also betwe5 po9ts 9 space & sets ( ?ree numb]s4 ,a s+le con/ant w s]ve to def9e ! position ( a po9t on a l9e2 two1 a po9t 9 ! plane2 ?ree1 a po9t 9 space2 etc4 ,9 ! same !ory a "o-to-"o correspond5ce is set up betwe5 loci 9 ! plane & equations 9 two v>iables2 betwe5 surfaces 9 space & equations 9 ?ree v>iables2 etc4 ,! equation ( a l9e 9 a plane involves two con/ants1 ei!r ( :i* may take an inf9ite numb] ( values4 ,f ?is x foll[s t !re is an inf9ity ( l9es 9 ! plane :i* is ( ! second ord] if ! inf9ity ( po9ts on a l9e is assum$ to be ( ! fir/4 ,9 ! same way a circle is det]m9$ by ?ree conditions2 a sph]e by f\r2 etc4 ,we might !n expect to be able to set up a "o-to-"o correspond5ce betwe5 circles 9 a plane & po9ts1 or planes 9 space1 or betwe5 sph]es & l9es 9 space4 ,su*1 inde$1 is ! case1 & x is (t5 possible to inf] !orems conc]n+ sph]es f !orems conc]n+ l9es1 & vice v]sa4 ,x is possibilities su* z ^! t1 give to ! !ory ( "o-to-"o correspond5ce its great importance = ! ma!matician4 ,x mu/ n be =gott51 h[ev]1 t we are consid]+ only cont9u\s correspond5ces4 ,x is p]fectly possible to set1 up a "o-to-"o correspond5ce betwe5 ! po9ts ( a l9e & ! po9ts ( a plane1 or1 inde$1 betwe5 ! po9ts ( a l9e & ! po9ts ( a space ( any f9ite numb] ( dim5sions1 if ! correspond5ce is n re/rict$ to be cont9u\s4 99#bb4 ,elem5ts at inf9ity499 ,a f9al ^w is necess>y 9 ord] to expla9 a phrase :i* is 9 con/ant use 9 ! /udy ( projective geometry4 ,we h spok5 ( ! po9t at inf9ity on a /raight l9e--a fictiti\s po9t only us$ to bridge ov] ! exceptional case :5 we are sett+ up a "o-to-"o correspond5ce betwe5 ! po9ts ( a l9e & ! l9es ?r\gh a po9t4 ,we speak ( x z a po9t & n z po9ts1 because 9 ! geometry /udi$ by ,euclid we assume only "o l9e ?r\gh a po9t p>allel to a giv5 l9e4 ,9 ! same s5se we speak ( all ! po9ts at inf9ity 9 a plane z ly+ on a l9e1 ! l9e at inf9ity1 because ! /raight l9e is ! simple/ locus we c imag9e :i* has only "o po9t 9 common ) any l9e 9 ! plane4 ,likewise we speak ( ! plane at inf9ity1 because t seems ! mo/ conv5i5t way ( imag9+ ! po9ts at inf9ity 9 space4 ,x mu/ n be inf]r$ t ^! conceptions h any ess5tial connection ) physical facts1 or t o!r means ( pictur+ to ourselves ! inf9itely di/ant configurations are n possible4 ,9 o!r bran*es ( ma!matics1 notably 9 ! !ory ( functions ( a complex v>iable1 q diff]5t assumptions are made & q diff]5t conceptions ( ! elem5ts at inf9ity are us$4 ,z we c "k no?+ exp]im5tally ab\t su* ?+s1 we are at lib]ty to make any assumptions we please1 s long z !y are consi/5t & s]ve "s useful purpose4 ,,problems #a4 ,s9ce !re is a ?reefold inf9ity ( po9ts 9 space1 !re mu/ be a sixfold inf9ity ( pairs ( po9ts 9 space4 ,ea* pair ( po9ts det]m9es a l9e4 ,:y1 !n1 is !re n a sixfold inf9ity ( l9es 9 space8 #b4 ,if !re is a f\rfold inf9ity ( l9es 9 space1 :y is x t !re is n a f\rfold inf9ity ( planes ?r\gh a po9t1 see+ t ea* l9e 9 space det]m9es a plane ?r\gh t po9t8 #c4 ,%[ t !re is a f\rfold inf9ity ( circles 9 space t pass ?r\gh a fix$ po9t4 7,set up a "o-to-"o correspond5ce betwe5 ! axes ( ! circles & l9es 9 space47 #d4 ,f9d ! ord] ( inf9ity ( all ! l9es ( space t cut across a giv5 l9e2 across two giv5 l9es2 across ?ree giv5 l9es2 across f\r giv5 l9es4 #e4 ,f9d ! ord] ( inf9ity ( all ! sph]es 9 space t pass ?r\gh a giv5 po9t2 ?r\gh two giv5 po9ts2 ?r\gh ?ree giv5 po9ts2 ?r\gh f\r giv5 po9ts4 #f4 ,f9d ! ord] ( inf9ity ( all ! circles on a sph]e2 ( all ! circles on a sph]e t pass ?r\gh a fix$ po9t2 ?r\gh two fix$ po9ts2 ?r\gh ?ree fix$ po9ts2 ( all ! circles 9 space2 ( all ! circles t cut across a giv5 l9e4 #g4 ,f9d ! ord] ( inf9ity ( all l9es tang5t to a sph]e2 ( all planes tang5t to a sph]e2 ( l9es & planes tang5t to a sph]e & pass+ ?r\gh a fix$ po9t4 #h4 ,set up a "o-to-"o correspond5ce betwe5 ! s]ies ( numb]s #a1 #b1 #c1 #d1 444 & ! s]ies ( ev5 numb]s #b1 #d1 #f1 #h 4444 ,are we ju/ifi$ 9 say+ t !re are j z _m ev5 numb]s z !re are numb]s altoge!r8 #i4 ,is ! axiom ,! :ole is great] ?an "o ( its p>ts applicable to inf9ite assemblages8 #aj4 ,make out a classifi$ li/ ( all ! inf9itudes ( ! fir/1 second1 ?ird1 & f\r? ord]s m5tion$ 9 ?is *apt]4 ,,*apt] ,,ii - ,,relations ,,betwe5 ,,fundam5tal ,,=ms ,,9 ,,"o-,,to-,,"o ,,correspond5ce ,,) ,,ea* ,,o!r 99#bc4 ,sev5 fundam5tal =ms499 ,9 ! prec$+ *apt] we h call$ att5tion to sev5 fundam5tal =ms3 ! po9t-r[1 ! p5cil ( rays1 ! axial p5cil1 ! plane sy/em1 ! po9t sy/em1 ! space sy/em1 & ! sy/em ( l9es 9 space4 ,^! fundam5tal =ms are ! mat]ial :i* we int5d to use 9 build+ up a g5]al !ory :i* w be f\nd to include ord9>y geometry z a special case4 ,we %all be conc]n$1 n ) measurem5t ( angles & areas or l9e segm5ts z 9 ! /udy ( ,euclid1 b 9 comb9+ & comp>+ ^! fundam5tal =ms & 9 g5]at+ new =ms by means ( !m4 ,9 problems ( con/ruction we %all make no use ( measurem5t1 ei!r ( angles or ( segm5ts1 & except 9 c]ta9 special applications ( ! g5]al !ory we %all n f9d x necess>y to require m ( ourselves ?an ! ability to draw ! l9e jo9+ two po9ts1 or to f9d ! po9t ( int]sections ( two l9es1 or ! l9e ( int]section ( two planes1 or1 9 g5]al1 ! common elem5ts ( two fundam5tal =ms4 99#bd4 ,projective prop]ties499 ,our *ief int]e/ 9 ?is *apt] w be ! discov]y ( relations betwe5 ! elem5ts ( "o =m :i* hold betwe5 ! correspond+ elem5ts ( any o!r =m 9 "o-to-"o correspond5ce ) x4 ,we h already call$ att5tion to ! dang] ( assum+ t :atev] relations hold betwe5 ! elem5ts ( "o assemblage mu/ also hold betwe5 ! correspond+ elem5ts ( any assemblage 9 "o-to-"o correspond5ce ) x4 ,?is false assumption is ! basis ( ! s-call$ pro( by analogy s mu* 9 vogue among speculative !ori/s4 ,:5 x appe>s t c]ta9 relations exi/+ betwe5 ! po9ts ( a giv5 po9t-r[ d n necessitate ! same relations betwe5 ! correspond+ elem5ts ( ano!r 9 "o-to-"o correspond5ce ) x1 we %\ld view ) suspicion any application ( ! pro( by analogy 9 realms ( ?\ght :]e accurate judgm5ts are n s easily made4 ,= example1 if 9 a giv5 po9t-r[ u ?ree po9ts1 ,a1 ,b1 & ,c1 are tak5 su* t ,b is ! middle po9t ( ! segm5t ,,ac1 x does n foll[ t ! ?ree po9ts ,a'1 ,b'1 ,c' 9 a po9t-r[ p]spective to u w be s relat$4 ,relations betwe5 ! elem5ts ( any =m :i* d g ov] unalt]$ to ! correspond+ elem5ts ( a =m projectively relat$ to x are call$ projective relations4 ,relations involv+ measurem5t ( l9es or ( angles are n projective4 99#be4 ,des>gues's !orem499 ,we consid] fir/ ! foll[+ beautiful !orem1 due to ,des>gues & call$ by 8 "n4 ,if two triangles1 ,a1 ,b1 ,c & ,a'1 ,b'1 ,c'1 are s situat$ t ! l9es ,,aa'1 ,,bb'1 & ,,cc' all meet 9 a po9t1 !n ! pairs ( sides ,,ab & ,a',b'1 ,,bc & ,b',c'1 ,,ca & ,c',a' all meet on a /raight l9e1 & conv]sely4 ,figure #c ,,fig4 #c ,let ! l9es ,,aa'1 ,,bb'1 & ,,cc' meet 9 ! po9t ,m 7,fig4 #c74 ,conceive ( ! figure z 9 space1 s t ,m is ! v]tex ( a trih$ral angle ( :i* ! giv5 triangles are plane sections4 ,! l9es ,,ab & ,a',b' are 9 ! same plane & mu/ meet :5 produc$1 _! po9t ( int]section be+ cle>ly a po9t 9 ! plane ( ea* triangle & !re=e 9 ! l9e ( int]section ( ^! two planes4 ,call ?is po9t ,p4 ,by simil> reason+ ! po9t ,q ( int]section ( ! l9es ,,bc & ,b',c' mu/ lie on ?is same l9e z well z ! po9t ,r ( int]section ( ,,ca & ,c',a'4 ,!re=e ! po9ts ,p1 ,q1 & ,r all lie on ! same l9e m4 ,if n[ we consid] ! figure a plane figure1 ! po9ts ,p1 ,q1 & ,r /ill all lie on a /raight l9e1 :i* proves ! !orem4 ,! conv]se is e/abli%$ 9 ! same mann]4 99#bf4 ,fundam5tal !orem conc]n+ two complete quadrangles499 ,?is !orem ?r[s into our h&s ! foll[+ fundam5tal !orem conc]n+ two complete quadrangles1 a complete quadrangle be+ def9$ z ! figure obta9$ by jo9+ any f\r giv5 po9ts by /raight l9es 9 ! six possible ways4 ,giv5 two complete quadrangles1 ,k1 ,l1 ,m1 ,n & ,k'1 ,l'1 ,m'1 ,n'1 s relat$ t ,,kl1 ,k',l'1 ,,mn1 ,m',n' all meet 9 a po9t ,a2 ,,lm1 ,l',m'1 ,,nk1 ,n',k' all meet 9 a po9t ,q2 & ,,ln1 ,l',n' meet 9 a po9t ,b on ! l9e ,,ac2 !n ! l9es ,,km & ,k',m' also meet 9 a po9t ,d on ! l9e ,,ac4 ,figure #d ,,fig4 #d ,=1 by ! conv]se ( ! la/ !orem1 ,,kk'1 ,,ll'1 & ,,nn' all meet 9 a po9t ,s 7,fig4 #d74 ,also ,,ll'1 ,,mm'1 & ,,nn' meet 9 a po9t1 & !re=e 9 ! same po9t ,s4 ,?us ,,kk'1 ,,ll'1 & ,,mm' meet 9 a po9t1 & s1 by ,des>gues's !orem itself1 ,a1 ,b1 & ,d are on a /raight l9e4 99#bg4 ,importance ( ! !orem499 ,! importance ( ?is !orem lies 9 ! fact t1 ,a1 ,b1 & ,c be+ giv51 an indef9ite numb] ( quadrangles ,k'1 ,l'1 ,m'1 ,n' my be f\nd su* t ,k',l' & ,m',n' meet 9 ,a1 ,k',n' & ,l',m' 9 ,c1 ) ,l',n' pass+ ?r\gh ,b4 ,inde$1 ! l9es ,,ak' & ,,am' may be drawn arbitr>ily ?r\gh ,a1 & any l9e ?r\gh ,b may be us$ to det]m9e ,l' & ,n'4 ,by jo9+ ^! two po9ts to ,c ! po9ts ,k' & ,m' are det]m9$4 ,!n ! l9e jo9+ ,k' & ,m'1 f\nd 9 ?is way1 mu/ pass ?r\gh ! po9t ,d already det]m9$ by ! quadrangle ,k1 ,l1 ,m1 ,n4 ,! ?ree po9ts ,a1 ,b1 ,c1 giv5 9 ord]1 s]ve ?us to det]m9e a f\r? po9t ,d4 99#bh499 ,9 a complete quadrangle ! l9e jo9+ any two po9ts is call$ ! opposite side to ! l9e jo9+ ! o!r two po9ts4 ,! result ( ! prec$+ p>agraph may !n be /at$ z foll[s3 ,giv5 ?ree po9ts1 ,a1 ,b1 ,c1 9 a /raight l9e1 if a pair ( opposite sides ( a complete quadrangle pass ?r\gh ,a1 & ano!r pair ?r\gh ,c1 & "o ( ! rema9+ two sides goes ?r\gh ,b1 !n ! o!r ( ! rema9+ two sides w g ?r\gh a fix$ po9t :i* does n dep5d on ! quadrangle employ$4 99#bi4 ,f\r h>monic po9ts499 ,f\r po9ts1 ,a1 ,b1 ,c1 ,d1 relat$ z 9 ! prec$+ !orem are call$ f\r h>monic po9ts4 ,! po9t ,d is call$ ! f\r? h>monic ( ,b ) respect to ,a & ,c4 ,s9ce ,b & ,d play exactly ! same rle 9 ! above con/ruction1 ,b is also ! f\r? h>monic ( ,d ) respect to ,a & ,c4 ,b & ,d are call$ h>monic conjugates ) respect to ,a & ,c4 ,we proce$ to %[ t ,a & ,c are also h>monic conjugates ) respect to ,b & ,d--t is1 t x is possible to f9d a quadrangle ( :i* two opposite sides %all pass ?r\gh ,b1 two ?r\gh ,d1 & ( ! rema9+ pair1 "o ?r\gh ,a & ! o!r ?r\gh ,c4 ,figure #e ,,fig4 #e ,let ,o be ! int]section ( ,,km & ,,ln 7,fig4 #e74 ,jo9 ,o to ,a & ,c4 ,! jo9+ l9es cut out on ! sides ( ! quadrangle f\r po9ts1 ,p1 ,q1 ,r1 ,s4 ,consid] ! quadrangle ,p1 ,k1 ,q1 ,o4 ,"o pair ( opposite sides passes ?r\gh ,a1 "o ?r\gh ,c1 & "o rema9+ side ?r\gh ,d2 !re=e ! o!r rema9+ side mu/ pass ?r\gh ,b4 ,simil>ly1 ,,rs passes ?r\gh ,b & ,,ps & ,,qr pass ?r\gh ,d4 ,! quadrangle ,p1 ,q1 ,r1 ,s !re=e has two opposite sides ?r\gh ,b1 two ?r\gh ,d1 & ! rema9+ pair ?r\gh ,a & ,c4 ,a & ,c are ?us h>monic conjugates ) respect to ,b & ,d4 ,we may sum up ! discussion1 !re=e1 z foll[s3 99#cj499 ,if ,a & ,c are h>monic conjugates ) respect to ,b & ,d1 !n ,b & ,d are h>monic conjugates ) respect to ,a & ,c4 99#ca4 ,importance ( ! notion499 ,! importance ( ! notion ( f\r h>monic po9ts lies 9 ! fact t x is a relation :i* is c>ri$ ov] f f\r po9ts 9 a po9t-r[ u to ! f\r po9ts t correspond to !m 9 any po9t-r[ u' p]spective to u4 ,to prove ?is /atem5t we con/ruct a quadrangle ,k1 ,l1 ,m1 ,n su* t ,,kl & ,,mn pass ?r\gh ,a1 ,,kn & ,,lm ?r\gh ,c1 ,,ln ?r\gh ,b1 & ,,km ?r\gh ,d4 ,take n[ any po9t ,s n 9 ! plane ( ! quadrangle & con/ruct ! planes det]m9$ by ,s & all ! sev5 l9es ( ! figure4 ,cut across ?is set ( planes by ano!r plane n pass+ ?r\gh ,s4 ,?is plane cuts out on ! set ( sev5 planes ano!r quadrangle :i* det]m9es f\r new h>monic po9ts1 ,a'1 ,b'1 ,c'1 ,d'1 on ! l9es jo9+ ,s to ,a1 ,b1 ,c1 ,d4 ,b ,s may be tak5 z any po9t1 s9ce ! orig9al quadrangle may be tak5 9 any plane ?r\gh ,a1 ,b1 ,c1 ,d2 &1 fur!r1 ! po9ts ,a'1 ,b'1 ,c'1 ,d' are ! int]section ( ,,sa1 ,,sb1 ,,sc1 ,,sd by any l9e4 ,we h1 !n1 ! rem>kable !orem3 99#cb499 ,if any po9t is jo9$ to f\r h>monic po9ts1 & ! f\r l9es ?us obta9$ are cut by any fif?1 ! f\r po9ts ( int]section are aga9 h>monic4 99#cc4 ,f\r h>monic l9es499 ,we are n[ able to ext5d ! notion ( h>monic elem5ts to p5cils ( rays1 & inde$ to axial p5cils4 ,= if we def9e f\r h>monic rays z f\r rays :i* pass ?r\gh a po9t & :i* pass "o ?r\gh ea* ( f\r h>monic po9ts1 we h ! !orem ,f\r h>monic l9es are cut by any transv]sal 9 f\r h>monic po9ts4 99#cd4 ,f\r h>monic planes499 ,we also def9e f\r h>monic planes z f\r planes ?r\gh a l9e :i* pass "o ?r\gh ea* ( f\r h>monic po9ts1 & we may %[ t ,f\r h>monic planes are cut by any plane n pass+ ?r\gh _! common l9e 9 f\r h>monic l9es1 & also by any l9e 9 f\r h>monic po9ts4 ,= let ! planes 1 1 1 1 :i* all pass ?r\gh ! l9e g1 pass also ?r\gh ! f\r h>monic po9ts ,a1 ,b1 ,c1 ,d1 s t passes ?r\gh ,a1 etc4 ,!n x is cle> t any plane ?r\gh ,a1 ,b1 ,c1 ,d w cut out f\r h>monic l9es f ! f\r planes1 = !y are l9es ?r\gh ! int]section ,p ( g ) ! plane 1 & !y pass ?r\gh ! giv5 h>monic po9ts ,a1 ,b1 ,c1 ,d4 ,any o!r plane cuts g 9 a po9t ,s & cuts 1 1 1 9 f\r l9es t meet 9 f\r po9ts ,a'1 ,b'1 ,c'1 ,d' ly+ on ,,pa1 ,,pb1 ,,pc1 & ,,pd respectively1 & are ?us f\r h>monic hues4 ,fur!r1 any ray cuts 1 1 1 9 f\r h>monic po9ts1 s9ce any plane ?r\gh ! ray gives f\r h>monic l9es ( int]section4 99#ce499 ,^! results may be put toge!r z foll[s3 ,giv5 any two assemblages ( po9ts1 rays1 or planes1 p]spectively relat$ to ea* o!r1 f\r h>monic elem5ts ( "o mu/ correspond to f\r elem5ts ( ! o!r :i* are likewise h>monic4 ,if1 n[1 two =ms are p]spectively relat$ to a ?ird1 any f\r h>monic elem5ts ( "o mu/ correspond to f\r h>monic elem5ts 9 ! o!r4 ,we take ?is z our def9ition ( projective correspond5ce1 & say3 99#cf4 ,def9ition ( projectivity499 ,two fundam5tal =ms are protectively relat$ to ea* o!r :5 a "o-to-"o correspond5ce exi/s betwe5 ! elem5ts ( ! two & :5 f\r h>monic elem5ts ( "o correspond to f\r h>monic elem5ts ( ! o!r4 ,figure #f ,,fig4 #f 99#cg4 ,correspond5ce betwe5 h>monic conjugates499 ,giv5 f\r h>monic po9ts1 ,a1 ,b1 ,c1 ,d2 if we fix ,a & ,c1 !n ,b & ,d v>y toge!r 9 a way t %\ld be ?or\ghly und]/ood4 ,to get a cle> conception ( _! relative motion we may fix ! po9ts ,l & ,m ( ! quadrangle ,k1 ,l1 ,m1 ,n 7,fig4 #f74 ,!n1 z ,b describes ! po9t-r[ ,,ac1 ! po9t ,n describes ! po9t-r[ ,,am p]spective to x4 ,project+ ,n aga9 f ,c1 we get a po9t-r[ ,k on ,,al p]spective to ! po9t-r[ ,n & ?us projective to ! po9t-r[ ,b4 ,project ! po9t-r[ ,k f ,m & we get a po9t-r[ ,d on ,,ac aga91 :i* is projective to ! po9t-r[ ,b4 ,= e po9t ,b we h ?us "o & only "o po9t ,d1 & conv]sely4 ,9 o!r words1 we h set up a "o-to-"o correspond5ce betwe5 ! po9ts ( a s+le po9t-r[1 :i* is also a projective correspond5ce because f\r h>monic po9ts ,b correspond to f\r h>monic po9ts ,d4 ,we may note also t ! correspond5ce is "h *>act]iz$ by a feature :i* does n always appe> 9 projective correspond5ces3 namely1 ! same process t c>ries "o f ,b to ,d w c>ry "o back f ,d to ,b aga94 ,?is special prop]ty w receive fur!r /udy 9 ! *apt] on ,involution4 99#ch499 ,x is se5 t z ,b approa*es ,a1 ,d also approa*es ,a4 ,z ,b moves f ,a t[>d ,c1 ,d moves f ,a 9 ! opposite direction1 pass+ ?r\gh ! po9t at inf9ity on ! l9e ,,ac1 & returns on ! o!r side to meet ,b at ,c aga94 ,9 o!r words1 z ,b trav]ses ,,ac1 ,d trav]ses ! re/ ( ! l9e f ,a to ,c ?r\gh inf9ity4 ,9 all positions ( ,b1 except at ,a or ,c1 ,b & ,d are sep>at$ f ea* o!r by ,a & ,c4 99#ci4 ,h>monic conjugate ( ! po9t at inf9ity499 ,x is natural to inquire :at position ( ,b corresponds to ! inf9itely di/ant position ( ,d4 ,we h prov$ 7 #bg7 t ! p>ticul> quadrangle ,k1 ,l1 ,m1 ,n employ$ is ( no consequ5ce4 ,we %all !re=e avail ourselves ( "o t l5ds itself mo/ readily to ! solution ( ! problem4 ,we *oose ! po9t ,l s t ! triangle ,,alc is isosceles 7,fig4 #g74 ,s9ce ,d is suppos$ to be at inf9ity1 ! l9e ,,km is p>allel to ,,ac4 ,!re=e ! triangles ,,kac & ,,mac are equal1 & ! triangle ,,anc is also isosceles4 ,! triangles ,,cnl & ,,anl are !re=e equal1 & ! l9e ,,lb bisects ! angle ,,alc4 ,b is !re=e ! middle po9t ( ,,ac1 & we h ! !orem ,! h>monic conjugate ( ! middle po9t ( ,,ac is at inf9ity4 ,figure #g ,,fig4 #g 99#dj4 ,projective !orems & metrical !orems4 ,l9e> con/ruction499 ,?is !orem is ! connect+ l9k betwe5 ! g5]al protective !orems :i* we h be5 consid]+ s f> & ! metrical !orems ( ord9>y geometry4 ,up to ?is po9t we h said no?+ ab\t measurem5ts1 ei!r ( l9e segm5ts or ( angles4 ,des>gues's !orem & ! !ory ( h>monic elem5ts :i* dep5ds on x h no?+ to d ) magnitudes at all4 ,n until ! notion ( an inf9itely di/ant po9t is br\ght 9 is any m5tion made ( di/ances or directions4 ,we h be5 able to make all ( our con/ructions up to ?is po9t by means ( ! /raight$ge1 or ungraduat$ rul]4 ,a con/ruction made ) su* an in/rum5t we %all call a l9e> con/ruction4 ,x requires m]ely t we be able to draw ! l9e jo9+ two po9ts or f9d ! po9t ( int]section ( two l9es4 99#da4 ,p>allels & mid-po9ts499 ,x might be ?\ght t draw+ a l9e ?r\gh a giv5 po9t p>allel to a giv5 l9e 0 only a special case ( draw+ a l9e jo9+ two po9ts4 ,inde$1 x consi/s only 9 draw+ a l9e ?r\gh ! giv5 po9t & ?r\gh ! inf9itely di/ant po9t on ! giv5 l9e4 ,x mu/ be rememb]$1 h[ev]1 t ! expression inf9itely di/ant po9t mu/ n be tak5 lit]ally4 ,:5 we say t two p>allel l9es meet at inf9ity1 we really mean t !y d n meet at all1 & ! only reason = us+ ! expression is to avoid t$i\s /atem5t ( exceptions & re/rictions to our !orems4 ,we ought !re=e to consid] ! draw+ ( a l9e p>allel to a giv5 l9e z a diff]5t accompli%m5t f ! draw+ ( ! l9e jo9+ two giv5 po9ts4 ,x is a rem>kable consequ5ce ( ! la/ !orem t a p>allel to a giv5 l9e & ! mid-po9t ( a giv5 segm5t are equival5t data4 ,= ! con/ruction is rev]sible1 & if we are giv5 ! middle po9t ( a giv5 segm5t1 we c con/ruct l9e>ly a l9e p>allel to t segm5t4 ,?us1 giv5 t ,b is ! middle po9t ( ,,ac1 we may draw any two l9es ?r\gh ,a1 & any l9e ?r\gh ,b cutt+ !m 9 po9ts ,n & ,l4 ,jo9 ,n & ,l to ,c & get ! po9ts ,k & ,m on ! two l9es ?r\gh ,a4 ,!n ,,km is p>allel to ,,ac4 ,! bisection ( a giv5 segm5t & ! draw+ ( a l9e p>allel to ! segm5t are equival5t data :5 l9e> con/ruction is us$4 99#db499 ,x is n difficult to give a l9e> con/ruction = ! problem to divide a giv5 segm5t into n equal p>ts1 giv5 only a p>allel to ! segm5t4 ,?is is simple en\gh :5 n is a p[] ( #b4 ,= any o!r numb]1 su* z #bi1 divide any segm5t on ! l9e p>allel to ,,ac into #cb equal p>ts1 by a repetition ( ! process j describ$4 ,take #bi ( ^!1 & jo9 ! fir/ to ,a & ! la/ to ,c4 ,let ^! jo9+ l9es meet 9 ,s4 ,jo9 ,s to all ! o!r po9ts4 ,o!r problems1 ( a simil> sort1 are giv5 at ! end ( ! *apt]4 99#dc4 ,num]ical relations499 ,s9ce ?ree po9ts1 giv5 9 ord]1 are suffici5t to det]m9e a f\r?1 z expla9$ above1 x ought to be possible to reproduce ! process num]ically 9 view ( ! "o-to-"o correspond5ce :i* exi/s betwe5 po9ts on a l9e & numb]s2 a correspond5ce :i*1 to be sure1 we h n e/abli%$ "h1 b :i* is discuss$ 9 any treatise on ! !ory ( po9t sets4 ,we proce$ to discov] :at relation betwe5 f\r numb]s corresponds to ! h>monic relation betwe5 f\r po9ts4 ,figure #h ,,fig4 #h 99#dd499 ,let ,a1 ,b1 ,c1 ,d be f\r h>monic po9ts 7,fig4 #h71 & let ,,sa1 ,,sb1 ,,sc1 ,,sd be f\r h>monic l9es4 ,assume a l9e drawn ?r\gh ,b p>allel to ,,sd1 meet+ ,,sa 9 ,a' & ,,sc 9 ,c'4 ,!n ,a'1 ,b'1 ,c'1 & ! inf9itely di/ant po9t on ,a',c' are f\r h>monic po9ts1 & !re=e ,b is ! middle po9t ( ! segm5t ,a',c'4 ,!n1 s9ce ! triangle ,,das is simil> to ! triangle ,,baa'1 we may write ! proportion ,,ab 3 ,,ad ,,ba' 3 ,,sd4 ,also1 f ! simil> triangles ,,dsc & ,,bcc'1 we h ,,cd 3 ,,cb ,,sd 3 ,b',c4 ,f ^! two proportions we h1 rememb]+ t ,,ba' ,,bc'1 =mula ! m9us sign be+ giv5 to ! ratio on acc\nt ( ! fact t ,a & ,c are always sep>at$ f ,b & ,d1 s t "o or ?ree ( ! segm5ts ,,ab1 ,,cd1 ,,ad1 ,,cb mu/ be negative4 99#de499 ,writ+ ! la/ equation 9 ! =m ,,cb 3 ,,ab -,,cd 3 ,,ad1 & us+ ! fundam5tal relation connect+ ?ree po9ts on a l9e1 ,,pr ,,rq ,,pq1 :i* holds = all positions ( ! ?ree po9ts if acc\nt be tak5 ( ! sign ( ! segm5ts1 ! la/ proportion may be writt5 7,,cb - ,,ba7 3 ,,ab -7,,ca - ,,da7 3 ,,ad1 or 7,,ab - ,,ac7 3 ,,ab 7,,ac - ,,ad7 3 ,,ad2 s t ,,ab1 ,,ac1 & ,,ad are ?ree quantities 9 hamonic progression1 s9ce ! diff]5ce betwe5 ! fir/ & second is to ! fir/ z ! diff]5ce betwe5 ! second & ?ird is to ! ?ird4 ,also1 f ?is la/ proportion comes ! famili> relation =mula :i* is conv5i5t = ! computation ( ! di/ance ,,ad :5 ,,ab & ,,ac are giv5 num]ically4 99#df4 ,anh>monic ratio499 ,! correspond+ relations betwe5 ! trigonometric functions ( ! angles det]m9$ by f\r h>monic l9es are n difficult to obta91 b z we %all n ne$ !m 9 build+ up ! !ory ( projective geometry1 we w n discuss !m "h4 ,/ud5ts :o h a slight acqua9tance ) trigonometry may read 9 a lat] *apt] 7 #afa7 a developm5t ( ! !ory ( a m g5]al relation1 call$ ! anh>monic ratio1 or cross ratio1 :i* connects any f\r po9ts on a l9e4 ,,problems 99#a994 ,draw ?r\gh a giv5 po9t a l9e :i* %all pass ?r\gh ! inaccessible po9t ( int]section ( two giv5 l9es4 ,! foll[+ con/ruction may be made to dep5d ^u ,des>gues's !orem3 ,?r\gh ! giv5 po9t ,p draw any two rays cutt+ ! two l9es 9 ! po9ts ,,ab' & ,a',b1 ,a1 ,b1 ly+ on "o ( ! giv5 l9es & ,a'1 ,b'1 on ! o!r4 ,jo9 ,,aa' & ,,bb'1 & f9d _! po9t ( int]section ,s4 ,?r\gh ,s draw any o!r ray1 cutt+ ! giv5 l9es 9 ,,cc'4 ,jo9 ,,bc' & ,b',c1 & obta9 _! po9t ( int]section ,q4 ,,pq is ! desir$ l9e4 ,ju/ify ?is con/ruction4 99#b499 ,to draw ?r\gh a giv5 po9t ,p a l9e :i* %all meet two giv5 l9es 9 po9ts ,a & ,b1 equally di/ant f ,p4 ,ju/ify ! foll[+ con/ruction3 ,jo9 ,p to ! po9t ,s ( int]section ( ! two giv5 l9es4 ,con/ruct ! f\r? h>monic ( ,,ps ) respect to ! two giv5 l9es4 ,draw ?r\gh ,p a l9e p>allel to ?is l9e4 ,?is is ! requir$ l9e4 99#c499 ,giv5 a p>allelogram 9 ! same plane ) a giv5 segm5t ,,ac1 to con/ruct l9e>ly ! middle po9t ( ,,ac4 99#d499 ,giv5 f\r h>monic l9es1 ( :i* "o pair are at "r angles to ea* o!r1 %[ t ! o!r pair make equal angles ) !m4 ,?is is a !orem ( :i* frequ5t use w be made4 99#e499 ,giv5 ! middle po9t ( a l9e segm5t1 to draw a l9e p>allel to ! segm5t & pass+ ?r\gh a giv5 po9t4 99#f499 ,a l9e is drawn cutt+ ! sides ( a triangle ,,abc 9 ! po9ts ,a'1 ,b'1 ,c' ! po9t ,a' ly+ on ! side ,,bc1 etc4 ,! h>monic conjugate ( ,a' ) respect to ,b & ,c is !n con/ruct$ & call$ ,a4 ,simil>ly1 ,b & ,c are con/ruct$4 ,%[ t ,a,b,c lie on a /raight l9e4 ,f9d o!r sets ( ?ree po9ts on a l9e 9 ! figure4 ,f9d also sets ( ?ree l9es ?r\gh a po9t4 ,,*apt] ,,iii - ,,comb9ation ,,( ,,two ,,projectively ,,relat$ ,,fundam5tal ,,=ms ,figure #i ,,fig4 #i 99#dg4 ,sup]pos$ fundam5tal =ms4 ,self-correspond+ elem5ts499 ,we h se5 7 #cg7 t two projective po9t-r[s may be sup]pos$ ^u ! same /raight l9e4 ,?is happ5s1 = example1 :5 two p5cils :i* are projective to ea* o!r are cut across by a /raight l9e4 ,x is also possible = two projective p5cils to h ! same c5t]4 ,?is happ5s1 = example1 :5 two projective po9t-r[s are project$ to ! same po9t4 ,simil>ly1 two projective axial p5cils may h ! same axis4 ,we exam9e n[ ! possibility ( two =ms relat$ 9 ?is way1 hav+ an elem5t or elem5ts t correspond to !mselves4 ,we h se51 inde$1 t if ,b & ,d are h>monic conjugates ) respect to ,a & ,c1 !n ! po9t-r[ describ$ by ,b is projective to ! po9t-r[ describ$ by ,d1 & t ,a & ,c are self-correspond+ po9ts4 ,consid] m g5]ally ! case ( two p5cils p]spective to ea* o!r ) axis ( p]spectivity u' 7,fig4 #i74 ,cut across !m by a l9e u4 ,we get ?us two projective po9t-r[s sup]pos$ on ! same l9e u1 & a mom5t's reflection s]ves to %[ t ! po9t ,n ( int]section u & u' corresponds to itself 9 ! two po9t-r[s4 ,also1 ! po9t ,m1 :]e u int]sects ! l9e jo9+ ! c5t]s ( ! two p5cils1 is se5 to correspond to itself4 ,x is ?us possible = two projective po9t-r[s1 sup]pos$ ^u ! same l9e1 to h two self-correspond+ po9ts4 ,cle>ly ,m & ,n may fall toge!r if ! l9e jo9+ ! c5t]s ( ! p5cils happ5s to pass ?r\gh ! po9t ( int]section ( ! l9es u & u'4 ,figure #aj ,,fig4 #aj 99#dh499 ,we may also give an illu/ration ( a case :]e two sup]pos$ projective po9t-r[s h no self-correspond+ po9ts at all4 ,?us we may take two l9es revolv+ ab\t a fix$ po9t ,s & always mak+ ! same angle a ) ea* o!r 7,fig4 #aj74 ,!y w cut out on any l9e u 9 ! plane two po9t-r[s :i* are easily se5 to be projective4 ,=1 giv5 any f\r rays ,,sp :i* are h>monic1 ! f\r correspond+ rays ,,sp' mu/ also be h>monic1 s9ce !y make ! same angles ) ea* o!r4 ,f\r h>monic po9ts ,p correspond1 !re=e1 to f\r h>monic po9ts ,p'4 ,x is cle>1 h[ev]1 t no po9t ,p c co9cide ) its correspond+ po9t ,p'1 = 9 t case ! l9es ,,ps & ,p',s w\ld co9cide1 :i* is impossible if ! angle betwe5 !m is to be con/ant4 99#di4 ,fundam5tal !orem4 ,po/ulate ( cont9uity499 ,we h ?us %[n t two projective po9t-r[s1 sup]pos$ "o on ! o!r1 may h two po9ts1 "o po9t1 or no po9t at all correspond+ to !mselves4 ,we proce$ to %[ t ,if two projective po9t-r[s1 sup]pos$ ^u ! same /raight l9e1 h m ?an two self-correspond+ po9ts1 !y mu/ h an inf9ite numb]1 & e po9t corresponds to itself2 t is1 ! two po9t-r[s are n ess5tially di/9ct4 ,if ?ree po9ts1 ,a1 ,b1 & ,c1 are self-correspond+1 !n ! h>monic conjugate ,d ( ,b ) respect to ,a & ,c mu/ also correspond to itself4 ,= f\r h>monic po9ts mu/ always correspond to f\r h>monic po9ts4 ,9 ! same way ! h>monic conjugate ( ,d ) respect to ,b & ,c mu/ correspond to itself4 ,comb9+ new po9ts ) old 9 ?is way1 we may obta9 z _m self-correspond+ po9ts z we wi%4 ,we %[ fur!r t e po9t on ! l9e is ! limit+ po9t ( a f9ite or inf9ite sequ5ce ( self-correspond+ po9ts4 ,?us1 let a po9t ,p lie betwe5 ,a & ,b4 ,con/ruct n[ ,d1 ! f\r? h>monic ( ,c ) respect to ,a & ,b4 ,d may co9cide ) ,p1 9 :i* case ! sequ5ce is clos$2 o!rwise ,p lies 9 ! /ret* ,,ad or 9 ! /ret* ,,db4 ,if x lies 9 ! /ret* ,,db1 con/ruct ! f\r? h>monic ( ,c ) respect to ,d & ,b4 ,?is po9t ,d' may co9cide ) ,p1 9 :i* case1 z be=e1 ! sequ5ce is clos$4 ,if ,p lies 9 ! /ret* ,,dd'1 we con/ruct ! f\r? h>monic ( ,c ) respect to ,,dd'1 etc4 ,9 ea* /ep ! region 9 :i* ,p lies is dim9i%$1 & ! process may be cont9u$ until two self-correspond+ po9ts are obta9$ on ei!r side ( ,p1 & at di/ances f x arbitr>ily small4 ,we n[ assume1 explicitly1 ! fundam5tal po/ulate t ! correspond5ce is cont9u\s1 t is1 t ! di/ance betwe5 two po9ts 9 "o po9t-r[ may be made arbitr>ily small by suffici5tly dim9i%+ ! di/ance betwe5 ! correspond+ po9ts 9 ! o!r4 ,suppose n[ t ,p is n a self-correspond+ po9t1 b corresponds to a po9t ,p' at a fix$ di/ance d f ,p4 ,z not$ above1 we c f9d self-correspond+ po9ts arbitr>ily close to ,p1 & x appe>s1 !n1 t we c take a po9t ,d z close to ,p z we wi%1 & yet ! di/ance betwe5 ! correspond+ po9ts ,d' & ,p' approa*es d z a limit1 & n z]o1 :i* contradicts ! po/ulate ( cont9uity4 99#ej499 ,x foll[s also t two projective p5cils :i* h ! same c5t] may h no m ?an two self-correspond+ rays1 unless ! p5cils are id5tical4 ,= if we cut across !m by a l9e1 we obta9 two projective po9t-r[s sup]pos$ on ! same /raight l9e1 :i* may h no m ?an two self-correspond+ po9ts4 ,! same consid]ations apply to two projective axial p5cils :i* h ! same axis4 99#ea4 ,projective po9t-r[s hav+ a self-correspond+ po9t 9 common499 ,consid] n[ two projective po9t-r[s ly+ on diff]5t l9es 9 ! same plane4 ,_! common po9t may or may n be a self-correspond+ po9t4 ,if ! two po9t-r[s are p]spectively relat$1 !n _! common po9t is evid5tly a self-correspond+ po9t4 ,! conv]se is also true1 & we h ! v important !orem3 99#eb499 ,if 9 two protective po9t-r[s1 ! po9t ( int]section corresponds to itself1 !n ! po9t-r[s are 9 p]spective position4 ,figure #aa ,,fig4 #aa ,let ! two po9t-r[s be u & u' 7,fig4 #aa74 ,let ,a & ,a'1 ,b & ,b'1 be correspond+ po9ts1 & let also ! po9t ,m ( int]section ( u & u' correspond to itself4 ,let ,,aa' & ,,bb' meet 9 ! po9t ,s4 ,take ,s z ! c5t] ( two p5cils1 "o p]spective to u & ! o!r p]spective to u'4 ,9 ^! two p5cils ,,sa co9cides ) its correspond+ ray ,,sa'1 ,,sb ) its correspond+ ray ,,sb'1 & ,,sm ) its correspond+ ray ,,sm'4 ,! two p5cils are ?us id5tical1 by ! prec$+ !orem1 & any ray ,,sd mu/ co9cide ) its correspond+ ray ,,sd'4 ,correspond+ po9ts ( u & u'1 !re=e1 all lie on l9es ?r\gh ! po9t ,s4 99#ec499 ,an entirely simil> discussion %[s t ,if 9 two projective p5cils ! l9e jo9+ _! c5t]s is a self-correspond+ ray1 !n ! two p5cils are p]spectively relat$4 99#ed499 ,a simil> !orem may be /at$ = two axial p5cils ( :i* ! axes int]sect4 ,v frequ5t use w be made ( ^! fundam5tal !orems4 99#ee4 ,po9t-r[ ( ! second ord]499 ,! "q naturally arises1 ,:at is ! locus ( po9ts ( int]section ( correspond+ rays ( two projective p5cils :i* are n 9 p]spective position8 ,?is locus1 :i* w be discuss$ 9 detail 9 subsequ5t *apt]s1 is easily se5 to h at mo/ two po9ts 9 common ) any l9e 9 ! plane1 & on acc\nt ( ?is fundam5tal prop]ty w be call$ a po9t-r[ ( ! second ord]4 ,= any l9e u 9 ! plane ( ! two p5cils w be cut by !m 9 two projective po9t-r[s :i* h at mo/ two self-correspond+ po9ts4 ,su* a self-correspond+ po9t is cle>ly a po9t ( int]section ( correspond+ rays ( ! two p5cils4 99#ef499 ,?is locus deg5]ates 9 ! case ( two p]spective p5cils to a pair ( /raight l9es1 "o ( :i* is ! axis ( p]spectivity & ! o!r ! common ray1 any po9t ( :i* may be consid]$ z ! po9t ( int]section ( correspond+ rays ( ! two p5cils4 99#eg4 ,p5cils ( rays ( ! second ord]499 ,simil> inve/igations may be made conc]n+ ! sy/em ( l9es jo9+ correspond+ po9ts ( two projective po9t-r[s4 ,if we project ! po9t-r[s to any po9t 9 ! plane1 we obta9 two projective p5cils hav+ ! same c5t]4 ,at mo/ two pairs ( self-correspond+ rays may pres5t !mselves4 ,su* a ray is cle>ly a l9e jo9+ two correspond+ po9ts 9 ! two po9t-r[s4 ,! result may be /at$ z foll[s3 ,! sy/em ( rays jo9+ correspond+ po9ts 9 two protective po9t-r[s has at mo/ two rays 9 common ) any p5cil 9 ! plane4 ,= t reason ! sy/em ( rays is call$ a p5cil ( rays ( ! second ord]4 99#eh499 ,9 ! case ( two p]spective po9t-r[s ?is sy/em ( rays deg5]ates into two p5cils ( rays ( ! fir/ ord]1 "o ( :i* has its c5t] at ! c5t] ( p]spectivity ( ! two po9t-r[s1 & ! o!r at ! int]section ( ! two po9t-r[s1 any ray ?r\gh :i* may be consid]$ z jo9+ two correspond+ po9ts ( ! two po9t-r[s4 99#ei4 ,cone ( ! second ord]499 ,! correspond+ !orems 9 space may easily be obta9$ by jo9+ ! po9ts & l9es consid]$ 9 ! plane !orems to a po9t ,s 9 space4 ,two projective p5cils give rise to two projective axial p5cils ) axes int]sect+4 ,correspond+ planes meet 9 l9es :i* all pass ?r\gh ,s & ?r\gh ! po9ts on a po9t-r[ ( ! second ord] g5]at$ by ! two p5cils ( rays4 ,!y are ?us g5]at+ l9es ( a cone ( ! second ord]1 or quadric cone1 s call$ because e plane 9 space n pass+ ?r\gh ,s cuts x 9 a po9t-r[ ( ! second ord]1 & e l9e also cuts x 9 at mo/ two po9ts4 ,if1 aga91 we project two po9t-r[s to a po9t ,s 9 space1 we obta9 two p5cils ( rays ) a common c5t] b ly+ 9 diff]5t planes4 ,correspond+ l9es ( ^! p5cils det]m9e planes :i* are ! projections to ,s ( ! l9es :i* jo9 ! correspond+ po9ts ( ! two po9t-r[s4 ,at mo/ two su* planes may pass ?r\gh any ray ?r\gh ,s4 ,x is call$ a p5cil ( planes ( ! second ord]4 ,,problems 99#a4 99 ,a man ,a moves along a /raight road u1 & ano!r man ,b moves along ! same road & walks s z always to keep sight ( ,a 9 a small mirror ,m at ! side ( ! road4 ,h[ _m times w !y come toge!r1 ,a mov+ always 9 ! same direction along ! road8 #b4 ,h[ _m times w\ld ! two m5 9 ! fir/ problem see ea* o!r 9 two mirrors ,m & ,n z !y walk along ! road z be=e8 7,! planes ( ! two mirrors are n necess>ily p>allel to u47 #c4 ,z ,a moves along u1 trace ! pa? ( ,b s t ! two m5 may always see ea* o!r 9 ! two mirrors4 #d4 ,two boys walk along two pa?s u & u' ea* hold+ a /r+ :i* !y keep /ret*$ tightly betwe5 !m4 ,!y bo? move at con/ant b diff]5t rates ( spe$1 lett+ out ! /r+ or draw+ x 9 z !y walk4 ,h[ _m times w ! l9e ( ! /r+ pass ov] any giv5 po9t 9 ! plane ( ! pa?s8 #e4 ,trace ! l9es ( ! /r+ :5 ! two boys move at ! same rate ( spe$ 9 ! two pa?s b d n />t at ! same "t f ! po9t :]e ! two pa?s int]sect4 #f4 ,a %ip is sail+ on a /raight c\rse & keeps a gun tra9$ on a po9t on ! %ore4 ,%[ t a l9e at "r angles to ! direction ( ! gun at its muzzle w pass ?r\gh any po9t 9 ! plane twice or n at all4 7,consid] ! po9t-r[ at inf9ity cut out by a l9e ?r\gh ! po9t on ! %ore at "r angles to ! direction ( ! gun47 #g4 ,two l9es u & u' revolve ab\t two po9ts ,u & ,u' respectively 9 ! same plane4 ,!y g 9 ! same direction & at ! same rate ( spe$1 b "o has an angle a ! />t ( ! o!r4 ,%[ t !y g5]ate a po9t-r[ ( ! second ord]4 #h4 ,discuss ! "q giv5 9 ! la/ problem :5 ! two l9es revolve 9 opposite directions4 ,c y recognize ! locus8 ,,*apt] ,,iv - ,,po9t-,,r[s ,,( ,,! ,,second ,,ord] 99#fj4 ,po9t-r[ ( ! second ord] def9$499 ,we h se5 t two fundam5tal =ms 9 "o-to-"o correspond5ce may sometimes g5]ate a =m ( high] ord]4 ,?us1 two po9t-r[s 7 #ee7 g5]ate a sy/em ( rays ( ! second ord]1 & two p5cils ( rays 7 #eg71 a sy/em ( po9ts ( ! second ord]4 ,z a sy/em ( po9ts is m famili> to mo/ /ud5ts ( geometry ?an a sy/em ( l9es1 we /udy fir/ ! po9t-r[ ( ! second ord]4 99#fa4 ,tang5t l9e499 ,we h %[n 9 ! la/ *apt] 7 #ee7 t ! locus ( int]section ( correspond+ rays ( two projective p5cils is a po9t-r[ ( ! second ord]2 t is1 x has at mo/ two po9ts 9 common ) any l9e 9 ! plane4 ,x is cle>1 fir/ ( all1 t ! c5t]s ( ! p5cils are po9ts ( ! locus2 = to ! l9e ,,ss'1 consid]$ z a ray ( ,s1 mu/ correspond "s ray ( ,s' :i* meets x 9 ,s'4 ,s'1 & by ! same argum5t ,s1 is !n a po9t :]e correspond+ rays meet4 ,any ray ?r\gh ,s w meet x 9 "o po9t besides ,s1 namely1 ! po9t ,p :]e x meets its correspond+ ray4 ,n[1 by *oos+ ! ray ?r\gh ,s suffici5tly close to ! ray ,,ss'1 ! po9t ,p may be made to approa* arbitr>ily close to ,s'1 & ! ray ,s',p may be made to diff] 9 position f ! tang5t l9e at ,s' by z little z we please4 ,we h1 !n1 ! important !orem ,! ray at ,s' :i* corresponds to ! common ray ,,ss' is tang5t to ! locus at ,s'4 ,9 ! same mann] ! tang5t at ,s may be con/ruct$4 99#fb4 ,det]m9ation ( ! locus499 ,we n[ %[ t x is possible to assign arbitr>ily ! position ( ?ree po9ts1 ,a1 ,b1 & ,c1 on ! locus 7besides ! po9ts ,s & ,s'72 b1 ^! ?ree po9ts be+ *os51 ! locus is completely det]m9$4 99#fc499 ,?is /atem5t is equival5t to ! foll[+3 ,giv5 ?ree pairs ( correspond+ rays 9 two projective p5cils1 x is possible to f9d a ray ( "o :i* corresponds to any ray ( ! o!r4 99#fd499 ,we proce$1 !n1 to ! solution ( ! fundam5tal ,,problem3 ,giv5 ?ree pairs ( rays1 aa'1 bb'1 & cc'1 ( two protective p5cils1 ,s & ,s'1 to f9d ! ray d' ( ,s' :i* corresponds to any ray d ( ,s4 ,figure #ab ,,fig4 #ab ,call ,a ! int]section ( aa'1 ,b ! int]section ( bb'1 & ,c ! int]section ( cc' 7,fig4 #ab74 ,jo9 ,,ab by ! l9e u1 & ,,ac by ! l9e u'4 ,consid] u z a po9t-r[ p]spective to ,s1 & u' z a po9t-r[ p]spective to ,s'4 u & u' are projectively relat$ to ea* o!r1 s9ce ,s & ,s' are1 by hypo!sis1 s relat$4 ,b _! po9t ( int]section ,a is a self-correspond+ po9t1 s9ce a & a' 7 suppos$ to be correspond+ rays4 ,x foll[s 7 #eb7 t u & u' are 9 p]spective position1 & t l9es ?r\gh correspond+ po9ts all pass ?r\gh a po9t ,m1 ! c5t] ( p]spectivity1 ! position ( :i* w be det]m9$ by any two su* l9es4 ,b ! int]section ( a ) u & ! int]section ( c' ) u' are correspond+ po9ts on u & u'1 & ! l9e jo9+ !m is cle>ly c itself4 ,simil>ly1 b' jo9s two correspond+ po9ts on u & u'1 & s ! c5t] ,m ( p]spectivity ( u & u' is ! int]section ( c & b'4 ,to f9d d' 9 ,s' correspond+ to a giv5 l9e d ( ,s we note ! po9t ,l :]e d meets u4 ,jo9 ,l to ,m & get ! po9t ,n :]e ?is l9e meets u'4 ,l & ,n are correspond+ po9ts on u & u'1 & d' mu/ !re=e pass ?r\gh ,n4 ,! int]section ,p ( d & d' is ?us ano!r po9t on ! locus4 ,9 ! same mann] any numb] ( o!r po9ts may be obta9$4 99#fe499 ,! l9es u & u' might h be5 drawn 9 any direction ?r\gh ,a 7avoid+1 ( c\rse1 ! l9e a = u & ! l9e a' = u'71 & ! c5t] ( p]spectivity ,m w\ld be easily obta9able2 b ! above con/ruction furni%es a simple & in/ructive figure4 ,an equally simple "o is obta9$ by tak+ a' = u & a = u'4 99#ff4 ,l9es jo9+ f\r po9ts ( ! locus to a fif?499 ,suppose t ! po9ts ,s1 ,s'1 ,b1 ,c1 & ,d are fix$1 & t f\r po9ts1 ,a1 ,a#a1 ,a#b1 & ,a#c1 are tak5 on ! locus at ! int]section ) x ( any f\r h>monic rays ?r\gh ,b4 ,^! f\r h>monic rays give f\r h>monic po9ts1 ,l1 ,l#a etc41 on ! fix$ ray ,,sd4 ,^!1 9 turn1 project ?r\gh ! fix$ po9t ,m into f\r h>monic po9ts1 ,n1 ,n#a etc41 on ! fix$ l9e ,,ds'4 ,^! la/ f\r h>monic po9ts give f\r h>monic rays ,,ca1 ,,ca#a1 ,,ca#b1 ,,ca#c4 ,!re=e ! f\r po9ts ,a :i* project to ,b 9 f\r h>monic rays also project to ,c 9 f\r h>monic rays4 ,b ,c may be any po9t on ! locus1 & s we h ! v important !orem1 ,f\r po9ts :i* are on ! locus1 & :i* project to a fif? po9t ( ! locus 9 f\r h>monic rays1 project to any po9t ( ! locus 9 f\r h>monic rays4 99#fg499 ,! !orem may also be /at$ ?us3 ,! locus ( po9ts f :i*1 f\r giv5 po9ts are se5 along f\r h>monic rays is a po9t-r[ ( ! second ord] ?r\gh !m4 99#fh499 ,a fur!r !orem ( prime importance also foll[s3 ,any two po9ts on ! locus may be tak5 z ! c5t]s ( two projective p5cils :i* w g5]ate ! locus4 99#fi4 ,pascal's !orem499 ,! po9ts ,a1 ,b1 ,c1 ,d1 ,s1 & ,s' may ?us be consid]$ z *os5 arbitr>ily on ! locus1 & ! foll[+ rem>kable !orem foll[s at once4 ,giv5 six po9ts1 #a1 #b1 #c1 #d1 #e1 #f1 on ! po9t-r[ ( ! second ord]1 if we call ,l ! int]section ( #ab ) #de1 ,m ! int]section ( #bc ) #ef1 ,n ! int]section ( #cd ) #fa1 !n ,l1 ,m1 & ,n are on a /raight l9e4 ,figure #ac ,,fig4 #ac 99#gj499 ,to get ! notation to correspond to ! figure1 we may take 7,fig4 #ac7 ,a #a1 ,b #b1 ,s' #c1 ,d #d1 ,s #e1 & ,c #f4 ,if we make ,a #a1 ,c#b1 ,s#c1 ,d #d1 ,s'#e1 &4 ,b #f1 ! po9ts ,l & ,n are int]*ang$1 b ! l9e is left un*ang$4 ,x is cle> t "o po9t may be nam$ arbitr>ily & ! o!r five nam$ 9 #e6 #abj diff]5t ways1 b s9ce1 z we h se51 two diff]5t assignm5ts ( names give ! same l9e1 x foll[s t !re _c be m ?an #fj diff]5t l9es ,,lmn obta9$ 9 ?is way f a giv5 set ( six po9ts4 ,z a matt] ( fact1 ! numb] obta9$ 9 ?is way is 9 g5]al #fj4 ,! above !orem1 :i* is ( c>d9al importance 9 ! !ory ( ! po9t-r[ ( ! second ord]1 is due to ,pascal & 0 discov]$ by him at ! age ( sixte54 ,x is1 no d\bt1 ! mo/ important contribution to ! !ory ( ^! loci s9ce ! days ( ,apollonius4 ,if ! six po9ts be call$ ! v]tices ( a hexagon inscrib$ 9 ! curve1 !n ! sides #ab & #de may be appropriately call$ a pair ( opposite sides4 ,pascal's !orem1 !n1 may be /at$ z foll[s3 ,! ?ree pairs ( opposite sides ( a hexagon inscrib$ 9 a po9t-r[ ( ! second ord] meet 9 ?ree po9ts on a l9e4 99#ga4 ,h>monic po9ts on a po9t-r[ ( ! second ord]499 ,be=e proce$+ to develop ! consequ5ces ( ?is !orem1 we note ano!r result ( ! utmo/ importance = ! high] developm5ts ( pure geometry1 :i* foll[s f ! fact t if f\r po9ts on ! locus project to a fif? 9 f\r h>monic rays1 !y w project to any po9t ( ! locus 9 f\r h>monic rays4 ,x is natural to speak ( f\r su* po9ts z f\r h>monic po9ts on ! locus1 & to use ?is notion to def9e projective correspond5ce betwe5 po9t-r[s ( ! second ord]1 or betwe5 a po9t-r[ ( ! second ord] & any fundam5tal =m ( ! fir/ ord]4 ,?us1 9 p>ticul>1 ! po9t-r[ ( ! second ord]1 1 is said to be p]spectively relat$ to ! p5cil ,s :5 e ray on ,s goes ?r\gh ! po9t on :i* corresponds to x4 99#gb4 ,det]m9ation ( ! locus499 ,x is n[ cle> t five po9ts1 arbitr>ily *os5 9 ! plane1 are suffici5t to det]m9e a po9t-r[ ( ! second ord] ?r\gh !m4 ,two ( ! po9ts may be tak5 z c5t]s ( two projective p5cils1 & ! ?ree o!rs w det]m9e ?ree pairs ( correspond+ rays ( ! p5cils1 & !re=e all pairs4 ,if f\r po9ts ( ! locus are giv51 toge!r ) ! tang5t at "o ( !m1 ! locus is likewise completely det]m9$4 ,= if ! po9t at :i* ! tang5t is giv5 be tak5 z ! c5t] ,s ( "o p5cil1 & any o!r ( ! po9ts = ,s'1 !n1 besides ! two pairs ( correspond+ rays det]m9$ by ! rema9+ two po9ts1 we h "o m pair1 consi/+ ( ! tang5t at ,s & ! ray ,,ss'4 ,simil>ly1 ! curve is det]m9$ by ?ree po9ts & ! tang5ts at two ( !m4 99#gc4 ,circles & conics z po9t-r[s ( ! second ord]499 ,x is n difficult to see t a circle is a po9t-r[ ( ! second ord]4 ,inde$1 take any po9t ,s on ! circle & draw f\r h>monic rays ?r\gh x4 ,!y w cut ! circle 9 f\r po9ts1 :i* w project to any o!r po9t ( ! curve 9 f\r h>monic rays2 =1 by ! !orem conc]n+ ! angles inscrib$ 9 a circle1 ! angles involv$ 9 ! second set ( f\r l9es are ! same z ^? 9 ! fir/ set4 ,if1 moreov]1 we project ! figure to any po9t 9 space1 we %all get a cone1 /&+ on a circul> base1 g5]at$ by two projective axial p5cils :i* are ! projections ( ! p5cils at ,s & ,s'4 ,cut across1 n[1 by any plane1 & we get a conic section :i* is ?us exhibit$ z ! locus ( int]section ( two projective p5cils4 ,x ?us appe>s t a conic section is a po9t-r[ ( ! second ord]4 ,x w lat] appe> t a po9t-r[ ( ! second ord] is a conic section4 ,9 ! future1 !re=e1 we %all ref] to a po9t-r[ ( ! second ord] z a conic4 ,figure #ad ,,fig4 #ad 99#gd4 ,conic ?r\gh five po9ts499 ,pascal's !orem furni%es an elegant solution ( ! problem ( draw+ a conic ?r\gh five giv5 po9ts4 ,to con/ruct a six? po9t on ! conic1 draw ?r\gh ! po9t numb]$ #a an arbitr>y l9e 7,fig4 #ad71 & let ! desir$ po9t #f be ! second po9t ( int]section ( ?is l9e ) ! conic4 ,! po9t ,l #ab-de is obta9able at once2 also ! po9t ,n #cd-fa4 ,b ,l & ,n det]m9e ,pascal's l9e1 & ! int]section ( #bc ) #ef mu/ be on ?is l9e4 ,int]sect1 !n1 ! l9e ,,ln ) #bc & obta9 ! po9t ,m4 ,jo9 ,m to #e & int]sect ) #fa = ! desir$ po9t #f4 ,figure #ae ,,fig4 #ae 99#ge4 ,tang5t to a conic499 ,if two po9ts ( ,pascal's hexagon approa* co9cid5ce1 !n ! l9e jo9+ !m approa*es z a limit+ position ! tang5t l9e at t po9t4 ,pascal's !orem ?us af=ds a ready me?od ( draw+ ! tang5t l9e to a conic at a giv5 po9t4 ,if ! conic is det]m9$ by ! po9ts #a1 #b1 #c1 #d1 #e 7,fig4 #ae71 & x is desir$ to draw ! tang5t at ! po9t #a1 we may call t po9t #a1 #f4 ,! po9ts ,l & ,m are obta9$ z usual1 & ! int]section ( #cd ) ,,lm gives ,n4 ,jo9 ,n to ! po9t #a = ! desir$ tang5t at t po9t4 99#gf4 ,inscrib$ quadrangle499 ,two pairs ( v]tices may coalesce1 giv+ an inscrib$ quadrangle4 ,pascal's !orem gives = ?is case ! v important !orem ,two pairs ( opposite sides ( any quadrangle inscrib$ 9 a conic meet on a /raight l9e1 ^u :i* l9e also int]sect ! two pairs ( tang5ts at ! opposite v]tices4 ,figure #af ,,fig4 #af ,figure #ag ,,fig4 #ag ,= let ! v]tices be ,a1 ,b1 ,c1 & ,d1 & call ! v]tex ,a ! po9t #a1 #f2 ,b1 ! po9t #b2 ,c1 ! po9t #c1 #d2 & ,d1 ! po9t #e 7,fig4 #af74 ,pascal's !orem !n indicates t ,l ,,ab-,,cd1 ,m ,,ad-,,bc1 & ,n1 :i* is ! int]section ( ! tang5ts at ,a & ,c1 are all on a /raight l9e u4 ,b if we 7 to call ,a ! po9t #b1 ,b ! po9t #f1 #a1 ,c ! po9t #e1 & ,d ! po9t #d1 #c1 !n ! int]section ,p ( ! tang5ts at ,b & ,d are also on ?is same l9e u4 ,?us ,l1 ,m1 ,n1 & ,p are f\r po9ts on a /raight l9e4 ,! consequ5ces ( ?is !orem are s num]\s & important t we %all devote a sep>ate *apt] to !m4 99#gg4 ,inscrib$ triangle499 ,f9ally1 ?ree ( ! v]tices ( ! hexagon may coalesce1 giv+ a triangle inscrib$ 9 a conic4 ,pascal's !orem !n reads z foll[s 7,fig4 #ag7 = ?is case3 ,! ?ree tang5ts at ! v]tices ( a triangle inscrib$ 9 a conic meet ! opposite sides 9 ?ree po9ts on a /raight l9e4 ,figure #ah ,,fig4 #ah 99#gh4 ,deg5]ate conic499 ,if we apply ,pascal's !orem to a deg5]ate conic made up ( a pair ( /raight l9es1 we get ! foll[+ !orem 7,fig4 #ah73 ,if ?ree po9ts1 ,a1 ,b1 ,c1 are *os5 on "o l9e1 & ?ree po9ts1 ,a'1 ,b'1 ,c'1 are *os5 on ano!r1 !n ! ?ree po9ts ,l ,,ab'-,a',b1 ,m ,,bc'-,b',c1 ,n ,,ca'-,c',a are all on a /raight l9e4 ,,problems #a4 ,9 ,fig4 #ab1 select diff]5t l9es u & trace ! locus ( ! c5t] ( p]spectivity ,m ( ! l9es u & u'4 #b4 ,giv5 f\r po9ts1 ,a1 ,b1 ,c1 ,d1 9 ! plane1 con/ruct a fif? po9t ,p su* t ! l9es ,,pa1 ,,pb1 ,,pc1 ,,pd %all be f\r h>monic l9es4 ,sugge/ion4 ,draw a l9e a ?r\gh ! po9t ,a su* t ! f\r l9es a1 ,,ab1 ,,ac1 ,,ad are h>monic4 ,con/ruct n[ a conic ?r\gh ,a1 ,b1 ,c1 & ,d hav+ a = a tang5t at ,a4 #c4 ,:]e are all ! po9ts ,p1 z det]m9$ 9 ! prec$+ "q1 to be f\nd8 #d4 ,select any five po9ts 9 ! plane & draw ! tang5t to ! conic ?r\gh !m at ea* ( ! five po9ts4 #e4 ,giv5 f\r po9ts on ! conic1 & ! tang5t at "o ( !m1 to con/ruct ! conic4 7,to con/ruct ! conic means "h to con/ruct z _m o!r po9ts z may be desir$47 #f4 ,giv5 ?ree po9ts on ! conic1 & ! tang5t at two ( !m1 to con/ruct ! conic4 #g4 ,giv5 five po9ts1 two ( :i* are at inf9ity 9 diff]5t directions1 to con/ruct ! conic4 7,9 ?is1 & 9 ! foll[+ examples1 ! /ud5t is suppos$ to be able to draw a l9e p>allel to a giv5 l9e47 #h4 ,giv5 f\r po9ts on a conic 7two ( :i* are at inf9ity & two 9 ! f9ite "p ( ! plane71 toge!r ) ! tang5t at "o ( ! f9ite po9ts1 to con/ruct ! conic4 #i4 ,! tang5ts to a curve at its inf9itely di/ant po9ts are call$ its asymptotes if !y pass ?r\gh a f9ite "p ( ! plane4 ,giv5 ! asymptotes & a f9ite po9t ( a conic1 to con/ruct ! conic4 #aj4 ,giv5 an asymptote & ?ree f9ite po9ts on ! conic1 to det]m9e ! conic4 #aa4 ,giv5 f\r po9ts1 "o ( :i* is at inf9ity1 & giv5 also t ! l9e at inf9ity is a tang5t l9e1 to con/ruct ! conic4 ,,*apt] ,v - ,,p5cils ,,( ,,rays ,,( ,,! ,,second ,,ord] 99#gi4 ,p5cil ( rays ( ! second ord] def9$499 ,if ! correspond+ po9ts ( two projective po9t-r[s be jo9$ by /raight l9es1 a sy/em ( l9es is obta9$ :i* is call$ a p5cil ( rays ( ! second ord]4 ,?is "n arises f ! fact1 easily %[n 7 #eg71 t at mo/ two l9es ( ! sy/em may pass ?r\gh any arbitr>y po9t 9 ! plane4 ,= if ?r\gh any po9t !re %\ld pass ?ree l9es ( ! sy/em1 !n ?is po9t might be tak5 z ! c5t] ( two projective p5cils1 "o project+ "o po9t-r[ & ! o!r project+ ! o!r4 ,s9ce1 n[1 ^! p5cils h ?ree rays ( "o co9cid5t ) ! correspond+ rays ( ! o!r1 ! two are id5tical & ! two po9t-r[s are 9 p]spective position1 :i* 0 n suppos$4 ,figure #ai ,,fig4 #ai 99#hj4 ,tang5ts to a circle499 ,to get a cle> notion ( ?is sy/em ( l9es1 we may fir/ %[ t ! tang5ts to a circle =m a sy/em ( ?is k9d4 ,= take any two tang5ts1 u & u'1 to a circle1 & let ,a & ,b be ! po9ts ( contact 7,fig4 #ai74 ,let n[ t be any ?ird tang5t ) po9t ( contact at ,c & meet+ u & u' 9 ,p & ,p' respectively4 ,jo9 ,a1 ,b1 ,p1 ,p'1 & ,c to ,o1 ! c5t] ( ! circle4 ,tang5ts f any po9t to a circle are equal1 & !re=e ! triangles ,,poa & ,,poc are equal1 z also are ! triangles ,p',,ob & ,p',,oc4 ,!re=e ! angle ,,pop' is con/ant1 be+ equal to half ! con/ant angle ,,aoc ,,cob4 ,?is be+ true1 if we take any f\r h>monic po9ts1 ,p#a1 ,p#b1 ,p#c1 ,p#d1 on ! l9e u1 !y w project to ,o 9 f\r h>monic l9es1 & ! tang5ts to ! circle f ^! f\r po9ts w meet u' 9 f\r h>monic po9ts1 ,p'#a1 ,p'#b1 ,p'#c1 ,p'#d1 because ! l9es f ^! po9ts to ,o inclose ! same angles z ! l9es f ! po9ts ,p#a1 ,p#b1 ,p#c1 ,p#d on u4 ,! po9t-r[ on u is !re=e projective to ! po9t-r[ on u'4 ,?us ! tang5ts to a circle are se5 to jo9 correspond+ po9ts on two projective po9t-r[s1 & s1 accord+ to ! def9ition1 =m a p5cil ( rays ( ! second ord]4 99#ha4 ,tang5ts to a conic499 ,if n[ ?is figure be project$ to a po9t outside ! plane ( ! circle1 & any section ( ! result+ cone be made by a plane1 we c easily see t ! sy/em ( rays tang5t to any conic section is a p5cil ( rays ( ! second ord]4 ,! conv]se is also true1 z we %all see lat]1 & a p5cil ( rays ( ! second ord] is also a set ( l9es tang5t to a conic section4 99#hb499 ,! po9t-r[s u & u' are1 !mselves1 l9es ( ! sy/em1 = to ! common po9t ( ! two po9t-r[s1 consid]$ z a po9t ( u1 mu/ correspond "s po9t ( u'1 & ! l9e jo9+ ^! two correspond+ po9ts is cle>ly u' itself4 ,simil>ly = ! l9e u4 99#hc4 ,det]m9ation ( ! p5cil499 ,we n[ %[ t x is possible to assign arbitr>ily ?ree l9es1 a1 b1 & c1 ( ! sy/em 7besides ! l9es u & u'72 b if ^! ?ree l9es are *os51 ! sy/em is completely det]m9$4 ,?is /atem5t is equival5t to ! foll[+3 ,giv5 ?ree pairs ( correspond+ po9ts 9 two projective po9t-r[s1 x is possible to f9d a po9t 9 "o :i* corresponds to any po9t ( ! o!r4 ,we proce$1 !n1 to ! solution ( ! fundam5tal ,,problem4 ,giv5 ?ree pairs ( po9ts1 ,,aa'1 ,,bb'1 & ,,cc'1 ( two projective po9t-r[s u & u'1 to f9d ! po9t ,d' ( u' :i* corresponds to any giv5 po9t ,d ( u4 ,figure #bj ,,fig4 #bj ,on ! l9e a1 jo9+ ,a & ,a'1 take two po9ts1 ,s & ,s'1 z c5t]s ( p5cils p]spective to u & u' respectively 7,fig4 #bj74 ,! figure w be mu* simplifi$ if we take ,s on ,,bb' & ,s' on ,,cc'4 ,,sa & ,s',a' are correspond+ rays ( ,s & ,s'1 & ! two p5cils are !re=e 9 p]spective position4 ,x is n difficult to see t ! axis ( p]spectivity m is ! l9e jo9+ ,b' & ,c4 ,giv5 any po9t ,d on u1 to f9d ! correspond+ po9t ,d' on u' we proce$ z foll[s3 ,jo9 ,d to ,s & note :]e ! jo9+ l9e meets m4 ,jo9 ?is po9t to ,s'4 ,?is la/ l9e meets u' 9 ! desir$ po9t ,d'4 ,we h n[ 9 ?is figure six l9es ( ! sy/em1 a1 b1 c1 d1 u1 & u'4 ,fix n[ ! position ( u1 u'1 b1 c1 & d1 & take f\r l9es ( ! sy/em1 a#a1 a#b1 a#c1 a#d1 :i* meet b 9 f\r h>monic po9ts4 ,^! po9ts project to ,d1 giv+ f\r h>monic po9ts on m4 ,^! aga9 project to ,d'1 giv+ f\r h>monic po9ts on c4 ,x is ?us cle> t ! rays a#a1 a#b1 a#c1 a#d cut out two projective po9t-r[s on any two l9es ( ! sy/em4 ,?us u & u' are n special rays1 & any two rays ( ! sy/em w s]ve z ! po9t-r[s to g5]ate ! sy/em ( l9es4 99#hd4 ,brian*on's !orem499 ,f ! figure also appe>s a fundam5tal !orem due to ,brian*on3 ,if #a1 #b1 #c1 #d1 #e1 #f are any six rays ( a p5cil ( ! second ord]1 !n ! l9es l 7#ab1 #de71 m 7#bc1 #ef71 n 7#cd1 #fa7 all pass ?r\gh a po9t4 ,figure #ba ,,fig4 #ba 99#he499 ,to make ! notation fit ! figure 7,fig4 #ba71 make a#a1 b #b1 u' #c1 d #d1 u #e1 c #f2 or1 int]*ang+ two ( ! l9es1 a #a1 c #b1 u #c1 d #d1 u' #e1 b #f4 ,?us1 by diff]5t nam+s ( ! l9es1 x appe>s t n m ?an #fj diff]5t ,brian*on po9ts are possible4 ,if we call #ab & #de opposite v]tices ( a circumscrib$ hexagon1 !n ,brian*on's !orem may be /at$ z foll[s3 ,! ?ree l9es jo9+ ! ?ree pairs ( opposite v]tices ( a hexagon circumscrib$ ab\t a conic meet 9 a po9t4 99#hf4 ,con/ruction ( ! p5cil by ,brian*on's !orem499 ,brian*on's !orem furni%es a ready me?od ( det]m9+ a six? l9e ( ! p5cil ( rays ( ! second ord] :5 five are giv54 ,?us1 select a po9t 9 l9e #a & suppose t l9e #f is to pass ?r\gh x4 ,!n l 7#ab1 #de71 n 7#cd1 #fa71 & ! l9e m 7#bc1 #ef7 mu/ pass ?r\gh 7l1 n74 ,!n 7#bc1 ln7 meets #e 9 a po9t ( ! requir$ six? l9e4 ,figure #bb ,,fig4 #bb 99#hg4 ,po9t ( contact ( a tang5t to a conic499 ,if ! l9e #b approa* z a limit+ position ! l9e #a1 !n ! int]section 7#a1 #b7 approa*es z a limit+ position ! po9t ( contact ( #a ) ! conic4 ,?is sugge/s an easy way to con/ruct ! po9t ( contact ( any tang5t ) ! conic4 ,?us 7,fig4 #bb71 giv5 ! l9es #a1 #b1 #c1 #d1 #e to con/ruct ! po9t ( contact ( #af4 ,draw l 7#ab1de71 m 7#bc1ef72 !n 7#cd1 lm7 meets #a 9 ! requir$ po9t ( contact ,t4 ,figure #bc ,,fig4 #bc 99#hh4 ,circumscrib$ quadrilat]al499 ,if two pairs ( l9es 9 ,brian*on's hexagon coalesce1 we h a !orem conc]n+ a quadrilat]al circumscrib$ ab\t a conic4 ,x is easily f\nd to be 7,fig4 #bc7 ,! f\r l9es jo9+ ! two opposite pairs ( v]tices & ! two opposite po9ts ( contact ( a quadrilat]al circumscrib$ ab\t a conic all meet 9 a po9t4 ,! consequ5ces ( ?is !orem w be d$uc$ lat]4 ,figure #bd ,,fig4 #bd 99#hi4 ,circumscrib$ triangle499 ,! hexagon may fur!r deg5]ate into a triangle1 giv+ ! !orem 7,fig4 #bd7 ,! l9es jo9+ ! v]tices to ! po9ts ( contact ( ! opposite sides ( a triangle circumscrib$ ab\t a conic all meet 9 a po9t4 99#ij499 ,brian*on's !orem may also be us$ to solve ! foll[+ problems3 ,giv5 f\r tang5ts & ! po9t ( contact on any "o ( !m1 to con/ruct o!r tang5ts to a conic4 ,giv5 ?ree tang5ts & ! po9ts ( contact ( any two ( !m1 to con/ruct o!r tang5ts to a conic4 99#ia4 ,h>monic tang5ts499 ,we h se5 t a v>iable tang5t cuts out on any two fix$ tang5ts projective po9t-r[s4 ,x foll[s t if f\r tang5ts cut a fif? 9 f\r h>monic po9ts1 !y mu/ cut e tang5t 9 f\r h>monic po9ts4 ,x is possible1 !re=e1 to make ! foll[+ def9ition3 ,f\r tang5ts to a conic are said to be h>monic :5 !y meet e o!r tang5t 9 f\r h>monic po9ts4 99#ib4 ,projectivity & p]spectivity499 ,?is def9ition sugge/s ! possibility ( def9+ a projective correspond5ce betwe5 ! elem5ts ( a p5cil ( rays ( ! second ord] & ! elem5ts ( any =m h]et(ore discuss$4 ,9 p>ticul>1 ! po9ts on a tang5t are said to be p]spectively relat$ to ! tang5ts ( a conic :5 ea* po9t lies on ! tang5t :i* corresponds to x4 ,^! notions are ( importance 9 ! high] developm5ts ( ! subject4 ,figure #be ,,fig4 #be 99#ic499 ,brian*on's !orem may also be appli$ to a deg5]ate conic made up ( two po9ts & ! l9es ?r\gh !m4 ,?us7,fig4 #be71 ,if a1 b1 c are ?ree l9es ?r\gh a po9t ,s1 & a'1 b'1 c' are ?ree l9es ?r\gh ano!r po9t ,s'1 !n ! l9es l 7ab'1 a'b71 m 7bc'1 b'c71 & n 7ca'1 c'a7 all meet 9 a po9t4 99#id4 ,law ( duality499 ,! obs]vant /ud5t w n h fail$ to note ! rem>kable simil>ity betwe5 ! !orems ( ?is *apt] & ^? ( ! prec$+4 ,he w h not$ t po9ts h replac$ l9es & l9es h replac$ po9ts2 t po9ts on a curve h be5 replac$ by tang5ts to a curve2 t p5cils h be5 replac$ by po9t-r[s1 & t a conic consid]$ z made up ( a succession ( po9ts has be5 replac$ by a conic consid]$ z g5]at$ by a mov+ tang5t l9e4 ,! !ory ^u :i* ?is wond]ful law ( duality is bas$ w be develop$ 9 ! next *apt]4 ,,problems #a4 ,giv5 f\r l9es 9 ! plane1 to con/ruct ano!r :i* %all meet !m 9 f\r h>monic po9ts4 #b4 ,:]e are all su* l9es f\nd8 #c4 ,giv5 any five l9es 9 ! plane1 con/ruct on ea* ! po9t ( contact ) ! conic tang5t to !m all4 #d4 ,giv5 f\r l9es & ! po9t ( contact on "o1 to con/ruct ! conic4 7,to con/ruct ! conic means "h to draw z _m o!r tang5ts z may be desir$47 #e4 ,giv5 ?ree l9es & ! po9t ( contact on two ( !m1 to con/ruct ! conic4 #f4 ,giv5 f\r l9es & ! l9e at inf9ity1 to con/ruct ! conic4 #g4 ,giv5 ?ree l9es & ! l9e at inf9ity1 toge!r ) ! po9t ( contact at inf9ity1 to con/ruct ! conic4 #h4 ,giv5 ?ree l9es1 two ( :i* are asymptotes1 to con/ruct ! conic4 #i4 ,giv5 five tang5ts to a conic1 to draw a tang5t :i* %all be p>allel to any "o ( !m4 #aj4 ,! l9es a1 b1 c are drawn p>allel to ea* o!r4 ,! l9es a'1 b'1 c' are also drawn p>allel to ea* o!r4 ,%[ :y ! l9es 7ab'1 a'b71 7bc'1 b'c71 7ca'1 c'a7 meet 9 a po9t4 7,9 problems #f to #aj inclusive1 p>allel l9es are to be drawn47 ,,*apt] ,,vi - ,,poles ,,& ,,pol>s 99#ie4 ,inscrib$ & circumscrib$ quadrilat]als499 ,! foll[+ !orems h be5 not$ z special cases ( ,pascal's & ,brian*on's !orems3 ,if a quadrilat]al be inscrib$ 9 a conic1 two pairs ( opposite sides & ! tang5ts at opposite v]tices int]sect 9 f\r po9ts1 all ( :i* lie on a /raight l9e4 ,if a quadrilat]al be circumscrib$ ab\t a conic1 ! l9es jo9+ two pairs ( opposite v]tices & ! l9es jo9+ two opposite po9ts ( contact are f\r l9es :i* meet 9 a po9t4 ,figure #bf ,,fig4 #bf 99#if4 ,def9ition ( ! pol> l9e ( a po9t499 ,consid] ! quadrilat]al ,k1 ,l1 ,m1 ,n inscrib$ 9 ! conic 7,fig4 #bf74 ,x det]m9es ! f\r h>monic po9ts ,a1 ,b1 ,c1 ,d :i* project f ,n 9 to ! f\r h>monic po9ts ,m1 ,b1 ,k1 ,o4 ,n[ ! tang5ts at ,k & ,m meet 9 ,p1 a po9t on ! l9e ,,ab4 ,! l9e ,,ab is ?us det]m9$ entirely by ! po9t ,o4 ,= if we draw any l9e ?r\gh x1 meet+ ! conic 9 ,k & ,m1 & con/ruct ! h>monic conjugate ,b ( ,o ) respect to ,k & ,m1 & also ! two tang5ts at ,k & ,m :i* meet 9 ! po9t ,p1 !n ,,bp is ! l9e 9 "q4 ,x ?us appe>s t ! l9e ,,lon may be any l9e :atev] ?r\gh ,o2 & s9ce ,d1 ,l1 ,o1 ,n are f\r h>monic po9ts1 we may describe ! l9e ,,ab z ! locus ( po9ts :i* are h>monic conjugates ( ,o ) respect to ! two po9ts :]e any l9e ?r\gh ,o meets ! curve4 99#ig499 ,fur!rmore1 s9ce ! tang5ts at ,l & ,n meet on ?is same l9e1 x appe>s z ! locus ( int]sections ( pairs ( tang5ts drawn at ! extremities ( *ords ?r\gh ,o4 99#ih499 ,?is important l9e1 :i* is completely det]m9$ by ! po9t ,o1 is call$ ! pol> ( ,o ) respect to ! conic2 & ! po9t ,o is call$ ! pole ( ! l9e ) respect to ! conic4 99#ii499 ,if a po9t ,b is on ! pol> ( ,o1 !n x is h>monically conjugate to ,o ) respect to ! two int]sections ,k & ,m ( ! l9e ,,bc ) ! conic4 ,b = ! same reason ,o is on ! pol> ( ,b4 ,we h1 !n1 ! fundam5tal !orem ,if "o po9t lies on ! pol> ( a second1 !n ! second lies on ! pol> ( ! fir/4 99#ajj4 ,conjugate po9ts & l9es499 ,su* a pair ( po9ts are said to be conjugate ) respect to ! conic4 ,simil>ly1 l9es are said to be conjugate to ea* o!r ) respect to ! conic if "o1 & consequ5tly ea*1 passes ?r\gh ! pole ( ! o!r4 ,figure #bg ,,fig4 #bg 99#aja4 ,con/ruction ( ! pol> l9e ( a giv5 po9t499 ,giv5 a po9t ,p1 if x is )9 ! conic 7t is1 if no tang5ts may be drawn f ,p to ! conic71 we may con/ruct its pol> l9e by draw+ ?r\gh x any two *ords & jo9+ ! two po9ts ( int]section ( ! two pairs ( tang5ts at _! extremities4 ,if ! po9t ,p is outside ! conic1 we may draw ! two tang5ts & con/ruct ! *ord ( contact 7,fig4 #bg74 99#ajb4 ,self-pol> triangle499 ,9 ,fig4 #bf x is n difficult to see t ,,aoc is a self-pol> triangle1 t is1 ea* v]tex is ! pole ( ! opposite side4 ,= ,b1 ,m1 ,o1 ,k are f\r h>monic po9ts1 & !y project to ,c 9 f\r h>monic rays4 ,! l9e ,,co1 !re=e1 meets ! l9e ,,amn 9 a po9t on ! pol> ( ,a1 be+ sep>at$ f ,a h>monically by ! po9ts ,m & ,n4 ,simil>ly1 ! l9e ,,co meets ,,kl 9 a po9t on ! pol> ( ,a1 & !re=e ,,co is ! pol> ( ,a4 ,simil>ly1 ,,oa is ! pol> ( ,c1 & !re=e ,o is ! pole ( ,,ac4 99#ajc4 ,pole & pol> projectively relat$499 ,ano!r v important !orem comes directly f ,fig4 #bf4 ,z a po9t ,a moves along a /raight l9e its pol> ) respect to a conic revolves ab\t a fix$ po9t & describes a p5cil projective to ! po9t-r[ describ$ by ,a4 ,=1 fix ! po9ts ,l & ,n & let ! po9t ,a move along ! l9e ,,aq2 !n ! po9t-r[ ,a is projective to ! p5cil ,,lk1 & s9ce ,k moves along ! conic1 ! p5cil ,,lk is projective to ! p5cil ,,nk1 :i* 9 turn is projective to ! po9t-r[ ,c1 :i*1 f9ally1 is projective to ! p5cil ,,oc1 :i* is ! pol> ( ,a4 99#ajd4 ,duality499 ,we h1 !n1 9 ! pole & pol> relation a device = sett+ up a "o-to-"o correspond5ce betwe5 ! po9ts & l9es ( ! plane--a correspond5ce :i* may be call$ projective1 because to f\r h>monic po9ts or l9es correspond always f\r h>monic l9es or po9ts4 ,to e figure made up ( po9ts & l9es w correspond a figure made up ( l9es & po9ts4 ,to a po9t-r[ ( ! second ord]1 :i* is a conic consid]$ z a po9t-locus1 corresponds a p5cil ( rays ( ! second ord]1 :i* is a conic consid]$ z a l9e-locus4 ,! "n 'duality' is us$ to describe ?is sort ( correspond5ce4 ,x is important to note t ! dual relation is subject to ! same exceptions z ! "o-to-"o correspond5ce is1 & mu/ n be appeal$ to 9 cases :]e ! "o-to-"o correspond5ce breaks d[n4 ,we h se5 t !re is 9 ,euclidean geometry "o & only "o ray 9 a p5cil :i* has no po9t 9 a po9t-r[ p]spective to x = a correspond+ po9t2 namely1 ! l9e p>allel to ! l9e ( ! po9t-r[4 ,any !orem1 !re=e1 t involves explicitly ! po9t at inf9ity is n to be translat$ into a !orem conc]n+ l9es4 ,fur!r1 9 ! p5cil ! angle betwe5 two l9es has no?+ to correspond to x 9 a po9t-r[ p]spective to ! p5cil4 ,any !orem1 !re=e1 t m5tions angles is n translatable into ano!r !orem by means ( ! law ( duality4 ,n[ we h se5 t ! notion ( ! inf9itely di/ant po9t on a l9e involves ! notion ( divid+ a segm5t into any numb] ( equal p>ts--9 o!r words1 ( measur+4 ,if1 !re=e1 we call any !orem t has to d ) ! l9e at inf9ity or ) ! measurem5t ( angles a metrical !orem1 & any o!r k9d a projective !orem1 we may put ! case z foll[s3 ,any projective !orem involves ano!r !orem1 dual to x1 obta9able by int]*ang+ ev]y:]e ! words 'po9t' & 'l9e4' 99#aje4 ,self-dual !orems499 ,! !orems ( ?is *apt] w be f\nd1 ^u exam9ation1 to be self-dual2 t is1 no new !orem results f apply+ ! process indicat$ 9 ! prec$+ p>agraph4 ,x is !re=e useless to look = new results f ! !orem on ! circumscrib$ quadrilat]al d]iv$ f ,brian*on's1 :i* is itself cle>ly ! dual ( ,pascal's !orem1 & 9 fact 0 fir/ discov]$ by dualization ( ,pascal's4 99#ajf499 ,x %\ld n be inf]r$ f ! above discussion t "o-to-"o correspond5ces may n be devis$ t w control c]ta9 ( ! s-call$ metrical relations4 ,a v important "o may be easily f\nd t leaves angles unalt]$4 ,! relation call$ simil>ity leaves ratios betwe5 correspond+ segm5ts unalt]$4 ,! above /atem5ts apply only to ! p>ticul> "o-to-"o correspond5ce consid]$4 ,,problems #a4 ,giv5 a quadrilat]al1 con/ruct ! quadrangle pol> to x ) respect to a giv5 conic4 #b4 ,a po9t moves along a /raight l9e4 ,%[ t its pol> l9es ) respect to two giv5 conics g5]ate a po9t-r[ ( ! second ord]4 #c4 ,giv5 five po9ts1 draw ! pol> ( a po9t ) respect to ! conic pass+ ?r\gh !m1 )\t draw+ ! conic itself4 #d4 ,giv5 five l9es1 draw ! pol> ( a po9t ) respect to ! conic tang5t to !m1 )\t draw+ ! conic itself4 #e4 ,dualize problems #c & #d4 #f4 ,giv5 f\r po9ts on ! conic1 & ! tang5t at "o ( !m1 draw ! pol> ( a giv5 po9t )\t draw+ ! conic4 ,dualize4 #g4 ,a po9t moves on a conic4 ,%[ t its pol> l9e ) respect to ano!r conic describes a p5cil ( rays ( ! second ord]4 ,sugge/ion4 ,replace ! giv5 conic by a pair ( protective p5cils4 #h4 ,%[ t ! poles ( ! tang5ts ( "o conic ) respect to ano!r lie on a conic4 #i4 ,! pol> ( a po9t ,a ) respect to "o conic is a1 & ! pole ( a ) respect to ano!r conic is ,a'4 ,%[ t z ,a travels along a l9e1 ,a' also travels along ano!r l9e4 ,9 g5]al1 if ,a describes a curve ( degree n1 %[ t ,a' describes ano!r curve ( ! same degree n4 7,! degree ( a curve is ! greate/ numb] ( po9ts t x may h 9 common ) any l9e 9 ! plane47 ,,*apt] ,,vii - ,,metrical ,,prop]ties ,,( ,,! ,,conic ,,sections 99#ajg4 ,diamet]s4 ,c5t]499 ,aft] :at has be5 said 9 ! la/ *apt] "o w\ld naturally expect to get at ! metrical prop]ties ( ! conic sections by ! introduction ( ! inf9ite elem5ts 9 ! plane4 ,ent]+ into ! !ory ( poles & pol>s ) ^! elem5ts1 we h ! foll[+ def9itions3 ,! pol> l9e ( an inf9itely di/ant po9t is call$ a diamet]1 & ! pole ( ! inf9itely di/ant l9e is call$ ! c5t]1 ( ! conic4 99#ajh499 ,f ! h>monic prop]ties ( poles & pol>s1 ,! c5t] bisects all *ords ?r\gh x 7 #ci74 ,e diamet] passes ?r\gh ! c5t]4 ,all *ords ?r\gh ! same po9t at inf9ity 7t is1 ea* ( a set ( p>allel *ords7 are bisect$ by ! diamet] :i* is ! pol> ( t inf9itely di/ant po9t4 99#aji4 ,conjugate diamet]s499 ,we h already def9$ conjugate l9es z l9es :i* pass ea* ?r\gh ! pole ( ! o!r 7 #ajj74 ,any diamet] bisects all *ords p>allel to its conjugate4 ,! tang5ts at ! extremities ( any diamet] are p>allel1 & p>allel to ! conjugate diamet]4 ,diamet]s p>allel to ! sides ( a circumscrib$ p>allelogram are conjugate4 ,all ^! !orems are easy ex]cises = ! /ud5t4 99#aaj4 ,classification ( conics499 ,conics are classifi$ accord+ to _! relation to ! inf9itely di/ant l9e4 ,if a conic has two po9ts 9 common ) ! l9e at inf9ity1 x is call$ a hyp]bola2 if x has no po9t 9 common ) ! inf9itely di/ant l9e1 x is call$ an ellipse2 if x is tang5t to ! l9e at inf9ity1 x is call$ a p>abola4 99#aaa499 ,9 a hyp]bola ! c5t] is outside ! curve 7 #aja71 s9ce ! two tang5ts to ! curve at ! po9ts :]e x meets ! l9e at inf9ity det]m9e by _! int]section ! c5t]4 ,z previ\sly not$1 ^! two tang5ts are call$ ! asymptotes ( ! curve4 ,! ellipse & ! p>abola h no asymptotes4 99#aab499 ,! c5t] ( ! p>abola is at inf9ity1 & !re=e all its diamet]s are p>allel1 = ! pole ( a tang5t l9e is ! po9t ( contact4 ,! locus ( ! middle po9ts ( a s]ies ( p>allel *ords 9 a p>abola is a diamet]1 & ! direction ( ! l9e ( c5t]s is ! same = all s]ies ( p>allel *ords4 ,! c5t] ( an ellipse is )9 ! curve4 ,figure #bh ,,fig4 #bh 99#aac4 ,!orems conc]n+ asymptotes499 ,we d]iv$ z a consequ5ce ( ! !orem ( ,brian*on 7 #hi7 ! proposition t if a triangle be circumscrib$ ab\t a conic1 ! l9es jo9+ ! v]tices to ! po9ts ( contact ( ! opposite sides all meet 9 a po9t4 ,take1 n[1 = two ( ! tang5ts ! asymptotes ( a hyp]bola1 & let any ?ird tang5t cut !m 9 ,a & ,b 7,fig4 #bh74 ,if1 !n1 ,o is ! int]section ( ! asymptotes1--& !re=e ! c5t] ( ! curve1-- !n ! triangle ,,oab is circumscrib$ ab\t ! curve4 ,by ! !orem j quot$1 ! l9e ?r\gh ,a p>allel to ,,ob1 ! l9e ?r\gh ,b p>allel to ,,oa1 & ! l9e ,,op ?r\gh ! po9t ( contact ( ! tang5t ,,ab all meet 9 a po9t ,c4 ,b ,,oacb is a p>allelogram1 & ,,pa ,,pb4 ,!re=e ,! asymptotes cut (f on ea* tang5t a segm5t :i* is bisect$ by ! po9t ( contact4 99#aad499 ,if we draw a l9e ,,oq p>allel to ,,ab1 !n ,,op & ,,oq are conjugate diamet]s1 s9ce ,,oq is p>allel to ! tang5t at ! po9t :]e ,,op meets ! curve4 ,!n1 s9ce ,a1 ,p1 ,b1 & ! po9t at inf9ity on ,,ab are f\r h>monic po9ts1 we h ! !orem ,conjugate diamet]s ( ! hyp]bola are h>monic conjugates ) respect to ! asymptotes4 99#aae499 ,! *ord ,a,b1 p>allel to ! diamet] ,,oq1 is bisect$ at ,p' by ! conjugate diamet] ,,op4 ,if ! *ord ,a,b meet ! asymptotes 9 ,a'1 ,b'1 !n ,a'1 ,p'1 ,b'1 & ! po9t at inf9ity are f\r h>monic po9ts1 & !re=e ,p' is ! middle po9t ( ,a',b'4 ,!re=e ,a',a ,b',b & we h ! !orem ,! segm5ts cut (f on any *ord betwe5 ! hyp]bola & its asymptotes are equal4 99#aaf499 ,?is !orem furni%es a ready means ( con/ruct+ ! hyp]bola by po9ts :5 a po9t on ! curve & ! two asymptotes are giv54 ,figure #bi ,,fig4 #bi 99#aag499 ,= ! circumscrib$ quadrilat]al1 ,brian*on's !orem gave 7 #hh7 ,! l9es jo9+ opposite v]tices & ! l9es jo9+ opposite po9ts ( contact are f\r l9es meet+ 9 a po9t4 ,take n[ = two ( ! tang5ts ! asymptotes1 & let ,,ab & ,,cd be any o!r two 7,fig4 #bi74 ,if ,b & ,d are opposite v]tices1 & also ,a & ,c1 !n ,,ac & ,,bd are p>allel1 & p>allel to ,,pq1 ! l9e jo9+ ! po9ts ( contact ( ,,ab & ,,cd1 = ^! are ?ree ( ! f\r l9es ( ! !orem j quot$4 ,! f\r? is ! l9e at inf9ity :i* jo9s ! po9t ( contact ( ! asymptotes4 ,x is ?us se5 t ! triangles ,,abc & ,,adc are equival5t1 & !re=e ! triangles ,,aob & ,,cod are also4 ,! tang5t ,,ab may be fix$1 & ! tang5t ,,cd *os5 arbitr>ily2 !re=e ,! triangle =m$ by any tang5t to ! hyp]bola & ! two asymptotes is ( con/ant area4 99#aah4 ,equation ( hyp]bola ref]r$ to ! asymptotes499 ,draw ?r\gh ! po9t ( contact ,p ( ! tang5t ,,ab two l9es1 "o p>allel to "o asymptote & ! o!r p>allel to ! o!r4 ,"o ( ^! l9es meets ,,ob at a di/ance y f ,o1 & ! o!r meets ,,oa at a di/ance x f ,o4 ,!n1 s9ce ,p is ! middle po9t ( ,,ab1 x is "o half ( ,,oa & y is "o half ( ,,ob4 ,! area ( ! p>allelogram ^: adjac5t sides are x & y is "o half ! area ( ! triangle ,,aob1 & !re=e1 by ! prec$+ p>agraph1 is con/ant4 ,?is area is equal to xy s9 1 :]e is ! con/ant angle betwe5 ! asymptotes4 ,x foll[s t ! product xy is con/ant1 & s9ce x & y are ! oblique cord9ates ( ! po9t ,p1 ! asymptotes be+ ! axes ( ref]5ce1 we h ,! equation ( ! hyp]bola1 ref]r$ to ! asymptotes z axes1 is xy con/ant4 ,?is id5tifies ! curve ) ! hyp]bola z def9$ & discuss$ 9 works on analytic geometry4 ,figure #cj ,,fig4 #cj 99#aai4 ,equation ( p>abola499 ,we h def9$ ! p>abola z a conic :i* is tang5t to ! l9e at inf9ity 7 #aaj74 ,draw n[ two tang5ts to ! curve 7,fig4 #cj71 meet+ 9 ,a1 ! po9ts ( contact be+ ,b & ,c4 ,^! two tang5ts1 toge!r ) ! l9e at inf9ity1 =m a triangle circumscrib$ ab\t ! conic4 ,draw ?r\gh ,b a p>allel to ,,ac1 & ?r\gh ,c a p>allel to ,,ab4 ,if ^! meet 9 ,d1 !n ,,ad is a diamet]4 ,let ,,ad meet ! curve 9 ,p1 & ! *ord ,,bc 9 ,q4 ,p is !n ! middle po9t ( ,,aq4 ,also1 ,q is ! middle po9t ( ! *ord ,,bc1 & !re=e ! diamet] ,,ad bisects all *ords p>allel to ,,bc4 ,9 p>ticul>1 ,,ad passes ?r\gh ,p1 ! po9t ( contact ( ! tang5t drawn p>allel to ,,bc4 ,draw n[ ano!r tang5t1 meet+ ,,ab 9 ,b' & ,,ac 9 ,c'4 ,!n ^! ?ree1 ) ! l9e at inf9ity1 make a circumscrib$ quadrilat]al4 ,b1 by ,brian*on's !orem appli$ to a quadrilat]al 7 #hh71 x appe>s t a p>allel to ,,ac ?r\gh ,b'1 a p>allel to ,,ab ?r\gh ,c'1 & ! l9e ,,bc meet 9 a po9t ,d'4 ,also1 f ! simil> triangles ,,bb',d' & ,,bac we h1 = all positions ( ! tang5t l9e ,b',c1 ,b',d' 3 ,,bb' ,,ac 3 ,,ab1 or1 s9ce ,b',d' ,,ac'1 ,,ac'3 ,,bb' ,,ac3,,ab con/ant4 ,if ano!r tang5t meet ,,ab 9 ,b & ,,ac 9 ,c1 we h ,,ac' 3 ,,bb' ,,ac 3 ,,bb1 & by subtraction we get ,c',c 3 ,b',b con/ant2 :5ce ,! segm5ts cut (f on any two tang5ts to a p>abola by a v>iable tang5t are proportional4 ,if n[ we take ! tang5t ,b',c' z axis ( ord9ates1 & ! diamet] ?r\gh ! po9t ( contact ,o z axis ( abscissas1 call+ ! coord9ates ( ,b7x1 y7 & ( ,c7x'1 y'71 !n1 f ! simil> triangles ,,bmd' & we h y 3 y' ,,bd' 3 ,d',c ,,bb' 3 ,,ab'4 ,also y 3 y' ,b',d' 3 ,c',c ,,ac' 3 ,c',c4 ,if n[ a l9e is drawn ?r\gh ,a p>allel to a diamet]1 meet+ ! axis ( ord9ates 9 ,k1 we h ,,ak 3 ,,oq' ,,ac' 3 ,,cc' y 3 y'1 & ,,om 3 ,,ak ,,bb' 3 ,,ab' y 3 y'1 &1 by multiplication1 ,,om 3 ,,oq' y#b 3 y'#b1 or x 3 x' y#b 3 y'#b2 :5ce ,! abscissas ( two po9ts on a p>abola are to ea* o!r z ! squ>es ( ! correspond+ cord9ates1 a diamet] & ! tang5t to ! curve at ! extremity ( ! diamet] be+ ! axes ( ref]5ce4 ,! la/ equation may be writt5 y#b #bpx1 :]e #bp /&s = y'#b 3 x'4 ,! p>abola is ?us id5tifi$ ) ! curve ( ! same "n /udi$ 9 treatises on analytic geometry4 99#abj4 ,equation ( c5tral conics ref]r$ to conjugate diamet]s499 ,consid] n[ a c5tral conic1 t is1 "o :i* is n a p>abola & ! c5t] ( :i* is !re=e at a f9ite di/ance4 ,draw any f\r tang5ts to x1 two ( :i* are p>allel 7,fig4 #ca74 ,let ! p>allel tang5ts meet "o ( ! o!r tang5ts 9 ,a & ,b & ! o!r 9 ,c & ,d1 & let ,p & ,q be ! po9ts ( contact ( ! p>allel tang5ts ,r & ,s ( ! o!rs4 ,!n ,,ac1 ,,bd1 ,,pq1 & ,,rs all meet 9 a po9t ,w 7 #hh74 ,f ! figure1 ,,pw 3 ,,wq ,,ap 3 ,,qc ,,pd 3 ,,bq1 or ,,ap ,,bq ,,pd ,,qc4 ,if n[ ,,dc is a fix$ tang5t & ,,ab a v>iable "o1 we h f ?is equation ,,ap ,,bq con/ant4 ,?is con/ant w be positive or negative accord+ z ,,pa & ,,bq are measur$ 9 ! same or 9 opposite directions4 ,accord+ly we write ,,ap ,,bq b#b4 ,figure #ca ,,fig4 #ca ,s9ce ,,ad & ,,bc are p>allel tang5ts1 ,,pq is a diamet] & ! conjugate diamet] is p>allel to ,,ad4 ,! middle po9t ( ,,pq is ! c5t] ( ! conic4 ,we take n[ = ! axis ( abscissas ! diamet] ,,pq1 & ! conjugate diamet] = ! axis ( ord9ates4 ,jo9 ,a to ,q & ,b to ,p & draw a l9e ?r\gh ,s p>allel to ! axis ( ord9ates4 ,^! ?ree l9es all meet 9 a po9t ,n1 because ,,ap1 ,,bq1 & ,,ab =m a triangle circumscrib$ to ! conic4 ,let ,,ns meet ,,pq 9 ,m4 ,!n1 f ! prop]ties ( ! circumscrib$ triangle 7 #hi71 ,m1 ,n1 ,s1 & ! po9t at inf9ity on ,,ns are f\r h>monic po9ts1 & !re=e ,n is ! middle po9t ( ,,ms4 ,if ! cord9ates ( ,s are 7x1 y71 s t ,,om is x & ,,ms is y1 !n ,,mn y_/#b4 ,n[ f ! simil> triangles ,,pmn & ,,pqb we h ,,bq 3 ,,pq ,,nm 3 ,,pm1 & f ! simil> triangles ,,pqa & ,,mqn1 ,,ap 3 ,,pq ,,mn 3 ,,mq1 :5ce1 multiply+1 we h b#b_/d a#b y#b_/d 7a x77a - x71 :]e =mula or1 simplify+1 =mula :i* is ! equation ( an ellipse :5 b#b has a positive sign1 & ( a hyp]bola :5 b#b has a negative sign4 ,we h ?us id5tifi$ po9t-r[s ( ! second ord] ) ! curves giv5 by equations ( ! second degree4 ,,problems #a4 ,draw a *ord ( a giv5 conic :i* %all be bisect$ by a giv5 po9t ,p4 #b4 ,%[ t all *ords ( a giv5 conic t are bisect$ by a giv5 *ord are tang5t to a p>abola4 #c4 ,con/ruct a p>abola1 giv5 two tang5ts ) _! po9ts ( contact4 #d4 ,con/ruct a p>abola1 giv5 ?ree po9ts & ! direction ( ! diamet]s4 #e4 ,a l9e u' is drawn ?r\gh ! pole ,u ( a l9e u & at "r angles to u4 ,! l9e u revolves ab\t a po9t ,p4 ,%[ t ! l9e u' is tang5t to a p>abola4 7,! l9es u & u' are call$ normal conjugates47 #f4 ,giv5 a circle & its c5t] ,o1 to draw a l9e ?r\gh a giv5 po9t ,p p>allel to a giv5 l9e q4 ,prove ! foll[+ con/ruction3 ,let p be ! pol> ( ,p1 ,q ! pole ( q1 & ,a ! int]section ( p ) ,,oq4 ,! pol> ( ,a is ! desir$ l9e4 ,,*apt] ,,viii - ,,involution ,figure #cb ,,fig4 #cb 99#aba4 ,fundam5tal !orem499 ,! important !orem conc]n+ two complete quadrangles 7 #bf71 ^u :i* ! !ory ( f\r h>monic po9ts 0 bas$1 c easily be ext5d$ to ! case :]e ! f\r l9es ,,kl1 ,k',l'1 ,,mn1 ,m',n' d n all meet 9 ! same po9t ,a1 & ! m g5]al !orem t results may also be made ! basis ( a !ory no less important1 :i* has to d ) six po9ts on a l9e4 ,! !orem is z foll[s3 ,giv5 two complete quadrangles1 ,k1 ,l1 ,m1 ,n & ,k'1 ,l'1 ,m'1 ,n'1 s relat$ t ,,kl & ,k',l' meet 9 ,a1 ,,mn & ,m',n' 9 ,a'1 ,,kn & ,k',n' 9 ,b1 ,,lm & ,l',m' 9 ,b'1 ,,ln & ,l',n' 9 ,c1 & ,,km & ,k',m' 9 ,c'1 !n1 if ,a1 ,a'1 ,b1 ,b'1 & ,c are 9 a /raight l9e1 ! po9t ,c' also lies on t /raight l9e4 ,! !orem foll[s f ,des>gues's !orem 7,fig4 #cb74 ,x is se5 t ,,kk'1 ,,ll'1 ,,mm'1 ,,nn' all meet 9 a po9t1 & ?us1 f ! same !orem1 appli$ to ! triangles ,,klm & ,k',l',m'1 ! po9t ,c' is on ! same l9e ) ,a & ,b'4 ,z 9 ! simpl] case1 x is se5 t !re is an indef9ite numb] ( quadrangles :i* may be drawn1 two sides ( :i* g ?r\gh ,a & ,a'1 two ?r\gh ,b & ,b'1 & "o ?r\gh ,c4 ,! six? side mu/ !n g ?r\gh ,c'4 ,!re=e1 99#abb499 ,two pairs ( po9ts1 ,a1 ,a' & ,b1 ,b'1 be+ giv51 !n ! po9t ,c' correspond+ to any giv5 po9t ,c is uniquely det]m9$4 ,! con/ruction ( ?is six? po9t is easily accompli%$4 ,draw ?r\gh ,a & ,a' any two l9es1 & cut across !m by any l9e ?r\gh ,c 9 ! po9ts ,l & ,n4 ,jo9 ,n to ,b & ,l to ,b'1 ?us det]m9+ ! po9ts ,k & ,m on ! two l9es ?r\gh ,a & ,a'1 ,! l9e ,,km det]m9es ! desir$ po9t ,c'4 ,manife/ly1 />t+ f ,c'1 we come 9 ?is way always to ! same po9t ,c4 ,! p>ticul> quadrangle employ$ is ( no consequ5ce4 ,moreov]1 s9ce "o pair ( opposite sides 9 a complete quadrangle is n di/+ui%able 9 any way f any o!r1 ! same set ( six po9ts w be obta9$ by />t+ f ! pairs ,,aa' & ,,cc'1 or f ! pairs ,,bb' & ,,cc'4 99#abc4 ,def9ition ( involution ( po9ts on a l9e499 ,?ree pairs ( po9ts on a l9e are said to be 9 involution if ?r\gh ea* pair may be drawn a pair ( opposite sides ( a complete quadrangle4 ,if two pairs are fix$ & "o ( ! ?ird pair describes ! l9e1 !n ! o!r also describes ! l9e1 & ! po9ts ( ! l9e are said to be pair$ 9 ! involution det]m9$ by ! two fix$ pairs4 ,figure #cc ,,fig4 #cc 99#abd4 ,d\ble-po9ts 9 an involution499 ,! po9ts ,c & ,c' describe projective po9t-r[s1 z may be se5 by fix+ ! po9ts ,l & ,m4 ,! self-correspond+ po9ts1 ( :i* !re are two or none1 are call$ ! d\ble-po9ts 9 ! involution4 ,x is n difficult to see t ! d\ble-po9ts 9 ! involution are h>monic conjugates ) respect to correspond+ po9ts 9 ! involution4 ,=1 fix+ z be=e ! po9ts ,l & ,m1 let ! int]section ( ! l9es ,,cl & ,c',m be ,p 7,fig4 #cc74 ,! locus ( ,p is a conic :i* goes ?r\gh ! d\ble-po9ts1 because ! po9t-r[s ,c & ,c' are projective1 & !re=e s are ! p5cils ,,lc & ,,mc' :i* g5]ate ! locus ( ,p4 ,also1 :5 ,c & ,c' fall toge!r1 ! po9t ,p co9cides ) !m4 ,fur!r1 ! tang5ts at ,l & ,m to ?is conic describ$ by ,p are ! l9es ,,lb & ,,mb4 ,= 9 ! p5cil at ,l ! ray ,,lm common to ! two p5cils :i* g5]ate ! conic is ! ray ,,lb' & corresponds to ! ray ,,mb ( ,m1 :i* is !re=e ! tang5t l9e to ! conic at ,m4 ,simil>ly = ! tang5t ,,lb at ,l4 ,,lm is !re=e ! pol> ( ,b ) respect to ?is conic1 & ,b & ,b' are !re=e h>monic conjugates ) respect to ! d\ble-po9ts4 ,! same discussion applies to any o!r pair ( correspond+ po9ts 9 ! involution4 ,figure #cd ,,fig4 #cd 99#abe4 ,des>gues's !orem conc]n+ conics ?r\gh f\r po9ts499 ,let ,,dd' be any pair ( po9ts 9 ! involution det]m9$ z above1 & consid] ! conic pass+ ?r\gh ! five po9ts ,k1 ,l1 ,m1 ,n1 ,d4 ,we %all use ,pascal's !orem to %[ t ?is conic also passes ?r\gh ,d'4 ,! po9t ,d' is det]m9$ z foll[s3 ,fix ,l & ,m z be=e 7,fig4 #cd7 & jo9 ,d to ,l1 giv+ on ,,mn ! po9t ,n'4 ,jo9 ,n' to ,b1 giv+ on ,,lk ! po9t ,k'4 ,!n ,,mk' det]m9es ! po9t ,d' on ! l9e ,,aa'1 giv5 by ! complete quadrangle ,k'1 ,l1 ,m1 ,n'4 ,consid] ! foll[+ six po9ts1 numb]+ !m 9 ord]3 ,d #a1 ,d' #b1 ,m #c1 ,n #d1 ,k #e1 & ,l #f4 ,we h ! foll[+ int]sections3 ,b 7#ab-de71 ,k' 7#bc-ef71 ,n' 7#cd-fa72 & s9ce by con/ruction ,b1 ,n1 & ,k' are on a /raight l9e1 x foll[s f ! conv]se ( ,pascal's !orem1 :i* is easily e/abli%$1 t ! six po9ts are on a conic4 ,we h1 !n1 ! beautiful !orem due to ,des>gues3 ,! sy/em ( conics ?r\gh f\r po9ts meets any l9e 9 ! plane 9 pairs ( po9ts 9 involution4 99#abf499 ,x appe>s also t ! six po9ts 9 involution det]m9$ by ! quadrangle ?r\gh ! f\r fix$ po9ts belong also to ! same involution ) ! po9ts cut out by ! sy/em ( conics1 z inde$ we might inf] f ! fact t ! ?ree pairs ( opposite sides ( ! quadrangle may be consid]$ z deg5]ate conics ( ! sy/em4 99#abg4 ,conics ?r\gh f\r po9ts t\*+ a giv5 l9e499 ,x is fur!r evid5t t ! involution det]m9$ on a l9e by ! sy/em ( conics w h a d\ble-po9t :]e a conic ( ! sy/em is tang5t to ! l9e4 ,we may !re=e inf] ! !orem ,?r\gh f\r fix$ po9ts 9 ! plane two conics or none may be drawn tang5t to any giv5 l9e4 ,figure #ce ,,fig4 #ce 99#abh4 ,d\ble correspond5ce499 ,we h se5 t correspond+ po9ts 9 an involution =m two projective po9t-r[s sup]pos$ on ! same /raight l9e4 ,two projective po9t-r[s sup]pos$ on ! same /raight l9e are1 h[ev]1 n necess>ily 9 involution1 z a simple example w %[4 ,take two l9es1 a & a'1 :i* bo? revolve ab\t a fix$ po9t ,s & :i* always make ! same angle ) ea* o!r 7,fig4 #ce74 ,^! l9es cut out on any l9e 9 ! plane :i* does n pass ?r\gh ,s two projective po9t-r[s1 :i* are n1 h[ev]1 9 involution unless ! angle betwe5 ! l9es is a "r angles4 ,= a po9t ,p may correspond to a po9t ,p'1 :i* 9 turn w correspond to "s o!r po9t ?an ,p4 ,! peculi>ity ( po9t-r[s 9 involution is t any po9t w correspond to ! same po9t1 9 :i*ev] po9t-r[ x is consid]$ z belong+4 ,9 ?is case1 if a po9t ,p corresponds to a po9t ,p'1 !n ! po9t ,p' corresponds back aga9 to ! po9t ,p4 ,! po9ts ,p & ,p' are !n said to correspond d\bly4 ,?is notion is wor?y ( fur!r /udy4 ,figure #cf ,,fig4 #cf 99#abi4 ,/e9]'s con/ruction499 ,x w be obs]v$ t ! solution ( ! fundam5tal problem giv5 9 #hc1 ,giv5 ?ree pairs ( po9ts ( two protective po9t-r[s1 to con/ruct o!r pairs1 _c be c>ri$ out if ! two po9t-r[s lie on ! same /raight l9e4 ,( c\rse ! me?od may be easily alt]$ to cov] t case also1 b x is wor? :ile to give ano!r solution ( ! problem1 due to ,/e9]1 :i* w also give fur!r in=mation reg>d+ ! !ory ( involution1 & :i* may1 inde$1 be us$ z a f\ndation = t !ory4 ,let ! two po9t-r[s ,a1 ,b1 ,c1 ,d1 444 & ,a'1 ,b'1 ,c'1 ,d'1 444 be sup]pos$ on ! l9e u4 ,project !m bo? to a po9t ,s & pass any conic ?r\gh ,s4 ,we ?us obta9 two projective p5cils1 a1 b1 c1 d1 444 & a'1 b'1 c'1 d'1 444 at ,s1 :i* meet ! conic 9 ! po9ts 1 1 1 1 444 & '1 '1 '1 '1 444 7,fig4 #cf74 ,take n[ z ! c5t] ( a p5cil project+ ! po9ts '1 '1 '1 4441 & take ' z ! c5t] ( a p5cil project+ ! po9ts 1 1 1 4444 ,^! two p5cils are projective to ea* o!r1 & s9ce !y h a self-correspond9 ray 9 common1 !y are 9 p]spective position & correspond+ rays meet on ! l9e jo9+ 7'1 '7 to 7'1 '74 ,! correspond5ce betwe5 po9ts 9 ! two po9t-r[s on u is n[ easily trac$4 99#acj4 ,application ( ,/e9]'s con/ruction to d\ble correspond5ce499 ,/e9]'s con/ruction ?r[s into our h&s an important !orem conc]n+ d\ble correspond5ce3 ,if two projective po9t-r[s1 sup]pos$ on ! same l9e1 h "o pair ( po9ts :i* correspond to ea* o!r d\bly1 !n all pairs correspond to ea* o!r d\bly1 & ! l9e is pair$ 9 involution4 ,to make ?is appe>1 let u call ! po9t ,a on u by two names1 ,a & ,p'1 accord+ z x is ?\ght ( z belong+ to ! "o or to ! o!r ( ! two po9t-r[s4 ,if ?is po9t is "o ( a pair :i* correspond to ea* o!r d\bly1 !n ! po9ts ,a' & ,p mu/ co9cide 7,fig4 #cg74 ,take n[ any po9t ,c1 :i* we w also call ,r'4 ,we mu/ %[ t ! correspond+ po9t ,c' mu/ also co9cide ) ! po9t ,b4 ,jo9 all ! po9ts to ,s1 z be=e1 & x appe>s t ! po9ts & ' co9cide1 z also d ! po9ts ' & '4 ,by ! above con/ruction ! l9e ' mu/ meet ' on ! l9e jo9+ 7'1 '7 ) 7'1 '74 ,b ^! f\r po9ts =m a quadrangle inscrib$ 9 ! conic1 & we "k by #ie t ! tang5ts at ! opposite v]tices & ' meet on ! l9e v4 ,! l9e ' is ?us a tang5t to ! conic1 & ,c' & ,r are ! same po9t4 ,t two projective po9t-r[s sup]pos$ on ! same l9e are also 9 involution :5 "o pair1 & !re=e all pairs1 correspond d\bly may be %[n by tak+ ,s at "o v]tex ( a complete quadrangle :i* has two pairs ( opposite sides go+ ?r\gh two pairs ( po9ts4 ,! details we leave to ! /ud5t4 ,figure #cg ,,fig4 #cg ,figure #ch ,,fig4 #ch 99#aca4 ,involution ( po9ts on a po9t-r[ ( ! second ord]499 ,x is important to note also1 9 ,/e9]'s con/ruction1 t we h obta9$ two po9t-r[s ( ! second ord] sup]pos$ on ! same conic1 & h pair$ ! po9ts ( "o ) ! po9ts ( ! o!r 9 su* a way t ! correspond5ce is d\ble4 ,we may !n ext5d ! notion ( involution to po9t-r[s ( ! second ord] & say t ! po9ts ( a conic are pair$ 9 involution :5 !y are correspond+ po9ts ( two projective po9t-r[s sup]pos$ on ! conic1 & :5 !y correspond to ea* o!r d\bly4 ,) ?is def9ition we may prove ! !orem3 ,! l9es jo9+ correspond+ po9ts ( a po9t-r[ ( ! second ord] 9 involution all pass ?r\gh a fix$ po9t ,u1 & ! l9e jo9+ any two po9ts ,a1 ,b meets ! l9e jo9+ ! two correspond+ po9ts ,a'1 ,b' 9 ! po9ts ( a l9e u1 :i* is ! pol> ( ,u ) respect to ! conic4 ,= take ,a & ,a' z ! c5t]s ( two p5cils1 ! fir/ p]spective to ! po9t-r[ ,a'1 ,b'1 ,c' & ! second p]spective to ! po9t-r[ ,a1 ,b1 ,c4 ,!n1 s9ce ! common ray ( ! two p5cils corresponds to itself1 !y are 9 p]spective position1 & _! axis ( p]spectivity u 7,fig4 #ch7 is ! l9e :i* jo9s ! po9t 7,,ab'1 ,a',b7 to ! po9t 7,,ac'1 ,a',c74 ,x is !n imm$iately cle>1 f ! !ory ( poles & pol>s1 t ,,bb' & ,,cc' pass ?r\gh ! pole ,u ( ! l9e u4 99#acb4 ,involution ( rays499 ,! :ole !ory ?us f> develop$ may be dualiz$1 & a !ory ( l9es 9 involution may be built up1 />t+ ) ! complete quadrilat]al4 ,?us1 ,! ?ree pairs ( rays :i* may be drawn f a po9t ?r\gh ! ?ree pairs ( opposite v]tices ( a complete quadrilat]al are said to be 9 involution4 ,if ! pairs aa' & bb' are fix$1 & ! l9e c describes a p5cil1 ! correspond+ l9e c' also describes a p5cil1 & ! rays ( ! p5cil are said to be pair$ 9 ! involution det]m9$ by aa' & bb'4 99#acc4 ,d\ble rays499 ,! self-correspond+ rays1 ( :i* !re are two or none1 are call$ d\ble rays ( ! involution4 ,correspond+ rays ( ! involution are h>monic conjugates ) respect to ! d\ble rays4 ,to ! !orem ( ,des>gues 7 #abe7 :i* has to d ) ! sy/em ( conics ?r\gh f\r po9ts we h ! dual3 ,! tang5ts f a fix$ po9t to a sy/em ( conics tang5t to f\r fix$ l9es =m a p5cil ( rays 9 involution4 99#acd499 ,if a conic ( ! sy/em %\ld g ?r\gh ! fix$ po9t1 x is cle> t ! two tang5ts w\ld co9cide & indicate a d\ble ray ( ! involution4 ,! !orem1 !re=e1 foll[s3 ,two conics or none may be drawn ?r\gh a fix$ po9t to be tang5t to f\r fix$ l9es4 99#ace4 ,d\ble correspond5ce499 ,x fur!r appe>s t two projective p5cils ( rays :i* h ! same c5t] are 9 involution if two pairs ( rays correspond to ea* o!r d\bly4 ,f ?is x is cle> t we might h deem$ six rays 9 involution z six rays :i* pass ?r\gh a po9t & also ?r\gh six po9ts 9 involution4 ,:ile ?is w\ld h be5 entirely 9 accord ) ! treatm5t :i* 0 giv5 ! correspond+ problem 9 ! !ory ( h>monic po9ts & l9es1 x is m satisfactory1 f an ae/hetic po9t ( view1 to build ! !ory ( l9es 9 involution on its own base4 ,! /ud5t c %[1 by me?ods entirely analog\s to ^? us$ 9 ! second *apt]1 t involution is a projective prop]ty2 t is1 six rays 9 involution are cut by any transv]sal 9 six po9ts 9 involution4 99#acf4 ,p5cils ( rays ( ! second ord] 9 involution499 ,we may also ext5d ! notion ( involution to p5cils ( rays ( ! second ord]4 ,?us1 ! tang5ts to a conic are 9 involution :5 !y are correspond+ rays ( two protective p5cils ( ! second ord] sup]pos$ ^u ! same conic1 & :5 !y correspond to ea* o!r d\bly4 ,we h !n ! !orem3 99#acg499 ,! int]sections ( correspond+ rays ( a p5cil ( ! second ord] 9 involution are all on a /raight l9e u1 & ! int]section ( any two tang5ts ab1 :5 jo9$ to ! int]section ( ! correspond+ tang5ts a'b'1 gives a l9e :i* passes ?r\gh a fix$ po9t ,u1 ! pole ( ! l9e u ) respect to ! conic4 99#ach4 ,involution ( rays det]m9$ by a conic499 ,we h se5 9 ! !ory ( poles & pol>s 7 #ajc7 t if a po9t ,p moves along a l9e m1 !n ! pol> ( ,p revolves ab\t a po9t4 ,?is p5cil cuts out on m ano!r po9t-r[ ,p'1 projective also to ,p4 ,s9ce ! pol> ( ,p passes ?r\gh ,p'1 ! pol> ( ,p' also passes ?r\gh ,p1 s t ! correspond5ce betwe5 ,p & ,p' is d\ble4 ,! two po9t-r[s are !re=e 9 involution1 & ! d\ble po9ts1 if any exi/1 are ! po9ts :]e ! l9e m meets ! conic4 ,a simil> involution ( rays may be f\nd at any po9t 9 ! plane1 correspond+ rays pass+ ea* ?r\gh ! pole ( ! o!r4 ,we h call$ su* po9ts & rays conjugate ) respect to ! conic 7 #ajj74 ,we may !n /ate ! foll[+ important !orem3 99#aci499 ,a conic det]m9es on e l9e 9 its plane an involution ( po9ts1 correspond+ po9ts 9 ! involution be+ conjugate ) respect to ! conic4 ,! d\ble po9ts1 if any exi/1 are ! po9ts :]e ! l9e meets ! conic4 99#adj499 ,! dual !orem reads3 ,a conic det]m9es at e po9t 9 ! plane an involution ( rays1 correspond+ rays be+ conjugate ) respect to ! conic4 ,! d\ble rays1 if any exi/1 are ! tang5ts f ! po9t to ! conic4 ,,problems #a4 ,two l9es are drawn ?r\gh a po9t on a conic s z always to make "r angles ) ea* o!r4 ,%[ t ! l9es jo9+ ! po9ts :]e !y meet ! conic aga9 all pass ?r\gh a fix$ po9t4 #b4 ,two l9es are drawn ?r\gh a fix$ po9t on a conic s z always to make equal angles ) ! tang5t at t po9t4 ,%[ t ! l9es jo9+ ! two po9ts :]e ! l9es meet ! conic aga9 all pass ?r\gh a fix$ po9t4 #c4 ,f\r l9es divide ! plane into a c]ta9 numb] ( regions4 ,det]m9e = ea* region :e!r two conics or none may be drawn to pass ?r\gh po9ts ( x & also to be tang5t to ! f\r l9es4 #d4 ,if a v>iable quadrangle move 9 su* a way z always to rema9 inscrib$ 9 a fix$ conic1 :ile ?ree ( its sides turn ea* ar\nd "o ( ?ree fix$ coll9e> po9ts1 !n ! f\r? w also turn ar\nd a f\r? fix$ po9t coll9e> ) ! o!r ?ree4 #e4 ,/ate & prove ! dual ( problem #d4 #f4 ,ext5d problem #d z foll[s3 ,if a v>iable polygon ( an ev5 numb] ( sides move 9 su* a way z always to rema9 inscrib$ 9 a fix$ conic1 :ile all its sides b "o pass ?r\gh z _m fix$ coll9e> po9ts1 !n ! la/ side w also pass ?r\gh a fix$ po9t coll9e> ) ! o!rs4 #g4 ,if a triangle ,,qrs be inscrib$ 9 a conic1 & if a transv]sal s meet two ( its sides 9 ,a & ,a'1 ! ?ird side & ! tang5t at ! opposite v]tex 9 ,b & ,b'1 & ! conic itself 9 ,c & ,c'1 !n ,,aa'1 ,,bb'1 ,,cc' are ?ree pairs ( po9ts 9 an involution4 #h4 ,use ! la/ ex]cise to solve ! problem3 ,giv5 five po9ts1 ,q1 ,r1 ,s1 ,c1 ,c'1 on a conic1 to draw ! tang5t at any "o ( !m4 #i4 ,/ate & prove ! dual ( problem #g & use x to prove ! dual ( problem #h4 #aj4 ,if a transv]sal cut two tang5ts to a conic 9 ,b & ,b'1 _! *ord ( contact 9 ,a1 & ! conic itself 9 ,p & ,p'1 !n ! po9t ,a is a d\ble po9t ( ! involution det]m9$ by ,,bb' & ,,pp'4 #aa4 ,/ate & prove ! dual ( problem #aj4 #ab4 ,if a v>iable conic pass ?r\gh two giv5 po9ts1 ,p & ,p'1 & if x be tang5t to two giv5 l9es1 ! *ord ( contact ( ^! two tang5ts w always pass ?r\gh a fix$ po9t on ,,pp'4 #ac4 ,use ! la/ !orem to solve ! problem3 ,giv5 f\r po9ts1 ,p1 ,p'1 ,q1 ,s1 on a conic1 & ! tang5t at "o ( !m1 ,q1 to draw ! tang5t at any "o ( ! o!r po9ts1 ,s4 #ad4 ,apply ! !orem ( problem #i to ! case ( a hyp]bola :]e ! two tang5ts are ! asymptotes4 ,%[ 9 ?is way t if a hyp]bola & its asymptotes be cut by a transv]sal1 ! segm5ts int]cept$ by ! curve & by ! asymptotes respectively h ! same middle po9t4 #ae4 ,9 a triangle circumscrib$ ab\t a conic1 any side is divid$ h>monically by its po9t ( contact & ! po9t :]e x meets ! *ord jo9+ ! po9ts ( contact ( ! o!r two sides4 ,,*apt] ,,ix - ,,metrical ,,prop]ties ,,( ,,involutions ,figure #ci ,,fig4 #ci 99#ada4 ,introduction ( inf9ite po9t2 c5t] ( involution499 ,we connect ! projective !ory ( involution ) ! metrical1 z usual1 by ! introduction ( ! elem5ts at inf9ity4 ,9 an involution ( po9ts on a l9e ! po9t :i* corresponds to ! inf9itely di/ant po9t is call$ ! c5t] ( ! involution4 ,s9ce correspond+ po9ts 9 ! involution h be5 %[n to be h>monic conjugates ) respect to ! d\ble po9ts1 ! c5t] is midway betwe5 ! d\ble po9ts :5 !y exi/4 ,to con/ruct ! c5t] 7,fig4 #ci7 we draw z usual ?r\gh ,a & ,a' any two rays & cut !m by a l9e p>allel to ,,aa' 9 ! po9ts ,k & ,m4 ,jo9 ^! po9ts to ,b & ,b'1 ?us det]m9+ on ,,ak & ,,an ! po9ts ,l & ,n4 ,,ln meets ,,aa' 9 ! c5t] ,o ( ! involution4 99#adb4 ,fundam5tal metrical !orem499 ,f ! figure we see t ! triangles ,,olb' & ,,plm are simil>1 ,p be+ ! int]section ( ,,km & ,,ln4 ,also ! triangles ,,kpn & ,,bon are simil>4 ,we ?us h ,,ob 3 ,,pk ,,on 3 ,,pn & ,,ob' 3 ,,pm ,,ol 3 ,,pl2 :5ce ,,ob ,,ob' 3 ,,pk ,,pm ,,on ,,ol 3 ,,pn ,,pl4 ,9 ! same way1 f ! simil> triangles ,,oal & ,,pkl1 & also ,,oa',n & ,,pmn1 we obta9 ,,oa ,,oa' 3 ,,pk ,,pm ,,on ,,ol 3 ,,pn ,,pl1 & ?is1 ) ! prec$+1 gives at once ! fundam5tal !orem1 :i* is sometimes tak5 also z ! def9ition ( involution3 ,,oa ,,oa' ,,ob ,,ob' con/ant1 or1 9 words1 ,! product ( ! di/ances f ! c5t] to two correspond+ po9ts 9 an involution ( po9ts is con/ant4 99#adc4 ,exi/5ce ( d\ble po9ts499 ,cle>ly1 accord+ z ! con/ant is positive or negative ! involution w or w n h d\ble po9ts4 ,! con/ant is ! squ>e root ( ! di/ance f ! c5t] to ! d\ble po9ts4 ,if ,a & ,a' lie bo? on ! same side ( ! c5t]1 ! product ,,oa ,,oa' is positive2 & if !y lie on opposite sides1 x is negative4 ,take ! case :]e !y bo? lie on ! same side ( ! c5t]1 & take also ! pair ( correspond+ po9ts ,,bb'4 ,!n1 s9ce ,,oa ,,oa' ,,ob ,,ob'1 x _c happ5 t ,b & ,b' are sep>at$ f ea* o!r by ,a & ,a'4 ,?is is evid5t en\gh if ! po9ts are on opposite sides ( ! c5t]4 ,if ! pairs are on ! same side ( ! c5t]1 & ,b lies betwe5 ,a & ,a'1 s t ,,ob is great]1 say1 ?an ,,oa1 b less ?an ,,oa'1 !n1 by ! equation ,,oa ,,oa' ,,ob ,,ob'1 we mu/ h ,,ob' also less ?an ,,oa' & great] ?an ,,oa4 ,a simil> discussion may be made = ! case :]e ,a & ,a' lie on opposite sides ( ,o4 ,! results may be /at$ z foll[s1 )\t any ref]5ce to ! c5t]3 ,giv5 two pairs ( po9ts 9 an involution ( po9ts1 if ! po9ts ( "o pair are sep>at$ f ea* o!r by ! po9ts ( ! o!r pair1 !n ! involution has no d\ble po9ts4 ,if ! po9ts ( "o pair are n sep>at$ f ea* o!r by ! po9ts ( ! o!r pair1 !n ! involution has two d\ble po9ts4 99#add499 ,an entirely simil> crit]ion decides :e!r an involution ( rays has or has n d\ble rays1 or :e!r an involution ( planes has or has n d\ble planes4 ,figure #dj ,,fig4 #dj 99#ade4 ,con/ruction ( an involution by means ( circles499 ,! equation j d]iv$1 ,,oa ,,oa' ,,ob ,,ob'1 indicates ano!r simple way 9 :i* po9ts ( an involution ( po9ts may be con/ruct$4 ,?r\gh ,a & ,a' draw any circle1 & draw also any circle ?r\gh ,b & ,b' to cut ! fir/ 9 ! two po9ts ,g & ,g' 7,fig4 #dj74 ,!n any circle ?r\gh ,g & ,g' w meet ! l9e 9 pairs ( po9ts 9 ! involution det]m9$ by ,,aa' & ,,bb'4 ,= if su* a circle meets ! l9e 9 ! po9ts ,,cc'1 !n1 by ! !orem 9 ! geometry ( ! circle :i* says t if any *ord is drawn ?r\gh a fix$ po9t )9 a circle1 ! product ( its segm5ts is con/ant 9 :atev] direction ! *ord is drawn1 & if a secant l9e be drawn f a fix$ po9t )\t a circle1 ! product ( ! secant & its ext]nal segm5t is con/ant 9 :atev] direction ! secant l9e is drawn1 we h ,,oc ,,oc' ,,og ,,og' con/ant4 ,s t = all su* po9ts ,,oa ,,oa' ,,ob ,,ob' ,,oc ,,oc'4 ,fur!r1 ! l9e ,,gg' meets ,,aa' 9 ! c5t] ( ! involution4 ,to f9d ! d\ble po9ts1 if !y exi/1 we draw a tang5t f ,o to any ( ! circles ?r\gh ,,gg'4 ,let ,t be ! po9t ( contact4 ,!n lay (f on ! l9e ,,oa a l9e ,,( equal to ,,ot4 ,!n1 s9ce by ! above !orem ( elem5t>y geometry ,,oa ,,oa' ,,ot#b ,,(#b1 we h "o d\ble po9t ,f4 ,! o!r is at an equal di/ance on ! o!r side ( ,o4 ,?is simple & effective me?od ( con/ruct+ an involution ( po9ts is (t5 tak5 z ! basis = ! !ory ( involution4 ,9 projective geometry1 h[ev]1 ! circle1 :i* is n a figure t rema9s unalt]$ by projection1 & is ess5tially a metrical notion1 ought n to be us$ to build up ! purely projective "p ( ! !ory4 99#adf499 ,x ought to be m5tion$ t ! !ory ( analytic geometry indicates t ! circle is a special conic section t happ5s to pass ?r\gh two p>ticul> imag9>y po9ts on ! l9e at inf9ity1 call$ ! circul> po9ts & usually d5ot$ by ,i & ,j4 ,! above me?od ( obta9+ a po9t-r[ 9 involution is1 !n1 no?+ b a special case ( ! g5]al !orem ( ! la/ *apt] 7 #abe71 :i* ass]t$ t a sy/em ( conics ?r\gh f\r po9ts w cut any l9e 9 ! plane 9 a po9t-r[ 9 involution4 ,figure #da ,,fig4 #da 99#adg4 ,pairs 9 an involution ( rays :i* are at "r angles4 ,circul> involution499 ,9 an involution ( rays !re is no "o ray :i* may be di/+ui%$ f all ! o!rs z ! po9t at inf9ity is di/+ui%$ f all o!r po9ts on a l9e4 ,!re is "o pair ( rays1 h[ev]1 :i* does diff] f all ! o!rs 9 t = ?is p>ticul> pair ! angle is a "r angle4 ,?is is mo/ easily %[n by us+ ! con/ruction t employs circles1 z indicat$ above4 ,! c5t]s ( all ! circles ?r\gh ,g & ,g' lie on ! p]p5dicul> bisector ( ! l9e ,,gg'4 ,let ?is l9e meet ! l9e ,,aa' 9 ! po9t ,c 7,fig4 #da71 & draw ! circle ) c5t] ,c :i* goes ?r\gh ,g & ,g'4 ,?is circle cuts out two po9ts ,m & ,m' 9 ! involution4 ,! rays ,,gm & ,,gm' are cle>ly at "r angles1 be+ inscrib$ 9 a semicircle4 ,if1 !re=e1 ! involution ( po9ts is project$ to ,g1 we h f\nd two correspond+ rays :i* are at "r angles to ea* o!r4 ,giv5 n[ any involution ( rays ) c5t] ,g1 we may cut across x by a /raight l9e & proce$ to f9d ! two po9ts ,m & ,m'4 ,cle>ly !re w be only "o su* pair unless ! p]p5dicul> bisector ( ,,gg' co9cides ) ! l9e ,,aa'4 ,9 ?is case e ray is at "r angles to its correspond+ ray1 & ! involution is call$ circul>4 99#adh4 ,axes ( conics499 ,at ! close ( ! la/ *apt] 7 #adj7 we gave ! !orem3 ,a conic det]m9es at e po9t 9 its plane an involution ( rays1 correspond+ rays be+ conjugate ) respect to ! conic4 ,! d\ble rays1 if any exi/1 are ! tang5ts f ! po9t to ! conic4 ,9 p>ticul>1 tak+ ! po9t z ! c5t] ( ! conic1 we f9d t conjugate diamet]s =m a sy/em ( rays 9 involution1 ( :i* ! asymptotes1 if !re are any1 are ! d\ble rays4 ,also1 conjugate diamet]s are h>monic conjugates ) respect to ! asymptotes4 ,by ! !orem ( ! la/ p>agraph1 !re are two conjugate diamet]s :i* are at "r angles to ea* o!r4 ,^! are call$ axes4 ,9 ! case ( ! p>abola1 :]e ! c5t] is at inf9ity1 & on ! curve1 !re are1 prop]ly speak+1 no conjugate diamet]s4 ,:ile ! l9e at inf9ity might be consid]$ z conjugate to all ! o!r diamet]s1 x is n possible to assign to x any p>ticul> direction1 & s x _c be us$ = ! purpose ( def9+ an axis ( a p>abola4 ,!re is "o diamet]1 h[ev]1 :i* is at "r angles to its conjugate sy/em ( *ords1 & ?is "o is call$ ! axis ( ! p>abola4 ,! circle also furni%es an exception 9 t e diamet] is an axis4 ,! involution 9 ?is case is circul>1 e ray be+ at "r angles to its conjugate ray at ! c5t]4 99#adi4 ,po9ts at :i* ! involution det]m9$ by a conic is circul>499 ,x is an important problem to discov] :e!r = any conic o!r ?an ! circle x is possible to f9d any po9t 9 ! plane :]e ! involution det]m9$ z above by ! conic is circul>4 ,we %all proce$ to ! curi\s problem ( prov+ ! exi/5ce ( su* po9ts & ( det]m9+ _! numb] & situation4 ,we %all !n develop ! important prop]ties ( su* po9ts4 99#aej499 ,x is cle>1 9 ! fir/ place1 t su* a po9t _c be on ! outside ( ! conic1 else ! involution w\ld h d\ble rays & su* rays w\ld h to be at "r angles to !mselves4 ,9 ! second place1 if two su* po9ts exi/1 ! l9e jo9+ !m mu/ be a diamet] &1 inde$1 an axis4 ,= if ,f & ,f' 7 two su* po9ts1 !n1 s9ce ! conjugate ray at ,f to ! l9e ,,ff' mu/ be at "r angles to x1 & also s9ce ! conjugate ray at ,f' to ! l9e ,,ff' mu/ be at "r angles to x1 ! pole ( ,,ff' mu/ be at inf9ity 9 a direction at "r angles to ,,ff'4 ,! l9e ,,ff' is !n a diamet]1 & s9ce x is at "r angles to its conjugate diamet]1 x mu/ be an axis4 ,f ?is x foll[s also t ! po9ts we are seek+ mu/ all lie on "o ( ! two axes1 else we %\ld h a diamet] :i* does n g ?r\gh ! int]section ( all axes--! c5t] ( ! conic4 ,at lea/ "o axis1 !re=e1 mu/ be free f any su* po9ts4 ,figure #db ,,fig4 #db 99#aea499 ,let n[ ,p be a po9t on "o ( ! axes 7,fig4 #db71 & draw any ray ?r\gh x1 su* z q4 ,z q revolves ab\t ,p1 its pole ,q moves along a l9e at "r angles to ! axis on :i* ,p lies1 describ+ a po9t-r[ p projective to ! p5cil ( rays q4 ,! po9t at inf9ity 9 a direction at "r angles to q also describes a po9t-r[ projective to q4 ,! l9e jo9+ correspond+ po9ts ( ^! two po9t-r[s is always a conjugate l9e to q & at "r angles to q1 or1 z we may call x1 a conjugate normal to q4 ,^! conjugate normals to q1 jo9+ z !y d correspond+ po9ts 9 two projective po9t-r[s1 =m a p5cil ( rays ( ! second ord]4 ,b s9ce ! po9t at inf9ity on ! po9t-r[ ,q corresponds to ! po9t at inf9ity 9 a direction at "r angles to q1 ^! po9t-r[s are 9 p]spective position & ! normal conjugates ( all ! l9es ?r\gh ,p meet 9 a po9t4 ,?is po9t lies on ! same axis ) ,p1 z is se5 by tak+ q at "r angles to ! axis on :i* ,p lies4 ,! c5t] ( ?is p5cil may be call$ ,p'1 & ?us we h pair$ ! po9t ,p ) ! po9t ,p'4 ,by mov+ ! po9t ,p along ! axis1 & by keep+ ! ray q p>allel to a fix$ direction1 we may see t ! po9t-r[ ,p & ! po9t-r[ ,p' are projective4 ,also ! correspond5ce is d\ble1 & by />t+ f ! po9t ,p' we arrive at ! po9t ,p4 ,!re=e ! po9t-r[s ,p & ,p' are 9 involution1 & if only ! involution has d\ble po9ts1 we %all h f\nd 9 !m ! po9ts we are seek+4 ,= x is cle> t ! rays ?r\gh ,p & ! correspond+ rays ?r\gh ,p' are conjugate normals2 & if ,p & ,p' co9cide1 we %all h a po9t :]e all rays are at "r angles to _! conjugates4 ,we %all n[ %[ t ! involution ?us obta9$ on "o ( ! two axes mu/ h d\ble po9ts4 ,figure #dc ,,fig4 #dc 99#aeb4 ,discov]y ( ! foci ( ! conic499 ,we "k t on "o axis no su* po9ts z we are seek+ c lie 7 #aej74 ,! involution ( po9ts ,,pp' on ?is axis c !re=e h no d\ble po9ts4 ,nev]!less1 let ,,pp' & ,,rr' be two pairs ( correspond+ po9ts on ?is axis 7,fig4 #dc74 ,!n we "k t ,p & ,p' are sep>at$ f ea* o!r by ,r & ,r' 7 #adc74 ,draw a circle on ,,pp' z a diamet]1 & "o on ,,rr' z a diamet]4 ,^! mu/ int]sect 9 two po9ts1 ,f & ,f'1 & s9ce ! c5t] ( ! conic is ! c5t] ( ! involution ,,pp'1 ,,rr'1 z is easily se51 x foll[s t ,f & ,f' are on ! o!r axis ( ! conic4 ,moreov]1 ,,fr & ,,fr' are conjugate normal rays1 s9ce ,,rfr' is inscrib$ 9 a semicircle1 & ! two rays g "o ?r\gh ,r & ! o!r ?r\gh ,r'4 ,! involution ( po9ts ,,pp'1 ,,rr' !re=e projects to ! two po9ts ,f & ,f' 9 two p5cils ( rays 9 involution :i* h = correspond+ rays conjugate normals to ! conic4 ,we may1 !n1 say3 ,!re are two & only two po9ts ( ! plane :]e ! involution det]m9$ by ! conic is circul>4 ,^! two po9ts lie on "o ( ! axes1 at equal di/ances f ! c5t]1 on ! inside ( ! conic4 ,^! po9ts are call$ ! foci ( ! conic4 99#aec4 ,! circle & ! p>abola499 ,! above discussion applies only to ! c5tral conics1 ap>t f ! circle4 ,9 ! circle ! two foci fall toge!r at ! c5t]4 ,9 ! case ( ! p>abola1 t "p ( ! inve/igation :i* proves ! exi/5ce ( two foci on "o ( ! axes w n hold1 z we h b "o axis4 ,x is se51 h[ev]1 t z ,p moves to inf9ity1 c>ry+ ! l9e q ) x1 q becomes ! l9e at inf9ity1 :i* = ! p>abola is a tang5t l9e4 ,its pole ,q is ?us at inf9ity & also ! po9t ,p'1 s t ,p & ,p' fall toge!r at inf9ity1 & !re=e "o focus ( ! p>abola is at inf9ity4 ,!re mu/ !re=e be ano!r1 s t ,a p>abola has "o & only "o focus 9 ! f9ite "p ( ! plane4 ,figure #dd ,,fig4 #dd 99#aed4 ,focal prop]ties ( conics499 ,we proce$ to develop "s !orems :i* w exhibit ! importance ( ^! po9ts 9 ! !ory ( ! conic section4 ,draw a tang5t to ! conic1 & also ! normal at ! po9t ( contact ,p4 ,^! two l9es are cle>ly conjugate normals4 ,! two po9ts ,t & ,n1 !re=e1 :]e !y meet ! axis :i* conta9s ! foci1 are correspond+ po9ts 9 ! involution consid]$ above1 & are !re=e h>monic conjugates ) respect to ! foci 7,fig4 #dd72 & if we jo9 !m to ! po9t ,p1 we %all obta9 f\r h>monic l9es4 ,b two ( !m are at "r angles to ea* o!r1 & s ! o!rs make equal angles ) !m 7,problem #d1 ,*apt] ,,ii74 ,!re=e ,! l9es jo9+ a po9t on ! conic to ! foci make equal angles ) ! tang5t4 ,x foll[s t rays f a s\rce ( light at "o focus are reflect$ by an ellipse to ! o!r4 99#aee499 ,9 ! case ( ! p>abola1 :]e "o ( ! foci mu/ be consid]$ to be at inf9ity 9 ! direction ( ! diamet]1 we h ,figure #de ,,fig4 #de ,a diamet] makes ! same angle ) ! tang5t at its extremity z t tang5t does ) ! l9e f its po9t ( contact to ! focus 7,fig4 #de74 99#aef499 ,?is la/ !orem is ! basis = ! con/ruction ( ! p>abolic reflector4 ,a ray ( light f ! focus is reflect$ f su* a reflector 9 a direction p>allel to ! axis ( ! reflector4 99#aeg4 ,directrix4 ,pr9cipal axis4 ,v]tex499 ,! pol> ( ! focus ) respect to ! conic is call$ ! directrix4 ,! axis :i* conta9s ! foci is call$ ! pr9cipal axis1 & ! int]section ( ! axis ) ! curve is call$ ! v]tex ( ! curve4 ,! directrix is at "r angles to ! pr9cipal axis4 ,9 a p>abola ! v]tex is equally di/ant f ! focus & ! directrix1 ^! ?ree po9ts & ! po9t at inf9ity on ! axis be+ f\r h>monic po9ts4 ,9 ! ellipse ! v]tex is ne>] to ! focus ?an x is to ! directrix1 = ! same reason1 & 9 ! hyp]bola x is f>!r f ! focus ?an x is f ! directrix4 ,figure #df ,,fig4 #df 99#aeh4 ,ano!r def9ition ( a conic499 ,let ,p be any po9t on ! directrix ?r\gh :i* a l9e is drawn meet+ ! conic 9 ! po9ts ,a & ,b 7,fig4 #df74 ,let ! tang5ts at ,a & ,b meet 9 ,t1 & call ! focus ,f4 ,!n ,,tf & ,,pf are conjugate l9es1 & z !y pass ?r\gh a focus !y mu/ be at "r angles to ea* o!r4 ,let ,,tf meet ,,ab 9 ,c4 ,!n ,p1 ,a1 ,c1 ,b are f\r h>monic po9ts4 ,project ^! f\r po9ts p>allel to ,,tf ^u ! directrix1 & we !n get ! f\r h>monic po9ts ,p1 ,m1 ,q1 ,n4 ,s9ce1 n[1 ,,tfp is a "r angle1 ! angles ,,mfq & ,,nfq are equal1 z well z ! angles ,,afc & ,,bfc4 ,!re=e ! triangles ,,maf & ,,nfb are simil>1 & ,,fa 3 ,,fm ,,fb 3 ,,bn4 ,dropp+ p]p5dicul>s ,,aa & ,,bb' ^u ! directrix1 ?is becomes ,,fa 3 ,,aa' ,,fb 3 ,,bb'4 ,we h ?us ! prop]ty (t5 tak5 z ! def9ition ( a conic3 ,! ratio ( ! di/ances f a po9t on ! conic to ! focus & ! directrix is con/ant4 ,figure #dg ,,fig4 #dg 99#aei4 ,ecc5tricity499 ,by tak+ ! po9t at ! v]tex ( ! conic1 we note t ?is ratio is less ?an unity = ! ellipse1 great] ?an unity = ! hyp]bola1 & equal to unity = ! p>abola4 ,?is ratio is call$ ! ecc5tricity4 ,figure #dh ,,fig4 #dh 99#afj4 ,sum or diff]5ce ( focal di/ances499 ,! ellipse & ! hyp]bola h two foci & two directrices4 ,! ecc5tricity1 ( c\rse1 is ! same = "o focus z = ! o!r1 s9ce ! curve is symmetrical ) respect to bo?4 ,if ! di/ances f a po9t on a conic to ! two foci are r & r'1 & ! di/ances f ! same po9t to ! correspond+ directrices are d & d' 7,fig4 #dg71 we h r 3 d r' 3 d'2 7r r'7 3 7d d'74 ,9 ! ellipse 7d d'7 is con/ant1 be+ ! di/ance betwe5 ! directrices4 ,9 ! hyp]bola ?is di/ance is 7d - d'74 ,x foll[s 7,fig4 #dh7 t ,9 ! ellipse ! sum ( ! focal di/ances ( any po9t on ! curve is con/ant1 & 9 ! hyp]bola ! diff]5ce betwe5 ! focal di/ances is con/ant4 ,,problems #a4 ,con/ruct ! axis ( a p>abola1 giv5 f\r tang5ts4 #b4 ,giv5 two conjugate l9es at "r angles to ea* o!r1 & let !m meet ! axis :i* has no foci on x 9 ! po9ts ,a & ,b4 ,! circle on ,,ab z diamet] w pass ?r\gh ! foci ( ! conic4 #c4 ,giv5 ! axes ( a conic 9 position1 & also a tang5t ) its po9t ( contact1 to con/ruct ! foci & det]m9e ! l5g? ( ! axes4 #d4 ,giv5 ! tang5t at ! v]tex ( a p>abola1 & two o!r tang5ts1 to f9d ! focus4 #e4 ,! locus ( ! c5t] ( a circle t\*+ two giv5 circles is a conic ) ! c5t]s ( ! giv5 circles = its foci4 #f4 ,giv5 ! axis ( a p>abola & a tang5t1 ) its po9t ( contact1 to f9d ! focus4 #g4 ,! locus ( ! c5t] ( a circle :i* t\*es a giv5 l9e & a giv5 circle consi/s ( two p>abolas4 #h4 ,let ,f & ,f' be ! foci ( an ellipse1 & ,p any po9t on x4 ,produce ,,pf to ,g1 mak+ ,,pg equal to ,,pf'4 ,f9d ! locus ( ,g4 #i4 ,if ! po9ts ,g ( a circle be fold$ ov] ^u a po9t ,f1 ! creases w all be tang5t to a conic4 ,if ,f is )9 ! circle1 ! conic w be an ellipse2 if ,f is )\t ! circle1 ! conic w be a hyp]bola4 #aj4 ,if ! po9ts ,g 9 ! la/ example be tak5 on a /raight l9e1 ! locus is a p>abola4 #aa4 ,f9d ! foci & ! l5g? ( ! pr9cipal axis ( ! conics 9 problems #i & #aj4 #ab4 ,9 problem #aj a correspond5ce is set up betwe5 /raight l9es & p>abolas4 ,z !re is a f\rfold inf9ity ( p>abolas 9 ! plane1 & only a tw(old inf9ity ( /raight l9es1 !re mu/ be "s re/riction on ! p>abolas obta9$ by ?is me?od4 ,f9d & expla9 ?is re/riction4 #ac4 ,/ate & expla9 ! simil> problem = problem #i4 #ad4 ,! la/ f\r problems are a /udy ( ! consequ5ces ( ! foll[+ trans=mation3 ,a po9t ,o is fix$ 9 ! plane4 ,!n to any po9t ,p is made to correspond ! l9e p at "r angles to ,,op & bisect+ x4 ,9 ?is correspond5ce1 :at happ5s to p :5 ,p moves along a /raight l9e8 ,:at corresponds to ! !orem t two l9es h only "o po9t 9 common8 ,:at to ! !orem t ! angle sum ( a triangle is two "r angles8 ,etc4 ,,*apt] ,x - ,,on ,,! ,,hi/ory ,,( ,,syn!tic ,,projective ,,geometry 99#afa4 ,anci5t results499 ,! !ory ( syn!tic projective geometry z we h built x up 9 ?is c\rse is less ?an a c5tury old4 ,?is is n to say t _m ( ! !orems & pr9ciples involv$ 7 n discov]$ mu* e>li]1 b isolat$ !orems d n make a !ory1 any m ?an a pile ( bricks makes a build+4 ,! mat]ials = our build+ h be5 contribut$ by _m diff]5t workm5 f ! days ( ,euclid d[n to ! pres5t "t4 ,?us1 ! notion ( f\r h>monic po9ts 0 famili> to ! anci5ts1 :o consid]$ x f ! metrical po9t ( view z ! division ( a l9e int]nally & ext]nally 9 ! same ratio7#a7 ! involution ( six po9ts cut out by any transv]sal :i* int]sects ! sides ( a complete quadrilat]al z /udi$ by ,pappus7#b72 b ^! notions 7 n made ! f\ndation = any g5]al !ory4 ,tak5 by !mselves1 !y are ( small consequ5ce2 x is _! relation to o!r !orems & sets ( !orems t gives !m _! importance4 ,! anci5ts 7 d\btless famili> ) ! !orem1 ,two l9es det]m9e a po9t1 & two po9ts det]m9e a l9e1 b !y _h no glimpse ( ! wond]ful law ( duality1 ( :i* ?is !orem is a simple example4 ,! pr9ciple ( projection1 by :i* _m prop]ties ( ! conic sections may be inf]r$ f correspond+ prop]ties ( ! circle :i* =ms ! base ( ! cone f :i* !y are cut--a pr9ciple s natural to mod]n ma!maticians--seems n to h occurr$ to ! ,greeks4 ,! ellipse1 ! hyp]bola1 & ! p>abola 7 to !m entirely diff]5t curves1 to be treat$ sep>ately ) me?ods appropriate to ea*4 ,?us ! focus ( ! ellipse 0 discov]$ "s five hundr$ ye>s be=e ! focus ( ! p>abola6 ,x 0 n till #aebb t ,v]n]7#c7 ( ,nrnb]g und]took to demon/rate ! prop]ties ( ! conic sections by means ( ! circle4 99#afb4 ,unify+ pr9ciples499 ,9 ! e>ly ye>s ( ! sev5te5? c5tury--t wond]ful epo* 9 ! hi/ory ( ! _w :i* produc$ a ,galileo1 a ,kepl]1 a ,ty*o ,brahe1 a ,desc>tes1 a ,des>gues1 a ,pascal1 a ,cavali]i1 a ,wallis1 a ,f]mat1 a ,huyg5s1 a ,bacon1 a ,napi]1 & a goodly array ( less] lights1 to say no?+ ( a ,rembr&t or ( a ,%akespe>e--!re began to appe> c]ta9 unify+ pr9ciples connect+ ! great mass ( mat]ial dug out by ! anci5ts4 ,?us1 9 #afjd ! great a/ronom] ,kepl]7#d7 introduc$ ! notion t p>allel l9es %\ld be consid]$ z meet+ at an inf9ite di/ance1 & t a p>abola is at once ! limit+ case ( an ellipse & ( a hyp]bola4 ,he also attributes to ! p>abola a bl9d focus 7caecus focus7 at inf9ity on ! axis4 99#afc4 ,des>gues499 ,9 #afci ,des>gues17#e7 an ar*itect ( ,lyons1 publi%$ a little treatise on ! conic sections1 9 :i* appe>s ! !orem ^u :i* we h f\nd$ ! !ory ( f\r h>monic po9ts 7 #be74 ,des>gues1 h[ev]1 does n make use ( x = t purpose4 ,f\r h>monic po9ts are = him a special case ( six po9ts 9 involution :5 two ( ! ?ree pairs co9cide giv+ d\ble po9ts4 ,8 developm5t ( ! !ory ( involution is also diff]5t f ! purely geometric "o :i* we h adopt$1 & is bas$ on ! !orem 7 #adb7 t ! product ( ! di/ances ( two conjugate po9ts f ! c5t] is con/ant4 ,he also proves ! projective "* ( an involution ( po9ts by %[+ t :5 six l9es pass ?r\gh a po9t & ?r\gh six po9ts 9 involution1 !n any transv]sal mu/ meet !m 9 six po9ts :i* are also 9 involution4 99#afd4 ,poles & pol>s499 ,9 ?is little treatise is also conta9$ ! !ory ( poles & pol>s4 ,! pol> l9e is call$ a trav]sal47#f7 ,! h>monic prop]ties ( poles & pol>s are giv51 b ,des>gues seems n to h arriv$ at ! metrical prop]ties :i* result :5 ! inf9ite elem5ts ( ! plane are introduc$4 ,?us he says1 ,:5 ! trav]sal is at an inf9ite di/ance1 all is unimag9able4 99#afe4 ,des>gues's !orem conc]n+ conics ?r\gh f\r po9ts499 ,we f9d 9 ?is little book ! beautiful !orem conc]n+ a quadrilat]al inscrib$ 9 a conic section1 :i* is giv5 by 8 "n 9 #ach4 ,! !orem is n giv5 9 t]ms ( a sy/em ( conics ?r\gh f\r po9ts1 = ,des>gues _h no conception ( any su* sy/em4 ,he /ates ! !orem1 9 effect1 z foll[s3 ,giv5 a simple quadrilat]al inscrib$ 9 a conic section1 e transv]sal meets ! conic & ! f\r sides ( ! quadrilat]al 9 six po9ts :i* are 9 involution4 99#aff4 ,ext5sion ( ! !ory ( poles & pol>s to space499 ,z an illu/ration ( 8 rem>kable p[]s ( g5]alization1 we may note t ,des>gues ext5d$ ! notion ( poles & pol>s to space ( ?ree dim5sions = ! sph]e & = c]ta9 o!r surfaces ( ! second degree4 ,?is is a matt] :i* has n be5 t\*$ on 9 ?is book1 b ! notion is n difficult to grasp4 ,if we draw ?r\gh any po9t ,p 9 space a l9e to cut a sph]e 9 two po9ts1 ,a & ,s1 & !n con/ruct ! f\r? h>monic ( ,p ) respect to ,a & ,b1 ! locus ( ?is f\r? h>monic1 = v>i\s l9es ?r\gh ,p1 is a plane call$ ! pol> plane ( ,p ) respect to ! sph]e4 ,) ?is def9ition & !orem "o c easily f9d dual relations betwe5 po9ts & planes 9 space analog\s to ^? betwe5 po9ts & l9es 9 a plane4 ,des>gues closes 8 discussion ( ?is matt] ) ! rem>k1 ,simil> prop]ties may be f\nd = ^? o!r solids :i* are relat$ to ! sph]e 9 ! same way t ! conic section is to ! circle4 ,x %\ld n be inf]r$ f ?is rem>k1 h[ev]1 t he 0 acqua9t$ ) all ! diff]5t v>ieties ( surfaces ( ! second ord]4 ,! anci5ts 7 well acqua9t$ ) ! surfaces obta9$ by revolv+ an ellipse or a p>abola ab\t an axis4 ,ev5 ! hyp]boloid ( two %eets1 obta9$ by revolv+ ! hyp]bola ab\t its major axis1 0 kn[n to !m1 b probably n ! hyp]boloid ( "o %eet1 :i* results f revolv+ a hyp]bola ab\t ! o!r axis4 ,all ! o!r solids ( ! second degree 7 probably unkn[n until _! discov]y by ,eul]47#g7 99#afg499 ,des>gues _h no conception ( ! conic section ( ! locus ( int]section ( correspond+ rays ( two projective p5cils ( rays4 ,he seems to h tri$ to describe ! curve by means ( a pair ( compasses1 mov+ "o leg back & =? along a /raight l9e in/ead ( hold+ x fix$ z 9 draw+ a circle4 ,he does n attempt to def9e ! law ( ! movem5t necess>y to obta9 a conic by ?is means4 99#afh4 ,reception ( ,des>gues's "w499 ,/range to say1 ,des>gues's immortal "w 0 heap$ ) ! mo/ viol5t abuse & held up to ridicule & scorn6 ,incr$ible errors6 ,enorm\s mi/akes & falsities6 ,really x is impossible = anyone :o is famili> ) ! sci5ce conc]n+ :i* he wi%es to retail 8 ?\ghts1 to keep f laugh+6 ,su* 7 ! comm5ts ( review]s & critics4 ,nor 7 8 detractors altoge!r ignorant & un9/ruct$ m54 ,9 spite ( ! devotion ( 8 pupils & 9 spite ( ! admiration & fri5d%ip ( m5 l ,desc>tes1 ,f]mat1 ,m]s5ne1 & ,rob]val1 8 book disappe>$ s completely t two c5turies aft] ! date ( its publication1 :5 ! ,fr5* geomet] ,*asles wrote 8 hi/ory ( geometry1 !re 0 no means ( e/imat+ ! value ( ! "w done by ,des>gues4 ,six ye>s lat]1 h[ev]1 9 #ahde1 ,*asles f\nd a manuscript copy ( ! ,bruillon-project1 made by ,des>gues's pupil1 ,de la ,hire4 99#afi4 ,cons]vatism 9 ,des>gues's "t499 ,x is n necess>y to suppose t ?is effacem5t ( ,des>gues's "w = two c5turies 0 due to ! savage attacks ( 8 critics4 ,all ?is 0 9 accordance ) ! fa%ion ( ! "t1 & no man escap$ bitt] d5unciation :o attempt$ to improve on ! me?ods ( ! anci5ts4 ,^? 7 days :5 m5 refus$ to believe t a heavy body falls at ! same rate z a light] "o1 ev5 :5 ,galileo made !m see x ) _! own eyes at ! foot ( ! t[] ( ,pisa4 ,c\ld !y n turn to ! exact page & l9e ( ,ari/otle :i* decl>$ t ! heavi] body mu/ fall ! fa/]6 ,i h read ,ari/otle's writ+s f end to end1 _m times1 wrote a ,jesuit prov9cial to ! ma!matician & a/ronom]1 ,*ri/oph ,s*e9]1 at ,+ol/adt1 ^: telescope seem$ to reveal c]ta9 my/]i\s spots on ! sun1 & ,i c assure y ,i h n[h]e f\nd any?+ simil> to :at y describe4 ,g1 my son1 & tranquilize y\rself2 be assur$ t :at y take = spots on ! sun are ! faults ( y\r glasses1 or ( y\r eyes4 ,! dead h& ( ,ari/otle b>r$ ! advance 9 e dep>tm5t ( rese>*4 ,physicians w\ld h no?+ to d ) ,h>vey's discov]ies ab\t ! circulation ( ! blood4 ,nature is accus$ ( tol]at+ a vacuum6 exclaim$ a prie/ :5 ,pascal began 8 exp]im5ts on ! ,puy-de-,dome to %[ t ! column ( m]cury 9 a glass tube v>i$ 9 height ) ! pressure ( ! atmosph]e4 99#agj4 ,des>gues's /yle ( writ+499 ,nev]!less1 au?ority c\nt$ = less at ?is "t 9 ,p>is ?an x did 9 ,italy1 & ! trag$y enact$ 9 ,rome :5 ,galileo 0 =c$ to d5y 8 inmo/ convictions at ! bidd+ ( a brutal ,inquisition c\ld n h be5 /ag$ 9 ,france4 ,moreov]1 9 ! little company ( sci5ti/s ( :i* ,des>gues 0 a memb] ! utmo/ lib]ty ( ?\ght & expression 0 ma9ta9$4 ,"o v good reason = ! disappe>ance ( ! "w ( ,des>gues is to be f\nd 9 8 /yle ( writ+4 ,he fail$ to he$ ! v good advice giv5 him 9 a lett] f 8 w>m admir] ,desc>tes47#h7 ,y may h two designs1 bo? v good & v laudable1 b :i* d n require ! same me?od ( proc$ure3 ,! "o is to write = ! le>n$1 & %[ !m "s new prop]ties ( ! conic sections :i* !y d n already "k2 & ! o!r is to write = ! curi\s unle>n$1 & to d x s t ?is matt] :i* until n[ has be5 und]/ood by only a v few1 & :i* is nev]!less v useful = p]spective1 = pa9t+1 ar*itecture1 etc41 %all become common & easy to all :o wi% to /udy !m 9 y\r book4 ,if y h ! fir/ idea1 !n x seems to me t x is necess>y to avoid us+ new t]ms2 = ! le>n$ are already accu/om$ to us+ ^? ( ,apollonius1 & w n readily *ange !m = o!rs1 ?\gh bett]1 & ?us y\rs w s]ve only to r5d] y\r demon/rations m difficult1 & to turn away y\r read]s f y\r book4 ,if y h ! second plan 9 m9d1 x is c]ta9 t y\r t]ms1 :i* are ,fr5*1 & conceiv$ ) _s & grace1 w be bett] receiv$ by p]sons n preoccupi$ ) ^? ( ! anci5ts4444 ,b1 if y h t int5tion1 y %\ld make ( x a great volume2 expla9 x all s fully & s di/9ctly t ^? g5tlem5 :o _c /udy )\t yawn+2 :o _c di/ress _! imag9ations en\gh to grasp a proposition 9 geometry1 nor turn ! leaves ( a book to look at ! lett]s 9 a figure1 %all f9d no?+ 9 y\r disc\rse m difficult to und]/& ?an ! description ( an en*ant$ palace 9 a fairy /ory4 ,! po9t ( ^! rem>ks is app>5t :5 we note t ,des>gues introduc$ "s sev5ty new t]ms 9 8 little book1 ( :i* only "o1 involution1 has surviv$4 ,curi\sly en\gh1 ?is is ! "o t]m s+l$ out = ! %>pe/ criticism & ridicule by 8 review]1 ,de ,beaugr&47#i7 ,t ,desc>tes knew ! "* ( ,des>gues's audi5ce bett] ?an he did is also evid5c$ by ! fact t ,de ,beaugr& exhau/$ 8 pati5ce 9 read+ ! fir/ t5 pages ( ! book4 99#aga4 ,lack ( appreciation ( ,des>gues499 ,des>gues's me?ods1 entirely diff]5t f ! analytic me?ods j !n be+ develop$ by ,desc>tes & ,f]mat1 seem to h be5 little und]/ood4 ,betwe5 y & me1 wrote ,desc>tes7#aj7 to ,m]s5ne1 ,i c h>dly =m an idea ( :at he may h writt5 conc]n+ conics4 ,des>gues seems to h boa/$ t he ow$ no?+ to any man1 & t all 8 results _h come f 8 own m9d4 ,8 favorite pupil1 ,de la ,hire1 did n realize ! extraord9>y simplicity & g5]ality ( 8 "w4 ,x is a rem>kable fact t ! only "o ( all 8 associates to und]/& & appreciate ! me?ods ( ,des>gues %\ld be a lad ( sixte5 ye>s6 99#agb4 ,pascal & 8 !orem499 ,"o does n h to believe all ! m>vel\s /ories ( ,pascal's admir+ si/]s to cr$it him ) wond]ful precocity4 ,we h ! fact t 9 #afdj1 :5 he 0 sixte5 ye>s old1 he publi%$ a little plac>d1 or po/]1 entitl$ ,essay p\r les conique17#aa7 9 :i* 8 great !orem appe>s = ! fir/ "t4 ,8 mann] ( putt+ x may be a little puzzl+ to "o :o has only se5 x 9 ! =m giv5 9 ?is book1 & x may be wor? :ile = ! /ud5t to comp>e ! two me?ods ( /at+ x4 ,x is giv5 z foll[s3 ,if 9 ! plane ( ,m1 ,s1 ,q we draw ?r\gh ,m ! two l9es ,,mk & ,,mv1 & ?r\gh ! po9t ,s ! two l9es ,,sk & ,,sv1 & let ,k be ! int]section ( ,,mk & ,,sk2 ,v ! int]section ( ,,mv & ,,sv2 ,a ! int]section ( ,,ma & ,,sa 7,a is ! int]section ( ,,sv & ,,mk71 & ! int]section ( ,,mv & ,,sk2 & if ?r\gh two ( ! f\r po9ts ,a1 ,k1 1 ,v1 :i* are n 9 ! same /raight l9e ) ,m & ,s1 su* z ,k & ,v1 we pass ! circumf]5ce ( a circle cutt+ ! l9es ,,mv1 ,,mp1 ,,sv1 ,,sk 9 ! po9ts ,o1 ,p1 ,q1 ,n2 ,i say t ! l9es ,,ms1 ,,no1 ,,pq are ( ! same ord]4 7,by l9es ( ! same ord] ,pascal means l9es :i* meet 9 ! same po9t or are p>allel47 ,by project+ ! figure ?us describ$ ^u ano!r plane he is able to /ate 8 !orem = ! case :]e ! circle is replac$ by any conic section4 99#agc499 ,x mu/ be und]/ood t ! ,essay 0 only a rsum ( a m ext5d$ treatise on conics :i*1 ow+ p>tly to ,pascal's extreme y\?1 p>tly to ! difficulty ( publi%+ sci5tific works 9 ^? days1 & also to 8 lat] morbid int]e/ 9 religi\s matt]s1 0 nev] publi%$4 ,leibniz7#ab7 exam9$ a copy ( ! complete "w1 & has report$ t ! great !orem on ! my/ic hexagram 0 made ! basis ( ! :ole !ory1 & t ,pascal _h d$uc$ "s f\r hundr$ coroll>ies f x4 ,?is w\ld indicate t "h 0 a man able to take ! unconnect$ mat]ials ( projective geometry & %ape !m into "s su* symmetrical edifice z we h to-day4 ,un=tunately = sci5ce1 ,pascal's e>ly dea? prev5t$ ! fur!r developm5t ( ! subject at 8 h&s4 99#agd499 ,9 ! ,essay ,pascal gives full cr$it to ,des>gues1 say+ ( "o ( ! o!r propositions1 ,we prove ?is prop]ty also1 ! orig9al discov]] ( :i* is ,m4 ,des>gues1 ( ,lyons1 "o ( ! greate/ m9ds ( ?is age 444 & ,i wi% to ackn[l$ge t ,i owe to him ! little :i* ,i h discov]$4 ,?is ackn[l$gm5t l$ ,desc>tes to believe t ,pascal's !orem %\ld also be cr$it$ to ,des>gues4 ,b 9 ! sci5tific club :i* ! "y ,pascal att5d$ 9 company ) 8 fa!r1 :o 0 also a sci5ti/ ( "s reputation1 ! !orem w5t by ! "n ( 'la ,pascalia1' & ,desc>tes's rem>ks d n seem to h be5 tak5 s]i\sly1 :i* inde$ is n to be wond]$ at1 see+ t he 0 9 ! habit ( giv+ scant cr$it to ! "w ( o!r sci5tific inve/igators ?an himself4 99#age4 ,de la ,hire & 8 "w499 ,de la ,hire add$ little to ! developm5t ( ! subject1 b he did put into pr9t mu* ( :at ,des>gues _h already work$ out1 n fully realiz+1 p]haps1 h[ mu* 0 8 own & h[ mu* he ow$ to 8 tea*]4 ,writ+ 9 #afgi1 he says17#ac7 ,i h j read = ! fir/ "t ,m4 ,des>gues's little treatise1 & h made a copy ( x 9 ord] to h a m p]fect k ( x4 ,x 0 ?is copy t sav$ ! "w ( 8 ma/] f oblivion4 ,de la ,hire %\ld be cr$it$1 among o!r ?+s1 ) ! inv5tion ( a me?od by :i* figures 9 ! plane may be trans=m$ into o!rs ( ! same ord]4 ,8 me?od is extremely int]e/+1 & w s]ve z an ex]cise = ! /ud5t 9 syn!tic projective geometry4 ,x is z foll[s3 ,draw two p>allel l9es1 a & b1 & select a po9t ,p 9 _! plane4 ,?r\gh any po9t ,m ( ! plane draw a l9e meet+ a 9 ,a & b 9 ,b4 ,draw a l9e ?r\gh ,b p>allel to ,,ap1 & let x meet ,,mp 9 ! po9t ,m'4 ,x may be %[n t ! po9t ,m' ?us obta9$ does n dep5d at all on ! p>ticul> ray ,,mab us$ 9 det]m9+ x1 s t we h set up a "o-to-"o correspond5ce betwe5 ! po9ts ,m & ,m' 9 ! plane4 ,! /ud5t may %[ t z ,m describes a po9t-r[1 ,m' describes a po9t-r[ projective to x4 ,z ,m describes a conic1 ,m' describes ano!r conic4 ,?is sort ( correspond5ce is call$ a coll9eation4 ,x w be f\nd t ! po9ts on ! l9e b trans=m into !mselves1 z does also ! s+le po9t ,p4 ,po9ts on ! l9e a trans=m into po9ts on ! l9e at inf9ity4 ,! /ud5t %\ld remove ! metrical features ( ! con/ruction & take1 in/ead ( two p>allel l9es a & b1 any two l9es :i* may meet 9 a f9ite "p ( ! plane4 ,! coll9eation is a special "o 9 t ! g5]al "o has an inv>iant triangle in/ead ( an inv>iant po9t & l9e4 99#agf4 ,desc>tes & 8 influ5ce499 ,! hi/ory ( syn!tic projective geometry has little to d ) ! "w ( ! great philosoph] ,desc>tes1 except 9 an indirect way4 ,! me?od ( algebraic analysis inv5t$ by him1 & ! diff]5tial & integral calculus :i* develop$ f x1 attract$ all ! int]e/ ( ! ma!matical _w = ne>ly two c5turies aft] ,des>gues1 & syn!tic geometry receiv$ scant att5tion dur+ ! re/ ( ! sev5te5? c5tury & = ! great] "p ( ! eighte5? c5tury4 ,x is difficult = mod]ns to conceive ( ! ri*ness & v>iety ( ! problems :i* confront$ ! fir/ work]s 9 ! calculus4 ,to come into ! possession ( a me?od :i* w\ld solve almo/ automatically problems :i* _h baffl$ ! ke5e/ m9ds ( antiquity2 to be able to d]ive 9 a few mom5ts results :i* an ,ar*im$es _h toil$ long & pati5tly to rea* or a ,galileo _h det]m9$ exp]im5tally2 su* 0 ! happy exp]i5ce ( ma!maticians = a c5tury & a half aft] ,desc>tes1 & x is n to be wond]$ at t along ) ?is en?usia/ic pursuit ( new !orems 9 analysis %\ld come a species ( contempt = ! me?ods ( ! anci5ts1 s t 9 8 preface to 8 ,m*anique ,analytique1 publi%$ 9 #aghh1 ,lagrange boa/s1 ,"o w f9d no figures 9 ?is "w4 ,b at ! close ( ! eighte5? c5tury ! field op5$ up to rese>* by ! inv5tion ( ! calculus began to appe> s ?or\ghly explor$ t new me?ods & new objects ( inve/igation began to attract att5tion4 ,lagrange himself1 9 8 lat] ye>s1 turn$ 9 we>9ess f analysis & me*anics1 & appli$ himself to *emi/ry1 physics1 & philosophical speculations4 ,?is /ate ( m9d1 says ,d>b\x17#ad7 we f9d almo/ always at c]ta9 mom5ts 9 ! lives ( ! greate/ s*ol>s4 ,at any rate1 aft] ly+ fall[ = almo/ two c5turies1 ! field ( pure geometry 0 attack$ ) almo/ religi\s en?usiasm4 99#agg4 ,newton & ,maclaur9499 ,b 9 ha/5+ on to ! epo* ( ,poncelet & ,/e9] we %\ld n omit to m5tion ! "w ( ,newton & ,maclaur94 ,al?\gh _! results 7 obta9$ by analysis = ! mo/ "p1 nev]!less !y h giv5 u !orems :i* fall naturally into ! doma9 ( syn!tic projective geometry4 ,?us ,newton's organic me?od7#ae7 ( g5]at+ conic sections is closely relat$ to ! me?od :i* we h made use ( 9 ,*apt] ,,iii4 ,x is z foll[s3 ,if two angles1 ,,aos & ,,ao',s1 ( giv5 magnitudes turn ab\t _! respective v]tices1 ,o & ,o'1 9 su* a way t ! po9t ( int]section1 ,s1 ( "o pair ( arms always lies on a /raight l9e1 ! po9t ( int]section1 ,a1 ( ! o!r pair ( arms w describe a conic4 ,! pro( ( ?is is left to ! /ud5t4 99#agh499 ,ano!r me?od ( g5]at+ a conic is due to ,maclaur947#af7 ,! con/ruction1 :i* we also leave = ! /ud5t to ju/ify1 is z foll[s3 ,if a triangle ,c',,pq move 9 su* a way t its sides1 ,,pq1 ,,qc'1 & ,c',p1 turn ar\nd ?ree fix$ po9ts1 ,r1 ,a1 ,b1 respectively1 :ile two ( its v]tices1 ,p1 ,q1 slide along two fix$ l9es1 ,,cb' & ,,ca'1 respectively1 !n ! rema9+ v]tex w describe a conic4 99#agi4 ,descriptive geometry & ! second revival499 ,! second revival ( pure geometry 0 aga9 to take place at a "t ( great intellectual activity4 ,! p]iod at ! close ( ! eighte5? & ! beg9n+ ( ! n9ete5? c5tury is adorn$ ) a glori\s li/ ( mighty names1 among :i* are ,gauss1 ,lagrange1 ,leg5dre1 ,laplace1 ,monge1 ,c>not1 ,poncelet1 ,cau*y1 ,f\ri]1 ,/e9]1 ,von ,/audt1 ,mbius1 ,abel1 & _m o!rs4 ,! r5aissance may be said to date f ! inv5tion by ,monge7#ag7 ( ! !ory ( descriptive geometry4 ,descriptive geometry is conc]n$ ) ! repres5tation ( figures 9 space ( ?ree dim5sions by means ( space ( two dim5sions4 ,! me?od commonly us$ consi/s 9 project+ ! space figure on two planes 7a v]tical & a horizontal plane be+ mo/ conv5i5t71 ! projections be+ made mo/ simply = metrical purposes f inf9ity 9 directions p]p5dicul> to ! two planes ( projection4 ,^! two planes are !n made to co9cide by revolv+ ! horizontal into ! v]tical ab\t _! common l9e4 ,su* is ! me?od ( descriptive geometry :i* 9 ! h&s ( ,monge acquir$ wond]ful g5]ality & elegance4 ,problems conc]n+ =tifications 7 work$ s quickly by ?is me?od t ! comm&ant at ! milit>y s*ool at ,mzires1 :]e ,monge 0 a draftsman & pupil1 view$ ! results ) di/ru/4 ,monge aft]w>d became pr(essor ( ma!matics at ,mzires & ga!r$ ar\nd him a gr\p ( /ud5ts de/9$ to h a %>e 9 ! advancem5t ( pure geometry4 ,among ^! 7 ,ha*ette1 ,brian*on1 ,dup91 ,*asles1 ,poncelet1 & _m o!rs4 99#ahj4 ,duality1 homology1 cont9uity1 cont+5t relations499 ,analytic geometry _h left little to d 9 ! way ( discov]y ( new mat]ial1 & ! ma!matical _w 0 ready = ! con/ruction ( ! edifice4 ,! activities ( ! gr\p ( m5 t foll[$ ,monge 7 direct$ t[>d ?is end1 & we n[ beg9 to he> ( ! great unify+ notions ( duality1 homology1 cont9uity1 cont+5t relations1 & ! l4 ,! devotees ( pure geometry 7 beg9n+ to feel ! ne$ ( a basis = _! sci5ce :i* %\ld be at once z g5]al & z rigor\s z t ( ! analy/s4 ,_! dream 0 ! build+ up ( a sy/em ( geometry :i* %\ld be indep5d5t ( analysis4 ,monge1 & aft] him ,poncelet1 sp5t mu* ?\ght on ! s-call$ pr9ciple ( cont9uity1 aft]w>ds discuss$ by ,*asles "u ! "n ( ! pr9ciple ( cont+5t relations4 ,to get a cle> idea ( ?is pr9ciple1 consid] a !orem 9 geometry 9 ! pro( ( :i* c]ta9 auxili>y elem5ts are employ$4 ,^! elem5ts d n appe> 9 ! /atem5t ( ! !orem1 & ! !orem might possibly be prov$ )\t !m4 ,9 draw+ ! figure = ! pro( ( ! !orem1 h[ev]1 "s ( ^! elem5ts may n appe>1 or1 z ! analy/ w\ld say1 !y become imag9>y4 ,no matt]1 says ! pr9ciple ( cont+5t relations1 ! !orem is true1 & ! pro( is valid :e!r ! elem5ts us$ 9 ! pro( are real or imag9>y4 99#aha4 ,poncelet & ,cau*y499 ,! ef=ts ( ,poncelet to compel ! acceptance ( ?is pr9ciple indep5d5t ( analysis result$ 9 a bitt] & p]haps fruitless controv]sy betwe5 him & ! great analy/ ,cau*y4 ,9 8 review ( ,poncelet's great "w on ! projective prop]ties ( figures7#ah7 ,cau*y says1 ,9 8 prelim9>y disc\rse ! au?or insi/s once m on ! necessity ( admitt+ into geometry :at he calls ! 'pr9ciple ( cont9uity4' ,we h already discuss$ t pr9ciple 444 & we h f\nd t t pr9ciple is1 prop]ly speak+1 only a /rong induction1 :i* _c be indiscrim9ately appli$ to all sorts ( que/ions 9 geometry1 nor ev5 9 analysis4 ,! reasons :i* we h giv5 z ! basis ( our op9ion are n affect$ by ! consid]ations :i* ! au?or has develop$ 9 8 ,trait des ,proprits ,projectives des ,figures4 ,al?\gh ?is pr9ciple is con/antly made use ( at ! pres5t day 9 all sorts ( inve/igations1 c>eful geometricians are 9 agreem5t ) ,cau*y 9 ?is matt]1 & use x only z a conv5i5t work+ tool = purposes ( exploration4 ,! "o-to-"o correspond5ce betwe5 geometric =ms & algebraic analysis is subject to _m & important exceptions4 ,! field ( analysis is mu* m g5]al ?an ! field ( geometry1 & :ile !re may be a cle> notion 9 analysis to1 correspond to e notion 9 geometry1 ! opposite is n true4 ,?us1 9 analysis we c deal ) f\r cord9ates z well z ) ?ree1 b ! exi/5ce ( a space ( f\r dim5sions to correspond to x does n !re=e foll[4 ,:5 ! geomet] speaks ( ! two real or imag9>y int]sections ( a /raight l9e ) a conic1 he is really speak+ ! language ( algebra4 ,ap>t f ! algebra involv$1 x is ! height ( absurdity to try to di/+ui% betwe5 ! two po9ts 9 :i* a l9e fails to meet a conic6 99#ahb4 ,! "w ( ,poncelet499 ,b ,poncelet's "r to ! title ,! ,fa!r ( ,mod]n ,geometry does n /& or fall ) ! pr9ciple ( cont+5t relations4 ,9 spite ( ! fact t he consid]$ ?is pr9ciple ! mo/ important ( all 8 discov]ies1 8 reputation re/s on m solid f\ndations4 ,he 0 ! fir/ to /udy figures 9 homology1 :i* is1 9 effect1 ! coll9eation describ$ 9 #age1 :]e correspond+ po9ts lie on /raight l9es ?r\gh a fix$ po9t4 ,he 0 ! fir/ to give1 by means ( ! !ory ( poles & pol>s1 a trans=mation by :i* an elem5t is trans=m$ into ano!r ( a diff]5t sort4 ,po9t-to-po9t trans=mations w sometimes g5]alize a !orem1 b ! trans=mation discov]$ by ,poncelet may ?r[ a !orem into "o ( an entirely diff]5t aspect4 ,! pr9ciple ( duality1 fir/ /at$ 9 def9ite =m by ,g]gonne17#ai7 ! editor ( ! ma!matical j\rnal 9 :i* ,poncelet publi%$ 8 rese>*es1 0 bas$ by ,poncelet on 8 !ory ( poles & pol>s4 ,he also put into def9ite =m ! notions ( ! inf9itely di/ant elem5ts 9 space z all ly+ on a plane at inf9ity4 99#ahc4 ,! debt :i* analytic geometry owes to syn!tic geometry499 ,! reaction ( pure geometry on analytic geometry is cle>ly se5 9 ! developm5t ( ! notion ( ! class ( a curve1 :i* is ! numb] ( tang5ts t may be drawn f a po9t 9 a plane to a giv5 curve ly+ 9 t plane4 ,if a po9t moves along a conic1 x is easy to %[--& ! /ud5t is recomm5d$ to furni% ! pro(--t ! pol> l9e ) respect to a conic rema9s tang5t to ano!r conic4 ,?is may be express$ by ! /atem5t t ! conic is ( ! second ord] & also ( ! second class4 ,x might be ?\ght t if a po9t mov$ along a cubic curve1 its pol> l9e ) respect to a conic w\ld rema9 tang5t to ano!r cubic curve4 ,?is is n ! case1 h[ev]1 & ! inve/igations ( ,poncelet & o!rs to det]m9e ! class ( a giv5 curve 7 aft]w>d complet$ by ,plck]4 ,! notion ( geometrical trans=mation l$ also to ! v important developm5ts 9 ! !ory ( inv>iants1 :i*1 geometrically1 are ! elem5ts & configurations :i* are n affect$ by ! trans=mation4 ,! anh>monic ratio ( f\r po9ts is su* an inv>iant1 s9ce x rema9s unalt]$ "u all projective trans=mations4 99#ahd4 ,/e9] & 8 "w499 ,9 ! "w ( ,poncelet & 8 contempor>ies1 ,*asles1 ,brian*on1 ,ha*ette1 ,dup91 ,g]gonne1 & o!rs1 ! anh>monic ratio enjoy$ a fundam5tal rle4 ,x is made also ! basis ( ! great "w ( ,/e9]17#bj7 :o 0 ! fir/ to treat ( ! conic1 n z ! projection ( a circle1 b z ! locus ( int]section ( correspond+ rays ( two projective p5cils4 ,/e9] n only relat$ to ea* o!r1 9 "o-to-"o correspond5ce1 po9t-r[s & p5cils & all ! o!r fundam5tal =ms1 b he set into correspond5ce ev5 curves & surfaces ( high] degrees4 ,?is new & f]tile conception gave him an easy & direct r\te into ! mo/ ab/ract & difficult regions ( pure geometry4 ,mu* ( 8 "w 0 giv5 )\t any indication ( ! me?ods by :i* he _h arriv$ at x1 & _m ( 8 results h only rec5tly be5 v]ifi$4 99#ahe4 ,von ,/audt & 8 "w499 ,to complete ! !ory ( geometry z we h x to-day x only rema9$ to free x f its dep5d5ce on ! semimetrical basis ( ! anh>monic ratio4 ,?is "w 0 accompli%$ by ,von ,/audt17#ba7 :o appli$ himself to ! re/atem5t ( ! !ory ( geometry 9 a =m indep5d5t ( analytic & metrical notions4 ,! me?od :i* has be5 us$ 9 ,*apt] ,,ii to develop ! notion ( f\r h>monic po9ts by means ( ! complete quadrilat]al is due to ,von ,/audt4 ,8 "w is *>act]iz$ by a mo/ rem>kable g5]ality1 9 t he is able to discuss real & imag9>y =ms ) equal ease4 ,?us he assumes a "o-to-"o correspond5ce betwe5 ! po9ts & l9es ( a plane1 & def9es a conic z ! locus ( po9ts :i* lie on _! correspond+ l9es1 & a p5cil ( rays ( ! second ord] z ! sy/em ( l9es :i* pass ?r\gh _! correspond+ po9ts4 ,! po9t-r[ & p5cil ( ! second ord] may be real or imag9>y1 b 8 !orems /ill apply4 ,an illu/ration ( a correspond5ce ( ?is sort1 :]e ! conic is imag9>y1 is giv5 9 #ae ( ! fir/ *apt]4 ,9 def9+ conjugate imag9>y po9ts on a l9e1 ,von ,/audt made use ( an involution ( po9ts hav+ no d\ble po9ts4 ,8 me?ods1 :ile elegant & p[]ful1 are h>dly adapt$ to an elem5t>y c\rse1 b ,reye7#bb7 & o!rs h done mu* t[>d simplify+ 8 pres5tation4 99#ahf4 ,rec5t developm5ts499 ,x w\ld be only confus+ to ! /ud5t to attempt to trace "h ! lat] developm5ts ( ! sci5ce ( protective geometry4 ,x is conc]n$ = ! mo/ "p ) curves & surfaces ( a high] degree ?an ! second4 ,purely syn!tic me?ods h be5 us$ ) m>k$ success 9 ! /udy ( ! /raight l9e 9 space4 ,! /ruggle betwe5 analysis & pure geometry has long s9ce come to an end4 ,ea* has its di/9ct advantages1 & ! ma!matician :o cultivates "o at ! exp5se ( ! o!r w nev] atta9 ! results t he w\ld atta9 if bo? me?ods 7 equally ready to 8 h&4 ,pure geometry has to its cr$it "s ( ! f9e/ discov]ies 9 ma!matics1 & ne$ n apologize = hav+ be5 born4 ,! day ( its usefulness has n pass$ ) ! inv5tion ( abridg$ notation & ( %ort me?ods 9 analysis4 ,:ile we may be c]ta9 t any geometrical problem may always be /at$ 9 analytic =m1 x does n foll[ t t /atem5t w be simple or easily int]pret$4 ,= _m ma!maticians ! geometric intuitions are weak1 & = su* ! me?od w h little attraction4 ,on ! o!r h&1 !re w always be ^? = :om ! subject w h a peculi> glamor--:o w foll[ ) delight ! curi\s & unexpect$ relations betwe5 ! =ms ( space4 ,!re is a correspond+ pleasure1 d\btless1 = ! analy/ 9 trac+ ! m>vel\s connections betwe5 ! v>i\s fields 9 :i* he w&]s1 & x is z absurd to %ut "o's eyes to ! beauties 9 "o z x is to ignore ^? 9 ! o!r4 ,let u cultivate geometry1 !n1 says ,d>b\x17#bc7 )\t wi%+ 9 all po9ts to equal x to its rival4 ,besides1 if we 7 tempt$ to neglect x1 x w\ld n be long 9 f9d+ 9 ! applications ( ma!matics1 z once x has already done1 ! means ( r5ew+ its life & ( develop+ itself anew4 ,x is l ! ,giant ,antaeus1 :o r5ew$1 8 /r5g? by t\*+ ! e>?4 ,,index 7,! numb]s ref] to ! p>agraphs7 ,abel 7#ahjb-ahbi71 #agi ,analogy1 #bd ,analytic geometry1 #ba1 #aah1 #aai1 #abj1 #adf1 #agf1 #ahj ,anh>monic ratio1 #df1 #afa1 #ahd1 #ahe ,apollonius 7second half ( ?ird c5tury ,b4,c471 #gj ,ar*im$es 7#bhg-bab ,b4,c471 #agf ,ari/otle 7#chd-cbb ,b4,c471 #afi ,asymptotes1 #aaa1 #aac1 #aad1 #aae1 #aaf1 #aag1 #aah1 #adh ,axes ( a conic1 #adh ,axial p5cil1 #g1 #h1 #bc1 #ej1 #ed ,axis ( p]spectivity1 #h1 #dg ,bacon 7#aefa-afbf71 #afb ,bisection1 #da1 #aji ,brian*on 7#aghe-ahfd71 #hd1 #he1 #hf1 #hh1 #hi1 #ij1 #ie1 #aje1 #aac1 #agd1 #ahd ,calculus1 #agf ,c>not 7#agif-ahcb71 #agi ,cau*y 7#aghi-aheg71 #agi1 #aha ,cavali]i 7#aeih-afdg71 #afb ,c5t] ( a conic1 #ajg1 #aab1 #adh ,c5t] ( involution1 #ada1 #adb ,c5t] ( p]spectivity1 #h ,c5tral conic1 #abj ,*asles 7#agic-ahhj71 #afh1 #agi1 #ahj1 #ahd ,circle1 #ba1 #gc1 #hj1 #ade1 #adf1 #adg ,circul> involution1 #adg1 #adi1 #aej1 #aea ,circul> po9ts1 #adf ,class ( a curve1 #ahc ,classification ( conics1 #aaj ,coll9eation1 #age ,conc5tric p5cils1 #ej ,cone ( ! second ord]1 #ei ,conic1 #gc1 #ha ,conjugate diamet]s1 #aad1 #adh ,conjugate normal1 #aea ,conjugate po9ts & l9es1 #ajj1 #aji1 #ach1 #aci1 #adj ,con/ants 9 an equation1 #ba ,cont+5t relations1 #ahj1 #aha ,cont9uity1 #ahj1 #aha ,cont9u\s correspond5ce1 #i1 #aj1 #ba1 #di ,correspond+ elem5ts1 #fd ,c\nt+1 #a1 #d ,cross ratio1 #df ,d>b\x1 #agf1 #ahf ,de ,beaugr&1 #agj ,deg5]ate p5cil ( rays ( ! second ord]1 #eh1 #ic ,deg5]ate po9t-r[ ( ! second ord]1 #ef1 #gh ,de la ,hire 7#afdj-agah71 #afh1 #aga1 #age ,des>gues 7#aeic-affb71 #be1 #bf1 #dj1 #aba1 #abe1 #afb1 #afc1 #afd1 #afe1 #aff1 #afg1 #afh1 #afi1 #agj1 #aga1 #agd1 #age ,desc>tes 7#aeif-afej71 #afb1 #agj1 #aga1 #agd1 #agf ,descriptive geometry1 #agi ,diamet]1 #ajg ,directrix1 #aeg1 #aeh1 #aei1 #afj ,d\ble correspond5ce1 #abh1 #acj ,d\ble po9ts ( an involution1 #abd ,d\ble rays ( an involution1 #acc1 #acd ,duality1 #id1 #ajd1 #afa1 #ahj1 #ahb ,dup9 7#aghd-ahgc71 #agd1 #ahd ,ecc5tricity ( conic1 #aei ,ellipse1 #aaj1 #aaa1 #afb ,equation ( conic1 #aah1 #aai1 #abj ,euclid 7ca4 #cjj ,b4,c471 #f1 #bb1 #ajd ,eul] 7#agjg-aghc71 #aff ,f]mat 7#afja-affe71 #afb1 #aga ,foci ( a conic1 #aeb1 #aec1 #aed1 #aee1 #aef1 #aeg1 #aeh1 #aei1 #afj1 #afa1 #afb ,f\ri] 7#agfh-ahcj71 #agi ,f\r? h>monic1 #bi ,fundam5tal =m1 #g1 #af1 #bc1 #cf1 #dg1 #fj1 #ahd ,galileo 7#aefd-afdb71 #afb1 #afi1 #agj1 #agf ,gauss 7#aggg-ahee71 #agi ,g]gonne 7#agga-ahei71 #ahb1 #ahd ,greek geometry1 #afa ,ha*ette 7#agfi-ahcd71 #agi1 #ahd ,h>monic conjugates1 #bi1 #cj1 #ci ,h>monic elem5ts1 #hf1 #di1 #ia1 #afc1 #ahe ,h>monic l9es1 #cc1 #cd1 #ce1 #ff1 #fg ,h>monic planes1 #cd1 #ce ,h>monic po9ts1 #bi1 #ca1 #cb1 #cc1 #cd1 #ce1 #cf1 #dc1 #ga1 #afa ,h>monic tang5ts to a conic1 #ia1 #ib ,h>vey 7#aegh-afeg71 #afi ,homology1 #ahj1 #ahb ,huyg5s 7#afbi-afie71 #afb ,hyp]bola1 #aaj1 #aaa1 #aac1 #aad1 #aae1 #aaf1 #aag1 #aah1 #afb ,imag9>y elem5ts1 #adf1 #ahj1 #aha1 #ahb1 #ahe ,inf9itely di/ant elem5ts1 #f1 #i1 #bb1 #ci1 #dj1 #da1 #ajd1 #ajg1 #aaj ,inf9ity1 #d1 #e1 #aj1 #ab1 #ac1 #ad1 #ae1 #ag1 #ah1 #ai1 #bj1 #ba1 #bb1 #da ,involution1 #cg1 #abc1 #abd1 #abe1 #abf1 #abg1 #abh1 #abi1 #acj1 #aca1 #acb1 #acc1 #acd1 #ace1 #acf1 #acg1 #ach1 #aci1 #adj1 #afa1 #afc1 #agj ,kepl] 7#aega-afcj71 #afb ,lagrange 7#agcf-ahac71 #agf1 #agi ,laplace 7#agdi-ahbg71 #agi ,leg5dre 7#ageb-ahcc71 #agi ,leibniz 7#afdf-agaf71 #agc ,l9e> con/ruction1 #dj1 #da1 #db ,maclaur9 7#afih-agdf71 #agg1 #agh ,measurem5ts1 #bc1 #dj1 #da1 #ajd ,m]s5ne 7#aehh-afdh71 #afh1 #aga ,metrical !orems1 #dj1 #ajd1 #ajf1 #ajg1 #ada ,middle po9t1 #ci1 #da ,mbius 7#agij-ahfh71 #agi ,monge 7#agdf-ahah71 #agi1 #ahj ,napi] 7#aeej-afag71 #afb ,newton 7#afdb-agbg71 #agg ,numb]s1 #d1 #ba1 #dc ,num]ical computations1 #dc1 #dd1 #df ,"o-to-"o correspond5ce1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #i1 #aj1 #aa1 #bd1 #cf1 #hg1 #dc1 #fj1 #ajd1 #ajf1 #ahd ,opposite sides ( a hexagon1 #gj ,opposite sides ( a quadrilat]al1 #bh1 #bi ,ord] ( a =m1 #g1 #aj1 #aa1 #ab1 #ac1 #ad1 #ae1 #af1 #ag1 #ah1 #ai1 #bj1 #ba ,pappus 7f\r? c5tury ,a4,d471 #afa ,p>abola1 #aaj1 #aaa1 #aab1 #aai1 #afb ,p>allel l9es1 #ci1 #da1 #afb ,pascal 7#afbc-affb71 #fi1 #gj1 #gd1 #ge1 #gf1 #gg1 #gh1 #ie1 #aje1 #abe1 #afb1 #afi1 #aga1 #agb1 #agc ,p5cil ( planes ( ! second ord]1 #ei ,p5cil ( rays1 #f1 #g1 #h1 #bc2 ( ! second ord]1 #eg1 #fj1 #gi1 #ha ,p]spective position1 #f1 #h1 #ce1 #cg1 #ea1 #ec1 #ga ,plane sy/em1 #af1 #bc ,planes on space1 #ag ,po9t ( contact1 #hg1 #hh1 #hi1 #ij ,po9t sy/em1 #af1 #bc ,po9t-r[1 #f1 #g1 #h1 #i1 #bc2 ( ! second ord]1 #ee1 #fj1 #fa1 #ff1 #fg1 #gb ,po9ts 9 space1 #ah ,pole & pol>1 #ih1 #ii1 #ajj1 #aja1 #ach1 #afd1 #aff ,poncelet 7#aghh-ahfg71 #agg1 #agi1 #ahj1 #aha1 #ahb1 #ahc1 #ahd ,pr9cipal axis ( a conic1 #aeg ,projection1 #afa ,protective axial p5cils1 #ei ,projective correspond5ce1 #i1 #ce1 #cf1 #cg1 #dg1 #ga1 #ib1 #ajd ,projective p5cils1 #ec1 #fd1 #fh ,projective po9t-r[s1 #ea1 #gi ,projective prop]ties1 #bd ,projective !orems1 #dj1 #ajd ,quadrangle1 #bf1 #bg1 #bh1 #bi ,quadric cone1 #ei ,quadrilat]al1 #hh1 #ie1 #if ,rob]val 7#afjb-afge71 #afh ,rul] con/ruction1 #dj ,s*e9]1 #afi ,self-correspond+ elem5ts1 #dg1 #dh1 #di1 #ej1 #ea ,self-dual1 #aje ,self-pol> triangle1 #ajb ,sep>ation ( elem5ts 9 involution1 #adh ,sep>ation ( h>monic conjugates1 #ch ,sequ5ce ( po9ts1 #di ,sign ( segm5t1 #dd1 #de ,simil>ity1 #ajf ,skew l9es1 #ab ,space sy/em1 #ai1 #bc ,sph]e1 #ba ,/e9] 7#agif-ahfc71 #abi1 #acj1 #aca1 #agg1 #agi1 #ahd ,/e9]'s con/ruction1 #abi1 #acj1 #aca ,sup]pos$ po9t-r[s1 #dg1 #dh1 #di ,surfaces ( ! second degree1 #aff ,sy/em ( l9es 9 space1 #bj1 #bc ,sy/ems ( conics1 #abe ,tang5t l9e1 #fa1 #hj1 #ha1 #hg1 #hh1 #hi1 #ij1 #ia1 #ib ,ty*o ,brahe 7#aedf-afja71 #afb ,v]n]1 #afa ,v]tex ( conic1 #aeg1 #aei ,von ,/audt 7#agih-ahfg71 #agi1 #ahe ,wallis 7#afaf-agjc71 #afb ,,footnotes #a ,! m g5]al notion ( anh>monic ratio1 :i* includes ! h>monic ratio z a special case1 0 also kn[n to ! anci5ts4 ,:ile we h n f\nd x necess>y to make use ( ! anh>monic ratio 9 build+ up our !ory1 x is s frequ5tly met ) 9 treatises on geometry t "s acc\nt ( x %\ld be giv54 ,consid] any f\r po9ts1 ,a1 ,b1 ,c1 ,d1 on a l9e1 & jo9 !m to any po9t ,s n on t l9e4 ,!n ! triangles ,,asb1 ,,gsd1 ,,asd1 ,,csb1 hav+ all ! same altitude1 are to ea* o!r z _! bases4 ,also1 s9ce ! area ( any triangle is "o half ! product ( any two ( its sides by ! s9e ( ! angle includ$ betwe5 !m1 we h =mula ,n[ ! fraction on ! "r w\ld be un*ang$ if in/ead ( ! po9ts ,a1 ,b1 ,c1 ,d we %\ld take any o!r f\r po9ts ,a'1 ,b'1 ,c'1 ,d' ly+ on any o!r l9e cutt+ across ,,sa1 ,,sb1 ,,sc1 ,,sd4 ,9 o!r words1 ! fraction on ! left is unalt]$ 9 value if ! po9ts ,a1 ,b1 ,c1 ,d are replac$ by any o!r f\r po9ts p]spective to !m4 ,aga91 ! fraction on ! left is un*ang$ if "s o!r po9t 7 tak5 in/ead ( ,s4 ,9 o!r words1 ! fraction on ! "r is unalt]$ if we replace ! f\r l9es ,,sa1 ,,sb1 ,,sc1 ,,sd by any o!r f\r l9es p]spective to !m4 ,! fraction on ! left is call$ ! anh>monic ratio ( ! f\r po9ts ,a1 ,b1 ,c1 ,d2 ! fraction on ! "r is call$ ! anh>monic ratio ( ! f\r l9es ,,sa1 ,,sb1 ,,sc1 ,,sd4 ,! anh>monic ratio ( f\r po9ts is sometimes writt5 7,,abcd71 s t =mula ,if we take ! po9ts 9 diff]5t ord]1 ! value ( ! anh>monic ratio w n necess>ily rema9 ! same4 ,! tw5ty-f\r diff]5t ways ( writ+ !m w1 h[ev]1 give n m ?an six diff]5t values = ! anh>monic ratio1 = by writ+ out ! fractions :i* def9e !m we c f9d t 7,,abcd7 7,,badc7 7,,cdab7 7,,dcba74 ,if we write 7,,abcd7 a1 x is n difficult to %[ t ! six values are =mula ,! pro( ( ?is we leave to ! /ud5t4 ,if ,a1 ,b1 ,c1 ,d are f\r h>monic po9ts 7see ,fig4 #f1 p4 99#bb71 & a quadrilat]al ,,klmn is con/ruct$ su* t ,,kl & ,,mn pass ?r\gh ,a1 ,,kn & ,,lm ?r\gh ,c1 ,,ln ?r\gh ,b1 & ,,km ?r\gh ,d1 !n1 project+ ,a1 ,b1 ,c1 ,d f ,l ^u ,,km1 we h 7,,abcd7 7,,komd71 :]e ,o is ! int]section ( ,,km ) ,,ln4 ,b1 project+ aga9 ! po9ts ,k1 ,o1 ,m1 ,d f ,n back ^u ! l9e ,,ab1 we h 7,,komd7 7,,cbad74 ,f ?is we h 7,,abcd7 7,,cbad71 or =mula :5ce a #j or a #b4 ,b x is easy to see t a #j implies t two ( ! f\r po9ts co9cide4 ,= f\r h>monic po9ts1 !re=e1 ! six values ( ! anh>monic ratio r$uce to ?ree1 namely1 #b1 =mula1 & -#a4 ,incid5tally we see t if an int]*ange ( any two po9ts 9 an anh>monic ratio does n *ange its value1 !n ! f\r po9ts are h>monic4 ,figure #di ,,fig4 #di ,_m !orems ( projective geometry are succ9ctly /at$ 9 t]ms ( anh>monic ratios4 ,?us1 ! anh>monic ratio ( any f\r elem5ts ( a =m is equal to ! anh>monic ratio ( ! correspond+ f\r elem5ts 9 any =m projectively relat$ to x4 ,! anh>monic ratio ( ! l9es jo9+ any f\r fix$ po9ts on a conic to a v>iable fif?po9t on ! conic is con/ant4 ,! locus ( po9ts f :i* f\r po9ts 9 a plane are se5 along f\r rays ( con/ant anh>monic ratio is a conic ?r\gh ! f\r po9ts4 ,we leave ^! !orems = ! /ud5t1 :o may also ju/ify ! foll[+ solution ( ! problem3 ,giv5 ?ree po9ts & a c]ta9 anh>monic ratio1 to f9d a f\r? po9t :i* %all h ) ! giv5 ?ree ! giv5 anh>monic ratio4 ,let ,a1 ,b1 ,d be ! ?ree giv5 po9ts 7,fig4 #di74 ,on any conv5i5t l9e ?r\gh ,a take two po9ts ,b' & ,d' su* t ,,ab'_/,,ad' is equal to ! giv5 anh>monic ratio4 ,jo9 ,,bb' & ,,dd' & let ! two l9es meet 9 ,s4 ,draw ?r\gh ,s a p>allel to ,,ab'4 ,?is l9e w meet ,,ab 9 ! requir$ po9t ,c4 #b ,pappus1 ,ma!maticae ,collectiones1 vii1 #abi4 #c ,j4 ,v]n]i1 ,libellus sup] vig9tiduobus elem5tis conicis1 etc4 #aebb4 #d ,kepl]1 ,ad ,vitellionem p>alipom5a quibus a/ronomiae p>s optica traditur4 #afjd4 #e ,des>gues1 ,bruillon-project d'une atte9te aux vnem5ts des r5contres d'un cne avec un plan4 #afci4 ,edit$ & analyz$ by ,p\dra1 #ahfd4 #f ,! t]m 'pole' 0 fir/ introduc$1 9 ! s5se 9 :i* we h us$ x1 9 #ahaj1 by a ,fr5* ma!matician nam$ ,s]vois 7,g]gonne1 ,annales des ,ma?matiques1 ,i1 #ccg71 & ! correspond+ t]m 'pol>' by ! editor1 ,g]gonne1 ( ?is same j\rnal ?ree ye>s lat]4 #g ,eul]1 ,introductio 9 analys9 inf9itorum1 ,app5dix1 cap4 ,v4 #agdh4 #h uvres de ,des>gues1 t4 ,,ii1 #acb4 #i uvres de ,des>gues1 t4 ,,ii1 #cgj4 #aj uvres de ,desc>tes1 t4 ,,ii1 #dii4 #aa uvres de ,pascal1 p> ,brunsehvig et ,b\tr\x1 t4 ,i1 #beb4 #ab ,*asles1 ,hi/oire de la ,gomtrie1 #gj4 #ac uvres de ,des>gues1 t4 ,i1 #bca4 #ad ,see ,ball1 ,hi/ory ( ,ma!matics1 ,fr5* edition1 t4 ,,ii1 #bcc4 #ae ,newton1 ,pr9cipia1 lib4 i1 lemma ,,xxi4 #af ,maclaur91 ,philosophical ,transactions ( ! ,royal ,society ( ,london1 #agce4 #ag ,monge1 ,gomtrie ,descriptive4 #ahjj4 #ah ,poncelet1 ,trait des ,proprits ,projectives des ,figures4 #ahbb4 7,see p4 #ceg1 ,vol4 ,,ii1 ( ! edition ( #ahff47 #ai ,g]gonne1 ,annales de ,ma?matiques1 ,,xvi1 #bji4 #ahbf4 #bj ,/e9]1 ,sy/ematis*e ,ehtwickelung d] ,abhngigkeit geometris*] ,ge/alt5 von e9&]4 #ahcb4 #ba ,von ,/audt1 ,geometrie d] ,lage4 #ahdg4 #bb ,reye1 ,geometrie d] ,lage4 ,translat$ by ,holgate1 #ahig4 #bc ,ball1 loc4 cit4 p4 #bfa4