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QlorttcU Uniueraitg Eihrari} 

Sttfata, Nrm $ark 



Tohn Henry. Tanner 



UAThUmATICJt 



OUVER, WAIT, AND JONES' fVlATHAIVtATICS'^ 



A 

TREATISl 

ON 



PROJECTIVE GEOMETRY. 



BY 



GEO. W. JONES AND ARTHUR ^.HATHAWAY. 



CF 



CORNELL UNIVERSITY. 



ITHACA, NY.: 
UUDUEY F. FINCH. 



CONTENTS. 



I 



SECTION, PAGE. 

IDuaWuf :... I 

R.Persbeciwe , 

o.rioTrioloQy o 

TYamislnina Poi-nis Q.y\6. Lmes ._ 1) 

5.FVojectiveF>3or6S_ 17 

(nCross Ratios* XX 

TH<xrTnoT)Ic Ratios ^ 

^.Projective RaTioes ox\h Pertcils 3^ 

9. Inuolation "O 

lO.&eneTal FVoberties o| Co- Its.. 50 

i |. CoTiic RaTiges aTnoL FeTDciis ol 

lis.CoTiic l-nvolatio-n --M 

l3, CoTjjugaie PoiriU on^ Lmcs- -- ol 

14. ReciVjTotcl Rqures 75 

15, Gtnters <xnh D/«Tne.te.r5 33 

16. Foci anb ^\Mi,c\:T\ce.s,^^ __, 94 



Projective Geometry. 

IW^ \v\ \\>*\s WV ftW Was ar* Wane ^ur«s.w\»s& «kVi«,W»6e sVoteA.-, atv^ bv^ 

§1, DUALITY. 

\n a UcYve tW« waxi lie tinro kin^s oj|^ \\(^\ires*. 
^w co-lines. 
v\i\x coy/bo'mts. 

lUo-We |Un«s |co^\>o\nt. 

J VWee -Unfc I ii*!"** I CQ -^omls . 







R Wmt-Uore oft^ alma-more are corrff/a^/t/e ^^z\\x^\iWtC <» 

H bom\: at\^u\in«» art, oorre/at/ye e^/emeffts. \n ^vo^tcVwe (jeevn 
dcn^y\\e WoyerVxes «Jj yo\«Ot-Jj^ur«s ©xvio^ Vv(t«-\t<\Ov-ts \na»\ Vie «*V ^t^ 
VA. coTrela'ti^ehrc/»o^thons,\.t}^ri^o%^\%'cci tWV r«\ftVe U j^oiaV- 



J. projective: GEomeTRY. fet. 

bomV c^k Vv-vi H Ipaw cAr c»rrt.\a]bW!& jwoWtitioniS Wvn «l Uua/ Brvoosmon . 
in ft^ttce W«Vs anA.Y\auMts art- Mrt«Xft.VwC tWmcukc oinA «o «ir« MMfiid 
t.Oi, kwoVf**^ aiireaUne: ttiree^f** Mve fl"|; . » ' 

S Z. PERSPECTIVE. 

\)^ AJ^4urc c»ns\sV oy\in.t» ^Wo a^xicti Wtn.\,«rcltur^»ec>»^y\ftnM 
(Mti CMM j ^cneraStti. Vu ftueW Uma, svcK 0. Vt^urC tS fr '^^tj'. TKe 
\mt»^y«\«^ vsV^ve centre, unA. \^«> \t(v.i« art. r-aus, 

\ fy- 2^0^ \ie ouV W Wo Vi\oAie», trunsverisekis, cKc accVvon* ore ^/o- 
vroB m, javzjiechiy^, ani ««x\\ a|^ tVvwxx \» CVf\ tfhO>Q€ fljr "ve oVVtT*, 

W( OBT* ^ /e«»^eff, H Wo^nsversoV cv^ft VkictWrtft <IJ[ aW^ \naytn- 



tttiaSl] 



HatV\OLOGY. 











TwtQi%^,/ftt¥ojfeur€& ire //t>kars^eU/re a tine cundrha 'mofW /M»^ 

S3. HOIYIOLOGY. 

1^ two \i^UT»» m yvrtyoicVtMCi U We\tiC^«^ Wotil tVv<w^ tg^tn* onObr^ 
aWrt ^W sarrvci f\9c<i9», cV» ^wo Cth^\(MvaT iriauYAc «o Wm«^ ore m /(wvr- 



MMLt. 



^ PROJECTIVE GEOftlETRY. Lfc^.vyS3. 

7heor,3. U(iroj7ai/rzs Infn kamtttojfj, toe miaje ^/^-/j,-,^ '«" ^f, •/ 

THEOR.^. ifUojiyjrcs U „v /lomJoy^.u-jf^^l^uiU firs *-»f^/^^^t ''*^*' 
eucic iitruij' frarri. o. kD^eoL 1 

For V bWc. twoJ^*' tkoJb ore imaoes oj 6ox<K oHvef i'n.iKe bnrworu 

11 « « * u u I L \ llieon ttlme^nro,, 

imtitje o| bh&J oj'bke Wrs|»e&lrive. TtU., I 

a]|r^'^k\.uV ;% \Vs own ;«»a^e IS a C^oullejf?'"'^- 

cknMT doulrlej , urJe&i hkel*^uYe^ co'tnc/'oCe Hrouokou,t, 



Q.E.D. 



cuibre 



tWfe..§3] 



HOWVOLOGY. 



JlttYvt tmaoes urvVess «\e cenbre oV |pro|8»cfc«orv ooirvcide ulvVW Bve cen- 
tre eJj ^«r«Vs«/c)tlvc; o.'^'viJL Hvwvs. VWa two ^«OTe.» \nVvor(\o\eQu co*v^c»dl0 

tiOR. ^ /f»a h'oures lr& in ItomolaQW, {i-n<£ /^ there i>& roar cl(tuUej j." 
770 three, of mz/wVA- cMre-/''^''*^^^''^' ,tke fw/z^t.-e^ caittCfde ihrvvvkotit. 



^ 



AB, AR BF^ re.}/eot into t/ie //kes AB^ AP/BRVt-ow/-*""^ <».3i'j S,i&r- 
&verijjaa,irQJjoaint^PjP,t^€'tyifoy^UTe^ a^-e iy(/ h&rsjje^ohi/^e, OT 
hoTyUiUgif, 5 

Ca) Me hiyo^iqures notco-hlarvQr: 

For '•' AB,/^V-mceb, 

.'. bKe-y are. coWlarvccr. 

So 8P, /I'p' are &o|j)an<vY, BP, B'P are- coj»\ouvvar; 
i', bW to-l«Vu6» Rm', BB', PP' oj' fclvese. bWee 
Maitas TnC'et'irv 5^ eke to-\3o"iKvl-oJ*f(^„8B 




Q.E.D. 



(u-j tke tiyv }fjvre'S oo-fth 



ayi/o^ 



iiciOY-«»%, dLrcuuUou VaoJiJc-^iQOre* fl"B ,..P... u;'k/o«e eo-Uv^t^lCB, 
tke.w''.' f[S,„?,..j fCvi',„V'.. aTe\>v j>Br5^.©U;»Vc (pmm sonxe c«kIt« Q jfc'^ 



PROJECTIVE GEOW ETR Y. 



[tK.7S3- 



• • /\B...R..,A 6...P... ofrtf mKoht* 

at.D. 

COR. IHm jioiktrjijares AB-.R.., 
/?'b'..P... icin.jiterifi9td.iifC arm btm- 
gioou. mmC otkigr ^i^ure le turned, 
oAfOvb eke ttXfS. ihey are gjyflnuu^ 
/*^j>erxiJ9eo'iire or- in, AantoU^w. 
t/i,e iCLyH^koiicbz arv in\,aj9G€ ^ 
ii tke seoMjeifoitds, atui tkeiarnc ^ 
li>ves ii ike 4ci,rH>e /tjves. 

HoMvo\a^i^ >«UMBr^|orc. a\l«v«-i^lv«J co.Se olj yort^eoVive, a.w<jLUqurQ>9 itv 







,. yWVMKrt . /f iiutiwf/mJ: /,7VC-},-^ttr^S o,ir...j,..., oLlr'..,)^.,. 6Ufroittts a\r. 
a,t>, Ifj^.^roj&c/t rrdo kh^eftonvts a'lr', <v}>', l^'^'.^royyu ew centre ^S. /o>- everi^ 
I'r af /n*ei b, b, f/ie twohovrei eare r*^ f>ersl»echir8. 

(Q'\ Che IrifYo l^'^vres noi ea-JtJAyutir: [jl. ter«L 



Ja/yva/T", 



Tor taka 0,0'aTv.tj cenUa* in s^a.c» ColitvMir wtVkS, oa\(& \«ik a , Ir", 



Cttoriv,*'' 0,0 are 



&oliirvdtt.r vAJiVWS , cuAcl «o ore. tke ipoi-nbs fliii.ckto ' [iiuk 



Pr.»533 



We»<\0LOGY. 



•*. ava\mes 0-a\r, 0-oSlr atcz eo- 

OiuiL'-' rniFc co-yio'inb li«& \ys. bke co- 
Ime oj' 0-a , O-oi ou»vct irv H\««l 

•*- it ts comfnon Co Ct , Ir ; 
i.e. a", Ir -in»et and are co>Manar 
So O-aib, Oixjo ttimV a-ntX tktirCo- 

hoxrk tS common U O' , )» ; 

b -me«ts a, . 

O-ljo, O-^lr'la -m««b ondiftetir q»- 

Doint »s CMnnxon jt» U' , jo"; 

b meebs Ir . 

a", Ir", jo" oxa> co'loVoarva.r. 



5o 



t.e 







fltvAV «.V...|o...,orlr"... JB*.. ore m jpe^rSJiAcbiy-e jprow 0, (wviJL ^rojw:V jpntm. 
•'> a\r...la..., a!lr'....k'o.Ye >»iK»«i»»Ua|a. Q £ t^ 

PROB. I . To e/ra,w Hef" ,"! . o/o/ qiYt-n/Jf^ nard Hxe t^j^worrw/ '^ 
llines; OYia.lr l.-W» rt^ujl*""^*^'^ 

. » . I D5 Ca-lr.E3 BC-a.Fs flft-DE , 
*^* *'Mol= ea.B,f»Urfl,.palr(U. 

|onj^ ■* i|»oinr 




8 



PROJECTIVE GEOMETRY. 



[S3 






ore m 



ttaMftrcc 15 c«iine«r 






Q.E.F. 






rcas 



are tb ^^._,^ «»«L rtiwr «nt j 

lverctcc.% ieo^Ii-nc 

jrerttac , ., yt/'des 

/. 




onA tlie:^-]""^'"^.' c €t,'ajQonol 



ie« 



-Une 




«..s.§2rJ 



HOMOLOGY. 



^ 



U»A«» ^ ll|o»r.\m*. lalr.ul. Ircjut-, oo-.trrf- ' ^ "^ 

lytTticcs; e.^.a. J lUt-a, Jef'^ J 



r J rr /verbot^ ^ ^^ heft^ he o?^ 

J vertex 11 e-s oc J /iiyteChrv /a-fioncC. u"*-/- 




i: PROitCTIVe CBOtftETRY CS3_ 

EXAmPLES. 

3r* jtthkr*5,«^^«lt»V»l»«ar«. ftBC... ., |)rainV« /1|b!&1 

B» ttnawaV oW """ ' 

f . In tf%. \, ^. B icA^ H\«r aiYvn \U<fr» Imt^ou oiroit*. 
Wr«U«\ Vmt«. 



W%,i>^ Midt Q^ o^^iYVn ^Y^W «Vki^ib$ OMibre, 



kka. S^3 VANISHING P MWTS A ND LINES . 1 1 



/J-A-'l^^V 






tuj3; 



1. Ir 0. Imt* Ctfrn, on A.^hicejl ^«iaV cw«JL caa^ a/ ^olmt o|^ itV»i 



-A 
§4.Y/WlSHiN& POINTS AND LINES. 

lyu ike tfM« fyvre "ikcitkure no /moyw m, J^ /fX^r- /f «/^•e 



For VKft I* CVS 0^ T*Ais blvflJb «U -HtV Ltorccr A. irflMwsreirs4 it a. 



12- 



PROJECTIYE GEOIYIETRY. 



IM,. 



^1 




y' I 



For, CutTv OTKtIIoure. a\>ovV vmj 9X\s lo ft*\v^ oWioxi^ PO««t\»T\,^V'n.<k cKt Va>v- 
VWwv *.' tW soanr\«/ bomts ocre i-mo^o/S U Vhe, &a>w«« ]|»omt9 tn Hie. oW\«ye «l% 
.\ the sao\Ae> Vaom^s KobVe. >vo i-yviauaC'S \vv.Hve> oWioue as xi^vm, co-U«w 

ae.o. 

Cor. i. 1^ two It'jur^K L^ it* l»mrsJae,otfif^, cufin^ ^ 9K€>^jvrC 
eti ifv^ ray J9ff.ya.lleJ ii its '/THtitfe^t orv the va^ikitiq tm* 



yrtzeti 



oj' tke-ji^ure, . 






tufi 



IIH9 



oc^ a 



S4-1 VANISHING POINTS AND UNES. |3^ 

For l»b a (fC amf liti«' «| one |^<|ur«/, cu tts imaoe'^ r a rau laaroJIcJ To a. ; 

then '■' I W imaa</ oV Ut«, laoint fct, i$lh«>|>oinl reJ, [<(|'.i)rcij. 

fc-nciL '•* J^, «'' «o-e jjarali*! , ikijji 

•■ Cha>))oi'nb reu nous iw t-mh(|«. Q.E.O. 

Cor. k. /f twol^'ouTzs I* mfursja^okiva.^ eattL iftrfa /m^s €if tntt, Umvr* met 
ujtffn, tnt, vttnisltiht Un€ »f nieJ'^'jvrt, chcir inmo»s are j^art^U/j tcnd^ can- 
irerae/y, if iw« Imtz y ff»€>-H«ifrt, vi ff.TO'l^t'l , xJit^r /nmaes m Cne onto' 
■fioffre fnazt v^n 0»e Vctjii's/iino /m» , 

It IS Ctrhx/a.-hivnV Co «)s»«Jk eji' Hi««. tno-yoi-Mt «l- iiwc/bbtn* J^ two |»artt.Ufti IrMS 

Hve.Ye^\«re. tWe' i-wvciaa ql bho^ viOk/vu$liiiiia lin^ mni«, olW j[i««n-e ', <M>u(. CK»t 
locus IS ^9, //irie'at-/TJ'/ruitu, bV tK«. -Kciure,. 

V/0R.5. /f(jy\,e, vvi7tf^h,/-ne /i9te 6e.tct mJrirMtxf, ^o /f tA^e rtwr'aJto, 

Cor 4. /f 5/^e, oenire 4 iter^Jieotivc be at tyJirtrtiu, uutfi^ all re^s 
are Lara' flaL 

This brojactioT^ is J!?o.rcu//e/ orojeot/arv. 

Cor. 5.Z^ tii^anfs oj- Jyer^mtiirt. u^ otin^iiuliu, cliM^jwary iiyte 
IS Isoisrot^lUi Co it$ ifvocua^. 

~^\, "2^|. 

IKJs LrojectioTi Kz/lornot/tettc projt^Xtoro, fliici Cnerv«i|totfrc» «t» itw- 

vOR.6. if Oie troj&ot/»i^ Ce vHlo haro^l/^l oo^eL AoTncinctic L:. 



i». ^ PROJECmVE GEiOPIETRY. [S4.<kjt-tl 

TheOR !0 i^inv^'our^s n< jtan^^cttwC have a. vuruzkrn^ Aitc « 

>«r J^' Ou. cowbre m gjb rnt^'-ni^u , Oie jraiAcVim. >s JMira^lUl . 1<<^. JMnr. jiroj . 

CteiM. '• ^ feitmc nirauoW uiC ce>dr& WrfrlM ti ^m. broLtt.SWr.Sul vmm,1» Hie 

- - Hie tiro urttiisvarsais on'* \MDr«'\\a),aMA H«e aqdis is o!b iH]i-iM\:u. 
•'< In. braiCicVion is homorkvrtic. Q-E..D. 






^_ -^ — — - 111 I 

<C«>i!T%. |iar»l'.&li to oiiVttir VTMvsreTSoJi is oA .m!^>t.ika , or it is \aox- 



r 



Hie. aiuSj ike. locos al Ike. cvhAre. oj- heriftaoUira /■s a, o/ro/e ^erL^nv 

)«r, lets( fl( lie I«iro ^imW' liaur«s m )wrsbBcJk«r«, S Oic c«nVre,« Hie 

avxtA. niackiiuf ^,si *m. eke tt-ncs a, 0^/ 
t\»tn >^ol revolve AirouC ^, ec re-mai-ns w Die Ua-m. fi^ {^cmm. 

and • ^,A rs-^ouYv i-maoes al eo.&k. oVWqa', Lni.J9, cn(. 

• • o remains it» Dm. mane a., ^' i.e. myo. 



§^3 



VAWSHING. POINTS AND UNES, 



IS. 




Throyuk a>\vi jaosiVJo-n o^>i dronu \\y\<t>i 5V, SV, para)!*! U cu,a.,ioryvt 



lltr.l. 



fr^ t/ai'' 



Uk- 



EXAIYIPUES. 

^ >Z, In cc. 11.5 oy §3 let tke last^^*'- fcU cU-la. U jV 

\-Viu 1 1 -AC 



16. 



PROJECTIVE GCOIYirTRY. 



lS4. 



4-. IWeuah a ciive-n Wnt U <ifMf a. Una* JMroiUl ka o. ovrvn lin* iiu «uiL <^ cu 
ditrwn |oox-a.lUUara.>vi. 

Ls/t a (re. |pa.Tak.(l«.l U ir a«*JL c U d. oaJL Uk Y, 6 Ur fckfr ^iven |s«iTib ••'H4. \iiic>; 

V- .1 I II 1 • I t < i • . L • I ll # > Ml II 



Ls/t a (re. |3a.Tak.(l«.l U ir a«*JL c U <! oaJL Uk Y, 6 Ur thfr ^iven |s«iTib •.'h4. hue*; 
r«>««'<S T^\8>ci: bW tr«oiv^ie oW n\to emu conwRiMit vriaiiale ol^o' WiHi Hie 



HANGES AND PENCILS IN PERSPECTIVE. 




fliUJiu U ens. «||>e«iA.\ c«v&» o^ -ra.vb<ic/S aW- b^Kc'iU m lpa.r$aa.tb'iYeiOcrc<(re<i|i«d 
toa<>niivr u»low: 

ij- fr bei«ci) Ue. out lu Cvro CransvvrstLls.ltiesactians 
oxt, Tojrujftfi in lB«rsba.ckiV«., co^d . omversciu , iL- tvvo 

TOOku lie atxM it'hf!' tin-btran tW co-JMu-nb tie Die aves 
oy cneranOfrS- 

Ir a.look W cut U^ two Cra'nsveT.SaL.U iiie %eci:ieits 

pencils u» ivi aoTsbeclive. tke ccniTe a|' paTsbaoiiVe 
mauiieavMi Wci-«vfc oYt. ^^e. edod a| ^la jirajcctiKo laok* 



co-<xi(io4 rawcies m ke 



nK« 



'*1t 



Ike iiMMv^ An a-nu tr^^srersai A^t^vo covcoibnc 
tUk-n- copccnar tKntciis tw |iier«p«c^ira. cnra c«nceHtric. 

Uiorc or* two dovUt.V'^ ftH^ yno 

. ''•"'' fits. 










cCTtlr 



SS.tWa^ 



PROJECTIVE HGURES. 



17 



Uii^yvfK^h ii% hvjbvrs^^iottirm, tfte rgtvi jtt^rttH^I ^o iftc Mies •f Ma 
artyoC *f ov^ rvun^e ticrt^ avv-u^f Ck% t/i$vliU koini, \^ 




DniyUi }« bkt TiJti^ J>p/'i%t oj in* (CH4/(rCe-i>o/nii aytft 6^0 centre of 
f>0r&ft%*^ttYt>, 

9 qJ L^r^Qictfre, lr» at />^/»»/^/ ^/e rayi^es aye irm/Jnr/ *k<^ 
If Ifotk r/rtse itf/wtj ue ai rtJrmCy, the rafioiS o-Tc 9fu»./> [«Mi»». 





i.0 



N 



T 



S5. Pf=^OJE:CTlVE FIGURES. 



11 one, |i(|t<iia> U|>rojeck«4 mU «nobKeT \i\^ <v s|iect|-, thi4 seum^ j[i^(nr« ihto 
h 4|i»cioJl cow* I5 lUb oj- > j»«neil a>ui Ut r«Ha6 cut|rnw «l; i;^ a«tj 

s».V^^ H»«, jnwe jalon*, a«i bW wwk ni«re«jtc;r c»nc«m3 M-fA«Mnr ^^tfr« oi*^jj , 



18. 



PROJECTIVE GEOMETRY. 



[iSlko. 



la«is HmuaK Ao! nl^ftct a! If'c' 

a ..u |T«H*e« flBV,/»BC 
tten . CKpi I.. ,. , , ««rx, IX 
I vMnuis tJr « ■ aire 




&& * CCr lu tc«tr«/, 



IABc.a'b'c' , . ,. ,. 



Q.E.D. 



r /»roeyo»»i 



Cor. 1. fivo ji^vre>i ^.eu^L ctrt\siHino 4»^/// i -I o^f^a 



cere 



•» are jbrojcc/Orc pot/re^. 



I 



cci'Urreni 



r , , jflBCD.fl'B'C'D', L lJ'H^^H''*^^' u 1 U^'"^! .1 

Tarr,[QX\ ' bfcomi two«uoK-|; I. anoLlebi , l/etirt 



e. 



anol. letl , 
i jour- 1 1 -n 6,8, 'iH 

,. mUt^Vs ,iH,Km iU,«tsBD-Erm-£F , lE'r/G^HjK' u 

^.x /kwiVts . jaa'cb', p . lJ/)'b'co'. , )A"b"c"d" u. i u 

k^kv.i'i. aV-/ ,if ..' rroiect-j „, ,,, wtof ^ „ , ,,/ So CKox th«» 
jtliret-i»«mtT«>»«j«» E F'H 1 • -I- * U iL lwr*«-V»omtr«Ti«^EFH; 



S5.kWJ4-i5j 



PROJECTIVE FIGURES. 



. I flBCD, fl^'B-C'D" • , , .. 



19. 



ftk.t. 

Q.E.D. 











ikrafl • . i^"B"c'C, , j . i il, (centre B'- •. iBWDC- \ 



I 






IHEOR. 15. /^ €oua^l -jfoure^ ore krvJz>ot/re. M'ouye^. 
-n« it aOout 'inrftc a-co>s: 



©r liv re-- 



i.^ PfigJECTiVE GEOMETRY. [|iM 

) . iKt, w^liitt. (^ &wTW\ocni.», WWi«^ con. iVaJb kVif )f>^u*» U« so Iwnti^ \\if^ »t- 
MTPU .Ao ^yioni. a^^«,\ly«^Uirre wi^ like^crU <| ^e l«ro !|^(^iir(.« m \|\t Mint ar^er. 

3./\ JMTbentiicular U tU CBir-wititA likfc \int» tttr«w<|k tke^ >vl«(i•|o•i^^^ o^iw 
iik* points. 

/InAv^e ViouTt. so rtvoWt^., "m al\ «V« ^•stbtovvs.ts ^>^ beTgjseclJre vutln 
Vkob wV\ioi^ lc>re6«.JLc<s aL-ri<i- H^a^ wVvUlrv Utiovrs ib. Q^D. UkS. 

Cor. Tiro S/miJarfffUrcs are jxrojeM^iv^. 

''/'• /n anu hleivtf , , a, cvrrc Htat ts «tn /m««s ol tn-fi'rKf antt tfi^ 
i>ther. 

For. leti U kli* •movm^^r'" eT>«lj ,«r. «iin«i.e ^r»t« ft-n^j c«ntr* S wjooitamt] 
)|b ' lime- '>» 

L' f-U ,|Pbr».t«,s aniline SPbrocfcS .iKfcroACS 

> I|}e«tf»wf6 i^MvS^ cnv«lol»s l]i«mnlo|ps 

Int fcCctun x^ tkiS tone, L QiC Mtnft m «lAlic4lJ^, !Mrc4; 

p 

a is «T^ ivKOkae oir QIC M.C'Cion -I , , • 

|%l»m»iiC i^ contbcC O b with IpiftTCCS 

ill i\ J tawacnU •t P, P. 

> 66tfU&u «^ K p - 

6o^ |Dro^««.^<«n& Worn ^ucccsStv^ 0Oitr«3. Q.E.D, 



|5> PROJECTIVE nSURgS. 2K 

IfrlOAS ^ 

e.a. 0. CAToie i« a. tufve' o^ n)e> ^cco^ndL vrdivc ay\<L the ^econ/ clois. 
Cor. ^cwrc and ih imaots art tStk ^me order and cf ihe seme, p/t^x. 

\nt maqts (^ou circle Qr& coTr/'or. 

t^^ iTit* yonisWiTiC| lm» lie uiitnoub Che circle, kne imiioe iias no ^nt «b 
tTiVi-vi'iW.CLYiJLis oi>\. e/Zf^se. 

IV in6 vaTii«kiiv<i ime toucK ike oir&le^ die i-mooe' ha( one real o«ixt atr 
iidi-n'iVvL, bnt^ vYvmuoo oi-Tne Vtomtol'Ca-nio.^t, anA. is o. h«a-oJrcla\ /{{| lint^ 
HirouvK n\«.^oiir\V •!- tcv^tach jarojex'i" i'>ibo b(M-oJl«4 Wcs ujhoJtc^nretii^n. 
it viie direct oriy tc frfU^iiu tL thejaaroirola.. TKe ConocTftt" Imer, 



mis 

at iniiiiiKj , the/ iiw»4|e« ^ uv<' co-lpoiTtb« ol Gne va>\!«lui««} Irne muI 




byyo tonoc-nbs ts bKe circle o^ Inese. oo-bomt^, bro)«rotmb» tnt- 
aents IJB bUe. m)«*Tlo\ft. ab Inj-i-nitxl, on j HiesB. ore \Mz ttS^n^otts 

n jiooT* Win«.dL oj' kYro lines or Wo j^oinlrs «• 0>«. ^rrtprvL^y 
coTf/'o: iw O'liYHibe^ $«m«« ^hase-JKiarec are tiie brejeotive ym,- 
QkoetE oioucircle* ayi«1 Hieu jaoc^sss Tnconij oi tke |pro|>enrbie3' 

oo>vtvM«-w. U all co>mc3. 

lomts wiHtiti Ck. cirole jtroject mVo pomts MCntn the c«?»/o> aw^ 
WnU wi^\toub ow circle Wo^ecb i-ntt Wmbs mtnovt i^e coyt/o. 



R PROJtCTlVE GEOMETRY. [SIM 

§6. CROSS R/ITI05. 

l>\oJt raft-c^ \Y\ w\«, oY\}oS)\:^ ».»t«/6Cio>v art- T\<6aa,\:»Y'«. s&OYV)«.'rvts. 

L-Q-, In Ik j^sV ^ur» flB, A^C, flD, BC BD,CD ^ 
ore o--.' bc^itive. ft » t 

in bW secoT^dl Vifluire AB,flC,/^D,OD are . . 

tossCiye BO, BD art Yic^gO'UY'e/. n o ^ 

m Cho thlrJl .^iciv/re, flB,/)C,/iD, &C are 



■^ 



T»^^a•uyi.., 3D, CO are. ipo$ifciv», ^fl o a 




r.. q.. i- AD bt cut sit B, b'ht 5fc£;we--rvU art /^B, BD, Q-ndL th«. rab^ 
'\iL ToJtia i5 ^€.r* , ^ ol' fhe wi-bovTtV.vhe ratio «« unity, i\ at W& btrtvii-noX 




•nftaacwT^. 



botrtt o-noL Out <yn€^. 

f«r, ieJt AD W a c^i/aTV Star(\ftint j tWou«» h icrmn ft \w»*v cmi u|>OTrv 



-43. 




bTSSbj CROx'iS RATIOS 

AxAW Eb t>are;\le.l CoFD cuttixo AD ix/B/ -^ ->» ^ 

a-wd ckcrc 15 Tjoon\cr5octt point, ^ -^-^-^ ^ \ '\ 

4-or >»o lit<6 Ihrpuoli £ T»ot barolle' fe R) tan cut" '^ ^ '- 

RQ m 'llie, (^ex ih»ii« . _ Q-E- D 

ll(X sutnent douTtC/iieolnridUi aiiwt ^lAs ,v^V9 jtrtnnM a jour-jxtiid rywoe- 
wko*t CAlreTne«s ore Hte vnh oV'&CrSaomoit,an^ imu metHS dTtf ihe. Lauu of aW'*", 

E.q. ;f /^D bt. fltiViede^ of B,C,ifie« /IB:BD,/IC:CD are«ifc.W 4i;v.W 
otnii abT^ ; AC : CD, or )tf eyw«J «B.CO :7irBD,»» bke tr«s«-Y»i»o«}U«|«r. 
powt Tw-K^e ABCD. rt is u/n'tten C/»BCDJ. 

■v^'mte, B>e joiTT le\:ters fl,B.C.D -mo*) U jaermwlre^ i>» W«.*ih»|oor l^'a»J«^ 
IVfir* art tweflku iouT crdS-S- ratios ioT Cne Some iowr points : 
(./IBCD) -h-(CB«D>-/ (BCftD) + (/tCBD ) ^« ( WBD > -\-(BflCO )t=:-\ 
(B/IDCJ / (D^BCl (CBDR) (BDftC) (DBflO (COBA ) 

fCOAB) (P.aCB) (ORCB) (CRDB) (ACDB^ ( DCftS ) 

\OCOfli (SCDfl) (flDBC) (DBCfl^ (BDCfl; ( f^BDCj 

ThEoR i o'. /^ S£f77ie->if of -ti ///»« /j c</^ tu Hie efttis aPahttrter- 
^eo?verit at One Jaryte lint /t* the <sayrie cro^^-rai/a tur ihe^ec^yiti 
j^ei^wcht /s out l/u tke otds <d- iffe-ffrsh, auhd /k- t^e ^oi»9 cron- 

Ta,bi0 ««J the ^eerryftryts reversed, cvt eoiAyk- ainer: 

J UKt ^ \\ - •' ' •, . 

Fqt, ( ftBCD) e /IB' CD •• flCBD = Bf^DC .BD'flC £ ( BADC) . 

= CD^: CT^ ' CCDBB ). 



= DCBfl:OBcfl = (ocBfl) Q.E.D. 

NOTE. In Vke uWva. UUa, ql' bro^j-cAios 4'Hit JtouT tomU R,BC,D,«<wh 4 



2A^ projective: GEOWETRY. [86.Ba.ig-jLa 

tno «i< columns convicts o^ oc itk *V ^6\ir touai cro«S.r atj OS; o-noL, >vi fltn- 

^m^t terms oi Hii iurytmttric "nloi/of^, iaktn tvyn (m\^ iW9, are t^kt 
F«wV AB-CD=(flD+D&)'CD= ADCD-BDCD 

:. AB- CD + BC-AD + CA^* BD == . -^^ C^E. D. 

BC-AP : CF§5 =-(CPAD); cpr&5\ BcTad^ - ( ca bp) . 
Be- AD : AB'CO=- (BCAD), AB-CO: BC'«D*-(BflCDj. 

C^.E.D. 
Cor i. IfrOe cunu croSS'Txi-t/o of fvur co/m&ar JaofTftr h,^,^,^, 

Bm/al It Oy crois- rO'tt'o aj- ine oilier, i/ie. Six. crtjii-rai/'os ojvrt eve set 
an it^eraJlj ewa^l ttilie J// croii- r-at/os- of- the dther: 

COR.5. Ij-r Ire. ], int ^t- crois-rodios- are 1, o, o, I, <»^ oa; /f 
t-id/^r of a.nv Jour ^c/Tti^ cure doy-o^l uri/e>5yS T^ej/ (re /■hC'Iuo- 

- --'" ^E^OR.Zo. ^ thr^jamis: ^ <i. rayiQe Ire jivcyi/, tnere /> one 
and. Out oivc ho/rit tna^T comi>lvtz^ inc rany^ /h- q^ ^ivt;*^ 
cross- ro^ioi. 



0»Ji Hi» GroSS->T*iio It kntvAi., ^ 

.'. iht rpklo ^ <lmW«i) of Oieqlven te^uit (m (W JDomt «gtfJit is known, 

«ui<^ ^Ut» ti-iw^«ite ^uei^ ItoiTik. Q.E.O. Dk.r7 

I I o ■ 

^^'J^^^^ <ro9i'r4iif'ot of Jour co/i'naar fioirtt^ 69 ^.C.'^, ^e>«- 



1*' . .blieolircctioTi oiiW ft,iCi«, 

So,tl»e «JL^ta-n« lr»w • \iWfc* 0. ^mt J« tU'Ww^ik ^-tiit 7tv«r»«i vfj- 

pit dUbftTnoe o/B u^ti^iy* ccni Bo. i« T>e^«iire. qL ^ 

l| tU axis QJ'-nncMUrMtviffir'ii* Im^Jiiwiiart* ike ^iven U«, Hie 4i«ini«» 

Ir o)c, oLj U lU MM of- Ult iiuM <k,«£, ikcK 

art} \o%tk B ^lyiaec «nft omelc m trie racio 
«xB : B<:6. 

Urrvv^ Che tio-jbonit cf ci¥9 /mta cUvftit eh^rr vmcfe tri Chk ^tt7tf9 

Let oL,i. Iz oriM hxo uncs, h a \iv%, DiroaJt Uttrr co-)ieln^ B, B jwtvU o>\ u, 

mm u\W »B, frO «re 4 «** ^X"** *'5'^/ cmi «e ore Bd , B'^ r 

9r <Uc a6), oSf' vr*. <L «^M«ik«r«it|i«t •«%) 9« ore b4,, B*^, 




2fc projective: geometry. [8k.tli.ar 

vnd ir uunvr co-st- olB : 3cL =* a& : B'd. . Q.E. D . |«\w . bri , 

cnwi B, B Ite ffiOm wt some. m\a\t/, 

if>r -not elliBr>y(i««, tnoM oA: BJL, aB':BU tiare Hia fatnc «i9n ; 

<mti cneij ite •» «. iitic< Cnr»«/ah poirxC ad 

r*»T \' aB IS U.r»l\»\ li 0.B' anjl BcL i* B'cl, 

^ Y *"^ pwtits, B,B are. c«ii'in«ai.r vui'tK Hie C6 poiv- (^ a,C. Q.E.O 

K" a, hint, Jiviil(! or, areola, the raVo of af'viSio-nr l/u tnivt im<, iiS-To jftye.7i owf ojr mws- 
UTtmMit, u tne rotis «! <l)v<tion oir u\e anqlt lu otiv poiVt ojr ^i^^- ii'ne # 

Ihe crxurj-rtutTiG oy ai^ pan oil oi iour -^u-ji f b,e,Tvin! 4^^ tm) roti'oj \^ 

d onu ^tifT-jatint ran** cvbf-ront it', oncL/s .rreie.jierteie7%tol ine cijies aj, mtQWrttnent. 

lor, Ifct aire- A, (r& a, 4our-"raw \ivr\C\\ out by ou I 

Craii«tvcrsa\ ••»iH\b t-amae A BCD.; 
\V^ V aB : aC ^ AIB -. AC , CA • BdL = CD -' BD / /^5^ 

1. 6. "U** Cross- Ta"tio Q}- the" bftncii = ' ^aCD); 

ancl tABCO) IS {|m «a'Me vir(iatev«sr the uyc& «> ni«a«urMiic«i(: . Q-£.0 • 

Lor, I. f^ cnss^rrd>o <tj aufwr-r«y jkenc// it yitt tJttmae/d- i^ ine rue 
ffC tTiterchov^eJ' /n />*>'r-s , 

Cof\X. i-tiirt'- r^jf^a^ouptnoil U 9iftrt, ther€ ii 9nt rv»j a.n^ out an4 ifitit 
ComUttcs ate k'^-'fcii i-n tc Q/vep cros^-rmittQ . 




§.b.tK.a:\ 



CROSS Rm-tos 



17 



Co.^. ^, 74e cross-roii/'os 0^ a?»y finro irojcoh'rc^yt/res ojrJoar e/e- 
Tntvts Cr«-»i9fcK or benfri/^) are eootJ, 

l HE0^.2>S . IHhcrc (ft Zrf9iry'e.c4^eJ*jto^s onJ ll an wrtfU m •»« 4ikcn 
ue d^ict^el Uf a> i/ar/c^ye jtoirri, the, ra^io ojf ^/Wsi'nt irzars Of tmsfiinf rmi/a tb 
are //ve rab'a of </msio-n />> "the otfiw-ji'mvrt. 

ror. let cl, (A. Ot/ byvo ImbSy B O' hv-c/^ poiiii and P ou' uari*«ut-le boiiitaii^ 
leV. ©.', dl', B , P' Ijfc the itnMe^ (y «/, <i/, B , P^ 
tkerv V to.BP«L)=CcL'B'P'cl') ' [fcktt.cri 

.'. aP:P«i: oL?'x?'^ = (x.B:Bd .' «JBTBVJ, a, consttoit. Q.E.D, 

Cot^. /V trfoit\iur»S it it> htmaJoau Hit 
mfSiemcft o/ > t/anai/t JLornt m otin, e^ 
^ew ^w its vtLW shiny /»« «*»r/'e« 
««" iltt. raZf'c of hhe ^tsianetx oj' th*i 
htint •n4 't<s irnaje jrom tnt ttnbrt 
of htmcJpqi/. 

Let S we Cne cunxre vV VM»inn*\oau, 
d, dU 0/ bo'\ ' o^ coi-nci<i.i»nb rouj», a.,ai' AWf 
/jjon- ill ut,o i-nasj B,B O/poir oJ[ ln*«'A. 
poivi'ts, P, ~ j. oojiY o^ variftVlCf bofints, V 
the van. 4-^ -5 ^sT.a ojfi<<Oure O'tW-j 
tnv^ . ^i . Pi' =SP: SP' Mij BJL: B'i'a: 

S&.SB' , rs'm.trl. 

ana. ouP: aB:;jfirft^'SUP^:B5^,W». 

«'. aP; ol3; «JP^eJ§« splsy : liTw'. 

_e.t a, ue. v; 
bliin '<* (t i« ub in^iTtiba wnL a'P*: affiji** I o" 

♦'. vP : vB = SP:SP' : SB: SB'. % E. D . 





Sy 


K 






7 


K 






/^v 


'. \ 


^ 




/ \v 


' \ ^ 


/ 

} 




/ \ ^ 


^BV 


..' 




/-»— f-lV- 


.vf-V^— -. , 


ifsr^~^ 




/■iM3::-::, 


nci 






.tt'Y>. ; 


\ 






> \\ / 






7 




-• 






V *^^/^y 




/ 


/-».—* —•-«-»-•— 


■M / 
f 1/- 




/ , 


^1^ 


i--»-«yi»-.-»-. 


^-.-.--. 


/ 


~ yr^^-^—^jrf- ' 


/ 


^,^--"'' 


/ 1 




^^ 


^»- • 


/ "^ 




"\ 




/ \ 


S V 






/ ^ 








/ ^ 








' i 






^\/ \ 






^ \ 






'^*. V 






T 


1 




♦v < 






-- » 









X '^ 


i 



pRajECT»/E a^meJ^r. 



i ^-^M 



SJh9 




Q.E.O 



,, ^, Ic'fc'S, 9, be i«& tcxtire awA, *ius o^ two iiocwc «n 

• •Chef, •'., _ , , _ ,.or« tnberebc«<riv«>Jr»iiiJ 

Cor. I . /%• crp5»- r*^/tf /-n uih/o/f "tif a%f^ and oe-nbrt d/'y/Ue the iff- 
ff«»t oj'«k/»^tr of /I'/ee J^Mits Js eova,l to t/tB croti-rtctio m \Atltlvri ihejj 
dtuldt the toi^t »Jf O' Lair QJ^ /i'km. /t'nes . 

For, UV A, Pi lie a- WJr o^ like jjoinbs 0T^ bke iJkt liwts o-, ou' , 
ken V bie rawae-flS^A' \s a ««,ofci«-r\ flt^tito. W-noA aSsa 

;, (/\S«fl') = CftS*ci) CJE.D. 

'his Concbft-nt crojji-raiio /« \n%t hara.'tne'ter oi-tne nflmoloGU . 
- ^ EXAMPLES. 

ir&ss I /jiftTicil, ' ip«i»ii 

i. It bhcr* itc two triaTKiUs, oiKil l-a^ifedLJ ^. on^ a r«i-i«ile jj ' m fcht blvne 

- eA/olf> ql iheTTt lie «o reiatei Qt«V Btese j'l • aivicic two j . t 4 «>^i^ trimole 
iiairc (^ eouai &ro«s-Tabios, Oian tVi« iw« ii«t>rfr« diccrtta^ uu Otetw* vijriftlic 



•jy art brojeotivt; axt^ ooTiverac-^j. 

5. U- there Wlwo s«is <»yioor caUft-narj.. bcired ir% fcnif «?iyen »vaij, Thv,-n 
JrQY o^ oivcn potTiC Or li-ne< iTt Cne- omtUct^Jre, ilic like, jsoi'nb o>r (me pr\ tnc otper 

f.Tviuo WroIe,&l:Ive,J-ioyre5 sotk Biai oo jt"""^- r>°'^'' o3r i^e one js m W*|j«:- 
tiva. with its iTno.oe. m -tfte. otlier ©.re, m faextbettive. rnrouj^oi/b. 

iiCfore are. cheir own trnaoes \n the. oHier, are m Kotnoioowi antL iv ~i^ 'fc^^- Its. 
I ) homes, Li ? I • L 1 col/ne^ir. i r > iv • 

tYir)o»-6 4t/g, CMe tvnj |i<ji/inss c.oi>v&ic^e trnrOu<}>i<Jt.'C. 

S7 HARMONIC RATIOS. 

f TL ', ., 4-« ^t J the wi''^04-n"fc 'V f- ^ ^■^'^ ^-^"^ '"^ '^•**'*' f o«'"*^' *^' 

t.o. Ihe ccy\ ooote cJf^ ..T' . -■> -^^ • ,1 .1 ' ^j 1 

3 ^J J O Jo^^T^ioL-Uy^e Q> «-»\ an-OJe li tke other -m.^*.' 

U-ne. 



u 



THEOR.25. lla.I^^-r'^'k dm4.el Ly^.r^/cJ/y L iU*^^^ 4 



30. PROJECTIVE GEOMETRr.t [Sl-fcu-an 



cano 



/J cC/'ridecL /ja.rryf<!'»ica.//y Ou tne athey- 'reversed'. ^.^. 

Cor, I he ^/)c cnm-rzd/o^ ol bit -four jsclnts o^ ou ho/T-Tyicrf^/c 



IHE QR.JZ.iQ . /n ciyJiarrrioritc rarxoe the d^zbiiJto& flV'm C^>f*l^ 



■jrffyyf c/ii's bolrtt' /» the oi^er Ir/o jbomts'. 

For let A l/e kht jjomt. Dibs eonjvjgabe., 3, C-, jaoinbs «A.iviol!>»a "i^ie 
Se<,.n'vfc>^b AD fiarme-rv tea liu, \<^4'.>vf| f- li > \ W r. -^ K T. f k^ 
fct^ey^, '.' AB-CD^-AC- BD [^Wa-iy/ 

;. ABCAD-ZiC)^ - AC(AP- AB) 
.\ AD=2AB AC : ABtAO. q.e.d. 




FoT, leb ot Che imi^-iooi-nt oP blie setprwewt AD, an^ B,C bet-wtfcs 
dr\vn '.' AB-CO = -- ACBD, a ^ ,"/i£i , 

.-. (ao+obUod~6c)=»(ao+oc)(o^h3D), '^^ 

1,6. (0D+0a)(0D-0C)=C0D-»-0C)(C'^-0O), 

Cor. I . /Ae. hivo ^e<pT?€7iit& oj- tu harmoK/o n my Pwrh^ »<n^ 



n^«H?a 



HflRlY10r/}& aflTIOS. 



*. 



FoT '•' w\t|>T«dloc't OSoO — OD*^ o*iA IS bosibivfc 
•'■ eitWeYBorC lies Le.bwc«nO,D aW bot^ In 



t, on Che socv«\e i\M i 



^E.D. 



Tnertt taj'Ci^narmoifC 'm>'Jt out.s Cit'rtjfftantflez | 

the orrcJe o^en Zhe other segment a.^ ol/tcmeter-/ Vb W c/ J\ 

an^,corn^erStily, /P hrp t^'re/es cvCeeufk other eJr ^^^\^^^ 

Tfonta^o/ej, etcher oL hhetn e/A^/'ties hmr?»fom'colfv 

Che d/ar>feter oP the otfter, 

IHEOR, X'Z. ffrn «su norrnenfc' he>ic-n ct^joouir aS ctrHfCjettt, ravs 
ha^ry'fht OLyio/e^^ theyfcutt^ffB mfd-^f'rrej oS 6fte o^er- /beu'r: 

'^r, ouC tne horvno'nic pcMMl alrccL tu o/ trainsyersal 
«'. 0. IS one. miU'lme v^\t c,^'»x\4' 4>- U bite ootcr. 




«.77( 



yy/ 






'M aiem farm * navmcn/c y 



3i. 



PROJECTlVtGEOMrmY. 



S7.IV;i. 



^c'1 


L ^^' h 


f 


"^ 1 






-G'-. 
.''"' 






D' A' 


/ 







'.hiiB 



vti,«iA'c:&'i:;&'F;BD' 



, ,,, I , ,/ I? nftriTiryirc 



Twrioe" o.^', oe, <jj[, iro(. 



, -' It 



q.E.D. 



/vertic, -=^^ olfo^on^/ Zr/atiiyk; and, crOTiircrse/z/ . 

I ratios I '-^^Jf 

Lei E1,F, H liewiechrfcfc poi-nW/VwwiH oLra-w ocnti cow- 
veniC'-nr \i't« «,v^ lom E,Fv i* two pomts fl,Ci \iMin Jc/iwA 

Ut B,D lie Ut6«l:lier t«-|>onits,- Uien K, iiie co-]pomb oj- BD, J 
EF ;« liie jiaint »ouy.t. Q.E.F. [ttva«J.cr. f 

lit Cbi's InroUcm, An<Llierea|t«i', wnvn aniy one «! •.^iir <^ coTreiaArivftj 




W K 



^^, ^v , 



HftHfyioNiC RATION 



rnQD.5. Gt'u&n iuA> sejmvinti of online,, -bojinl -iwo Ji>o/htS ^imi 

Sha^ll Un/idt* eouoh (U- hh^m harrrtonicajl^', 

Let Afl , BB^ be two segTrfe-^-t-s ^ a. 
\mz s;u|joii flfl',BB' as ckoT^S »ro,w two 

OTUes c,a', -m«.ofc;n3 m E,F; let EF cot S "^ ^Jif^T ^ 

m 0, wJi'tK a.5 centra &mc wviU* tboIuS 

Circle, I.e. 5uci> tfnoJt t** = 0£0F, droAA/ 

U bW se^-meYibs ouer ia^/c^e jpro^uuc^ OE-OF Is -rjejatioe aiii 




B. 



ere. iS no so 



Ivjotf 



ov*-- 



EXAMPLES. 

CtiWt* tvjo tarallel tm&J.,bi) (ii'sectT a- s«^-rneYtb (U- cvte si fcv^eim, 
Jl.Gt\Jc« ou se«TnaTifcCE9),an5> Its mio-tomt H, to bwuMt a. li'ine J3*.r«ilcl fe this 
saQTTKydt tKr^vA^ii ec Qiwen boint ft , ^ 

5. (riven OL^eoTnenb aiii itstrMO-poinlj 

l)«T of equfibi Vwrti; covtbTvA<<l~o,li'v>e80 
laocTftUel tg the Qii;ev) segnneT>: a'nS or> 






A' 



I > 111' ' y '"^ ^■^^^ x \ ' "^ V" « '"i^^ ^ 

J V) I, p f / / 1 }\ » — — T > ; 

'\, Given o^ urcle, aii^ its ceritre,. | / / /' / ^x ">. ^^ v 

ui '• \ I ' / ^\\' 



o. Giiien. OAv (Aivxalc ownb awe In- i//,^'' 
lecbor, to drowu' Hie ooier Ijisector-, 






'J 



h^_ PROJECTIVE C-£QIY|ETR Y la^.Hi.ao. 

SS, PROJECTIVE. R/^NGES AND PENCILS. 

iHnOnoO - tfV ' ,, , Ocbfursolf Jorminctwfi ^. 
' — ^ -'la,4<lj>, 0,C- ' "^IrwfsJ -^ lf?er?cik 

^Y ' T , , Hiher&n the-/. J,, , , , ' / ,4ormed frvm thret 

hairs oj^f' oofi a/vuf J-ourtn b^ir arv aqu-i cross, then. cne. iu/o 







„esA8(::..A'Bt'...,^, ^^^^^ 
ibcncils oU^-v'*'^*^'" ' 

For,bn>Jaottlifc4iTstTange. so llwt ABC JQnye^J^S mlo APC «Tiix mto x 

liien VCABCXJ'W'B'C'X') IWsj^. 

= (A'B'C'x') [tlrM.cc3; 

tncir imAaes m Uie ether, bhe '^**^ji "^ ., cwncice trrrov-jhin^, 

Cu9,X.U;r, ivuo j>roje,cUvJr^^-3^* a-j^'"'''^ oj ant. comcik uJibh /A 

.M«« mtit€ uthtr, tile ^'^'O/T*''^ vj «*^ /» /aeri/>€ti{ri't/e. 

For 4 A=A>en ABC- "jn-ojects |rom 5, = BB'-CC', iUo A'B'C'" • 

/cptjcorrent -^/ray J liiencil 

tie ht/m-IP*^*^' ^ so ienme.h cure in herslteotive; arti tft^irf ^ 



18. .t)\lg| 3R.] 



PROJECTIVE RAN6£S» fiW PEINCILS 



3£ 



/ir rn heTsbeoc/i/& ou't'lrh eojcA. of ^ 

Far •; 0A,0'A' comciia, 

;. O-ABC-; O-a'B'C'-arc in )iii^|«eKv« 

^0 wii mC- OctA i^nm6'y^f(b'0'-. 





Q.E.X;. 



InE0R.3l./>7 ^w;/7Tp/e.c6t/e/ ^,, £^e/ ^/'^ of cnn! iwir ol- 

jimints , . irxKnoe ( j,- fco-Jmz , .■ 

/ray's Ibertcii /Of-po/nt ^ 

«t9«5// J, . nrh) iZ4U>h obfier-Jnnm cuMy^^c \ 
-^ /Jbroje.ct ^ ^ ■* 



Gerftm . 






f^, let ABPQ-, A'b'p'q'- (ieUjecthtranjas 
Wnereivi /\ connuoes wi'tK B^ ani P,Q'«.Te omj two liofiits 
«p« |rwrn eooH Twoe; 
then V tt»e bepcHs P-^A'BW" ', P- ABPQ ocre \^ ^-^^L-1% 



V W 



jjersbaotii/e £H). 3o tr. 1. 

^ \ PQf PQ T«|Je.cfc mb ^acl^ ^er iri»v*\ iiic &vis tf bersbtbbnie, tWt cl»- 

Wo^PB'~P'B a/vH;.P/^'-P>\, 
I I.e. -the |i-Keil liW A'B . ' - ' ;i ■ 

I- 
THEOR. ^Z.jf a6a/r of vm'ebUe boinis so ^/Ue t^Jxei ^^c > 
ihat }jh^broSuK.t aJ- thtir 6/ stances Jrrrn J/Kei Jjt/ttii on if^e axes /« 
Comia;ni,if?eya^'n9ratejbro/eci/tfe rarjyeJ of-tanieh iffejrre^ p»t,tfj 
ve. i/fM x/aaUshma homki^ oni, contfenttfy, any Jba/r o^ ^rof^tt/'t/e rv^jfS 
-may hsom^^r^el. 



;:S. pffojECTrVE C£or(^E.TRY. <3 [S8|4i.3i, 

For, let V,V' lie Uo iixei ^Inbs w^ Hie Ajces, F? P W> v»*"'-N <^ 



.'. PP' gtr^eoAes the ^icbuf. rtrngt*, V, A,l; • ■ , (h^ " ■, \^B,D. LH^.ao. 

^^«^erj^, Iftb Vy U Uie. DaT^ishirg ^mb onk IJ the jsolnbs at rT>fiV»tj 
d h»n jmuech'ue Tomgw, anS kb ft.ft'U a- [lair c^ Ijl(e!|»xc6 poirh. ani P, P'(^ 

Hien ^• CVftPl) = a'AW'^ IWi.^cj:3 

/. VP-V'P'=yA-V'A'. Q.E.D. 

Iht mj^bvint ol-llje i/ori'sln'Tig jpet'r^ts oj-twt) coaxi'o-l [)TX)je.cfci\)e 
ratiQCS «« tioe centre. oU the romaes . 

Cor. /P/7t «?u/o coaicial hr^wtfx/e- rctrioes mei-r cenirre. l/e hx^ken 
as a^ponri of vne rxxT>^e>, it cuis the seomertt {/eiu/eer? /c3 im^je 

ojfi Uk tfanishirty ja<n'nh ^ Bk, obheTrmge: ext^^^^lH > '} ^^ ?t»-2^w 
Acu/e tiA/o doalrlehointsj <xi ikima.^2,lf{Jf€yhavebtAi: cm& domfrfe 
fjo/Tft; mierna/Jy, ij ihey have rto dou.lrle point. 

Yor, )«t V,v'l;e, tlie vonishirij ^joir^ts, t^eJr Tn«i-|!>«i^t ft-ni I'b 

tlienv vo=ov; VOVo'= VS'V'5, ^ \r O '^' ' [hy)9.trv.: 
/. V0CV'0+0Qf>*CVOi-OS)Cvtos-O5), 
.-. YO-00'^05'" 
,\ OV'.00'=-05'' 
an^ iVVjO'lie ot\ tiie somne, si"^«< o||- 0, liKeri Ov'^OO' is ||»«^Itiv/« , aif>^ 
Httf^ ore hvo toiTvls 5 eoui^is^ont VroYn 0/ 
IJ-O.O' comciie, ihcYi 0Y'00'»9 3»ro otS S i5 o>tO; 



St.VlR-343 PROJEICTIVE RffNG£S fl^fD PENCILS. 37. 



itTe- 



\\y yO lie or, oli>V>o*iy:e si^es qJ- O, \;lie.-n OV' 00 Is Tiega-tlve-, antl tl 
ts Yio ^oukiU looivit- QE.D. 

1HE.0R.33. llj.L^diTCL-rUUh'^'^f'l ''^■*^«'/; «, co»a^««tcr„«. ' 
I t J / roAfS turn 

»ny ^our oJ Jarojec''itire^i ^^ 'T^* ,/ r<4A^X«i/-c twoJLovlrle 

■r ' /conomrtir-i'c oenc/'s 

iU V lSPP'T)=(SAflT) J :v S> '-"''al'- 

.-. (5P/\T)-(SP'A'T) '^ 

.'. P, P'obTtcra-iK -W I^TOjecbive Tomqe^ SAT" •. 5 A T* • • , 
dmdL 3,T «.Tc oUuUe ]ooint«. QE.D. 

Cartt/trs^/if, l»tS,T be tke JlouUe pomts 4 t^* cottincl ran^»5 vn^w- 
«UecVivc>/ P, P a. jawr ol- vwriaUe-poiTits anJLfl.fl any -lixttdLbositions oj-P.P'i 
yter^ '.' CSP^T) « fep'A'T) j 

iLol kVfo co|»lft.T«ar benciU.OTieTnoj l»-m«Je U eo'»ioiAe M»»t^ tlie- <AV»r bjj 
s\Uit«« *nd.iomI-y '«t m iU oirrTa U«uTf»fc.tlie two |jaTje'il» *Te»^<»/; »f- wj 
tarWmo it oyer, the»| arc <^/«c/'C» . ^ -v ^ 

tulmr «/ ///(« vritJrlw rmifS /Ir censttmt; in two ojt^oiite ^A 
LmdU tkt ntU-Jines ^ o^/itiir <?/ /f^e T0,y9 kawa^jit' . 
•^ <^^e/7^. CGreow. ^^ 



l'\S 




^a PffOJECTtYEl GEOMETRY. \silW^ 

Cor. I . Irro corice-nirfc tfi/mJ f>e,yioih Attve nombirlrn rw, rnrtd. 

Cor./-. ~h>/o corrcenirt'c. oja^9.sri& JL&nKftfS n^v9 ^Mt «^#iA»4 ruyj^ 

at r/tjht ayt^sBi to eotch. oiAer- eindr Vnen- ^eoiioy-s uu a, trayts- 

II ^ I III 

Vvr, ieb V, y tre the vani^i^ma |>c«ri& flj'inft two coo^i'obl nm^s, itie 

jjoim si's 

>'. tt'c«Utv<e8»i 0!v' [HiSIer 

TTse^-n tj o'o, OY* 
linen OSV 15 ouTfoKx a>v«l». C&e«>»i. 

'kroutfij 5 flMr«.vy IS\ b«.ra^i*l t* Che avjS", aT.<il «c i^evin'-n* t^'m 

for \" ZjSV'«=-i^Q50 -<^VSJ , tGwm. 

• i| klie jfTsV jaRYic •! ■« Cam«(JL W«u^n an « (i- t^c«e •»«)«« ,tW ^r** 

ra.vjS SI, SO SV coiy^ttci* vyiH 'iV, SO.'Sr, 

.'. SP c*titc<dLes v»iVh 5P, l^.^c«i 

.'. Wit two |)«noiic «re c^fr\ Q.E,D. 

Cor. fl fituir y 09HC€nirfC jjuy^mtjUv ftmwU is m. r9^mt,it*n 



%t.fa.'V-s 3 



PROitCTIVE RRNftES fiiNO PENCfLS 



39. 



dUik^lrlft poI-nbSiOTi*. oTTj'^na-, bus ho^vJr.o ty-'t 



t, pocnCs ,OTi*. OT Tj'^Yia., bus hft«v<r.fl T>'v'o i-e.aJ a-ni s»i?«*ro.Xfc aouvK. pom'b^ 
two r«w a-nJL coincvjiie-viir (^coye- poi'nts,OTr--l:>v(i )VriAanio.rWiloiiW6 }>ointS/ 
lvii«« flfct Binlt«ctcmce.s ±^ VOV' £>0' Jrom 0,-klie ce,T> tre. of t Hfc ravine? , ]%^^, 
0V«00' I'o ncooiive, Hie i>vo boi'-nl* c^i'siari iV— OV-'oo'|rom Oere-tii-it 
/Ubu^s vLcnt' ^oulrlz' ibotTtis, 

Oo Xvfo conca.irflric' proic»cxiY<& penC'iis navi' br/o vcm a'nd, sejiere^W 
't^'fO T«.fti ftyi^ cotnoae-nt, or Xvvo jYn«am».Ttj dowlz. rtvvi 



Z. T<Wi S , 



pROa.4". ^'rcTT Three Jo<u'rs9jy io consirvct iw« Jorvjuhye^ 




on S/ A! B'C/, three liW«. JMmts o-n s'; 

I. ftn "Bie co-li-ne cil- «>»^ ^ouVo^ Vike 
b«mrs, fl/fli tflJke. 5,S a-n«/ C(mvc7^i»Tit cen- 
tres. Icb B" be bkc co-(>oi'T,lr •} SB, S% C * 
t1h<Jb oj-SC, S'C', Ok^nct fl"fcfa+ of AA; bV 
bli»n ,rom S, A,B,CjjTojett'^ it4»A"B& 

throu«h <S wtolW si jaaraikt SS cyttiTifl «" ww V , thcnV tiie 
co-|jo»ntoj'-SV"-«' 1*5 He vUYii«h«'-no bomi ol A'B'C-", 

C mto A'^r^"/ iKen |>voi««/t Al B, C mte A'B,C cu airov'e. 

t. DTa.w ktit coUe i". •(- AB'-A'B.AC-AC; 
iUm A'P wttn^ «• m P" ««A Al»'wtti«^ s' .W P.' 

Pp03 5 T» ji'yid, ike. d^uU^j^mU 




40 PROJECTIVE GEOMETRY- [S9. 

CQiisbrvck U\c va-'ntsKinq |3oi-ni&V,V, Ine cvp^rttO, its rmo^eO; onOV as 
Utocmeier; J^o.y4 «l cirolc/ Cnen tne otrcle mioce ceinire is ani( T«.0(ii;s m«/oI 

boi-nts ^ouaht. 

l> lie witnifi Qie ciTcic'OV.uicn cne circJe wne^e cemtr* TsOondL radius 
Cdual Co Bit r\oil\^ Giiorjl |»ei4)6n(UculaT to O'v cubs cne oius in bne iJ««.ic oj- the 
itncccmaru cLouvie poi'nts. 

EXAIVIPLES. 

I. Bu -meliiocls nvrt, snown Wrovc Irrolrs. ),Xf. 

A E c 

iSiyUc ^ lcolo»c»r. 
IvcrticCX I concurrent. 

3. b' one tnaaiAle OTCUrnccriu* a-noUicr, an m|iviii'&-nuTnv«r oLcnonjics 
iMkH oedLrkyrn ctrc<rni«crib*i. olrowt tne itrs^ mnd. mccriliccL m Die •Bfer, 

lnrou«yi •oxJ) vcii«e •|-Bift')trdb Tn«i\a)e JbmM matis c'rot/ak Qic cnrec vcr* 
Decs «t| tile dttier; nieLene'ilc s«|«mi«^ ore.cwrobu hff», inberKJUeciivc, ovik. 
a-nu orree tike tms itrm one <J- iAc tnentlec 'Couvnr. 

S9. INVOLUTION. ^" 

K»« ktte cam* n««5c wU»4i|cr >^ U kmkvn &» «cn el«vn»nl- o^ Uie om |«'3 vrc or 
Aj^^o^a iks rnioj* ore. e^r/vymln/A^^^ ^ Bm mvolwt^n. 




' ' ' ^ f CoricertCr/'c f>e7ic//S ^ ' ToAf 

are f-n. ffrx/o/vu/vyi^. 



For, lit A/\'''P<3;,A'A'P'P Wivfo coo-pcial j)roj&ct»ve Tu^^qM where- 

m vi\t laomt A W* Vnc -sa-me- nnaoe, A, lo wK('cKev«/r ra-nqe Jt IdoTiq^; 

tkcn '.■ (.APQA') = (A'P'P/^) ril»-rL.cF3 

= CAPP'A') ^^'^ 

.•. QsP' ttk:* 

J.e. every bomi \\<kA tt>e *ame. •wage' t* >v1k»c4vever rw-ngc »b lt«ion^. 

Cof?. /. CoosyLfa/ jarojediye vayiy&S iwosc iranlsfif'rt^faointicomcg'/t, 
involution^; (XTtoL, emHrenchf^ m f>*o ctuKTCfo./ j»Traje,ct'UemMts 



rn nvwotuvtayir utc vcLrmmrfo 



CORl'/^ baJr^^i '^^^^'^T^. 4.€hiirfnirt£ ati intolvb'm^. 
' '^ J coTictrrren-t Ones 

Let A, a', B,0' U b^ra jjom ©j €<mj't««A;e. |oo»Y»ts \r\ «» m«»lutf«i^, 
P ax VI otlxer Wcmi o>\ the' ayiS; 

.'. P y»a^ «n«« oundLbob o-ne 5ocVi coyviuctale< QE-D. 



JIBA/^'C).CCBB'A)AACC'B) , , , 



iwu projective: GEOMETRr. [S8.tt.3t. 

f«yr, LBhPi'QLQBQ'hyCACC'B)^ A'CBV^C'B : ^a'-OB'-AC' 

= - (A/l'c'B) :(/\'A CB^— I. Q. E.Dlif,jat^ 

cmi 1,fl', B,B, C,C orre,cai^yt{g»:\:e|>omls m i-nvolwtiorvx'. Q.E.D.liHt:50,3t. 
Cor. ^ /\, a' B,B,' C,C l^e yrrK^fmirs oj cnijv^cutc jtwrftl «/ «w 

tm rvTtoes nt ntmJt/tiwt wiose centre, /s i;fizju/uRd^ jmn't/ Oj^i convene- 

For '.' «^ wwge^ ao IowkI w* jnvjccbve roTMies, wicrtirj tne»r voni'tii- 

,". owtf are »» inyialufcfOTi. Q.CD. Crfi.3W 

ConuerseJy, li ue/ Hie. centre oi Hw. Vnv/o^okloTij 

lite coTisJ-oT»i |np»Avot 0P< OP'i's \ir\K,J>ourer of l^e I'noolobW. 

Cor. J . U thchour^-r oj &yi. rnuoluh'o>\ Oefjmr9 //fee tiwvhftfmi JhiS 
' fttm^»i.f're A 

For, lit le the c*»itre, 5 tiie olouUe |3ioinb, k* tt* jaower ol'Hiciwolwt'oTu) 
ti'.en, '.' 5 is "tS owirt co-nioqotfc, 

/, OS"- k" 



QtkL Cner* ore, \:viro<)-p»ft.l an3. camcid^nl: pOm^S S yvwose diihjr.ces Jttjw. 

I yiejO'tiV*'. 

Yi mvolutioTi Hiot no.5 tvyoKTne-O'l anrf. comc.»<^e-nb doulne ^tetncnts »i 

ri-maQl*naru. 

J^fe • \ u- 
0.^ improper mvolutio-n, 

COR.,vL. ^; on /mhro^er fnvoh/Utv c/ie a<aU/e elentent/i are cfKjvj*^ 
oj- svry Pcfttr e/emeni e/ i/re mi/oli/h'oi^. 

lHEOF^,33. Cf'rofes HrreuQh ZVn ^jteeL ^r?t^i cvi ettiy irtnuttrerm,! rn, 
tfit C9iyvif».te Jaoirtts oj- 07i /nvolvti'oyt 1>v7tose cenire /'i bfie co'jfe/nt'^ 

t^ l\>r, UtS,S ue two Y^i^dk bombs, an«l letonv^ tnmSweTsal 
But l^*- o^ord SS «iT»iL a t»rc|e Bitou^ S, S"w«, 0, P, P'; 

ttevi. yyfiose ce-xirc is 0. 

Cow. A' ^oU/m» '^f^Uw ***^ ^* J^^^e^nJr 0jim 




44 PROJECTIVE GEO<yi£TRY. [SS.tt.v^4tf4i. 

mcLit'o tAMOi iwnpTceid^y 'Men deh^rmine.. cnj mvo/ut/tir? tA^0ie eUul-le 

UcUonf y»^oM <re so QencratenC, 

lor cne-i ' ,1 so d«he.-r»wrwe<A arc faTroiechVe. 

and tl,t Wed \ ^t^ax^ ^'^ ^'o"^!'^ I ^'"^ W' » 

' ^^^1, JT( ^ <^ew«Tute.«l are "m wxTdlfyticr-. Q-E.D. 

I beincii JS J 

c^/'a-nre^ar- ^T of u^e^'jfxd- <:.rrcJe: ^'71 an /ni/o/v^i'in^ wftoze t/ov/r/e 
joomts a.re ^ T. Ufe-iS<xJ. 

THEOR/H fl ^Mi^^ t^iuJ concf^ftkric pencils oe/tende/'lytilnur 
oJ rm.tfy at TiQnt errafet 60 egicA -^Ontr are /rt rf9f€Cti\ft fvuc/utior^. 

rcrr. xne, benciU ore- prpieofiinz,^ 



5^Vhv^-H8.1 INVOLUTION. 4:5, 

mA. \' e.Y&Tjj Tflij Kfts tW satufr \m0.9e', o-C rtqlit atioles bo ib, to vy/dittiaveY btnc^ll it 

and vnvre, m ho c^ouLlcrou, 

.'. vnc ]atY\cnt> are m iic^ooitiye/ mvolytioirt. (j^.E.D. 

\j^.J^veru irvtnsvericJ cuts €»,JocJy~ oj' z.qye^/ f^o-nct^nirrrc loen^t.iJS 
qentt^raicd. ly *^i>o./'r~ d rtxus a't rioM Oi^njles Co eaoA. otnenr in cu-harr 

riQkt cnto/ts ii eacA otfterr 

Tor, leb — K?" UK- the. bowenf o^ om moalv;bov«., ibs Qjentre, V, Powv« "taiVcJ 

U vOP.OP'=-k" _ 1\_.---- - 

.'. lifcj oe.-fcvy«.e.vu P, P 
DtoJH OS l>e.Trt>en<A'»cola-r li PP a>\oL fcotlce S iAAti> "thcrf OS =- K.; 
ilio> '.' OS IS tiie QcoTYiecino •wvea"»^ oi PO^OP 

;. PSP' (5 a. T-i'ak-t w-nj/e. Q.E.D. Secwi. 



\ (-v\ U., ■^ /"^!>-i-«-*'.i- ^ '■.'■■, I. -5, "-<-i . 



Cor, -^</Byy />«//" ft^y^e^e/A" />r mvo/otfoy^ /s a. reJ/cci/on^ of a^^/T 
of effva/leyic,//s tt^^oje like rtnyJ arc oJr-rtJit ary\tj/es. 

IMEOR. 4-3 . U ivyv he-ncikic m mtrofahffytr there /'s tme^arr oj conju^oie, 
rtius a-i Tiofft ayio/^S} attti ij- iftere It "b^ inch jaairs, tfieyi, evcrtfitorr 

fbr, let S-<»-b'--,S-aIu''--l;e,bwo joei^lls m »7iUoli;bo-w,a»J lei"Outvwfiswer- 
S»l 5 cub witse pencils m ^B"", A B " •, ivvo ra-K\oe.s m rv\.volubiow. 

"TTiTOuoh 5^"cl M-Icn ce-nbre on S^olraw- ct oivJe^ oubbm« s 'n\ G, G-; 



kh. projective: geometry. [sgntt.»»fw 

vWYi '.' G,& IS a dio^-mfbitr aj'feii^ c.'\ruv ar\t^^ «v Uomfc uboTt it, 

.'. {jtfy ^aMvN <f ct' ore. ok n'glit (x-nojcs; 
or>eL '.' G, G art; <»,. Icvair flt- c-PHft/Qate. polwlTS [thi<|ori/'<iJ|. 

nyxd, \\ anolrttr ipair oj- c«mj jaale ra^s be al •n'»)it andlu, 

S i\s egyftlkj tJlisbftnl: |r*vvi S,T, T&ewH.' 

Theor w Ti. ^.,>« jf 'hHM^^^ H^t^^ 

a-t«olv, & oin^ b^ns>ji»r&6.l HiQil' e-t/bs the fl|>|*os«le' (irff&BCi 
hi .r. L, \L , lilt sidei Cf>, BD In M,m' bi>e. «»4«,s ftB,CO m 
N.fy'i |rotr. C, Jrrvj^.tl' lU itttta/sA^EBM , EfclAft mto bi)» j--* 

- - Q.E.D. 

f^v^ Ut^ de^^ ^m^j/*"* <^c..^,^^/W« if^Ut^yii^fx, 




^*s 




UiJ tNVOLUTlON. UJL 

For, l*k ABC b& ar\i^ iriav.q\t,, \tk BC,Cft,/^B ly^cyk bvj a, 
bro»sueTSal -8 m L,f/I,N and. Iw \\7\t,i cVvroavVi <s [wiiy^'t D «m(i< 
iW Jwr^ictS m ft,B,C, arvflt lei s be. c^k uu tlieje. lmtsmL,IV\,N; 
ihtri '.' L,t*, M,M, N,N' art jjairs eV CoY«j«<>ftt» joomis i-« n*- 

,'. tf»e|pe-ncil D-UL! WIm'nn'i's w I'wuolubiow 

.•. D-(M'UL!N!)«D-(M'Mnn'u').D-(L'Nr/W)=-/ l4h. 

,•. (B/.AtKc//IB'fl)-(/irvcB;«~I [la2. \|. 

^tVKS^, Let CBLA'cHeiMB'A).(ANc'B)--l,aW let L,iv1,rV ^|co- 

,', CBl.A"C)»CBi-A'C) 

(Cor [^**'®-^ theorem. Chrrctrrtrft /t\te^ '^i/^/'^*'^"e/^ 5f>f'.- 
tm/e t/wtlff Me Jfi/es in ?*tC«j %tA«it^dvU U-J ' ' vneL, f-9nims«f\j 

for V CBL/^'C)(CMeV))(/»fVC'B)«=-l £*. 

0>'ill '•' Hita Wvtdwoi- (s bl»« ratt'c ol" the bnroJUd" ol 0»t Totus i^ oIiviiv«y» (fcil^e cars 

«wdL '.' ll»tl»4ib»n oj-ine tolinMT paniil « liwle^nienr ol^-ZW" (f ^« cwcuTTtntlwel. 

'■' ilietwo j)r«d»)*.h ore St|>t.-rckV«lvj cimsVomi 
«nA- ',' »l Utt l»«tV>t5 be Uljen »w tV»e l»»«. «H mji-nilu tlit |>rfl(*v»c.V' qI Iheir Yftti»s 

,\ m Lwiiwcl^ »« alwuMS -I; 0T«^ ^ oH)«r \\-^\> Q£,D. 

^■^ 11 tB* baTfcw»t»r o^ ort\ noTM»\o4ij be — I, the no>n«/oo«^ u neur7rfo-r»fc.. 



tfg. PROJECTIVE GEOmETRY. [53.ft..%IV,t). 

AaJ ifie yjajyte /7*i«9e co lAyi/'cmei/er- j'/'oure ih Mo-y%q^/artdf crnnfertehf^ 

o»ni'<yheverjtQvre /h oe/o-yias ^e vvyoJi'aor-es a-ne m narmo^u'c ^o>no/oQ^. 

Y^y o- poinX Oky\(k. tts rwta.ae (xre coli-rvea.T LoirU cne centv^ av^d wrve 
noTTrvtcrwit c«rv\ juao-'tes oui CO cUe, ce-yvtre <vv\eL' (Ao^I-S, 

fciCr ej- bn«. ^ovv^ele()u. 

f Rob, fo , G-tvey\y ivya ^irs <)/ cvyyoja-te^r^'^^^ ^ ^»,«^ mvo/u- 
-o/a>u,tb coTtJ'/ru.c/' 6'ne KonjojeciiC- 'f '^Jij^^jfv' 
I. LttL,*-, iYl,fVl' be twx) Waits c> co-nit/o ».ie IkiitiU 

otrtiw orwv to-nucnu^i^t l-v-.* ^g , otv»i let C-, D vt 

■liien I'i W', tde cuo-^oi-vtt oj- Lrw-CD, tKe ^.'-wt Sovjfit.tthff. ^ e».E.F. 

/ \^ ' ^ 




"tneeti-ntt rw S^THieirv the tiVt^le STN cuts 

Trie do'jllfc boivts »\ay Ire -looviii' Ui Fy-oIt 3, 




j^ii^ .«_-___ ' INVOLUTION. ijg_ 

EXAIMPLEIS. 

! If tUeJ^»''^^=o| c,^ ir.vo\ui\or.\l''' ^H tU ^,ve;ei^'^' oi 0, Uv,,l«. lire 
I llT'MS ' IV)fl.si thro luerticcs > J 

lco-mi6S I iL -^ ' V. . \.L iL UL 1-- |y*^'t« ( :| L ' J . ic«»ntirtTcinfc. 

ico-poirtts" J J "^ IsiolftS 1 J IcoliTKWnT. 

^ .Sides i /^,or~|»oii<b fco-^e.arrwt. 

3, I ne, mi'id-'ooi'wts o| cW? 'iAiaqo-wob ol- oo jo-^-a-V^IiviK' <xr«- coliv\eaYT 
,, , , . , icooymi ,1 .; ira-mQe^ • v o- • i ■ • 

S li- t\i«. W<llt.s «|- o- ^ nft-ng/c (a cut ok cooa( ony/es Lmtta-uj throu^k' 9^ 
^i», c.«yvty« ol- ty»e cJrt^* Tn«. tnvee se.«vy\«.7\ts cure cot\c.i 



^t/TW-ii r, , 



(»»d Cewft.. 



50. 



y - u 



projective: 6-Ettf»1£TKy 



_5!2*» 



- \ 



SIO.&EN£:R/^L PROPERTIES OF CONICS. 



/ / / *J I'r-eK-noe.S. 



xyyt 








5 






is We. iwattC »v. Oie Jfrsn" oj-j, m the SCCOTii; tnroMn 4 «-r«(. iaiv 






H- • I jlL l^^s5T,SP,S<J,"; 
this cmJe OTiol tnel- . , , 

• • WCV ' , . , , ore proifc&Ci</iti, 






Itfl.fckMTt^ GENERAL PROPttTTlES of cqnicS- 

lTar>«es t - sp q <« • , t- sbo^<i' ' ' 

, u .. iSTPQ'-. . , , , . .,,.,.. , .STW-V 

.. the+iQOrej ,, li mperstec^n/e with pia uTtle-^ ,/ , > 

' J I «c|»o/«" ' ' I sc par '"' 

IL - LL j|>«"tiUS-P'Q'.«-,T-P'Q'-- L ' .- .Hi5>PQ-,T-PQ-; 
cne>\ . chef «« , ,, , , ocra Wec/xiv^. w«tW , '. , 

wncL wir.\ eft&h otiner, [htj|).^nv 

,, f , . , are broiec/bv/e. Q.E.D. 

co.,|flhV3 4 twojj^i"; tUj^^^^o)- ^^■r.\^^l.^ «T.d tf,e^.|'^.p^j. of tfte 

Com, -u. -e?"5f'«'Xr;ji.y^' V^^"^' 

l/MmiA df-^niae^ of- ^e ».j(0J, 



ss 



5L 



PROJECTIVE GEOfYlETP-V(, 



[SiftbkiM, 



Q.E.D, 












1 co-H'6'trrrertt 



JntTTi «. .irxe^/ ' and.. cunVHrre/^j , 




<Wj "-VS 




for. lei db CA &C it« su, hei(-«-<lovv •Ti*C'rvlff;^'«rv «, CowiC,' 

o«^ |Tin|e4;b!\if ,01,(1 iKt biro Trti^jS i« ftVM \*n*. JmHi^ «m fi iwa««&, t^, M-i 
,-. AB; AC' t..* fch* |»^wU B.C, 1^ Ttk-n^*! ABtoti*, A Belter llniJt- «»« 

.', y^e'-^'s, 8C-b'c, (^H-th i*% *^.\.%«r. q. ed. 



tUkwjc . ^ - &Ei<ERAL PROPorrtEis or cowjcs. ss. 

pyr, le.b ABCD be aTOj|our-poiwt mivnutt^ n\ a coyii'c- avct \cb a, [{-nt 5 &y6" 
cl\e kaiVs ol obbosibe sides AC,BD, BC, AD m M,m' />tN,' einoL c«/t ifie conc't m 

and-.- A-PP'BC, D-P'PCBATe^ftohWl»e7.6;U, 

.'. PPWm,p'PN'rYl' are l^njjtohivt rft-njti; 
fln<A. ".' P Hols P lor iV»-ft.9e to wfn'chcwer" nmoc 
it ue)on«S/ 
.'t f^P ore A^ir <^ c<nii«i|aai& jtoiVtis o|- tlie 

iwvolut'on fAtA'NN'. QE.D. 




n sustem ai- conies theLt<r /I«ut<' is e^tf «4 cffwfts. 

Cor. ^ «/^^^ V '^-^^ ^'//"^i.j h .-y/^'"^'^ i^yUr, </ 

J , ,, I -^ one id the co-n/t.^,thBj. /i ^.dsoue 

i ''A 




S± PROJECTIVE CEQMETRY. [Sl»FVrT 

J trrro /t/'er&iccs. ' v/ • 

F0Y7 leJ: /^BC ba o^ T;r»*-ir»<jl6, a.-n^ let" a. corxio 

coir B C m D, D' CA m E.E', /»B ^ r f, aY,J la 
DE,D'E'cut/\BmG,G'; 

u«*i \* A,B, F>F] G, G- ore. conjuooire t>omis ol 

om i-nuolubo-n Qii/fyi. j/u Hic -MK/r-toitft 

DD'EE' ClK.H<| 

.". (AFGB)=(.BF'&'A) = (AG'rB) 

ic AG AG'= AF.AF'- 

GB g'B FB F'B' 

OTid- '." I4ic |>««T»'ts D,E,G> D'F'C' so .in/.Xc ti-e ii«ic5 bh^C 

BD.Cg. M ^ BD'. CE'.^G'^ _ 1 ^_i = i ^(is-. cy: 

DC ER GB D'C E'« &'B 

' BD CE AF_ 3D' CE^ Ar'_i 

" DCEAF8 D'C E'A F'B 

» 

pr /-ii fto L LL L BD ae'CE CE'.AF AfT^i . 

BC,C^f^B^ suck Wiafc 5e§£-F/V e^Fb'p^-'' 

tnrouali Q, d', E,E'^ rk»«s, a. coTn.ic v»v,osc see«v\^ co-JDOint w»'tK AB I's F" 

DC D'C Efl E A FB F*B 

,. AF'^^'a^,iP"^F' Q.E.D. 

F'B F^ 



SlO. Pr a.1 &ENERAL PROPEf^TIE-S OF CoryicS 55_ 

]t**^^''^ q.Verv U tVi«, other lUe, tJ,ef°>'-t5 ofJ^^ ^U ^U,V >W>aA«! 

lUjjo-n ' ' J /+a-n4e>itsa,lr'c,a', I- -^ I |.oiVil^^^ 

I,, ll: liTiflL B^zi^^J,''^ , oi- {ke \\v.yi(KQor\. 0.-S '<r\ bfie converse to] V;**^*^* 
I thro ' tSrcKz c' ' J (BrtoT»cJtoi6 

d, y one qj the Qive-n-^ T ' ' tuke ayvu j . .tneobbosibei ,, 

.. iUWoM ' ^ /UT\5»T>tvo^, ^-tjoof-nt "r- luerficts 

, u l4t»A--boiTf>b ; , ,, ., ., ibo/nbi , > i t < 

0^ '-.lef' ' ui . i?,7\.0u the c»tnenr«urV' , a»ve ovirv *Y\volubis>v 

iNOTE .These cxanstracHoTiS Tir)oi\i ue 'hiacle uWie-n bwT> c4 '«^« qiuenl' 

-* ' '-' /innoents 



; ', Hits], '^, '. ft (.I the co^^mQ^te OM •' a|f*"^ T uide- CftT^ic, 
jo.!" Ijiiomt, J -^ ' lo- IboiTioentw 









^' PflbB. S". Cii^r? M'^ej^'"^' "f-.a, conic, ti cc/n^^cf /A*/^-'!;^'^* '^U^, 
I foartqeytti CB Kontacii oj- 

\ X\ M lPoL.ica.1 line. 5 1 »| 1 lAABCflB j , v. v- 

»nnan.boiT»t'0 ' J | a a. Ire a w" 

'o-cof llmennno » coiita.tr a o, . 



a. PflOJ CCTIVC GEOMETRY rSlO.frtf.iii. 

|)»eirvt 0-n /^A 



^°^*-ii.w H>- "«' J^'^-'^''' 



me thro o-a 



1 tt^TtcenCS f hpfnX 



: nU>osit:K4 . . nir me. , 

/fov 






a- ire a-' (raM ^ lfc»'rt«|e"»it-oifc B; 

U Hie inwoluiiOTi be twsi'llve CWte toWcs ot« boasiU*; ^ntqalive^ Yicm«, 

PROB. jo. (^e^ ^i^ree/f"'"^ ^^•^ , Or.J. «W/^*^^ A T' £:,, 
deteT-mi-ned Ou thc-j . ■» , i n ll «-bo>v^ ,' ' i>v tKese 

/ uotli invalah'«ns l/e joosibve ciiere onre inurcaWc^* but 4* of- ^ iiirce 
[jjoJ'nbs ico-/rne , 1 • icofiCMrent .il n J t > 



ex>- 



fcwioe, 

EiOi "lT)T«e boi-rta dLettvDxi'ne <v civile; cvvo JDoi-rtts o-t»<A. v^ Ta.-yvqent 
tiM) olrwefc; ^ boiYvtr (vt\(^ two bo>.-nQeTr»xx,<7r' ck-ree ■taT^S**'*^/]?''^' wtp^ 
dec. CiVe/les tnTovLoK. ■twx) boi>v4;S (^^te-r-miTie ft-Yv )Tr\yol«/ti'i»v uljo-n 
ttr-v b)n».v\sv'«fV^o.I. —^jr, 

' <»-ff7re% ' /5t»,lro I bomb b: 

FmoL Che <iou(r/e |»oi7>l5 of i;he rrmjM ivc wVvie/fv tl)e joenti'k S-ABC, 
"WBC (vrc cut Uj s. Thwne a.re arwoyx two svoK b»)V-ts.,Tneool ay- »-»>-.- 

Lei ABLMN, ABPQf^ U t^ic Uo co-n.'os, ci>7ot in /^BPQR (umsfcruct 
aW-fxnufc ABl'jv^' iAA.J, H,at AC— >^L, BIYl'=Bry1,(vyxoLlci H- 

Iwro c«v^^cs coicoft eocA oblner vm&w cwti ojl meir j .f ton-JrA? 

/ (co-canbeMb 



EXAMPLES. 

/5Mm«nt,enV«lojof. '{aomt^ Ico-li-ne fenueJoJK 



O/ CiTVMC-. 



' A/.l]- i:vw» I^TteiJL cwvoenti Ito <v kortvlroltv he. cub Uj o*' vofln'alrle 
tii>:hoe>y\t , tne Ywarxaes So 4min«L o-t«- Si-m'ilonr/aTxi, conv»er5«-ltj,Hie cnve.lo|> 
cAcfte CB-lines oj^ liKe Uomts o|- Si'toMox trtt^bs is <x^ panra-ln) I «- , 

3. The co-boi-nts o|- like ttwa o|- ■l\yd obbosfc*, be-noi Ls \ie> odx 

^KfcU ci-n kub&rlrola. is ou 7-«c!t».Ti^ulo.y- Ayf>mn^6cf/au. . _. . 

4.11- Wo triou-noUs U i-n ^orr.oloQU.ttiej'^+T'^ oV ttief '*'" of- 

jSides torn. 'bomtS /S/de envetobs 

/ h . jciTX0rv»ScrH»6 d. ioor-Woint |( . , , „ 

U- a. conic j , , ,.l r the redtanqle ol,the di^ioi 

i lenueiop a .MuY'->li'ne J I 



acoTiiC. 



.v\c&& 



to tli« Te&t*vialf o|- ibs flUsto.-ncfcS -[t»T*» wi« •tner janMy. 



5i«J 



ftENERai- PFtOPgHTiES or corf I OS. 



OT. 



(5, C>to.'t6 a.-r\<^ ovas/^ the. coTrelatiW o|- E>-.7, 

lO.Hie mid-jsoi-nts oj-tiie 5ides o}- o^Uor-toiVt lie fl>\ «-cowi'c, -ynroo^K- 
ilia di'ouoTial jaoiVU , [lii 4-?«ril).H!rct;th.J0. 

1 1 .The Tni'^-lao»7\U oj- Mit ^des ol a, tria"n«i^* a-^ii tiie |eet o|- Uie ber- 
kendtcwiaTS -jro-m U»e objoosibe verKces li'e 

ci'We. tn-rojgli -tne tKree •mi'oL-jaomts 
ol tne co-toi-n-t o|- •fcn* cnrrr* ber^en- 
dicv/lcxs (Wlii tt»e tnT6e i/erti'oc-i C.6<t.|0- 

TTiuS dixit »« ti)C -niyte- jooin b" 
ctrxJc ttj- T3ie tnaiiqle. 

3. "The circle ciVoinmtcri'lreiiy 
o/lroub «• -triavvgle jsAfttes liiTovAgb vc\t 




o(.tii 



C■«^-t 



/l5lSo, ijiucn thT«e(\ ,a-n(^ bfte-j ^, ^, , -fcwT) «t zk&n^. 



I(i).T; ^Tt^MT ^ ^-'C'|te,j!*,;ya5;ve»v^j?'^*^o*voL^, 



ic *AntcnOW>^ 



ho, PROJECTIVE GEOMETRY. [SU'. 

Ico -Co.T^cil^•'r^t5 

17. ^. dm-o. o. .<xv.io->^^^",. il'.e vxr.k«0«vr,|«'h''"''-^ , of two M.'ven 

^ , itoorhinq , illW. _, , ,, 

itnTTj J JjooinC. 

Wloe;»\ lio the|o*n- ^scilu-tienz l/Bco-wig two sol»/i?»ov\X OL-h«C or>\. ♦»«- 

con^d ..;iW<j;v*w ^(Tv.,'c vu«i Wiab t/.el^'''7 oj-^vtat^^sses tU 
a.^iv6Tv^|,^^ fiwo wIvtievTS or -note. 

file j, J . erne ei- irne-v)^ , 

jaevlrolok, Ui-r-ouok tlie cewire ownc^ o>\* ©wM' fl^ wi'* ♦re, 



!0. To coTistrucb ox\ liubervt>ffl- (i:r,<k^ its t^-S.inrvo'totas, Qi\je;y\'\ 
-TOUT poi-nis an(A/ Hifc dire'ct lOv ol-'cvie as^Tntiuocre; 

Two Woi'ntiiQ-ne aSiMvaJatoirG, tvrMtl li"i«; c^ir^t'ti'irvL c!^ cne cbner: 
/W^ 'o tOYiScroc/t c^Tv .n-v.jbe-rl-oltA.y viluey. 
one boii^t a.Y>(A- UtK u.5^;yv^.l>to!^ec, 

,,2('|, 10 cij'n.sfcrctt i,C/ po>.r(A.lrolc«^ i^.tt() its cliVecti'tyvv U (•yvJUyuCvj, oiuen^ 

L, ^ jpofviti. \lWo 5olutitfnS or noTrte,. 

SI I. CONIC RANGES AND PENCILS, 

i^ the lioltt oj- Tneor. ^+7 tne "mea-ni'TTj ol ^ne kmvt^ f>rx)Je.t.hive mo^ 
u^noiei .1 i ^^ tlxe co-nio. 



rbr, Ic/b" CTie come, h^e bewcJi, at»<1 <« i/omtj oj- Cvnto^ U yieiwaje 



iT 



PROJECTIVE G£:0;Y>LTHr. 



[sM.H».a 



tneVr coTilro.txs n, B,C •■ '. 

L»ul Cn& penc/il a.\ro"- I'M ftn« Zayyavnt 8, 
and Anrri S , tne contacl- o-l- ^ i jsrojec'c tne vonae 

f\SCi a-nd -jrorn tne ce,-nhr& brojei t we- 

Hien ".' tiie-mus C-5<!L, air, SC^- cire |jcr|>e»v- 
di'ci/lftr Lo the rcivi S-/^,B, C"'; 
these two laerAtiia a-re, cjHrfccbki eoc/o-l; 




[£ei 



om. 



."- the TaT>oe «-abx>"- arid rhe i'eyic/i! 5-ASC*"oTcbroj&cti»'e, 

uTHCOR. 5Jl. >^ ^w aycon/'c hrojeA.-tiiiej '^ , l^^]!^' i oj- «wy 
' >'///77e« / ff>€Ttc// feo-bomt 

Farr, let A PQ " */ A' P ' • • Itc anv ivrt. co -coni't bT«ic>ctive T»»i_je», wwern'-n 
f?Q o-Tt a*vw joo.iV- oi k)«i"*iH toJreTv «■»»«■ -Wtnrv e** 
T»noc,o-n(t R IS a. IiV&d/ bomb; 
" *• HiefcewiUA-AyQ'"'-,/l-APQ''<»r&|»n»JBctti/fe 

,, I ft m' • •! Ch'^f* 

cfie ra« fln 1% its ovm »Ttwje 

^P'. fl'P, AO', fl'O AT* ri««^ and, rt^*«t rr»to e«Ji ctW jrim »i!l $ 
PQ\ P'(5 Tiijeot mta e<u^ ottier JtTwn -S , Q.E.D. [Hi *« 



bneii 
and 




i)A(.9;trk. 






Tbt Sri«n boint ol two JTrojeotlut conic bfcYicik and ^te Pascoi llw^tc^^ifcr 
bnieckt^je ranaftS |anrie><i Uj ilieir jaomts o}' conbuira^e Z^/^ emhAo/ar as b(i<ti e«Tii((; 

Cor. I. y^e ^'"'^*]j^^^ of a^ptu'r oJ- co-con re, jornjecbi'dei. ^ -i 

/Cfhtaiwewts ' lanan Joo/rtt; 

far, jet Q oe. cnz oi tne co-jwrnU •name.d,, ar\tiQ I'bi iwa^e; 
Oien V PQIP'Q rij^letb wto eAcfj etKerjnrms otkI Q it oviS, Cbli.hij|>. 

,', FQ' joasse* yrrouoh Q; 

/. Q'sQ. Q.LD. 

.^v 'I ' 

lien'.'POjP'Q-weet'mQ. ' 

.'. Q lies uAovi 5 cn^di ist x- co-point oj- s. flwrf, fhe corn't, (^.E.D. 

LOR.JC. /^<? >^«»/'y- (^ haiyfQSnht ajt the. &n^ of cu c^ar-d- oLo.CQ7h 
/o meet rn tn<e,poJe oi tne.t vhoreC 

For '.' bne cc-boinl:s oLrfie Pas&ai line ore (ioi^irje |}omts o{. bfie tu>o 

.'. bne bo^ttcmti chereat' are ci«vbt^e. tuij« oj- kite Te6i|>rt)Cral |«ndi ovd 
Tfltet ttl? Hie Brifl-n, bomb. iHi.si. Qi. cr I, 

Cor. 3 M »y //Tftf^ovCfiC eu cen/cf, i'{;i^o/e, aA hi bbgi 

corttOffS hn€ forrzTwlr of- cfie cortfO) ^^cL/ covver9e/\f. 



(otf-. P.^ai£CT! VE C-EOMtTTR y. \;St»..t{>.53. 

PROB. 13. O^'ven «/ oiro/e, tb^'r<C ^/>» doMeif^'" ^o^-A*^ Jltrv- 

• J.- JcoanuaJ -mrnie^. 

(Con c0TiCrrfc Jo&nofls . 

Pruiec/(r bfje Tow^es S-ABC'" s- A'B C--- -from omj- ]bomt5> oj- Wie 
aVole.. mU Uie ciVcuUr romoes LlVlN**", L'M'N"'. Let blie- co-lme 
4 UV!-lIlV\ , LN'-l^N. CtUTeuSccul liT.a)-m^eb bU circle, m H,K 
and bake s-SH, S-SK Jor Hie recoi'reol c^ooUe jioinbs. Ltn.52..cv.l. 

S.i;i. CONIC INVOLUTION. 

Ij "pi two co-co-ni'c jaroje^ctiveJ^^^J*^ evertf-l I**"" Jio-S the -Some 
rma&e. U miiofiev/er Ugore. it uetonos cne two j, ^ ., are m con/'c mvaiuHon. 

)HE0R.S3 . '/7 i^H^ co-coTtto-j J J in mvo/uib/ort ei/^e/^y 

Of own muolu'irierty^ \ 

For; let A,A'. B.S; P.P' lit U^ ^^ _ , -^ 

ol c<mju.<{«>-le }a»mbs o|-a, coWc m - i ^^ ^-^-jy^. 
•uolwtion; i^^^" ^,*». *| 

liicT^ \' AB. AP. BP T^»cb ;,*o ; ~~;'-^- - - ---^ 

A'B:A'P;B'p'WnHt,Ii$- /.'_' 

co-l liTte of tlie bT^'acbVe ' 

Twujes ABB'P, A'B'BP', 




■ ' tt>e tno-ngUs/VBP.AB'P tiL-re. m V>er«lie,ctive, 
.'. flR', BB'.PP'-meet m a ca-l^omt S 

LwT aal.blr be Orej^wVs oJ^ttqajxS qI cm Teoijorocal mvolutian oj- taTH)€^is 

Bleu,', ao-, ^/^, wnd Irlr, - B.brojecir mb a!(L,= A' o-tiX U^'= B,'^ 
tne oWoLTV. bomt; Ltks"?-. 

.'. Ss AA-3p, i's bjic Bnan-^xnrvfc, - ED. 

CoT?x^TStMj^ lets be. a.->v^ Uoi-nh a-nd throvoK S c^ra-vy \m^ tiT>es 
CutklnQ Die conic i-r /t, fl, B,B- 

CTlCn- tii^ C<rv\i'c iV\.uol-u.l lov.. cielx-nnirted. ba the ivuo |>a4T5 ol bo"nit"5A,ft, 
9,B'Kc\s 5 M)T Bma-w-.i-oinL", (^.E.D. 

vjOR. Iv «- coni'c fni/o/uo/oM. '-i//'on vo oCouJr/e ele-tnernhs, bne^ 

Vox, it IrKJre 06 no clouMe e. I em "it its tne BnVTv. i3oml:-o l\es nYithiyv^ 

bne cffviic, LctiS'i.cTT/ 

fitndl ofiords cnroucill it cu)r tki cqiho rn arts m«t ui/erlcuja.. Q.E.D, 

Oorttj^y^^kf, ijL eke arts averlcJs, die Miord-S -meet wtbrn liKe 
CoT>i«.jM Crie B'rift,-n>l3oi-nt, and bnens urt^ no drAl-Je eUmtrtts , Q.E-D. 



IHElOfl.5'^, /^ CO -come involuirKntS. /?av€ n>ie f^ajr of- con'ji/^atB 
(ll%tntnLi in Qcr^mryr vm/uss efi-C/hy qL thcm have, t\fyv cCou^bi^- kiltnitni^ 
^(M^h i^h&h ilie nctiV /nw/tct/'tm oCeiiemn'TteeC Oy Vtese vw^ohairi /s »e^^'ye. 

For, icb 5,T lie cjic Br)'ar>.pomcS at vwc co-c«rMic, m-wofutiovt^s; 
n^tn '.' lines CT\roy»ri S, T c-v^-t cne corn'c m coruajaie bo'-nbs o^ tvi 
two mVoJ-u-ti'oTis C^vi; 

.'. we li-ne STcv-ts cne c<rviic iT\< c»- bouV «*-boi>tfcs thoroTe con. 



i:3 



(A PRQJi:CTIVE GEOIVICTRY. [Sq. 

jucare m born mvolvtbionS; 
ffn<t ',' this line ouls Hie comic ij- ei'tfier S or I lie witbln It, 

,' d- cichcr \r\vo\\Jc\jr) nave no douWe joomt, fchc two m volvti ms have 
rr commorn o- jooir 0+ c«vt[i>tQflU%^, . G.E.D. 

ueb l-ocn mvolu-tions hcwt, douifle jao\r\hs H,K, /V1,^'; 
i[*en ', c'r.e tcw^eT^fcs |i,k, -m,Ti, ab M,)^, M,N joi'yt bw'rs cfconjuoahe 
jxuVits.Yl^: the Im? )3omb t^l^ihe o/«^c joomtH,i}iosci>^K,o[-M,i;f N, 
.'. hk^5, "nrn = T^ 

.', ST IS ^^a RxscoJ Ime oy crio mvpruiriovi oUtcrmi'Tieti bu cne WO 
ioo.i>s o| jaomts H,K, iVl,N; C^Pous. liW. 

ond ST ov>.t5 bhe conic vmUss ciie m'volu'tisvi HK IY)N U/negft'tiTre, 

Q.E.D. [t^.52fr7J. 

f conceyitrjo ' \/ ^ 

aa-ircs -^(^.tiUss the^ Jti^en ^^^^^^^ i/jedbui/e ^emenix over/o-p. 

NOTE. T^He-o^f^ •mo.-u be |>rov6'cl bw cJ«L qL tviis tv>eoreYn olS 

La-b oe Okie co'Cewtve o»- bwe cv>rt) oen&ils a-nii tVnnn*.aK ^roM 
a coYii'c cutting cv\©yyv rvo ■b^T> vvw^rUAtrjoviS vimase Bti<x>v, bovvii't'i^. 
and/ tuiix O (vs cc-wtir* >oww atiobvie/r (nvU^kbtrovt uyhose conja^oite 
Ttvus are a,t -norvt" anoleiS^' 
OWM ',' tVii? lo^t m.voi'vjri'o-n hous -no <i«uirle r«Aj, 

<'• lis Tinis cub cne come m mn mvolubion wntie Brio.w.boi'nt li'es 
witnin tn«r come. Leb T b« ibis koiTib; «nd l*t «ic line ST cult 
tne conic kx H^K; 
ooul OH, OK «-'>^' conju^cibe. Ta.\js al- l/om irwolwlionj, 

IJ- tnfc con/o b^ oy oircle- I iS o»e ectibrc-. ' L^eovn'. 



?ii.t!n.yy.3 conjug/^te pqiists amo lines. {^^ 

EXAMPLES. 

1 . Given o-oittJc and a.in 'mMoluri'oD of JP"'^ •^'to consfcruct trie (AouWe 

elements , . 

fi, G-iUe-n a, circle a-nd two mvoluVi'rnS ol-l 'to const-ru/ct the^ 

Cowman coTUiAaatTeA . 

3.&iU6T\ ou ci'rtle omii. a-w i-w.-uolviiri'jryu or "Twy^; l« co-w-strrt^ot fc><e 
Co«J'vuja'b«3 VhoJtroert ah nortt (Mi«/e5. 

rn oil cheSe, eita-mUes di'str'w-qaisli VECween. Bie •cw© c«.ses; 
(lO xAmtn bne cnciej, *"* it Uie-j ' C^> lA^cn itoloe^ -wot. 

S 1^. CONJUGATE POINTS AND LINES, 

THEO 1^.55. // ar -psMT loomis of O^ conic Cvoiaenis Ire drvA^^oi , tht 
JdM^hoirrt O-y*^ ^freJijur- /me So firnfieeL nave hrte Jormt djaoonai CnemfU 

For, let K;d,C, D U tde W jtwmb, K? 

anil leb EFG le. tde di&jonal mangle ' \ sdf^l^f^ ^^\ 

iABCD cmcL ^5 ^at o(^ o'lrccL, P A^^\ " ] 

en '.' {\yz ^eicaQons A A BCCD^ABBCDD <fj7U?^-^^ / 

nQY«. two Loirs «4 coTDtnon oj^osiie ^ i/ y^^^sol v ^^^^^^ ^^ 

sides AB,GD, BC,AD Xep/" c^^^B 

and '.* ttie Imxi'ts o| of>|»«'^« *'<*«* AA.CC, 'i^-^"^'^^'^ ^ 

BB, DD me^t m Che [>Qmts ac, (r(^ i»nose co-lme,tfie Bkscal line i'% o, ^.iff 

/.gsEF: 
So c»*FG, ^sGE. Q.E.D. 



e 

'e. 



t>2S PROjEcrtYE GEOMETKY feafart-fl 



THEQR.56 . l/o/a, y«yi^/»/>«^-/«'''^;"''^''^'^ '" 



Ci-' oortKbr 




% A 

loT, let A 8 BB 'jB 01. \ML.Ti0.lr)e «n4cT)te«L4<jtfr-jjoitit ^"'^^^ 
Suc-n fch<<i S, — AA ~0B, is|<xect; 
then. '.' n,/^, B,B tjlwe, tne conto mVolaiion im/iojC 
Bri««i.|ooiinb IS S, 
, , t^e. d«ag<ma.l boi'-nb. A B- AB , A B-/\'b' jie. on 
bhe Fo-ficcJ Ime/, 5, oj- Bie mi;o)ubiOTi . 

Convcrsejy, iel P be. tAinj Ijoi'nt v\ Hie PctstoJ li'ne., s; Hirouoli 5 cflTRW o, 
oIlordL A/^; |tt AP Cwit Hifc CoYiTc »<^am m B, anol. SB >V» B; 
tnen ','A A,B8 £«.re coTiiu.cia.re ooiVics ol ((te n'ivo'u;^ior) wiho^e Po-scal liisftiss, 

.'• fl'B' incets AB on s, ar^ei. so oLfc- P; Ltti.iTjL. 

.'. S, P M'c dio^^-rvLoJ jaciiits oj- H,e Lo.: |?omt AA'BB', Q.fc.D. 

V OR.2.,^ a. /ww^ »>.7irf M^iw rt»i «xis, Utc'tMilwr d the P0mt curni vJiortt^cmttt^ 
oricL axis and ctrtiy^ ve h«/a.-r 0o*J.t)»h^ CtfuCf e<»nJ*-rSii/\f. 



^^'Jlhfb "^ ^* '^'^V^*^*//^*''*^^- *» t*fvo/*U-t9n,. 



ffataUn 



For, le.1 S, C|^Y tti.ybj, Ire- a^ii^zA. line-/ F^ P oc JdcxiV of Uavialr/e tflr'a;^- 
onal bomts VMborv s; ie-tS Ire. bne- bole o|^5, anct L-f"\-x)u^A S olr-aiw o- 
d/non^ /] fl , a.v<L cffvustTTM.t/- the -kiar-tomi ^^ BB c-S /-»t. tf c ccn- 
Ve-y^e o|- tnc/or;^, ancC "no-viVi^ SPP *or its C^iajoyial tn'ciY|o/6; 
BieYv',' AJB, /IB aencra'te. "bi/vQ t)TDJe<l/"ve jbe/nc'iU, Cyi.w-7. 

.'. fJP qener-otte- twtJ ijTTOj'e.c'Cu/e r»rw&S/ 

,', P71C1.S tne same iwaQe to M^ii'c-he.T/e.r' ra-noc it leloYio, 
a'n^ H?a two Tti-MQCS ort, i-w iVwolvction. Q. E.D. 

-TT J- fjjbowts , I OY) in scTTi/re^cLJo u?r- to I'-nt . , ... 
vrt Confouf^bc^ <y.h to tne co>»ic-; cxruX Hi*:*, m- vol-vAtion. a|-c*AcK cOtv- 

' jti^OTVil . I line, 
'tv»ro 'point. 

Cor. I. 7he/'^:f""''^^ _^4 Ue oiue-KJf'",\a.-ncL 6Ae <u>r„\ar^ 

For, le/b s t.\U: cvxc tcvMc tvi- n, Kj 
Bievu',' ttifeTBuuJS /^H, A'H oiA^ S in. t^ft Savv\e jami^tH, 

,', M 11 ]tS O^ATW C0rMjULQ«.t«.j'tX>V«tto|fl>rK' Q.E-.0 

Cor, 51. >?>6«r 4jf^ tftoi «»« c»ryu^tl az i-9 * cW «?^ harmmic 



70 . PROJECTIVC GgQNieTITY CSHl. 

two an<l Uo, zo art, Uenr-Z^^f '',''*; C/^. ^. 



} .//• the come artc^orte /fe*i ,i_i j 



Vcr '.' i|- <*• tieA-tejt lie u/Itvim Che csv\tc tvic- locui cff! iCS Cfln- 






er- twt) verKcM lie- tvlrfiout; 




^. d.EL.D, 

oHut henc/f,' and,, Centrer^ely. 

For; \*)c P bmce ctTaviae s - P- " uboyc /5; 
.«»x *.' P.Ure c0MJu.«»le af^ f) trutfc^ aviin«e s-p... IjUfDfectivt wiWv »-P"v 
.'.SP/ Bic jiolfrv erf P, iAVee^i tfvter cl, |a*ncil bnijecti«/€ Writfi 

rf»e beiTilri 0^ the J^ric "mnae. 



CoB.7. '^«/f ^^.''f/V '^^o/.'^l y i^if^-'^''^u.., a. 



eu Cij/^fC/r, 



V ^^fi Q/tmrt haiy mco/r^'" ^ ^a-ir are coriftMcitz as Iro tf>e 

con/c 

For, let B, B ^t eLjivvn bair oj- bom Li ov* a, 
fVrMtr, n, any obher'bomb «»v\ Che conic, S (X. 3 
-«v*jA^rt.tfc o, BB; a-nri-ffvu BB take's we tole '^^/^^'yr^^ 
1 &; O-n/l lei; S/^ ftut bhe coTKC mfl 
uhttu •.' AB, A'B) /IB,' A'B -m^t on « in tfie ciiaj- 

onal jsoinirt R P'ftJ Wift|«or»^omfr A A'SB ! 

• '. /IB, AB meeb S fn fcjre conjaeo.te laomks f?P. ().E.O. 

Cor. i^/""''/'''" ^/ a.>arr <r//M^ '«- a.a,r„'cand»,Uir' 
iff*'^/. s I L t^ I Che eon^o. 




:iix 



PR QJEf-.TI VP &£0 M EiTRY. 



Ssxijhk. 



Cff?r/u<fAt^ e/enre?T6i «.j ^any ^ortf'c />* bne, k/tct^e nf t^^ 




RiT, lat A,L, BiM, C,N Utoirrt (([(^bosiU uertit** ^w , 
oj- a iear-liTie |«rtn«,d[ lu o- krlaiyl* ABC a-nA o*, fcnems- /S^i»1 

VwsaI UMf*i; l«iA,L^ BW be Wo boiir^ wt conjujuU 
ipivtbs, antil D. aii^ conucnienklgoint, l^e bde bole op 
.WN; let AD,BD,CD cAtfc LMN m L'm'n'; 
oien '.' L,L!, |V)«M, NfN are b«Lir4 «^ co^juaebe bei'iits o}^ 9ie mvolutivn'otc- 
-turmiTiedL Uj ktie i«aT%. |>oii«t A B CD . E^M 

.mid '.' ADL" ,BDW' oTft Vde |)olar« of u.m W^-H 

■'. L,L, IV|,IVI are Wlr« ot «onlvA4efce bembs oVdbe mvoLbian et«t»TT)il'n«dl' 

.', N,N are co'njVoateiS «.% )?o i:>e ««T)it/, 
«T,dl CDM'.t bt,e>darof N, 

.',C,N are coitlu^atttS 9\ be Bie coTii&, Cjf.E.D. 

ThEOR.60./^// co«.« ^«^4^*'^*^''"«'/x»«^^'J^*^^ 
i^a«/f J«^ ea»juj^t8 *J ^ oA'y ^t^^^^'^^'^i^j^^' 
^orMh U ^ |««^ JKimt, EFG- (U b»-««^ie.l%t A£.A^ A&Mel}FQ>jSf JSrtn I', 

r,' Qr'i axii. c*nt\sr\A*^ B, D, C ri«rmoni e c«nj<M«U» fl|^ «< to E,E, ^ G|iC/ >^ 

iKn ' '.' ET i« biie |»«k af F& OS b tm «m« ^dU 

ondi '.' 6 u Ute liemii-iitc c«n|u^l« J>A*« hi EE 

.\ B k ou |»«iVb of i(i« «iviio. ^f 4>< iM 
^ D,C ore l^omh 4 1^ cmio. Q.E<D. 




Sin.l>>.M.l Con J uG-^ TEL Pqin'T^s asnd line-s. "JS^ 

^au<i Cnt-eC other-/ ".'^"-^^ 'ofrc c*. s/nf/e -^elj- Cory'u<fate. hn'o^rv^U. 
J/'fi-c/t. co/y'****^* *-» to c^// ii,e coni'cs <4 Che// '*'"""'' c.^ ..06 <^-^e /^v'r- 

fir, littPU tlieqi'ueTx |ot»inb ujid l&L-,A,B,C,D 
U v\i »«jr lAtin'ts ui' Hit ioav-benil-'ol. Hie IjeJicil ot-cumcs _ Jii 
Q-n^. So ntt».iit«.«t Uikkls Cite avuil&s APB, CPD tU imtott, LL 

Blftn '."^ CK bviTB inuoLtittiiS Whost ci-uAle vukji &.r«^ Pft, 

PS, PCiPD ,i6kV/e. *. iai'r <r- co-Vft-u* ^iL anol P'' 

tu': AB, CO m niVali»i'nii»s »Ylio»*tikov*Arl* tidnbj ave.fl,B, C/J, 
1,6, in Ojt rti\j«lwtiaiil OflAB,CD y'l^Ti L e«/«.r^ comV a|^ Big. joanoil; 

•' I p, b t*k AB , CD •'■" jsc iTits L , L , lYi, lYl H)fth ore, CimjU»atc «kJ to euery come 
.', f^'s LIV|**LIY1, i« CoMjwtQute to P»» lo ti/trj toviic, Lik-S*!. 

. . Pr is CM.b hom^eTii'callu yu tu*r<« covti'o et Hie ibtmcil u-no m b«h-fci'cu.l»ar |m 
cue ojjljoJiibt SiiHas 4 t^e ^r-jsoint «| Hit jtet^til, G-E.D. L'<I<-S"i,cr,l^ 




/'iw/eJ '' I /me 



't conrcs Of O' g fuen^ 



1' , <f o^HTuev^T o^ to cte conies oh o^ g 

^jsoks ^ '^ I/me ' d 



I 

iHi^0f^>^^.Zui»C«'*/(^//^ie'''.^de^V6 0^ i^tjamc mv»Lti0» ¥f6«n(/iei' i 

K»r '/ bk&dt«^OT«Ai |)omts o^ like lotMr-bombs core nmoges, 

Qo-tfcbowts oJi U Hi« like ccy)»u QED. 

lite c\ttu LoniU «^t mj^mib^ ^m»wLJt uvht'cfi c^K ciVcl&s bas$ <t^^ bvtc 



fS ••i oenidrb . 






F«r; lei: Gi G' U yLckftles al 4 <a U ^ U<e ovyii^ •»<! Itir H (m> e. jaomt eo ^ 

aA ij^Hit muiLb'»A ^% JMti^ivt^H lies wtl^tm lioU\ e«y\(CS, 

aiii .\' S it c«w|(M*be b« ^» tf«»r<|j| AB, A'b'm U (Itttir toviitS^ 

.*. Hie t»4i«et «^ A,B, A,B* et\i omw L|tV tf' evovjwuteh* jiimb E,F» o^v A 

t.i,. AB,'- AP, AQ -' r^ttb W « mb VB/- /(p; A*Q/- 



Sn.fr.iH^H coiNJJ&.'iTt: POINTS AHr> UNES. ^r 

.'.tine ceWcS /Id" P<J", A'^'-PV-cmnemte^^Peotiie uil-') j a', ayt/: &>.t, 

C/wfl, I, /- bwc cottt'cs o«S-s tnrOu^h c/te Jayrre^u/o rmaoiyaiy joomiz air 
ittjivtihffOrtni i»7n» pvf9 retJ hai'»6s a.i inj-ml-iy j-p i^at <^' Loi-rrt act 

UoR. *-, nil u>nics inrtnt^h w« ctrouJ«r/to/n6s ai- rnJ/hJ^ u>^ crrxJcz. 
In* ce«tt»^ fli p«fl-st*cllue. oLtwo •i-mllckr con* 05 are CBiio-^ ol 

^.Pvi +i»/e|L ,vJi -B^«_ th« corio«T»M toYistrucl- two| , ' " 

J I Iboi-wb* ^,B»QUE <^ 'on -S |lioi>.M?(l. 

, 11 , (UmRQ; li (bokr iPa. 

EXAlVlPLtS. 

' Iau OL.uivJifc«j . to conScrvLcCbne muolutiin oiiien uuacontCf . ' 
'At J l|»mt, • J -J |«4lr,oA«'. 

ti„ y V^vn L ** 'to eo««tpubb tfi* muoluBov thutm \n h com'tr' . V'""* 
IU|>ow ' Hmt, J J lo,ir,c,d,e. 

aGlW-Yi rourl'^ «ni ft. JKiir J- ctM^tttV*' 'b ctnjWci- oie ovimc. 



7fe. P ROJECTIYL GEOfYlfcTRY. , [Sl3. 

Ac- a llwmts t « I I.' \i*i>iry\ WTiftjciS. 

T. Oo, Qivferi tnreeJr ar.o «.n »Tiuolw.ti«mv < 

J. JO, oivjcn cvifo|i , a.-n<iL o^ £el|-co-n|«A<4n(-~e bnarvole. 

^ C ^- (bomb f , (»%«»( . . , , 

"Oo. Oii/em ot».t|r .o.-n%tvni , ,iiivrolvtb<m5. 

the. itiuolubbiQ-ns «>e -ne.Qo>.tiu< 
bomV L'u Hie TneHiod oT- trmoTT^I ' it li'w w-porx tfie turx^ 
confu4a-te CnftTvale, <Knt iiTio tVic mvolw/t/oyi^ tJao-n Vn 



11- on*, or bntlx o|- thfc itiuolubbiQ-ns «>e -neaoLti we , iiTifi a, cu*«n.vt|cJri of- Hie Qicen 
iiiV L'u tflc TneHiod oT- trmoTT^I ' it li'w \Afor\ viz turx^ent". n'm •- S^'j"' 
confu4a-te CnftTvalej <\Tic iiTio tVic nTUolw/t/'oyi^ tJao-n Vn^ cWg ^ides Cr\»* 
b»SX Ehrovt.al) the co-bomb ci the aiuon «.TCeS; Oien <na^ OkS in nte.«>'. iTt, vT, 

-J -T- , , . itbrouJi ibtfi'iiti ( , . , • , 'tui»b 

/• 10 con sbru^oh <A tonic (i ,V /OtuPfJ, Cn^o <*■ ofrn* cl- Miifoq(At*j' 

ibOuJiiiiCj. l!iii«;S I «> -^ IlineS 

Or (>- QIUtY*. m.UO I *A»tl OVt . 

o. /\ conio IS m Uersbecbiue VAjith itseli Jrom atiu centre ar»d ^>tii 
Uiah ore bole qti^ boiar- iks to bh» camtoi ar^ Cne Ua'niSnifia li'ne^ comc'i'flle 
^I'dwuxt (letweeiv t(i» cC'Titre, (mi^ (^^ClS, 

^ Iwn «i(ipjr& I-0II5 sfiort op 

*/. nil arc or tA- jso-Talrol a- i.o otornis a line JMX»llei Lu ici o/nords anii 

on huberWU tttts 

imiwZni UehvcfcT) chat cnordl an^ Oie cu-bomi: J- Ui-noC'TtU at me. CviM 0f Cn£ onorc^i 

,, ibomtsliJe , ,, J jliirie y jlociu 1 'L .__ ' U 

IO.Ua.|,-. , tcbov u ,ye.clf , tbei , ,0+ it» CflTOTnor) coniuofttl 

I lime town • ' <|j«mb |eRW«lob ' ^ -J 

03 to bwro coTii'cs 15 a, cotii'cJ , fclireA U-iX-d.]}^ 'WT»tre(»f ot»« i4 oluiatfS r*»l< 

Icouwtino I limes, I -J 

11. me HirecLVe^j., |«u,t»^ w e/-7 are wej .. 0^ a. tri'anq/i Ijiet 
IS seJj- cottju^ftlre c^s b« fco^fi ol tjie di'u»-n conicjyan^ tC tvtro <4- tfiettf- 



il23 GONJV^M^ POINTS ANiP UNES. JT^ 

I \. lies '^J J' Ito-bof-nt L^JM swirtcnU tficreot 

\ju t>»fe o'nJcn cottlcS (wre/ r«ft.| o>.t\\ ff«S,rla.b. 

1/^, If fcvvo Qiuevx coni'cs hskvc cloier jow-T" or r>cj r"^ HietV cor.>- 
J «^ ' »co-€HTioe»tt5 

•mom sell co-niw.Qoirc CriaiKfl-e \% rz,Q}i yy tneir hcwv |j\A/h bvvuj ' 
Hiob tri'a-nAle' haS livJr cv^e re<il vertex <wii «yie(H\e obtojit^reo/! jr<j(e. 

13. |t- k.vyx> co-nic^ l»e 'lYv Wmojoay cmiu are also iti nowoloQW as Co*. 
Seioi^i, o-f-ii OiVtt «* *ec«ii^ tentv^e Of nc»'»iolofl«^| oir>v) they are. itv notn-* 
oio^u fts (to eitVier ov ave two ct>\ters Joi-nedL vy^lMi eibfitr of- fclTe "bviB 

(^)Lm; ocno uht co-loon*b of- tiie bwt> aites >'« b^e |>oU d. fcli(« cx)-liVe 
oi ni« two center^ as to t'otfv. tde tonics. 

' *' nnu rvvo CO -pa-noor corn'c^s a^e m }irmolo<ijf,ani were iidHier 
one j»ftir oj- cA'^e.s om'i) tenters, (as m Ex.lO^or bliree SvAeA. pair;. 

'fi^Tvw co-jlilanar to-n I ci l^ave *!o«Ar/^*'7"'" (rc«.l oriTn«amory 

iv\be^ir.s) a-ni ft^ talr o* j^ (at in. Evt.to) ore obboiite]^' *^ 

J ' I Icei^rs ' ' hjertites 

1^ i|o«.-bo;,,b-^^^j^ » ^^^ .co-^o.Vfc^. 

' (i«vv.r-l««e I ' Ico-tftTtje*^", 

lL.Gi\/ev\. ttxio i<Ve.(L jaoI-ntA S,S, tw^o -I'xtjct Imes X,t' arxl 

j_ 1, lL i J8eqwc*ibj ,|A,A',S /co/meaY- 
tbto bUct fc+»e tvy«j J I thaH ,^(xrt) , 

r la.'WQ/es |««-i«i-/3 /coyjcvcTrenc 



./-.< 






yS PffOJECTiVE CEOMETRY. fSfl. 

' 'f)omH A,A'4on ^ lime o 'SA.SA 

.Li'' 

^ T/V.TA'. 

A,8 Ike iniMfcS ojoon «,t cjt/l[B') Hj«.»\ ^S «b \jan65, CA,OB jc.^«.^».U 

'ill ]irtfc of^ Bie 7% Imes; jrom W aj-f>o»nk' so |<rvwV dr»M/ e^/ U »»6 
tiirouiqK tlte, sccotio boint tio'*y*«'*kr Hie s»con<i- lin<, an^ to a>^b1l "«.ll 
of tli€ T» Lomts liflkvfe been k»«isse<t R-^.i o-ll "f- tKe n l«'wM nftvtiretn 
Tne-t. Mie co-j>omt ©j- the last l»T»e »o oroAAm. «Y»t lKc|ir«b uitlliiob 



"•3, 

SI 



ddkollU Tv-liTie m Ac wcrtejc $<wAjki, 1%to(ajK — ^ ,. 

n^SWUnq jrom a- >iVed point, (M»m^ o^ li'-nt. u ciJirflbLVe^ Itnr^ 



l ' ■ 



XtNio ^ tectTMiS I^Vc^ ^1''*^ ^>*^ '<' 0^ ^''1 '^ ilr^ts ore. oroMA(yw nirovLoK. 

ou qlutivs. oct^file wi^ bfie-jtVst. 

DTlSCtA'U me s|»ee.i«J ta.st vt^ereiA bhe|1T<^ aviS tecsvi'b boi'nfx are 
*uwmtW<» at tp we jVnlr liVit, \jiie itc9n\ o-vi tfViri pcmts w.» bo Hie 
SecftiiV Irnc, 011^ ^ On. 

rlO.Tl>»-tiaqn a. QruerK-jbombto otcmm a- line j>ftrtt.!le! 1o"(a. qt'oew liSie, 

jLJI.To l»i&«J^ to aiid»Yv reQmewtor- a. Ime. 
vr3.Tw IjiSeot «v- Qiueyi a.vc n- a-oo-cle, 

iSTGiv/en ot Se^cwt" AB,to bro(iM/ce 'it" Co C so tnft't AC iKcill (re 

S14. RECIPROCAL FIGURES. 

Kuo ii'owres lumo na fche UAne o|- ou coni'cr, Hie one. ^^ bomb-t»'qur€, 0)6 
oli»«r a iine^iiaMre CM^t i$ wmifc vil» ot 0»e polarS, as b» CT)« com' t, <* the 
jjoiTib w|- bije jirif-iiovBre, ftTc r^cj^rvca/J/ot^ores.lJie. (xm'c •« tAe 
^J«, Ab«rnt cnrit «* toloT ore rec/%rve»/ e/ef7fefi^ ttf Oieivvw I^Wres. 

^•Q<; if a. jaomt cro«.e o^ curve, tne|)«loT-(;|. the bofmb ey^oele^j twi- 
oliicr cuTue, ooni tKe jaoiiwts * cne jiVst curve onn\ tf>e bmaenb /ro>e 
obKer foT'^TV "Etuo »c&i^r»C'«^ jiou-res. 

iHE-Off. ^^7 /S cff-Z/^e (^f- iwQJsomil of ayf>o/nt'^'^^urc an[the co- 
JI)ti)itQfde likt irwo l/hes d tire recjlfrtca/^efure are TtajtrouUlemenii. 



S9i PROJECTIVE Geo/V)ETffY ;>JH^>fe 

• or, !el.- ABC*'- Le cc boiV t'l-iWre Anc cJic •• • Jtre ^*c:■c^^»r«c.oi 'mc-^iaarC 

token to orW co»<c cus lrasc>; 

^CT A Bis tt>e War oj- alr/AC oj- «c^ •• ■ . Q.E.P. [tft.rV«^ 

C OR / ctvc run'-es ue. so rc/att.tiL iheet Hie-ponttSi <j/ h^e Uni t^re 
te*i//aroeu./s Le tht {etnjents tt iht second inen ih^Jiemis of tht: 
it.con'^ o.re rt.cf^roea/ to die i^^e.hi% i* tht. prfi 

fbr tK* "tanoiMRts t» tk* liVat /^Ju.re fire c9-imeA 4r c(msecictl»/e 
jsoirt^ ft-nflL the re^^ro^ol tJo »nis m. me lecuni o-re tKe Cy-j^wlnis yv 
ijit a-^clprocai coriS«,cc*.t>ue Cft-n«|e'Tr\tS. Q. E.D. 

tH^OKloS-TZe yf!c/i>roca^ aJ i.<- con/c /s <^ c«'^/c- 

for, jet b, b' he a. IsatV ol uon'^Urle 'C'-^s <^v<M«m frw«« cuyu [)V«-tl kotn^ 
fliyie Ciio&-n cottic a* u ■viuTttwye poi.>t pl> cv w.tt otli«:t-, tan' ..-b fyPlct Ctt 

tne.1 y.e reciprocal Cu.nie. it e^velojoBi IniThe liiicPF; t.»i.fe^c^.T«MJ}.|ifl. 
a;n5> '.' p,i?' cje-iierule tuJu pruje*'-tue Pfnclli tch.^i 

.•.a,^en.elo}> 4PP';&o.corM'c. . Q.E-D. tl+i.^T 

Cor. 4^ ^e We fl/ i^^e .me ah inl^nl^ lieiuhm a eonlc-. i/>$ 79^ 

Toca/ Ok the carttc /S u7ifha.r*«yr»f»^. 



iltffV».iy.«k3 t?EC H» KOC/lL ftOU RES. 

For.le^ABC./l'B'c'ittwo ^ 
iri(Knql«& Su£n Utoir ABC ore . 

l«a}B'C;C'A;A'B',aT,il«.t \ 
lYJ,rVsBC-B'c;cA-C'A', \ 
AB-LM; s» 




e 

tlien '.• iv» Ifne Wr-bomk ionmb 

Ui liifc criomqic ABC ) 

Lli'esonB'c'kkcfwUr-;^'^ \ 

oVA.fltniWonC'fl'll.t \ \ 

jpolaruj- B, 

:. N l.es «m A'B' ^e bolor o| C , \}^. S'] 

,', bSe 'trtatxqles ocre rn |3cr5|3fc«>fci«/e GL.E.D, LbWfe 

I ROB. 15. Gi^nn a^J^tur of- criarto/e^ rt<.lj>rocoJ as to a, t.ornio,iocorv- 

Lefc- ABC, A'B'C' lie tiie [»ir o|- Teci|>rocoi| tn'ftTv^les, S tfic aicix o|- ^ler- 
tbMbNic,, cwJbtm} BC, CA, AB m L, M, N, cmi S tde tifcTttre, t^K-tt 

?m\ P'o. tomb «« C'B'L stwJ, lli«t Hie nrnje C'b'LP'.s [.nyWr;* vwtfi 

X P amy hwnt, iS consbruxt UteftJar tf P : 

Consbv*^ tKe Wes of- AP, BP; Uicir- co-live, is kljt jme sou^fitr. [B..4>t. 

Pf?OB 16. ^««» ck^jbafT of ahjtittnar inemqlc^ m j^rs^ct/'ve, to ctmsh^ud 
He c§akjfary, *i latuhio^ the tyt'otis/es ore rec/'fnoca/. 



gX. PROJECTIVE &gomETffY. [Si'f: 

Let ABC, A'B'c' (;e tlie. oiW-n brjar^les, S the <aXiS,o the ceT)bre.; 
Co7)S>rT\A.t(: bde 5usb<;m oi Iboles a-ni bolars Ly JDTX>lrl5"/ o^nh hvke CTje dou: 
Ilia boi-nts cl tHe iWoluti'ffn detenmi'-neb k' twt> jj^irs of C(m|Uj«cnB 

ij-o- smgie jaosixive inuolvXiov\- -mau (;e so +inA;MO,tne cotiic \i Tvaf 
^M}'i \njjt ij- «i/ery ^*joc|» muoltAijoTi be negoiHiJe, tne coyiio Is mraynarnf. 

h susbe-m (^ Jaombs ani Ij'-ncs thoit-are jooles an^ jbolors o^s to kkt*^ 
cxy\\^i v/heH>cr rtfl^t covmo Ic reo-l oir i-m oiQi-na,-'^, is a.^o/or stjsterrt. 

TTifc welViodL flj-recibrDcal jsolaxs mov ^ w^ect tb jorove cne cxjrrelft- 
lioes. ot Hielmnexibive lorOJbcTtie-s of cor»i'cS)W liie Tecibrotal as t« 






virliile, tiie jjTojectiveTeio^i'wis are wnohoiyed, 

EXAIV\PLE5. 

Co-nstTu.ci- tWi>o)or susbem, oT)^ w covjio w4>eT» f^ftl, oive^n', 
I, A seU^- coTrjivaia.bt trjcmQJs «nflpd- nejdxi'we j . , mvoivariffw , 
^.A 6eit-cflniw»j»fttfc moTifie^aWe, otjO its bolair. _ ' 

^ Imdial ' ' ''^ . Mmes. 



8\y.^.faTl CENTERS AND PIAIYIETE-F^S ^ 

S 15. CENTERS AND D|A(Y)ETERS. 

me cerrire, dr a conic is tne bole, otS'to irnxb conic, ol- tn& ItVie o>'t mtini- 
b/. An axes oj o_ conic is o^ coTUw^o^e, q|- the line otinUrjiba^i.e. icrs ».-mj 
Ime oiroKoh me centre. Twt> a.y.es ctkkI" ome cenjixDotte Ttiajs oj n»e r«4)ta| 
mvoiulr^b-n Qiuen Uj o^ cofii'c ot its centre are co7?^c4^ette aocas ^wc com- 
ic., ^ c/f'arrretBr oj- o^ conic i* a- cnoro Hvrouon the centre. 

~mE.OR.fcl. (a), T^e cc»^re oj-artljaarviOv/eL. iiesjed: mUn'ih^ on ine ujrve^ ar.> 

Clr). ITjt jaoi'rs ef Imes ho /'rtJinlvf hfrrwan the cenire, e^a- corilc are ine <*■- 
stpnlatvttJ (jf-tht covlc, Qni th&t art me t/oulfe rays af-i^e rhvo/ui/'p^t <^ 

(&). 7?e OLpceJ (d- <K Io0krp>^o^^ ore /rs(lMi.Txi/lelJ /mes io ^'"'b/ 

(jd.).7^e Mumf>ba'tas oJ- an huberlrol^^ -hrm ^ narmoTfC hertdj ujlth 
avv h»'ir of canju^Qatt a.)C&s. 

(A)Jn omJ I ' a/-o.iA/ro/ con/ocqatc<^yescUi t/>^ 



^xA^rvB, 



fb-r tifie Ima. »b mj-inlbi J jrrws a- JcU-Cyn'jiA,<jeJre fcriocnole with ever ; 
Ukit* ol- coT«(u.o(Kbe aKes,oj- m\oS«. «iie,s two onLi out bh« cfMJc. Ii^-5=t • err 

l^od ovi'& of cm liubotolo-uihiJi cuts Uie cuive »'& the irtmsverse axi^s. 
(if). (^ o^borOi/liloQroyry cjrcu,7»scri6e a, tefnic-, 'W dt^^nmU *r-€ 

Br tiie li-ne oit rrvj-mliu jorms o. seU-c«nfu^'te iWomo/e wtrtk tiie 



r.n PROJECTIVE frE-OlAETRY f^lgtifc».it'>^. 

THEOR, 62. Z^ /we* j5W are confuamiBf nS t» <fc CffWCv ^ ».qfvtn 
uyus ^ thai cow'c, are, JmitoJI^I ^ the coiyUoate aX/'s. 

For '.* li«e|iole oiana)Li& It'es tm it& coijM^».te,l:^ line ahmjfnicj, 
an«(L ',' oil lines coTij»*gatiE. io cdeQJVW <v)u$ toss BoTJugl* "'•^M*'fc«'^%^ 
.'. Mie<e irnes acre' bom i)ef. Q.E.D. 

Rn; 1^ HK be. oyie oi Ui«se ckor^ XX'Uie aWs coWu^oeU fe tnese 

poie ot Xa , 

Qien \* I, M ore c«>iju4»te JEMmts as io uie «oym'o Wi* St. 

/. M i% t)ie cenb«.o|-Uie muoluit«n viliose dovUt |>0iViU(<n~ their 
ii«als} areH,K« 

Cor. I. 7^ tjerttrt, cf ^canro ai^e iniiJt0mt of eveiy StomeieK 
COR.JLi^ mnla ftf-^^ ^M« £ta7i^e^ra are i^t i/tr^/oe^^a 

jrmm ntS^rii^sL tin m^ C997/C are ctuttrrfters tj- tfte cpn/o. 

C0R.3. 7^ i»rMents to a, cortt'c «t i^ tn^^s tf- a. ii'mtnebertirt 

le <i^«v>f ^ •.-mf JMOit^ ih9 ciw/c, tnqt^ c^m^is *re^rn//e/ 



&lg.ftl.70-l CEISTERS AND OI/VIYIET&RS . 'g^ 

(HEOR.T'O. "V coyy/c may nai't (htc a.vi 6ui on6hei/'r (d rront co-nju-oate ^y.es; or 
ei/erif ha-ir moy ^f at r/ont orio/eS/ c».7?a iffe cotj/c /s tbefi Ou c/'rc/e. 

For; eiLner tKc mvola'tiffn oV coWiAaJXte ocy-es f»(vs <me o-ticI uitt one- jpair oj;^ ro-vis ac 
rialnt omqies, 

or, i| two tcjr Ire ou- tki--,C ».Tflies. and sOjebie-nj b»Jr; 
Lnen'.' cWos drawn jroTri tlie en^s cj (xtwx ijUaweteT Iq aTT\j samne |3oiT»t"o[ tne 
ConJc-, ci:re at rig W flOK^Ias , [kyja. m- 1*) . tx: f- 

,'. tne cimic \& a. circle, Q.L..M 

iTi tW V>OLTclrcla. OT>e ol tVie. rtoKt aites is tfie, line ok rmmiVu a^oLir/ie c'tbav iS 
tW locu-S o[ bine Tni'cl-bomts of c/ior^s lDcrbe7)Sicw.!aT to the baTO-llel <NVeS 0»|- 
L'rui- buTu-liolix, 

cop, I, JT/e rialita.yc.es da. cc"?/c are axes f^ ^u/ttmetry: cLndfCorto'cnek^ 
av ^n/^ of 6ifmmeiry /J one ol tne rfjntaKe.s. 

for, Hnew Insecir oM c-fiorks WrbejD^iculoir fe tnem, 

QOf^.R^.The r7'o/)t ay.es oj- am hifl>&r(rolo^ ar^ the •mi(i-li?}e-r. cj/Hieajmp- 
tohes. 

\yOHS y^e '5e.OTT7eni& of-a-lme moluit,oi veiween «. ci?r?/c a»i t'^s 
aiUTntjiote^ are eou^o-i. 

For feb blic oi\;eTi fiVje cu-t ^ e cotk'c m fl,B, ccnA tfje £»SuTn/>tobcs -ri H,K 
tJien '.' Hie aici's conjuoafce fe Hiat Ime basses Hirooo// H7e centre lYl (^ ine 
iTiwolukioTy oivew by bKe dLoolrle fjoints A, B, [Hi-fo?. 

on^ iiJiTouflh ^^ cenirc a|- ir?e mi/olutris'n aJx/cti lu Hi€ douWe potTfls Kl(, 

.•.AH=KB,AK=HB. Q.^X). 



tb, PROJECTIVE GEOmETRY. [SlSJliM. 

1HEOR.7I- ^/V»e /J7 6hej!>/ane oj- a. con/o /'& conjvugai^ bbkhc axu thro 
fU jiole, ant fth c<tJ3on^ with the. MC/'s k coniuqate la /is jw/e. 

KyT~OL. line is coniuoa-te co o»ii (iVie cKrPuC|n iis pol&, 
(icn6 tne tole I'g conjuxtoie. (^ oriu pomboY) its kolar [cn.Sfc. 

Lr'ieds the cAor^. 

me- haiX cnoro is ati cn6^/77^^^e to its coWtcaate o/Vi-s, j.e. to Hie <x.>ti'< 

an) li)e^oh«|- Hie tcoiffeyil" ot tlie cn^ o^ tlic ori/nat-e, i« a^juJr'tari^&»'t. 

\.AJn\- iC- if SuJi^anoerfi /'s oiAlt nar-7no>t/'ca//u lu cne. coirO-. 
^.o, cu|»aT0»-wx)(«, iii*«e.ofcs «cs c.u.(f(a>ia entr. See S"t3.Ex'2r. 

LX)ff-3. ZJc ano/e Ire-iyreeyi bum /fr>&s /Is &ou^ Co ^e Cavo/e' ottcnf 

twa /m€S. 

Var, o. line ftno oit o I'am e c er" ctrrD vtan its Ljle are. a/~TioKhaYi«/eS. 

~JHE0R.7^.^if«wye7?i^ & ec contc cuts the rx».di'»/ mi/oluh'im cf the 
CbMiotiie ax^s tJ- the. conio itt an ayc/a/ rnvp/u.t'/o'n U/nose hoHfer- is 
aic imbms/Ce el- the pouver ^ ine /nuo/utiort oive,yi ty ine cofjf'c uMon 
tAe uxishan^ d t^e tanaearrt. 

Itar^fe^ /\B oe <un) oiameber dF- a/ c^rwi'c asnh OTOT'ci.|»ouir »^ ceYMtM«ii.efl4M 

•Meter bocnoJIel ioTT; Irom P a- L«fiii- oj- Hie ^C^"*""^ 7 ~"^»1 

atm%, %uAt 1|mi6 /iP is JMnxiCe/ to OT, brvjett 

^8 t"li /I, B am A B; 

mi} V A n is coiiju«ale U AB,ac to Bie cvtmo, 

.■./\ is ttie centrtot- tite mi/tluti»n7^T o^ ^~. -.r~~~r:=r^«iC^:;v^* 




i\a CEIVTERS AHD DIAMETERS . 77. 

OLnJLAT-AT'is iUj^worj 
and '.' n,0 ore c»n|a«oibe> \>ontt& a« bw nte ci»niu,ani ic Uie ce>ttr«^ tA.5f. 

•'. Oh ' 06 1% Hie IpoWlor o[ tile muolution qive<-n Im the conio om A^. 
Bul- '.• AP I* |!>«ralleJ te OT, to»n«lr. 

.*. OrfMi' \s A/pwrallciooraTn, omiAT^O/l. 
So \' IS Hk. 7niVi[-f}omt o|. AB, ani BP .s (wrollel to OX [}ku^m\^. 

.'. tWon^le-s 0AVBOB"aTe, e^ool, anb AT'«0B" 

.'.ATAT'--OA-QB': Q.E.D. 

OOR. ). ^5e yreroUbcb of- ^e *e03ne.n-i^ c^ o/ ^fmgsn&^a^ cn-fsh <f«~ 
ri*Te» its oonttuUr ani cluv ha,/'r o4 comU^ecie aaCB^ /s C0»^»»tmiiLe 

Foi; OA'0B"=-|+ OBT*; fcke s«i«re <a W^Jr^^ , Kott <k-|meber m ii^eif fe 
I— I- ' Imaqinarv ' trMfavrwou 

ti 07C Jiomeit:^" haroffe/ w vhe vonoerftj ani /'ts r»iS-I>omi /t fts hmntof- t»T~ 

0(itt,ani y?e oyrer- dJMoncJ n beirti./fe/ "^ c6e ot/>erasujwAio^ , 

u>'T;leb mo. l;aTK|enb at A meek tne asumjabotes 
r» H,Kj 
Bien •,* HA *'AK,ccn<)OA'i« bara|l«.| ttP^eOM*! / ^^^ 

b^:|i«tTt, [tr:;L. f^^^ W" 

.'. OAKA! OA'AH <»t«^ (s^Talklojroms . R. E .D . 

Cor. 4: TJic lirvctutt of ihe seanfertts ofocj^ed. ieoToertt betu/e^n th 

90Uitn d tne half diameter Jaai-a/M tp cheffifedt tan^tffft. 




?y. 



PROJh'CTlV£ GEOMETRY. 



[siS.tiisJVTt 



[Vr; tneji'vei lomQeDt oJ'^, its bocrolie/! tomq&rth o^Q,axih c^ jaair (rf VftW- 
a>Mc |)aiY-a)lel rorMQc-nts TV,T'V'-|oTYn oc WYXU/Yyvscri bi-no baroil/eJoar oi-rv\. 
TtW i*f\ase dliooimais VT,' V'Toure covj agate- ff/te-s . rii iw/ 

J:^.c7x>//e./ c-a-TToents ve-ivL^eef} we/T-corj^'/icts er??l! \ / 
o. z'curioiJrfe. aurrqerrt /j eoueJ ko cue seuarf: ov 
i^-t nalfdioiTi^et'e.irjjaralfe.liolneff'xe^bmQeTtk. 



V 

\ 



\\ 






c 


^A 


f 


\ 




k 




x; 






— ^p 



!-or V BV=-AT' 

/. AT- BV = ~ AT- AT'. 

~THE0F}.73. fl hair oj- cowUoate. dieane^ers oj- a. com'c broject lruboixi//e/ 
charhs iirtc any oiher peJr of conjcMote ciar^erers ^ tnd' cotj/o , 

ftrr, leb .^B, A B' te ocnjj ^«uV tf^-covyw^abc ft- 
vmtttiBTS of Wie covii'cO, CD«riw omerwowietei; 

c-jTxws A C/ D D bcLTtxOel to AC, BD; bmw OX can- 
'i^aotetome-laaTaUcl cAorbsan^ OX )aaTal/«/ to 

Chen •.• O^XACX\O-XXcXarc|HmT»Wc|>enJsX0<,0)f'6.•5eci|•TJ5OT6^lle/^»/iC,/^^^ 
.■- OX,0X^ O/^, OA', 0C,0C ore. joairy 0/ cnHuaabe toys tn miioWriorv; 

avA '.' we, Ursi two jscwtsare cotijuaofce a;ces, *-^}iJp- 

.'.OC^OCare C(nTJuQat& a%es QX.D. [ol^.CDiij.oyes. 

1HE0R-7^ 7Irel»mv.lMo9rerm «w ^«^o ctmJMaitt haU^ d/'enrfeterS of a. om/cy 
.has CL corrsttait arecu. 

FoT;lctAB,fl'B( CD,C'D'lieo<n>jtwo(Kur5ofc8Ti- C 
juLfloLe diotnetcTS <i[ tiie coT>ic O, anh let OA,OC,Tneel 

>Wn ••' AC, A'C' a^^ ^nrollel, [H,.B. '^' 





ftffiiaSl CENTERS WD PIOWETEHS. ^ 

.'. Uiair TnidJMmb X, W. cent colinewmiU) 0, wb HtV^ "* MK, 

,\ Aft OHflT » OC'K; A'H A - KC'C, Le<{. Wes aT»S oIK 

.'. As OAfli -OCC', ay>h s«}ot tfieir SotcWes. Q.E.D. 

OOR. I. Tne JoetFet//e/o9rajn on iurtt neJf ott'om^^TS «/ e* Ctf7»/c /i eon^/ 

FoT '.* As OKA', A'KC - OG'H, HC'A [a<«>i;€ . 

.'. t^S OA'C "OC'A; aT>§ so ^or tAenr ioo^Wes. G.E.D. 

(XndL t^ tfmoe>-ni has cu ctrytsicent atco, 
6qu«J to ch^ karaJ/fi/aqram ov a,f>air of 
con/u,ytuti, halj- e(/'ameie,rz. 

THE RIGHT HYPERBOLA. [SlO.Ex.?. 

The OR. 75", ^ co-/»om^s of Itke. rwfs o^ oUjo^'lre l>er^h It'eonan 
fiyherOvlcu u/^o^e. OLSumhirffies are oct Tfjht ano/es apii hfie h/uo 
centers ctTre at Ihe turo enhs t4 ^ t^amei^erj' arui, conv»T*eJy, arty 

Cor. I. TJte. o.5utnh6oiei ol a^ n'^hi hyp&-lrvlaj irf^^dhfte ajtr- 
a/es ^&^c4/eev any f>a^r Of can^UfOfi-^re aryLes. 

COR.il. 7^e anq/ej biiu^err emu iu/o a.ye4 (^ cun'oAi^ ^v/*^' 
vol^ A7-e e^oW' to tiiQi onyxes tfci^/tert 6Aetr- conjugates. 

COR.3.7^^ c/'rc/e crrcuTrt^scninio anu seff-conjuJjate in^Ttqm^ 
a^ rfo/rt hyberlrolcLfl^ssei ilrrvM^ the, cenh^ i^Cne hyferMtu. 

C0I?.4. ^€ /aou,5 of- ifte ctnhz of ariJii hyfterlrttla, tir- 
Oumscr/Oryjf a, cjiwn tn'any/^ /J /If* rthie point ciro/^. 



SQ: PROJECTIVE GEOMETRY- r5«5:ri».7frTI.Pr.g 

TheoR. Ho. /fnyba/r of con/uaait> fOamtiert ol a. noht huberMti^ 
are KouycJ irt Iz^nobh; anS, coni/ersehf, d A. iirtofe ieJr of c/mi'i^ate 
d/'tLrneirerz of- art /mJ:)eiHnlgu i^e eoLueum lenqhh, the hrul>er(rtJia, /J a 
riQlti kuJDerlmlay. 

inEQRYJ. ^ cu c^r?/'o cirowm^CT/lrt a^hnemole unb he.iJ i:^ 

^o-Si-te ^rdej) /& /5 O- riq/it nyjaeriro/euj (Vfri, CorfWeriefy, », n'oht 
hvfterlrole^ i^-ro inree bOfnts,h»^V ifrro "the ortng center tff- 
their tn'onqle.. 

For ■." die ojjjjcslire si'bes o^wie mscritei Jour-jjoiwt; ABCD, jW»iei tvtfie 
y«rbce.s c^p fcne mainale cwi its «m)ocea»fcre ore n'^nt" c«yi|UoabB 

T»di'o.l mvolatian of Ti0 /it" ceiiju^alre Imes. A^'^ 

/. QIC •k$u'm|>h»tea ct a»e cirttcmscrilf c^ ctvirV arc erf no/rf«inolej[^^ 

^. 7lm)/\BC jxiss <xnw Twht n(/|9erlfo)a' ou£6mQ tfie li'nc at nvj-inJI^ 
Chen ".' l:nc copic /\BCDJ ts-anyttr fiuberlK'la. Ewi. 

,'.ABCJKcomc;<Jesvw-th ABCDJK. Q.E.D. tH,.««.crX 

Cor. ^ CL Tjqhi hiaiMie iie mitn'Oe^lL m a rr'M /ty/tef^irola. 
i6e i^snqcrd at me vertex ^tfie riaktanjfe is berhertti&vJ^i'r 
if the TtuffoH^nusf. .._____ 

iROB.iy. "^-fi-ni the cewdre mxh a* pair ef- C0jnii4Mtiit ^6- 
aantftmrs ^^ C4B7/*a 

fex» GtOen Jiue j^inti dtke conic- ^3^C>D/E: 



K^TneCer 



SlSJ CENTERg AND DtAMeTEt^S [ 91. 

HirovAaii A|B, ^TAAA) barAliels to BC, Cfl -mtzbiv^ tne covm'6 »r\ D, D, \hen 
Hie dliftoawol bn'«m*lM J- tf»e.|our-l3o»i»fc* /IBCDJ /^BCD*Uve tfie tjerb'oes 

ilie «m»c, t^eiV Co-jpomt i* tiie centre, wnl l^i^ir cerMJujafce^ are 
|)ftmlleJ hiAD'BD'; 

ToJtc 0/\' the oJ»to»lbc (rl>0^; tdet) /lA 1 5 cl Sm 
Covislruci; HE Hie. cor^iA^ate/ al-f\h. 

ubon tne conic an\ ciAHin4 f £ iViboints ^P ^<fb orz cornu^Oibtaihii 
Biecowo) biien OE -theliajl- dlKATneter coruvLoabe/ to 0/) Ts com*' te 

yop'op; 

~Io -iT>^ fcl>e Woht a./e«, m uie*mvolu.ti'c*» o|-co>ijvA«a'te iKXts, cv)n- 
5fcTi«^t7)e b«ir ol coT\|u,oabe TAV^ tfiocf ore c»tTi"a)rt ano/cs.. 

0/; Hjrouoli A,B,C ^raw/ oc circle, am'^ jtn^ »H -Wmi co-jomVjt 




...i binVjts at nvlm'ifca 

an^ '.' dieu ore also coiuw/flotte as to Hre circle ABC, 'lVa.7o. 

.'. H>eu lie 07) H)e (jisettors «>}■ tiie 07^oleS nanmed, Q.E.R [li>.6l;Hl»2l 



i) Given -litre iavoznti ho hh^ conf'o, eu, Ir, c, eC, e * 

Rxrallel fe Oi}if>^, ^^'o tcmocnti a^lrj t! ocn\i tolre tJie ^Ift^si^l* <i^ 
blic fwrftDelooTaw^ Icl/c', cflicV;a><«iU';as.a)T)jvt5ote. ©.xesn 

ITIe avi's c<mjtuj«te lo o-, a.' cwis Vpem oT /),/) i4ie eni« o|^«4i«inck«t 

"We centre alone i«4«un^ as blie co-borni ol^ the bisacbors o| ^e (^laqotiak^ 



Sg^ PROJECnVE CEOWETKY. L^s. l>i«.H'iq. 

PRQB. )^. MM97 a transverse heiM-dj!n»feter OH oa^ rl^ cont/^fc^ OB, if 

eoTfsiruct lxu'r& oj-carnu^ate hoU eC/amet*T^ of an J, J", . 

M ft ^(XVL) a |para\le) and a berterniicwJajr to OB, lemi^ tne tonoent 
and "noTtnol fe tnc» conio »i:fl; on 6 «m Hie TDcmrMxl bDJI<e bffints C, D 
%Kij^\ t^oii- 0C=-0D*OB; 

H,eTTanvioiTxJleBiatJK«"^^5'^, ^ . , ^^ c. ^ 
■^ icM.ts at TiQht onolei the oiTx.le on CD a\ diameccr 

cwls tVie baTioents fn jaoiiiisX"^'*"*^ ^^^^ AT-AT'= ^lOB^an^ OT, 

OT'oT*. conju.oabe ayes. 

rroYvt /) ^rtMiV ordinabes bo tiiese oice s weeb/n q oiem m V/V«n& 

take OE , OF «cLch bhot OE - >&TOV, OP 'VoT^OV; tfien OE, OFam bU 

conrvkoabe lial^ diaTnoters sought. [Uull.c):!. 

Hie cfrcle|^^*'^Qive5 blie naht aices^viAerein OO'/s Gisccbciat 

Tt^ angles U^ bhe brOLTioe-nk om^ OO' is ou segment onO^ bl)at i« cuJt 
n«3-mo'ni'callw ij[<f Hie circle (m CD ct« diaTneber* Ltn.i7/Cai:^. 

Pf?OB. l9. G/ve.1? o, Leur ol ctmfu,o«te oC'am&te^rS <J- a. covfo- bo 

COnsiruct'lJ^ ' ', hhe, cow/c. 
fcunoenti co 

Web AA be. Ck cransverse cAiosmet^r, BB ibs caniuQaie/Oyciu oie 
kvn^enbs ob A/A . 

Hinia.ah ocnu Uair oi coYUUQobe JDoiTits vn B B' drouui Imes 



EXAMPLES. 

It Qmxx wojf^ t A ^ ccmic ati& a, borr aC coititAotiulc axes, to 
ConstrvM-h iKe eoniC' 

^. So, ortfen a. kgbnaent, iU kwrnb oj- conboLct,<«nd o. Ibmt oj- coTijuoAic aices. 

4-. (ti bne dicJoot, alraue, lek (v tanoenb ite, iMe li-ne o4r mjVniiv. 

0. lo C0»«trru.6t (K pair rf conjvt^afce oiyes o^ a, conic whose wyie 
IS Qiven, 

Solve/ w co>f)ornbs wibh a circle bnroubh nie ends <» abvcnsuers^ «(XTT?eler. 

7. Ij-o, jjoir ol c0nju^(tbB diomeierft 4 "^ii if ^ i ^^ jmyeetci ia|>ot> 

ami -Uyei cda-mefccr inj or<&(tifi>-te.8 fe »t, ^ne] j*!!" oj- oie 3oaftr€« «>v 
Bie JTTpiecb'oyis eoueJi tne Square on qie|rxe^ d-ioanmher. 

B . In lliel I'^'t . t^t'ci , ,7 4 t^e. souatcs on (v bair 4 cwfuoafe 
'nyberl/ola »d»jjere.TKe ' *' i i u j 

diomebeis i& ctmsbonb. 

001 

W eeqineni tdktn m ^e a^tr direcb'an . 




91l 



PRgiECTlVE GEOmETRY. 



[SHiA.Ti. 



10. 11 in>m a uanaUe poi'^t cm ocn ajstinn|>tobe. IS oti hvke.rliolA' imc-s lie biWMt? mA, 
Ute^ ^iTe^tion Qie Isro^uxJ: ol- liie IvM «e<pnents between Ins jwrnt on^ ne Iwjtt^ivU' I't 
C<n»»tonr>lr. 

I i.U- on nujoerirola cpt^ VY\ ositwibfcoht be cut Ui a Iransucrsal iVi Imnts A,A, H, 
Die tnelubct AH-HA' eouoU tiie Sfl«M>T£ «[- the Imnilici ka^- hiomtkfjr. 

I^.lj- on nii\s«riro1a on^ iks awrnbtbte^ be caa.V um a bTKntirerio* ni'|)ornts 
^,H,H' Uie |itxy8u.cb HA-AH'e<jU4>ls the uuAre o^ '•'^e i>»TaJM h«^- 6i'aweter. 

13. from ey.lJL,4-ino ou coTts^ru.cb'wi J«i«- tfie -(<iqk^^iaTnalb8,Tft <ti on\ KubeP> 
jwia,, oi'ucn O'bWr oj c«niMobt nolj- oi'onnrteters <wn4^:n«aSui«abto^e«- 

§16. FOCI AND DIRECTRICES. 




a/ anijv>ai£t Ln'nis id »rf mvofuhart u^0rt the «>yj 
ftS b the coWc 0, P, P'fchej'r cojjcmb wnU one 



m*\\ 



Dn V F^,P'A, PI, P'R PP/P'I «», lUe |b«ir» tf 
Ti^ ciniu.««te Imec Qirt f^P, 1^|>. 

.'. Uw (>M Wenuk ore Mu«(, omi ewci^ t^m ^^ P -mcc^s '^^^j'"^ cnj/es 







•n 



laJbai Qrro P 



Q.ED. 



«.^ciK>il «^ |w^ll«J i^« J-PP'Q9'"V ahV (Mw^«r such |»e3fiftll K-I^'Q" 
wWce T«j3 «re oJb ri«ht ATialc9 ti )!»« tv^c 9^ the l«rtfr; 



SlfcHkrW.] FOCI AND PIBECTRIOES. 957 

tlwn V J-PP'OQ'" •, K- P'pq'9"' «*»■«• ^jecbiot pencils, lAlrmn, iit.Sft^ 

*'F,^ Q^Q'"' ore bopTS of coTiiao ate boiTitj i^ m volution , G),E.D. 

me <^oabl«. BPmlrs o\- cneie muoluti'op* are- tWe /?c/' oj tJie con/c^ fclie nUt 
OOtu are yoear auces, o-'nb me mvolwiiaws wi>oii Hiem are Je/eci/ muolvuhfomz. 

C/OR. l.'^e cerrire. of- a^locoJ mvo/(A'ht'on is eii y>e ceyftr^ i^ t/?9 tmi'v^ 
and^d we itA/ofOcal mt/o/wt/ert-s, ffne. is hai^ive tuni i^e other neocti/ut . 

rbr either ni tne Wqlj't a\ces ani v* tarftlle-l (j ihe otter ar^ n'o/tf cotd'k- 
aotfe lm«, (tn^ tfieu cut Hie otner ftWs oi^We centre o|-i/i« cotiic oni of injiTiifj; 
onJi Went b«ir oj- nW liViea cuts two omw n'ofjh ltV»es, owe »« iie lawe 
Vie d^ meir ((h|oieiTvi, cmk tne oWier ovi ojbbojike «»'^ts. 

mafc-iocal aitis u|)oyv mAjcK lie tne reAl^ci j's tne ^7»tf/<?r ay/^;t{je oiW 
/« tne rnirmrr axi'js. 7T>e |D9iay-& oi tne|oc( ore the eii'r^cir/ce-s . 

v^OR. /?,, 7%e rec^/ c/f'ret.'tr/ces wne //hes oct n'ant a-no/es cotna m^r 
AVj's 9.17^ eou^e^lh^ cCf'stpait Irtrm ihe centre, vlthe, cort/o. 

For '.* tn» tolftTS of- ihe ru>i wci , F, f 'a-rt coTrTiuoat66 ol the Yiiojpr awi; 

.'. Weu are berbeTri^icwJar W Ihftt <v>Li5, 
flTilk ",' liie|eet^ P,D' o| tncse jjolors are c«mcaoa.i6^ f^ ^i^ / 

.' . OP' OD = OA*" =^ OF '■ Qd' v»|»ereiYi 0/\ •« the Ulf w^yor ^ameteT, Ibfi.^. 

.'. DO»OD' Q.E.D. [cr.l 

C0R3. The /Htirrbranh line at htfiniiy of- a. poralrolo' are -fi/cut av't df'~ 
re<try/i hhe terrbrt am rrnr>0T Axis^ apoirft ami we Tanoerfk ihereat^ oi/ rn one . 

THEOR.^^. /^ ouJ0ou,S avh oi rt9 rrtffer fh/Trt^ «, a/rtioQives o. rvSia/ 
mvolui-tov of riqkt conjUMaie. Ji»eJ. 

Ftrr, ftUfrnjpaiV w- WflJil" Imes throM-ah et^+ocoLi o.v«. ctmlu-ffa-te. eu fco tne 
Oi'i,^ Coituerseiu ifF (jb/ a toinb such tnot tveru kair c| riajif Imea uitp 



96. PROJECTIVE CeOWETFtt- t&»6. 

Ic are tonivLQotc. lines, 
own trie C(mjuaat«. \>«r-nt& ojr a^ccal mvetvi.fion ttAt^nh (k.Ti«nl~ •Ti^lC' 

atF; * ' [tl..T^. 

I.e. fcueru circle tliro a. |>o<V<H c»i»j«*«ate fcoiwta as diV-metnc poiS-tis 

an^ '.* lK< i^c-I oj- Hie oHier 4«caf mvolu^i'onii •fr-tiie onlji *Atl» Wmts, lo^'^^fc- 
;.ri^w.e ol-Hitjoc;. Q.E.D. 

LfOf^.l./fye.Jocf ol a- came lie, wihhin vne. curw. enni euro. Ore co-j)omi^ of Ok 
fmnr9 d CmrMmary) ■icaioertt^ Id ihe. cemc -jrvm ate crrcuJeor Jaofnii atmfinily. 

COn-Z. //re. rrt^or a.y.is ^^ amfO cuJt^^ Me Co7t*'o m r9,«J ^mts. 

/)ies» Mmi^S oa-e the z/^ri/ces oj- Uie comic/- 

00Fl3. /I ban- of rrjlft conitufohi, ImesL e,q. afca'njent and nvrmal) ecc^ 
a4« -moMfr- ajcf's harnTorr/ce^lJy of x» eneJocf, ani ikev cut hne miTtvr «%/x 
m eke. *l/a.mm.-t*r of ihe c/rv/e (nroMf^ their eofioint ani vftefuci. 

/fie Jocus <d a- paraOp/a. U^ects hne jeOTnent an We mater- aja'* 
btttfe^eft a^hmir ti tfitfit eonfUfod-z. Jmes. 

CQf\-^ ^ Cttneetf^ Co a, can/C/ cuts tne. tcntfen'ts at ici oertice*. rn 
the oL'ame'irr'C' ho/rrts f^P'i>f »> c/rc/e ihntuqli Utefocf; o-yri P, rat MBr= 
l^es awJ F,F^'tK9 oUaQOrra/ hof'rrts, dm^erynnte a.fincn-Lotnt u/hote tidQ 
cj>fto*ite. ifie inrtjcni PP'i* T^e norma/. 

VoT '.'the. co-lines oj- P, P «*»Ilii anyj taint oy\ Die -major aici's ore c««in«afc« liiwsUv. 

.'. PFF; PF'P' ore. Ti;5l,tcm3(«; Wi.7». 

and */ i-n line okion JotMp|>Qmb two Vwrrs ot \\^ts vece. ri^W^ csiMMtflfc Imc^ 

.'. %c ilnirdl l>AJr ore. Ti«lih conjugate liTtCA* ^evm.lii.SS' 

ai4 '•' oTtt of Uiese iinftft Is outia«ac*it is Hie cmnic. 



%I1..B».^0'«1 FOC» ANP D|RECTP>C£S. 97 

Ttoctti (/ Me/r co-jtomt ani (^ ate temoenti jrpvf (J*etr co-^'ni. 

COR-6. Th^joGol raJ/us ef^he co^rtt ^-Itvt ^noeyris to a cow'c-aifi tit 
.foca/ rti//'us of Utt. c^-jjoivt tf thtir chori 4 cotrtfict lAfiU ^is i»ree6rM 
are. ifre. m/d-//nes (tj wc i^caf rad» to chm /fOfMs ^ co»6ttct. 

<JUrm.ctrf}(. JulnKr?9 a. rio^t emf/e at 6h« 4^e**-S. 

THEOR.'SO.Z^e Uus e/U^^offtaraUas Ufat ipuJt Ur^t f'xU 
/me.s /■s a, e/rc/t cnroupA cfia vnr«.e. co-^/rtis o/ ^e Jmt^. 

For; let F Xhl Ifieiocus, a>»J i tnt joomb at" iVU-mfU o| a bar0(io/a. lov&iii-na 
Oie lme.s a,,\r,z, wfiO'SC co-tointje cere C, A, B; 
fclien '.'Zs BAC, FAI have tiie. «OT»e -mfd-l/ioM ItJiTlaS"- 

.'.Zs C-Al = ^l5FAC, 
SO /Cs c-B( =Z^rBC, 

:.£. FAC -Z FBC, 

«»^ Fli"es«rrvtbe c*VcJe A3C. QBtD. 

^^■Uur ctnJtz ifiectjitiSi iJrro the co-^^i'vts oJ- ihe U'ne.s, iroiker hhree anlbtm. 

TREOR^!.^ senpntnH ot a^ianoentl'ehreejf twoU%eh ianqenti io^ 
conic svltcni' o/'aJocus one ,v^me tmt/e iubien^tj: lu ihe chtrS d anttaet 
duK Imto Cemjartti; an^ cnn/erst/y^ a. iMment l/eiwztn 4^eh /mes {htt ' 

touches Htz4i%€.i IroBZi ani Aa^ iltz jr*.zh hoint jor a,4o<M^ . 

I. Rw, let brrioertts at A/B o|^ cu conic meet mToni cut" a. ia-nQcnt* o^ 
Pm rVI,fV; ttni ]«t F le o- Jocus; 

rtcn V/.MFP=5:^AFC, Z.PFN = iZ.PFB Lt»,.71.o:3. 

.•,Z.MFN-iZ,AFB. QH.D, 



ai PHQJECTIVE &€01Y\ETRY- C5lW.tf>ll. 

a. '.' cn« 'Seamenb o}- even^ ^amaevib to Oifr cot)iV azkvrmirtt^ Ui the T<%t6 
Imes a.T^^ ovie V)ojii:ion o|-fy) N, as bwioenls, ecnh 9^- rioiii^ T»6t*f 
mvolvitiovx oL-F u/ill 6e a. bosl{:ran o[-IY)N, [bli, Kyis. 

oni *.'tlie WeTiuU F-IYl"', p-N ••■ are eouoJ [hy|>. 

/- M'", N'- cam, jirojecbnie TonniQes 

anh mN bou,oKes Hi* conic* iri e^eT«) to«itiOT\. Q.E.D. DK-fT 

ThEOR ^^.Z?«y6<7M^er <^olJoccJ muoliAtim it i:h€ f>pu/er oj-iltt 
mvo/udr/'vn oiveri ly iho. cprt/'c «n the ».%i5 AjJ" we power of tne mw- 
fut/'art Q/uen iy &h€ cort/c art th* oin^rj^ca/ ajt^'^- 

fbr, let b& O' lioniC' oni T oi. bomb uJbm\ it/ U^ ^a »o«iae.Trf"«m^ Tiot>- 
TruJ ahTcut tilt ai^ocatXacal o.X«S cfchP, p awi ileo^fceriocoJ o^i !>»•(?, (J, 

tiwn •/ OP:OQ = Q,T : Q^Q =^RT: P'P, = OQ, : P'o+Ofj , [jiWinanjU 

.", OP op'=op.ofi>-oq.oQ,. Q.E.D 

OOR. 1 . 75e squAm of ihe seontent ^ tite ittajer axis Leiwceri che cen- 
c»r «nj a, focus is iJfa fitf/fervnce of- i^g jauares of hh^ ha^ mMor ttni -me 
JieJf rnimn- diamehers. 

V^OR.il. ihe. nmUrrtintn- Jimtnt^er is intewoTne-tTtc mean of tfieft>c»,l 
jejmenis of the inafoTr aUowe^r: 

fir V hf'VK = CAO +OF)(A0-0F) =« OA*"- OF*" U).7%.cr. I 

am\ •: OF^*OA*'=p OB*" {c^c, I 

/. AF fa'-=±ob" Q.E.D. 

~nieT«lio qi- l»e se«yn«i«b oLoie nMuor g^iCiS /lebweerv Uie-Uc'r^ibiTM 



>>b.»AilLte-M'l Foot AND PmECTRlCES. 55 

CO^.% 75r ndto of Int. dittctncti ^a,\/eaiiu. of-a eemlcjr^m a^Jocus 
^ncjrrm ivi c/irectn/C efM.h the. eccenirt'c/k d- tne cowc, 

fT)r'v OF'OD « OA*" 

;, e«0F;0A-0A;OP ='A0->-QF.'D 0+OA=>AF:PA, 9-E.O. 

lnE,0R.^3. The refi*b d^^ c/tstances ^a,JI)mft^ei, cenicjrvm a^Jeaami^ 
■Jnm ftz d/ractrfjc zqwk iJie eccerdn'c/'^ U Uia cantc; em6,C9fnfeaa^,ituhMd 
Jo Tttove: atotche nd'/c oj-iU ^itiartc^jroTn ayJijtei jnu'pt omijnm »Jrjfei 
line U amitArti, tnepotrti inues a, cvnic, w'dh ihe.jni€ih0mt cm\ live oxJ^cut 
ani JirtctriiL. 

I.For, let A"*P be o. conic, TX olJccus axih drrecbny, A-'P om rmoM, «tA"-P 

jproYn Fa« cc-Yibrv. on^ uiibh-j' o^ ucmKlimq lin«^; 

men \* ^ tstA ine Ime ot iT4mihj ore Iwle cni^ jao^ar OkX t« A **'P, D^aM ^ r>^> 

,'. f« tl»« cewtcf (d- A'"' P ; 
OnV -*,* oic racial inuoluti'oTi ok F is cne ^ainejor et'Bicr' ooni'o, Cui'blL 

«'.A"'P is ol circle, anb FP I's con^boni:, 
But ',• FP'.^P^ FP ' C , wKereiw c i"« ^ome ewiTbatit, IHvS^**: 



cwkV. 



;&. VFP:jP = c, [K^f- 

.'. FP = C, wnb A"'P I'x Wie. irnaoe J a. civxie A«'r wiH F«« cewtcrj 
anb ',' F t« tne c«-nt"er ot l>«LTs|>ecfciVe ft^b-f Hie vaWshiTif Ime, 



I jao 



'• F,/ owe jaele vtih jaolor aj (9 A '"P/ [m««B (title m^JMU 

on^ tKe racial mvolation at F W an'obt rm»olwi>'m. Q.E.D. 

fs consLxMr ami ^f^ tfihe m^W c/zimttier,' tm^, canvenefy,^o.Ji>9lttt 



too. pROJEGTtYE GE.OMETRY. feib.Mi.'B 

I. For, let r, F be Hie Joci oJ- cu ^y\\t,, PQ, PQ' tlie olrrecirix c(i«ta-n&ti o^a 

Bofnt'P OTi "tfje conic; 

Jien\-FP5PQ=-F'P:PQ'-e. ' Cl«3. 

.'. FPiF*? =PQ± P0'« «; 
ort^ •/ Pa±Pqf ^(^a', a,cpn«taTtt, 

.•.FP±F'P is const awt 

= FA ± F'fl = -major ^igu-mc^cr. 0= ED. 

^.R>v, ewcry bomb (i- W'], . . , BrrotAqh one japiib'on o^ P ttT>^ vWfcb F, F 

a:s4oci,isoL|aosit}oyt irt P EtF»-. 

ocn^ tiie iflcal oisbaixj^s o| a jaom^ P rnt)>out' or uiiOnn bni's conic nave 

(jOR-2.. Tiiv CftUocmJ cuw'cs fto-v* y*« reo/ cnrrrntn htmovt^,anh iney 
CUbL 9»ok oHkt <d: rnfkt ccnm/^x m-jour ratj ^'tttsC^tm*. cwiic lie. ocvt. 
eli|a» «ni ^e otjicr a-r^ Viij|»enH»lcu) or m none. 

CftUgBu/ JLartJra/a9 cut «««( oihvf cd. -rrJtt onVex m ktmojAnliz. 
pmntSf mr-rr* inrrte. 

/I enck. cutis 9M0mry jtmnr ^ fnt^s tinvwo^k ihx oofht. mtrifM^mitf^B*. 

^Jpcos flf a- C0nif c C9 Or v^crimite t'ayi0eanit' is a» ctre/e unmJ* 
dMrntm^t^r- iS ihe Jnofor xiiarmmtmr «/ the, Cfif/e, 



itf.l>.^6:\ Foa AltP DtAMETE-BS. I0>. 

foir, tet lie Hie centre, oj- o, coniC; Aft' \h& -mojor oiometear; F,F tjie W', 
P a |winb m\ Hie tonic, PQ a- brnaent, Fq iW ^w^'^icuUr jrmn F fe PQ, 

RAt co.)„mVoJ.fQ,F'P; 

iU V PO U \Uin;A«rme ^ PP, PR cn^ is (.eT^ni;tu)at- lit Ffij-t* Tf.cr, h^^. 

.•,fq'*qo,fp=pr 

Oni VFO-OF' 

,'.Og-iF'R =i(F'P±PR)=i;)fl' Q.E.D.[tfw»1. 

QXiHXTh*. inv^uct d Uie, iittntncesjrom ttjocvs to a ^eur .^ harmi- 
CQR.^. tfie hroduti td Me tUsiamct* at in^-UciJrvm » tmwQrn&rs 

Tfie ciTcJe on tfie major ^lonnekcr «L». evniu I'l tf»e mofor- cantrnttr 
etrJe. U liic barolio\«. liiia crrJe l»r«oik,« ub mU Hie bmMenh ohUf •ertices. 

a. amio is a. circ/9 c«7fcm?iirrc- vm'ih ore c«»»/c miilsn nttiuLS *wA#4e a^mrz 
A ^€ Jow ^ Ui€ sfuceres U-^im Imlj-TiMMr ani hm^-ynhfr dt'amettri. 

i^, let a. boh- oj- nAt brnjcrfts [),^ Um>T cut die twajV c«nt»cf cirtJe 

inP,p; Q.Q'i 

«.en VTPTP'-OT-O/^ t^e*^- 

oni V TP = f>F> TP'= |>F' LttKl^3««». 

.•.TP-TP«f»Ff.P =±op*^ [Wi.tr,cr.i. 

*;,OT"=OA"-±OB*- Q.E.D. 

Tfiis circle is tiie c/f'r^ctcr^Cfrc/e o[ Hie crvtio. 

T^Un'K. anS ihe Jrne atmf/'m'ljf. 



\0l PROJECTIVE GEOMETRY. felb.Pra.JmjU. 

Me cUreciroK o'nJe. of- o. na^t m/^e-rMei, reSiooas ho a, l»wrtt, the e»nt»r; 
ana f rna- hYbesr^alo. OB'''^ OA, ^e ci/rec^or crreJe. is m?aamary. 

PROB . RJO. Gi/bn ihe irwjcci «wJ o- tanqent^ to conshvcb c4e centc : 

lake 0, ^hfc -mti-jaomt ©^ FV^ o-s cenber fltni |oo(t <^ ifie jjerlscTt&reuJar 
^jrom «il4i««- iocuft to tne tonaen^ ojq o^ bomb qI Uie Tnojor ca«vl«c^ eiTcle, 
ar>^ ^-n^ A,A U1& va.rVHcc4 iX Uw. conio, awd a, a.' Bie ianaenbK ok A, A; 
yirowLjh r,r AroYV owwj circic CM^V:ivi0 0l,oJ m ^i»n*ebri& jaoinis RB; R3 is 
(vtrawoent to ttic carnicj oct»c Hie na-rmonic Coniubqobe. o,\ toFfF'of- tf»& 
Uflfc 1^- B B IS ou boii«t iL oie norrmal . 

TVie normal -mm al-vft W iownd a« in Uieor. 7<), cr.^-. 

PROB.^1. ^iwn tLJ^cvs etni tinrtm t^-naerjts, la awjtruct the camel 

Let riie llie ^iwen locus on^ L,ni.N lie Uie co-borats ot )4ie cM^enis. 
&eb«ec»LM,LN constmob o. segment WV wch Uwi Z^'PN^Z^MFN^- 
then M'N IS ionqe^ (o tiie cimic. 

1^ F'sucAi Uiat / LMF'»ZJFWfV£LNF'«Z FWM;F'iS Uie ccmmA^ocmk 

EXAMPLES. 




llw <iJl*c^ JL tW ^iheTV- ookL Uaic •« tlacwa ol Ike «eeti«n «l^ tie C3«ac 
^ a. (iL^e. oniUe ..J^ 1^ lU |^«e .A ae )4«i« 4 tte cnxle ^ c^ 



SlUJ roCi A wo DlRECTHtCt:S. |g3, 

byJr ^ tKft cone and •fsKere^ i» A^direchrric. 

T. Kt e-K..3, ^ Hic mdiWiiom ol tiie culb'nj jJone to Vim qkis a|- tiic coac \m 

jmoicr nian ^ , , ^ , ianellij»»e. 

4*Sflu»l CO Uic halt- uerticot oinaie ol «ne cone Uic secban ts-J^iKT^Wton. 
hh«ttian ^ -^ ^ U'^^vWU. 

S^.Att •lti|»9C IS Uie m^ino^onai ormtcbitn «f a circle hanmi (kro««(i «■ 
on^e tvmse cosine i« ijic mbo ^ llie mm«r t9 Mic w^*** ^«i««ter7