INTRODUCTION.IiHEartsand havebKomefa ulenfivc.that to facilitate their acquirement is of umuch as co extendtheir boulldarics.TI1ul'lr.nion. if il dcl not f!lortcn the time offiudy, will at Ieafl makeit mOre agreeable. THISh1l a greateraimthanmereillul1r.nion; wc do not intro- coloun for thepurport of entertainment, or10amurr'"m-la;" {_"illal;DIIJ of till' "lid far... , bUI to a!1i1l themindinits rdearchesaftertruth, toincreafc thefacilitiesof inftrutlion, andto diffurcpermanent knowledge. If wCwantedauthoritiestoprove the importance.lIdufefulndsof geometry, we might quoteevuy philofophn finet thedaysofPlalo, Among the Greeks, inancient, asin the(chool ofPellaloni and others in roxent times, gcometrywas adoptedas thebeft gymnatlicof themind. In fad,Eudid'l Elements havebecome. by commonconCem, theMfii of mathematical (dellCC: all Over thecivilized globe.But Ihil will nOl appear eXlraordinary. if we confidumalIhi. fublimefdeneokligns .rt: lhe bell which effed their purpofel with thegrt:ltell precifion .nd difpatch. Such for all common pur-potn an: the .udible lign' called word,. which an Ilil!COO'IIidered IS audible. whelher addrelred immedi.atdyto theelt, or throuCh the medium of lette.- 10 the cyr:. Cro-metrical di.gnma .te /lO1 fign', bUI the mneri,,!, of geo-melrical fcience. the obj.d of which i, 10 Ihow the relativeqw.ntitie' of their pnu by " procef' of rt:"roning calledIXmonfln,tion. Thil rn.bning has been geocl'lltly carriedon by...-onb, kllen, and black CC unc:oloured diagram,;but u the ufe of roIoural fymOO1l, lign', and diag....m, inthe linear arlS and !Cience,. tende.- the proccf' of reafon-inK more precife. IIJ1d the attainment mort: upedi'iou', theyhafe bedc mlYbeCOlloCC'ived 10be plac:cduponanother, ro 11endly10C01'cr it. Or Co dut ucrypanofeach/hall endlycoin.cide.Alineil faid10 bepr,JuuJ, when it isexlended, pr0-longed, or ha. ill lenglh inc:rcared, Ind the inc:relre oflengthwhichit l'ttCivet i, calledillpart, or illprqJ"Oim.The emire lengthof theline or linel which endofcafigure, i. callediu ,"mutrr. Thefirft. fixbookl of Eudn.il raid10Thus: lhe right angkdparallelogn.mbe containedbythefidel and ---;Or ilmaybe: hrieflydcfignatcdby _ ----,tbtn ... hi.tIrts i tlu HlUlrt if tko.t/t,1t liN: ,j tlJuaf t ~ tilt pm ifIlIrI rtOa"slts ""tai"t" !It tilt t D I I ~ t hift uitadDj'it.jNU1s.-'= f+==:=(D. I, pr. 46.)DcfcribciDnw _____"-_.-"..-_.---_.-1=-.--.'. --'= -_.-+ -_.-.Q. E. D.BOOK ll. PROP. /11. THEOR.ss--:=::..:':- =-' + - -,or.. _ = _t+ _._Ddi:ribc(pr. 46. B. I.)ComplclcI (pr. 3'. B. I.)--.-and --- ,1_-.-- ,-.-=---* + -.-:III Iimil.or lDI.ll.... ;1 ""'1bc readily!howlltha, _ = _' + -'-.Q. E. 0Defcribc,od {...... ~ ::S6 BOOK If. PROP. W. THE-OR.__'=_"+_.+twice _.-.D (p'. ,6. B. 0.)""". -__ (poft. I.).==} (",. 3'.B. ,.)4 = ...(p'. 5. B. '.).4= 4(p,. ". B.,.)...... =4BOOK11. PROP. 11'. THE-OR.57_. ::;;1== _. -Cnr. 4). b. I.)--'= -'+ -'+twice _,_.Q. E.D.S8 BOOK1I. PROP. JI'. PROS.E'il( tC U1Itil _ =_ ;tau _(pr- 10, B. I.).-' (poll. j.).and_ 10mecefromIhe eentre :draw and---Then.....="" (D. I. pr. 5,)but A.c ~ or C '"(ll. I. pr. 16.).'. C (B. I.pr. 19.)bo,, _----_..... ,.....--;-----c----:::J.'. nerypointin ___lies within the circle.Q., E. D.,6nOOKIll. PROP. Ill. THEOR.Dn. __..,d __10the nlre of the rirc:k.--common, and__= '. = (B. '. pd.)and.", .L _ ...~ (9. I. dd. ~ . )Againlet J.. _ _Thenin ~ .nd ~~ = .....(8. I. pr. 5') =. (hyp.), ~ ~ -___ (B. l.pr. 26.)and bifelb _ ' ~ " .Q. E.D.BOOK //1, PROP. IV. 1'HEOR, 77OF" tituir{kr-jIrilizlltlinu{lit .IUtUr""", dU-AJ."" 60th /"1ft tltrtllllAthun, tit? J. ""Diftl1 Qlt-"',Ir onc ollhcHnet pa6through lhccenm,;t is t:vident muit Clnncnbebifcded by the other, whichdoesnotpars throughIht: centre.BUI if neilher of IhelinC$ or ---plft throughIhecenlre, duwwfromthe "nne10 their;nterfedion.If bebifeded, __ .Lto it (11. 3, pr. 3').. = D Indif---bifcded,.. __ w ..1. (B. 3 pr. l).'.~ - DI n d . " ~ = ;:lplnt:quIIto thewhole, which;s Ibfllrd,.'. and --do1'101 biredQDC anolhcr.Q. E.D........ -78 BOOK1/1. PROP. P. TIlEOR.supporc;1 poffiblc lhutwo interCedingcirclqhave common ott",cnu." ---;"",&\'i3Ff.--"'7riM... nuu 0g ...hidir MI INWit,..,. Iwr {==="r, '1'''''''' INrirc.... /rrrtrtt; the lretlltjl .j t'-ftlino;' rhat (__.) (Chklt ",ff" Ihrr;ulh lhe mu,..,tin' /h, 1,#is,Ir, remainingparr( ) if 'h'iaM""'.Of Ih, .lh,rI, tlral ( ) whirlris lI(ar""4the lill( !"jJingIhrollt,h IIr, untrr, it t,r,at". titan,/oar( ) ""Aid.is r,_e.Fig. 2. The 1_liN' (-- aM )vW.-lt male tIJ",,1 anlUSorilh tlw "';;"1t"'-t,lt I'"wu,." ....!/"'fiu#r of it. iIrt tlJltIIl t. eulrflilttr; aJtlwrt t_,....I NJrtJ"t"" lAid litrt lfu./ ,. Ih,/'"IIrt f- poisuttI tire nrnmrfrrnltt.FIGUREI. __ __and -_ -jthen_- = _ (B. I. def.IJ.)---=-+.. _C (B.I.pr.2o.)in likemanner ..__nlaybe/hewntobe greater thlll____ Or any otherlined",wn fromthe fame point10 the eil'n the famelln.ighe lillCInd at me fame fide ofit, mulltlfo coincide (B. 3- pr. :I).), andan thereforeQ.. E. D.,oB BOOK Ill. PROP. XXY. PROB.From "'Y pot0l in the kgmc:1'IItlocm, andfrom thepoinu o f b i ~dnw..>ddnw.L.L... --"..wheretheymeet ilthe cenlre of the cirek.Benufe terminatcd in thecircle il bifeBedperpendicularly by il po.m:. through thentre (B. 1. pr. I.). li!r.e",i(e ----palTcl throughthecentre, therefore .henlreil ,n lhe interfdl;ionofthefeperpendicula.,.Q.: E. I).BOOK JII. PROP. XXP!. THEOR.'"RN~ . . J ,;,k. 0 .,.{ 0,MtM pC' ,,-/, ""-"""' .... d .4J-I "uI all", ./In,,"fIt tlwcnttrt.,. tire ...frr-. pC "uI.f'IIft, lier _ ;\1 the crnlrt'.- - - = = _.,.........' ' .. ". hlvc..................,,'"do" _,od.=.,.._ ~ ~ (D. I. pr. 4-.) .Bur..= ..(D.). pr. 20.);"o and 0 arcfunibr (B.]. dd.10.);rhcy an.110 equal (B.). pr. 2+.)110 BOOKHI. PROP. XXYI. THE-OR.If thcrefOl'C the: equal f t ~ n u be tUen fmmdwequal cin:b, dxremainingkgmenu .ill becqaaJ;~ ~ = ~ (n.].);....'. '-" =,-".But if t!M: gi..cn eqlla.l I.llglc:s be 11 the cirnalnkmu.it la""idth,"YI; fromthe ccnlre of the gi"en eircle drlw anyradiUl.Make' =., (8. l.pr. 13')..,".=.'At Ihe ell"tremilics of IheIheradii, drlw ,and ""ngenu10ti,e,i..en circle. (S. 3. pr. 17.)The fouranglcl of , takcnlogcthu, areequal to fourright anglC!. (D. I. pr. p.)128 BOOK W. PROP. Ill. PROB.I""and areright angles (ronft.)= ill. lwo righl ""gles""j .'... =.A.In!heCame it anbe .. 4.= ,4(B. I.pr. p.)and IM: IlUngkcircumli:ribedaboul I"" glnn i, eqllw.gubr toclw: !:Iyentriangle.Q.: E. O.BOOK JY. PROP. JY. PROB. 129Fr;;''' tire/to,...(B. I. pr. 9.) by,, ..Ind;fi'omtM point when lhek linexmeel draw, - __lJId _. !l'(pcdiwdy pa_porxlieuI..r 10, _"d-..... '1

In , :( and J ......... - "".common, .' _ =(8. I. pr. pr..h6.)In like manner, it may be (hown aleoIhu . =,... --- = .._=.. _ ;hence ...ith any Onc of .here lines u ",diu., dekribeoand;l will pars Ihrough the Ulrem;(;el of (heotber!>wo, and (be 6dn of the given 1Il.l/Iglc. being pn-pc,reforeif a circlebe defcribedfromIlKpoint ...lKrcIllde(i,c line.mcet. withanyonc of lhema5aradiu5. ;t .... ill circumCeribcthegivenprnugon."""Fromany point in theofthe "Yell (iKle ddaibc 0 paffingthrough itsand drawthe di:unc!cL"I::::' and ; dn.w, _, .&C. andtherequimi hengon ;. inrcnbcd in the givencircle.BOOK W. PROP. XV. PROB. 143in/mM011 l1uil4ural and''luiJI,, .D lw/ar Itt""SMin Q KMn tinkOSince___ palI"es through the ttntrclof ,h. .od'" ",il"..,trianglel, hence ..... =-.. =onc_third of tworightanglel; (B. I. pr. jZ) butQ. E. D.(B. I. pr. '3) .""== = of ffi(B...pr. 31). and the: angle. vCrlinl1yoppofitc tolhefeuc .nequal to OManother (B. T. pr. '5)' and!bndoneq....l.rthcl(B. 3. pr. :6).whichare fublcndcd by equalchords (8. 3. pr. 29); andfilK'Ceach ofthe: '"gin of diebccngondoubleof theangleof an eqlliLm:r:a1 U'iangk.il ;1.110 equiangular.144 ROOK IY. PROP. XYI. PROP.Let and belbe 6desofan Ujui!.ato:ra! pml1p!inlCribcd in theciIrk, and___the 6de ofan imcn"bcd equi-lueral triangle.The arcfubto:ndc:dby J {=t =rr-_.... _-The arcfubteooedby J [=.. =rrTheir difference =IToflhc....holecircumference.of thewholecircumference..'. Ihe.rc rubtended by ._ = TT difference ofthewhole circumference.Hencc if llraightlirn:1 __be pbooi in thecirrle(B. 4. pr. I), ancquibteral.ndqui:>-d(c:lgonwillbeIhul infaibcd inthe circle.2. E.D.BOOKV.DEFINITIONS.Li1LESS magnit\lde is faidco be an aliquoc part orfubmuiliple of aglUIer magnitude, whencheIcft mcafura the greater; that is, when theIcft is coota.illed acertainnumbel' oftimc:a u-.air ill cbe: greate".11.AGUATr.1. magnicude;1 raidtobea muilipleof aleft,whentheis mcafurcd by theleft; that iJ, whenthegreatercontainJ the Ie6a ttrtain lluoWer offirmlexaaly.Ill.RATIO il the reb.tionwhichoncquantity bean to anotherof thefamekind, with to magnitude.IV.l'olAGNnVDaare raidto havea nria to one IlllIXher, wbenthey are of lbe famekind; andthe: 0I1Cwhich is001 thegmter canbe multipliedtoallto uceedthe other.Tht tUji";t;",,, VJ;II kliw" IItNuIMIlI Iltt Wurt INir "iJ;1 fir) rtfllirti.,,,,6AXIOMS.~J.UIMULTJPLESor Cljllifubmoltipks of thef.amc:. or of c q ~ magnirudn. ue equal.rfA =B, thenI....;.;.: A =tw>ee B. that is,1A=28;3 A=38 ;4"'=4 8 ;&C. &c.andtofA=tofB;.ofA=+of8;"". ""-11.AIoIllLTIPU Qf .. grater magnitude;1 greater thanthe famemultiple of. Ids.Let AE:: B. thenaAE:aB:]"1:]8;4AI:... 8;&C. &.c.Ill.THAT.....gnitulk. of which.multiple is greater d=rM&memUltiple of allOlhct, is gmtcr thanthe other.Lcl1ACaB,IMnA c: B;or,let]A ClB, thenA CB;01'", kt. A C. B, lhenA CB.BOOK Y. PROP.1. THEOR. .47~ I 00000bf: the fame multiple of O.that isof .that00000is of O.Then is eyide..1 thuOOOOO} 10 is the lamemllltiple of 00000 lowhich thu 00000il of0 ;becau(c there Ire IJ many magniwdea{OOOOO} {Ot, =00000 0Illhere Irein 00000= O.The tamedemo..lln.lionboi.1I uprdi Ihis definition gcnulIlly, thul :If M C. =or::JIll ..bc,n M.C. = or::JIII 1$4 BOOKY. DEFINITION r.Then we inferthat ., the 6rft. huthe fame ratioco thcfccood, which the third, rn.1 to thcfourth: uprelTcdinthefuecccding dcmonftrations thUI:.'." .'.;orthlll,.;. =.;.;..... _.or UlUI,. _... as e;0 to 10 is 10 ."And if e:. ::. :. we/hall infer ifM eC, =or :::J _ thenwillM. C.=or:J_ .Thl!i" if theIirfl be10thcfceond,.. thelhirdil10 tI..fourth; thcnif Mtime' the61fl begreater than, cqulllO.orlcf. than III time. the fond. then llul1 MtilllCl lhe:thirdbegreater than, eqUl.I 10, or lcf, thanIII time. thefourth, in which MandIII Ircnot tobe confidercdparti-cular multiplc., but cvcry plir of mUltiplcs whalcvcr;nor Irc fuehmlrks as et .t. t&.c. co beeonfidemlInymorc thinrcprcfcntltivc. of gcomctrial magnitude.The(tudcnt lhould thoroughlyuoocrflandIhi, dcfinitionbeforeprocuding further.BOOKr. PROP. 11'. THEOR. 155[IF IMp) Y"./-r-KlJitJuk, h.wfM/_rtUiof#fhl juJ, ....wh fN fhirJ Nil f# fN/MlTfh, fN.61IJftj";"lIlti!'u '!f fNfir.!tmJ fltid,NJ1 haw INftm. rllf., 1#II"Jftjlln-Itipfu yINftNlI4 tIlIJjo"rth; ,>iz., fN.'{"imllki!/. ifINJiff /hll/fIttsw lit. fawu ralio f#fh"ttifthe fir#IIJ, Vlhirlt fit, ('{ui-'''''/Iipl, tif fit, thirJ h.., 1# llusf tif the jourlh.Let.:. ::. :.,lhen3.: 3. :z "ery e.g-nitudclarcproponionaJ..Thi. ddinilion will in futuro: be exprdkd thlU:-If M' C'" 0, bUI M' =or::J IIt'. 'then.:OC.:+_In the ..bove genC'r.al expremon, M' and'"an: fO beeconlidered parlicular mu1ripJ.cs, nol like themultipksMBOOK V. DEFINITION VU. 167tndIIIintrodud inthc fifthdcfinilion, whichIrCinthltdcfinilion >nfidcrcdtobe cvcry pair of multiplcl thll nilbetlIkcn. Itmull Ilfo be here obfcrvcd, rh:u 0, .,and the likcCymbals are10 be confidcKd mcrdythercpn-fcntlltivcl of gcornctric;:al rmgnitudcs.Inapanial arithmeticalthis maybe(ct fOtth 11lUllo1rJ ;Let ul lIkcthe fourS, 7, 10. andLL1:jTlttrJ.IFm."It., I 7""'.""" "I",s ,0'",.l5;0H".'60i;C

i:;:"'""JO'009""P.' ",11of>

'"9'1;0/17", 9',,0,,'&,. /. /&,.Amongthcabove multiplcs we find 16 C 1+ and:-C rha' is, twice the firll is grealer than twice lhefcamd, Ind twice the third., greater tbm twice the fourth;and III a, and>"th>.t is,tWK C S6and l' Cdut is, 9tima thcfirAis gralu Uun8 limathefecond, :md 9timu thc thirdisgrc:aIUUun 8times the: foonh. Manyother equimul-168BOOK Y. DEFINrrlON Y/l.liplel mighl be fClected, which would lend 10 !how thalthe numbe" 8, 7. 10. WC'l'C proportionalt. bUI they:uelKM. for wc can find a multiple of the.6tft C multiple oft ~ fccond, bul the Wnc multiple of the third tN.t ha! bntaken of lhe firn not C lhe f,me multiple of the (ourthwhich hu been taken of the fecond; for innancc, 9 timeathe firft iI C 10 times I ~ fccond. but 9 limca the third i.lKM C 10 times t ~ fourth. thal il. 71 C 70, but ~not C or 8 ti....,. the /irfl wc find C 9 limes theCC>IId, bUl 8 time' the lhird i, nol g",aler lhan 9 time.the fourlh, that i,_ 6+ C 63, bul So i. not CWhenally fuch multiples u thek Cln be found. the firll (8, it(,id 10 hue 10 the fC.cond (1)' grencl ratio Ihan I ~ third(10) h.. to the fourlh (9) 'nd on the contrary the lhird(10) is faid 10 h,ve to the fourth (9) a Icfs ratio lh,n thefirll (8) h.. 10 lhe fccond (1).BOOK 1'. PROP, nIl, 11lOR. 169gmDF _,..J -I.." ,'" I'M'" ,", I " ~ " rllt,# tD tArf-' tlo"" Ik Iif' 6#1: "lfd t", ja""mIlglfitllM ",,' "grillt" ",tu. tD tlu lift t"/11 it.v, tD ,"" K"'''Urut and be tWO unequal magnitudes,and. :lily olher.We /hall firll prove that. which is the greater of the'wo unequ:LI magnitudes, has. grnter ratio to ,hln !he lees. has to ;thatis,.:. C.:.;take M' "' M' a. and "'.;Cuch, that M' .. and M' /hall be each C ;alCo takc '" the leaCt muhiple of .,whkh will makc '" C M' =M' .;,', M'. isnotC"' but M'. isC.,' for,u'" il the firfl muhirlc which firn becomes CM'.,thln ("i minus I) or m' minus. ;s not C M' and is not C r.... .,,', '" minus. + mull. be :::j M' +M' .. ;llat is, ..' mull be :::j M' .;:, M' is C .,' ; but it has bn lhown above that-'70 BOOK. Y. PROP. YlIl. THEOR.M'. is note11 ., thuefon:, by the fcvcnth definition, hu 10. a grater n.ao tban : .Next Wclhall ~ that. hat a g'C;UC1" n.tio m ., rhccIcC., thanithasto ., the greater;or :. C : .Tab: ...oM" .....o and M' theCame u inthe lira eak. fueh. thatM" .. and1\1' ....i11be eachC and .' the bJlmultiple of., ..hich lirn becomes gra.tulhan M'. =M' ._,', ..' min.....;, oot C M' and is not C J',1' .. ; eonfequcntly.,'.minu +. i.:::JM'. +M' I:... "" i':::J M' ., and... by thefcvcnthdefinition, has10 greater "'Iio th.n hasto .,', Of unequal nugnitudcl, &c.The rorurinncc cmployw inthis propofilion for findingamong the multiples taken, u in the fifth dclinitin mul-tiple of the lirft greaterth.:an the muhipk of the lood. bulthe: fame multiple of thethird ..hichhas beentd.cn of thelirft,not greaterthan the famemultiple of the bunh whichhat bttn ukcn ofthc fccond, may beilluftnted numerica1JyI'follo.... :-TfM, number 9 Iw greatern1tK>10 7than huID 7:lbal ;'.9: le '7:or, 1+ I: 1 C :7BOOK Y. PROP. YIIl. THEOR, 17'ThemUltiple or" whichfirll gmtcr thao 7-is 8limes. the"'fore we nl2Ymultiplythefirll and r.hirdby 8, 9, 10. or any other srellcr number; in this cafe, leluSmultiply thelirAandIhird by 8. andWeh....e 6,+ 8.nd 4: agaill, the lirll multiple of7 which bccorocIgreater than 6. is 10limes; then, by multiplying IhcCecondand fourlhby10,"'e!hall ha"e 70 and70; men,Ilrangingmefcmultiplcl, "'c havc-._ 1';_ '0 ,i_........ ... ........... _...........64+ 8 70 64 70ConCfljuently6+ + 8, or 72, is grealcr than 70'but/)is not greater than 70, .', by !he: fcvenlh definilion, 9 has agreatcr nlioID 7 llun has to -.TheIboYCis merely illullnti1'e of !he: foregoing demon-llnl>on. forthilpr0fll'ny could be Jhown of thd"c or othernumben veryreadily inthefollowinglIWlllCr; bcaufc, ifananlCCCdenl containl ill confcqucnt agreatcr numberoflimel lhananother anlcfitionperfttLl)'before ptoc:eeding funner, inorder fully!O rom-preneDd the following propofilionl of Ihif book. We then:-for,: llrongl)' rrc:ommend lhe: learner 10commenceagain,andreadup10 lhis Ilowly, and nrefully reafonu eaeh Rep,alhe proceed., p.rticularly guardingagaioRthemifchiev-OUt (yRemof dependingwholly onthememory. Byfol-lowingthere infuullion., he will findthal tN: pans whichufually pn:renl o;onfidel'lbk dilflc,ulties will pn:fent no diffi-culties what"er, in prol"uting IN: ftudy of Ihif important""'*.BOOKr. PROP. IX. 'TlIEOR. 173Let.;.::. ; then. = .For. i(not.1eI.C., the:nwill ; C. ; (B. j.pr.8)....hichib(urd aord.ing10the: hypothc6s...ilnotC._Inthe famemannerit maybe fhown, that isnotC.,.'.=eAgain, let ; ;: : tben ,,ill. _.For (invert.). : :: : therdore, bythefirl1cafe, = .. Magnitude. wbichhavethefamerati", &c.This may bel'hownother,..ife, IS follows:-Let : El = :C, thenB =C. for, ISthe fraaion- =thef""aion(;, andthenumerator of oncCoet... " ",","",",", ""...- lot Wn... " .... C. I_loo. ''''BOOKVI.DEFINITIONS.I-mECTILINEARligula arc &idtobe: /imiLu, wbenrmlhavetheir fc-"enl anglesequalJtach lOad.andthefidet about theequal '"gin proportional. .,11.T,,'o01 oneligure Ill' raid toberttiprocallypropor.lional to twofidel of Inotber figurewhen onc of thefidelof thelirll il tothe {ccood, IS thercma.ining fide of the{odcs. XX. "l'HEOR.--,..--- .and : ---C011acrount of thefim,lar polygons," --_._. , --..'. _.._.- : ----eroequ:ali (8. 5. pr. n), antl:l.l thefepropoflion:al 6dclront:ainequal :angles, thetri:anglcJ; and;'arclimilar (B. 6. pr. 6).Inlikemanner it ma), be(hownthat tlx,triangles ..IIld Tarelimil:ar.But ...tItIIIf ii to ...tIIin theduplicate"tio of..u._to_(B. 6. pr. 19), llnd.;. is to;' inlikemanner. intheduplicateratio of_ , to.._. ;(B. j. pr. 11):w __lhe duplicate nlioof-IIQOK Pl. PROP. Xx. THEOR. 145theduplicate ulio of _'" ---.. and u onc of the:lmedcnts is10 onta t'wll arta. _ _ "" nuttJillt IIwf1"art if bM.[.ct tlJ.e gi"en ar be = _ '.BiCell _... , orlndiC ----.theproblemis BUI if ...----. * thenmul!C (hyp.). .1. ... _-=....-.... :-_..... =....-or :with s""dius deCeribc I circle CUlling lhegivenline; d""wThenX -"'+--'-(B. 2. pr. s) --'..... _.Boo ---'--'+(8. I.pr. +7);-',,'BOOKPI.PROP. XXYlll.PROB.X+ ,+-iromborn, lake,,'"X-Bo, __ _._... (contl.).:md ._--__- il CodiyideddUI _... X -----. _ __.Q. E. D.BOOKVI. PROP. XXIX. PROS. ~ S 7110 poJuu"giwnjlr4ig1rtlim(_).fltMt lIu "StJIIgkc.,,-taiMJ '"tN !rptnIIs6rtUlUIf tlur"tr..i J ~ , fj tN giumfiM ami tN pWtt to. " ~ , , it is Jms-thsuJ.1N1] 6r rf",,1 NI" giwft ....u ... r. ''1",,1 tll tNf'f"..... lit. ---Make ---.......-, "d-'+ -dnw --' _.-...L .- = ----,'nw-===:".odWilhthe l"Idiul " defcribca circlemeeling _ ... produced.Then-_X__' (B. 2. pr. 6.) _no< ' = ......... + (B.I.pr'!-7.).......... '.. _ ..... - X ---._ ' ++ _ -......... "frombornlalteand -_.. _._ X;;:::.::, ;bul _. ' _ the given un.Q.; E. D.:1.58 BOOK n. PROP. XXX PROB.--=..........(n. I.rr. ,.6); andproduce ,foIharX- _ ._........ . -(B. 6. pr. 19);n0tll'" giwllji"'it j/raigM "'" (_... )~ intKfrtltU "Nd m,..n '''10..On _ ddi:ribc lhe{quire11-.... ,11 _ - (B. I. pr. ]1).ok,anddnwmeeting _n . , ~ __ X .andi,; and if fromboth,hefe cquz15hetakenthe commonpan I 'n. whichit the iquuc of will be =I, "'hichis :;;:: X....._... :llur i,.. -is di.idcdine:uremc andmean ratio.(B. 6. okf. ]).Q: E. D.BOOK n. PROP. XXXI. THEOR.'59OF "11] j_illr ul1iJi1it"r. jKlfl"tI kj_il4r'J Iklau",d."Ilu jtks if"riglrt ( l J f ~KidIri."gk ( ... _). tN!l.'SUbftribtJ ." IIr, jb( ... _) I.m-w,JiJrgtNright ."gk u'1",,1l ~ INI." if INfigllr",n tk (lflyr ftks.fromIheright angle dra'" perpendicularID _;Ihen __-,(B. 6. pr. 8)..._:: -: ---(B. 6. pr. 20).-" ...... -(B. 6. pr. 20).: ..Hence - + : -,,-+-, -;but +-=......_;...... _+--_.Q.. E. O. BOOK n. PROP. XXXII. TilE-ORSin _11 , = (B. I. pr. alro11 . =&(B.. p'. '9);... &= &;:: ..... : --. (hyp.,(B. 6. pr. 6);....",,&-.;

(B. I. pr. 32). and.'. Ind _...... ill thefamefiraighl line (B. l.pr. 11.).Q., E. D.BOOK Vl. PROP. XXXIII. THEOR. ON (0, 0), tUIg/tt,.kfMr tllIN UlffN",. lirni_ftrlMt, tlU;" fN ftllM ,tit" tf Me tilt,,"fMc.=.:::::J 6(or,. limel J)""" -.__..... (or _ timea _)Co=. ~..................... (or,.timel _);.,: ,,:: - : _. (B. 5. def. 5). or the.ngles ot theCenlre .r. 01 the UCI onwhichIheyfl.nd;but the.ngle[ the circumferencebeing h.he5of Ihe.ngles ot th. cenlre (B. 3. pr. ::0) are in the Came ratio(B. 5. pr. 15}.lnd therefore ore as the.1U on which Ihey"'od.h is eYident, that kdori inequal cire1e:s, and On equ.1ora an: equal (B. I. pr. 4; B. 3. pr$. 24. 27. and def. 9)Hence. ifthefcdonbe fubiliwlro for the angles in theabcno. demonl'tntion. theCttond put of the propofitioo willbe efl.b1ilhcd. thu is. inClJ.uo.l circleathe feaonhave thefame.,.tiotoOtIC IlIXllber2.1 the Ira on which they lbnd.Cl.:. E. D.BOOK n. PROP. A. 'THEOR. :63OFrlNr;ghrlinr(. )., lIifraing iln rxunllll... " " g / ~ , if tlu lTi-__,k d ~ , ,," "1'fi"fiJr{ ) pnJIIUJ,tlt"t